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Computational Studies of Correlated Electronic Systems

Permanent Link: http://ufdc.ufl.edu/UFE0041480/00001

Material Information

Title: Computational Studies of Correlated Electronic Systems
Physical Description: 1 online resource (147 p.)
Language: english
Creator: Kemper, Alexander
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: correlated, cuprate, dca, dft, impurity, pnictide, rpa, spinfluctuation, superconductivity, superconductor
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This work presents several studies of correlated electronic systems, in particular focusing on superconductors. All the studies are done with forms of computational methodology, and we shall describe the links between them. We use Density Functional Theory (DFT) to study cobalt dopants in the iron-pnictide superconductor Ba Fe sub As sub 2, and use the band structure determined by DFT to form a model Hamiltonian. We find that Co is an intermediate strength magnetic scatterer with long-range effects visible in the local density of states. The model developed using DFT is then used, within the spin fluctuation pairing mechanism, to study the symmetry of the superconducting order parameter in these systems and the anisotropy of the quasiparticle lifetime. In the iron pnictide LaOFeAs we find that the order parameter is highly sensitive to a number of key aspects of the electronic structure, in particular to the presence of a Fermi surface pocket near (pi,pi). In Ba Fe sub 2 As sub 2, we find that a full three-dimensional model is needed to properly describe the order parameter symmetry and magnetic susceptibility. Finally,the spin-fluctuations can account for the observed scattering rate difference between the electron and hole Fermi surfaces in LaOFeAs.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Alexander Kemper.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Cheng, Hai Ping.
Local: Co-adviser: Hirschfeld, Peter J.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041480:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041480/00001

Material Information

Title: Computational Studies of Correlated Electronic Systems
Physical Description: 1 online resource (147 p.)
Language: english
Creator: Kemper, Alexander
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: correlated, cuprate, dca, dft, impurity, pnictide, rpa, spinfluctuation, superconductivity, superconductor
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This work presents several studies of correlated electronic systems, in particular focusing on superconductors. All the studies are done with forms of computational methodology, and we shall describe the links between them. We use Density Functional Theory (DFT) to study cobalt dopants in the iron-pnictide superconductor Ba Fe sub As sub 2, and use the band structure determined by DFT to form a model Hamiltonian. We find that Co is an intermediate strength magnetic scatterer with long-range effects visible in the local density of states. The model developed using DFT is then used, within the spin fluctuation pairing mechanism, to study the symmetry of the superconducting order parameter in these systems and the anisotropy of the quasiparticle lifetime. In the iron pnictide LaOFeAs we find that the order parameter is highly sensitive to a number of key aspects of the electronic structure, in particular to the presence of a Fermi surface pocket near (pi,pi). In Ba Fe sub 2 As sub 2, we find that a full three-dimensional model is needed to properly describe the order parameter symmetry and magnetic susceptibility. Finally,the spin-fluctuations can account for the observed scattering rate difference between the electron and hole Fermi surfaces in LaOFeAs.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Alexander Kemper.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Cheng, Hai Ping.
Local: Co-adviser: Hirschfeld, Peter J.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041480:00001


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IwouldliketoacknowledgeDr.Hai-PingChengandDr.PeterJ.Hirschfeldinallthiswork.Noneofitwouldhavebeenpossiblewithouttheirpatienceandteaching.IwouldfurtherliketothankDrs.ChaoCao,SumithDoluweera,SiggiGraser,MarkJarrell,MaximKorshunov,ThomasMaierandDougScalapinofortheopportunitytoworkwiththemonvariousprojects.ThefacultyattheUniveristyofFlorida,andinparticularDrs.DmitriiMaslov,JimFry,andGregStewartprovidedinvaluableinputandideasformyresearch.Last,butcertainlynotleast,IthankmyformerandcurrentfellowstudentsGregBoyd,WeiChen,TomoyukiNakayama,HridisPalforlettingmebotherthemwithmydailyquestionsanddiscussions. 4

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page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 14 CHAPTER 1INTRODUCTIONTOELECTRONICCORRELATIONS .............. 15 1.1OverviewofCorrelationsinElectronicSystems ............... 15 1.2Magnetism ................................... 16 1.3Superconductivity ............................... 17 1.3.1CuprateSuperconductors ....................... 18 1.3.2Iron-Pnictidesuperconductors ..................... 21 2IMPURITYEFFECTSINCUPRATES ....................... 25 2.1BriefHistoryofImpurityEffectsinSuperconductors ............ 25 2.2TheDynamicalClusterApproximation .................... 29 2.2.1HistoryofQuantumClusterAlgorithms ................ 29 2.2.2DetailsoftheDynamicalClusterApproximation ........... 30 2.2.3ImpuritiesintheDynamicalClusterApproximation ......... 32 2.3Results ..................................... 34 2.4Discussion ................................... 38 2.5Conclusions ................................... 39 3COBALTDOPINGOFBaFe2As2 41 3.1Introduction ................................... 41 3.2Method:DensityFunctionalTheory ...................... 43 3.3CalculationalDetailsandStructure ...................... 46 3.4ComparisonoftheSpin-DensityWaveandParamagneticStates ..... 47 3.5EffectofCobaltDoping ............................ 47 3.6Three-DimensionalityofBaFe2As2 54 3.7Conclusions ................................... 55 4SENSITIVITYOFTHESUPERCONDUCTINGSTATETOTHEELECTRONICSTRUCTUREINTHEFERROPNICTIDES .................... 66 4.1Introduction ................................... 66 4.2SpinFluctuationPairing ............................ 71 4.3SpinRotationalInvariantCase ........................ 75 4.4BrokenSpinRotationalInvariance ...................... 78 5

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..................... 80 4.6EffectofaSurfaceonPairState ....................... 80 4.7ApproximateFormsofScatteringVertices .................. 81 4.8Conclusions ................................... 86 5TIGHT-BINDINGMODELANDPAIRINGSYMMETRYINBaFe2As2 102 5.1Introduction ................................... 102 5.2Tight-bindingFitoftheLDABandStructure ................. 103 5.3FittingParametersforthe5-OrbitalTight-BindingModelforBaFe2As2 105 5.43DMulti-orbitalSusceptibility ......................... 106 5.5PairingSymmetry ............................... 108 5.6Conclusions ................................... 110 6QUASIPARTICLELIFETIMESINLaOFeAs .................... 126 6.1Introduction ................................... 126 6.2Model ...................................... 128 6.3DerivationoftheInteractionLines ...................... 130 6.4Intra-OrbitalCoulombInteraction ....................... 132 6.5Conclusions ................................... 133 REFERENCES ....................................... 137 BIOGRAPHICALSKETCH ................................ 147 6

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Table page 3-1ProjectionsofthescatteringpotentialontodorbitalsonboththecobaltdopantsiteandthenearestneighborFesites,inboththechargeandspinchannels.AllvaluesareineV. ................................. 57 4-1Interactionmatrixinthereduced1(dxz),2(dyz),3(dxy)basis. .......... 83 5-1TheinterorbitalhoppingparametersusedfortheDFTtofthe5orbitalmodel. ............................................. 112 5-2TheintraorbitalhoppingparametersusedfortheDFTtofthe5orbitalmodel. 113 6-1AveragescatteringratesforeachFermisurfacesheetat!=10meV.TheinteractionparametersareU=1.55eV,V=J=0eV. ....................... 134 7

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Figure page 1-1ApieceofYBCOlevitatingaboveapermanentmagnet,demonstratingtheperfectdiamagnetismknownastheMeissnereffect. ............... 18 1-2BSCCOcrystalstructure(notethatmultipleunitcellsareshown) ........ 19 1-3Schematicphasediagramofthehole-andelectron-dopedcuprates. ...... 20 1-4CrystalstructureofLaOFeAs ............................ 22 1-5PhasediagramandstructureofCo-dopedBaFe2As2asdeterminedbymeasurementsofheresistivity,heatcapacity,susceptibility,andHallcoefcient.[ 1 ] ....... 23 2-1Left(Right):Schematicofthesuperconductinggapinans-wave(d-wave)superconductor,plottedalongtheFermisurface.Red(blue)indicatesthephaseofthegapaspositive(negative). .................................. 26 2-2SchematicofDMFT/DCAalgorithm.Frequency(andmomentum)indiceshavebeensuppressed,dependingonwhichmethodisillustrated(seetext). ..... 30 2-3Coarse-grainingprocedureshownforNc=4.EachcelliscenteredonaclustermomentumK;theaveragingisdonewithineachzoneoverthemomentum~k(reproducedfromREF) .............................. 32 2-4Inversepair-eldsusceptibilityfromMaieretal.(Fig.1),calculatedona4-sitecluster.SolidlinesaretstothefunctionP1d/(TTc),whichisusedtodeterminethecriticaltemperature.Inset:criticaltemperatureasafunctionofimpurityconcentrationx. ............................... 33 2-5CriticaltemperatureasafunctionofimpuritypotentialVfora16-siteclusteratimpurityconcentrationsx=3%andx=6%.Inset:blowupoftheregionofsmallimpuritypotential. ............................... 35 2-6Criticaltemperatureasafunctionofimpurityconcentrationxfora16-sitecluster,atimpuritypotentialsV=t,V=4tandV=20t.TheAbrikosov-Gor'kov(AG)resultisattothecriticalconcentrationforV=20t. ............ 36 2-7Dynamicalspinsusceptibilityfora16-siteclusteratimpuritypotentialV=4tandimpurityconcentrationx=3%.Inset:EffectiveexchangecouplingJeasafunctionofimpuritypotentialV. .......................... 37 2-8Squaredlocalmagnetizationfora16-sitecluster,asafunctionoftemperatureforvariousimpuritypotentialsandconcentrations.Thelinesareguidestotheeye. .......................................... 38 2-9IllustrationofenhancementofJebytheseparationoftwoenergylevelsE0byasmallenergy. ................................. 39 8

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........................... 57 3-2DOSforBaFe2As2intheundopedPMandundopedSDWstates.TheFermilevelsforbothsystemshavebeenalignedat0. .................. 58 3-3UndopedPMbandstructurealonghigh-symmetrylines ............. 58 3-4UndopedSDWbandstructurealonghigh-symmetrylines.Duetosymmetryinup-anddown-spin,theindividualspinstatesaredegenerateandthusonlyoneisshown. ..................................... 59 3-5CongurationsofBa(Fe1-xCox)2As2forx=1 16,intheSDWstate.(A)40-atomunitcellwithasingleCodopant(B)80-atomunitcellwithtwoCodopantsofoppositespin(C)80-atomunitcellwithtwoCodopantsofsamespin.Baatomsarelightblue,Asatomsareyellow,grayandredballsdenoteFeatomsofupanddownspin,andtheCodopantsaredarkblue.NotethatcongurationsBandChave2Codopantseach,whilemaintainingthesameconcentration. 59 3-6DOSforBa(Fe1-xCox)2As2intheundopedanddopedcongurationA,SDWstates.TheFermilevelsforbothsystemshavebeenalignedat0. ....... 60 3-7SDWbandstructurealongwith6.25%Codoping,plottedalonghigh-symmetrylines.Black(green)indicatesthemajority(minority)spin. ............ 60 3-8Localspinpolarizationinthedopantplane.Arelativelylargepolarizationisinducedaroundthedopantsite,inadditiontoachangeinpolarizationonnearbyFesitesoflikespin. ................................. 61 3-9Projecteddensityofstates(PDOS)foratomicspeciesCo(dashed)andFe(solid).TheFestatesbelongtoatomsintheunitcelllocatedasfaraspossiblefromtheCo. ...................................... 61 3-10Top:Linecutsofthepotentialchangeinthecharge(left)andspin(right)channelsuponCodoping.Bottom:CutsthroughtheFeplaneofthesame. ........ 62 3-11Top:linearintegratedchargedensityforthedopedandundopedsystems,innumberofelectrons.Bottom:differencebetweendopedandundopedlinearintegratedchargedensities ............................. 63 3-12ChangeinchargedensityupondopingasafunctionofdistancefromthedopantalongboththeFe-FeandFe-Asdirections. .................... 63 3-13PlanecutsofthelocalFermilevelDOS(seeEq. 3 ).Red(gray)ballsindicateFeioninthespindown(up)state.TheCodopantisindicatedblue,hasbeencircledwherevisible.FourunitcellsareshownforcongurationA,andtwounitcellsareshownforcongurationsBandC. .................. 64 9

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3 )throughtheverticalplaneFeAsplaneforLaOFeAsinthePMstate,BaFe2As2inthePMstateBaFe2As2intheSDWstateandBa(Fe1-xCox)2As2(x=1 16)incongurationA.Colorsarescaledfrom0%to2%ofthemaximumlocalDOSinthehorizontalFeAsplane.ForBaFe2As2(undopedandincongurationA),red(gray)ballsindicatespindown(up)Featoms. ................................. 65 4-1Fermisheetsoftheve-bandmodelforn=6.03(top)andn=5.95(bottom)withcolorsindicatingmajorityorbitalcharacter(red=dxz,green=dyz,blue=dxy).NotetheFermisurfacesheetisaholepocketwhichappearsfor1%holedoping. ........................................ 88 4-2Diagramscontributingtospin-uctuationinteractionsasnotedinthetext. ... 89 4-3Schematicplotofthespin-uctuationmediatedinteractionforatwo-dimensionalsystemwithshort-rangeantiferromagneticinteraction. .............. 89 4-4Top:pairingvertex`1,`2,`3,`4denedintermsoforbitalstates`iofincomingandoutgoingelectrons.Bottom:representativeexamplesofclassesoforbitalverticesreferredtointhetext:intra-,inter-andmixedorbitalvertices. ..... 90 4-5Thegapeigenfunctionsg(k)foraspinrotationallyinvariantparametersetU=1.3,U0=0.9,J=J0=0,0.2. ......................... 91 4-6Thegapfunctiong(k)onthe1pocketforn=5.95,J=0.2(redsquares)andn=6.03,J=0.2(bluecircles).Heretheangleismeasuredfromthekx-axis. ........................................ 92 4-7OrbitalpairingverticesalonghighsymmetrydirectionsinqspaceforU=1.3,andJ=0.2forn=5.95(bottom)andn=6.03(top),spinrotationinvarianceassumed.Solid(green)line:2222(intra);dashed(blue)2332(inter);dashed-dotted(red)2233(mixed).Notethattheverticalscalesinthetwopanelsaredifferent. ..................................... 93 4-8Thetotalpairscatteringvertexij(k,k0)forn=5.95withparametersU=1.3andJ=0.2asafunctionofkwithk0settothepointontheFermisurfaceindicatedineachpanelbytheblackdot. ...................... 94 4-9Thegapfunctiong(k)onthe1pocketforn=5.95,J=0(blackcrosses)andn=5.95,J=0.2(redsquares).Heretheangleismeasuredfromthekx-axis. ........................................ 95 4-10Theinter-orbitalpairscatteringvertex1331alonghighsymmetrydirectionsforn=5.95withparametersU=1.3andJ=0.0,0.2. .............. 96 10

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...................................... 97 4-12Theeigenfunctionsforthehole-doped(x=1%)compoundwherethepocketcharacterhasbeenadjustedtobeofd3z2r2type.TheinteractionparametershavebeenchosenasU=1.3,J=0.2. ...................... 98 4-13TheDFTbandstructurecalculatedforaBaFe2As2slab(graypoints).TheredpointsshowthebulkcontributionsfromtheFeAslayers,whiletheblackpointsdenotethecorrespondingsurfacecontributions. .............. 99 4-14Therstorder(a-d)andsomesecondorder(e-g)scatteringverticescorrespondingtointra-(a,e),inter-(d,f),andmixed-orbital(b,c,g)scatteringprocesses. .... 100 4-15Noninteractingsusceptibility0`1,`2,`3,`4denedintermsoforbitalstates`iofincomingandoutgoingelectrons. .......................... 100 4-16Thenon-interactingsusceptibilities0`1`2`3`4forn=6.03. ............. 101 5-1SketchoftheBrillouinzoneoftheI4/mmmcrystalsymmetry(a)andofthelargeeffectiveBZcorrespondingtothe1Fe/unitcell(b).Thebluelineshowsthetwopathsinthe1Fe/unitcellBZthathavetobefoldedbythereciprocallatticevectorT=(,,)(redarrow)togivethecorrespondingpathinthe2Fe/unitcellBZoftheP4/nmmsymmetry. .................... 112 5-2TheparamagneticDFTbandstructure(fullline)andaWanniert(crosses)ofthe10bandsinthevicinityoftheFermisurfaceontotheFe-3dorbitals(a).The5-orbitaltight-bindingt(coloredpoints)ofthe10-orbitalWanniert(blackpoints)withacolorcodingofthemainorbitalcontributions(b).Thecolorscorrespondtodxz(red),dyz(green),dxy(blue),dx2y2(orange),andd3z2r2(magenta).AllenergiesaremeasuredfromtheFermienergyEF=10.86eV 114 5-3Thepartialdensityofstatesofthe5-orbitaltight-bindingt,usingthesamecolorcodingasinFig. 5-2 b. ............................ 115 5-4ThemainorbitalcontributionstotheFermisurfacesatkz=0(a)andkz=(b)usingthesamecolorcodingasinFig. 5-2 b. ................. 116 5-52Dsusceptibility:TherealpartoftheRPAenhancedsusceptibilityRPA(q)asafunctionofthein-planemomentumtransfercalculatedin2Dforasinglevalueofkz,kz=0(a,c,e)andkz=(b,d,f)foraholedopedcompoundwithhni=5.9.For(a)and(b)wehaveusedU=0.65andJ=0,whilefor(c)and(d)wehaveusedU=0.55andJ=0.25U.Inpanels(e)and(f)thesusceptibilityisshownalongthemainsymmetrylineswithU=0.65,J=0(red),andU=0.55,J=0.25U(blue). ....................... 117 11

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5-5 ................................ 118 5-72Dsusceptibility:TherealpartoftheRPAenhancedsusceptibilityRPA(q)asafunctionofthein-planemomentumtransfercalculatedin2Dforasinglevalueofkz,kz=0(a,c,e)andkz=(b,d,f)foranundopedcompoundwithhni=6.For(a)and(b)wehaveusedU=0.7andJ=0,whilefor(c)and(d)wehaveusedU=0.6andJ=0.25U.Inpanels(e)and(f)weusethesamecoloringschemeasinFig. 5-5 ........................... 119 5-83Dsusceptibility:TherealpartoftheRPAenhancedsusceptibilityRPA(q)asafunctionofthein-planemomentumtransferfortwodifferentvaluesofqz,qz=0(a,c,e)andqz=(b,d,f)foranundopedcompoundwithhni=6.For(a)and(b)wehaveusedU=1.1andJ=0,whilefor(c)and(d)wehaveusedU=0.9andJ=0.25U.Inpanels(e)and(f)weusethesamecoloringschemeasinFig. 5-5 ................................ 120 5-9Fermisurfacemeshforthecalculationofthepairingfunctions.Hereweused2410k-pointsforeveryFermisurfacesheetwith1(red),2(blue),1,2(green),and(yellow). ............................... 121 5-102Dpairingfunctions,J=0:Theleading(upperrow)andsubleading(lowerrow)pairingfunctionfortheholedopedcompound(hni=5.9)plottedalongtheFermisurfacesattwodifferentkzcuts.Thepairingfunctionsareshownintheorder1,2,,1,and2runningcounter-clockwisearoundeachFermisurfacesheetwiththerightmostpointasthestartingpointoneachsheet,exceptthe2pocketwheretheplotsstartwiththeuppermostpoint.ThecalculationswereperformedforU=0.65andJ=0andtheeigenvaluesare=0.005(d-wave)and=0.004(s-wave)forkz=0,and=1.268(s-wave)and=0.354(d-wave)forkz=. ........................... 122 5-112Dpairingfunctions:Theleading(upperrow)andsubleading(lowerrow)pairingfunctionsfortheholedopedcompound(hni=5.9)plottedasbeforefortwodifferentvaluesofkz,calculatedforU=0.55andJ=0.25U.Theeigenvaluesare=0.014(s-wave)and=0.011(dx2y2-wave)forkz=0,and=0.62(s-wave)and=0.352(dxy-wave)forkz=. .................. 123 5-123Dpairingfunctions:Theleadingpairingfunctionsfortheholedopedcompound(hni=5.9)plottedasbeforefortwodifferentvaluesofkz,calculatedforU=1.1andJ=0(upperrow)andU=0.8andJ=0.25U(lowerrow).Herethemaximumeigenvaluesare=1.005and=0.862,respectively. ....... 124 12

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.................... 125 6-1Fermisurfacefor8%holedopedLaOFeAs.ThecolorsindicatethemajorityorbitalcontributionstotheFermisurface.Intheundopedandelectrondopedsystems,thepocketfallsbelowtheFermisurfaceanddisappears. ...... 135 6-2Firstcontributiontothequasiparticlelifetime.TheRomanindicesdenotetheorbitalcharacter;theGreekindicesdenotethebandindices. .......... 136 6-3Inverselifetime1 ..................................... 136 13

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2 ]E0=(62)5=3 10nEF(1+)5=3+(1)5=3+gn2 16

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5+1 3(1gF)+O(4) 1-1 ). 17

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ApieceofYBCOlevitatingaboveapermanentmagnet,demonstratingtheperfectdiamagnetismknownastheMeissnereffect. Mostelementalmetalssuperconduct,albeitatverynearabsolutezero.TheeffectwasdiscoveredbyHeikeKamerlinghOnnesin1911.Hewasthersttoliquefyhelium,whichallowedhimtostudymetalsatsufcientlylowtemperatures.Ittooknearly50yearsbeforeBardeen,CooperandSchriefferconstructedafulltheorytoexplaintheeffect.[ 3 ]OneofthebuildingblocksofBCSsuperconductivityistheexistenceofpairsofelectrons,knownasCooperpairs.Naively,thismeansthatelectronsneedsomeattractiveinteractiontoformpairs.Inelementalsuperconductors(BCSsuperconductors),thisinteractionisprovidedbyphonons.Anelectrontravellingthroughalatticeofpositivelychargedionsdistortsthelatticetowardsitspath,whichisfeltasanattractionbythenextelectrontravellingalongthepath.This,togetherwiththerequirementthattheelectronbetime-reversedpairs,allowsfortheformationofCooperpairs.ThestatementthatthephononsaretheattractivepairinginteractionwassuggestedbyFrohlich,[ 4 ]andexperimentallyconrmedbyobservingthatthecriticaltemperatureisaffectedwhentheelementisreplacedbyanisotope. 5 ] 18

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1-2 ),andthusareknownasthecuprates,withvariableotherlayersseparatingtheseplanes(seeFigure).Upondoping,whichisusuallydonechemically,theysuperconductattemperaturesfarabovetheelementalsuperconductors.AfewexamplesareYBa2Cu3O7(knownasYBCO,Tc=93K),Bi2Sr2CaCu2O8(knownasBSSCO,Tc=92K)andLa2xSrxCuO4(knownasLSCO,Tc=90K). Figure1-2. BSCCOcrystalstructure(notethatmultipleunitcellsareshown) Theyfurthermorehaveanapproximatelygenericphasediagram,whichhasanantiferromagneticphasewhenthesystemisundoped.Uponholedopingadomeinwhichsuperconductivityoccursappears,andathereissomeevidenceofanormalFermiliquidwhichappearswhenthesystemisdopedfurther.AthighertemperaturesthereisaphaselineknownasT,orthepseudogaptransition. 19

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Schematicphasediagramofthehole-andelectron-dopedcuprates. Thecriticaltemperaturesofthecupratesaretoohightobeexplainedbytherelativelyweakphonon-electroninteractionthatisthecauseofBCSsuperconductivity,andthuscondensedmatterphysicistshavesearchedforotherpairingglue.Amongthesuggestionswasthepossibilitythatelectronsexchangedabosonofantiferromagneticspinordering,orspinuctuations,whichservedasapairinginteraction.[ 6 ]TheideaisthatanelectronexcitesanS=1particle-holepair,whichcanbeaninteractionbetweenitandanotherelectron.Acleardistinctionbetweenthisandthephononcouplingisthatthisinteractionisusuallyrepulsive.However,thisstillallowsforpairinginthefollowingway.Theinteractionappearsintheequationsofsuperconductivityask=Xk0V(k,k0)k0 20

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7 8 ]InLaOFeAshowever,whendopedwithF,thecriticaltemperaturewasdrivenupto26K.[ 9 ]Withintwomonths,varioussimilarcompoundsweresynthesized,eventuallyarrivingatamaximumTcof55KforSmOFeAs.Inadditiontogrowingnewmaterials,thelargenumberofexperimentaltechniquesdevelopedtostudycupratesuperconductivitywereimmediatelyapplied,atleastwhenpossible.LaOFeAsformspowdersamplesratherthanlargesinglecrystals,whichmakesexperimentsthatrequirelargesampleswithatomicallyatsurfacessuchasangle-resolvedphotoemissionspectroscopyandscanningtunnellingmicroscopydifcult.ItwasneverthelessquicklyestablishedthatLaOFeAsisfoundinatetragonalstructure(spacegroupI4/MMM)whichhaswell-separatedLaOandFeAsplanes,withtheAsandOatomsalternatelyaboveandbelowtheFeandLaplanes(seeFigure 1-4 ).Asthetemperatureislowered,ittransformsintoanorthorhombicstate,whichisclosetotheformationoflong-rangemagneticorder.Thisisreminiscentofthecuprates,whichhaveanantiferromagneticphaseathalflling.However,themagneticstateinLaOFeAsturnsouttobeastripe-likephasewithQ=(1 2,1 2,1 2)(inthefoldedzone).ThismeansthattheFeionsarecoupledferromagneticallyinonedirection(theb-axis),andantiferromagneticallyintheother(a-axis),andtheplanesarecoupledantiferromagnetically.[ 10 ]Upondoping,boththemagneticandorthorhombicphase 21

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CrystalstructureofLaOFeAs aresuppressed,andsuperconductivitysetsin,reachingamaximumTcaround10%Fdoping.[ 11 ]ShortlyafterthediscoveryofsuperconductivityinLaOFeAs,Rotteretal.reportedsuperconductivityinarelatedsystem,potassium-dopedBaFe2As2.[ 12 ]Thiswasquicklyfollowedbythediscoverythatcobaltdopingworkedaswell,causingsuperconductivityat22K.[ 13 ]Thisclassofmaterials,knowncolloquiallyasthesweregrownfromux,andformedlargercrystals.AlthoughinitiallymanystudieswereperformedonK-dopedBaFe2As2,thefocusquicklyshiftedtoCo-doping,sinceitdidnotrequiretheadditionaldifcultyofdealingwithpotassiumandmadehigherqualitycrystals.Althoughthecriticaltemperaturewaslowerthaninthe1111s,thelargersinglecrystalsallowedformoredetailedandvariedexperimentstobedone.Fairlyquickly,itwasdeterminedthatthestructureofthe122sisnolongersimplytetragonal.Instead,thepositionsoftheAsaboveandbelowtheFeplaneareoutofphasefromonelayertothenext(seeFig. 1-5 ),which(amongotherthings)causesincreasedthree-dimensionalitywithrespecttothe1111s.ThealkalineearthhassubsequentlybeenchangedfromBatoCaandSr, 22

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PhasediagramandstructureofCo-dopedBaFe2As2asdeterminedbymeasurementsofheresistivity,heatcapacity,susceptibility,andHallcoefcient.[ 1 ] andevenintothelanthanides,formingcompoundssuchasEuFe2As2.The122s,likethe1111s,haveastripe-likemagneticstate,orspin-densitywave(SDW),atthesameorderingwavevector.Upondoping,thismagneticstateissuppressed;howeveritisyetunclearwhethersuperconductivitycoexistswiththeSDW(seeFig. 1-5 ).Initsundopedstate,BaFe2As2doesnotsuperconductatambientpressure(althoughitdoesbetween28and60kbarinbothBaFe2As2andSrFe2As2[ 14 ]).Thedopingbehavioronthehole-dopedsideandtheelectron-dopedsideisquitedifferent.UnlikeK-doping,whichhasamaximumTcaround40%,CobaltdopinghasamaximumTcat8%.Additionally,theSDWstateissuppressedmuchmorerapidlyuponCo-doping;thisparticularfeaturewillbediscussedinchapter 3 .Boththe1111sandthe122sarecollectivelyknownastheIron-pnictidesuperconductors,shortforferropnictide,becausetheycontainelementsfromGroupVoftheperiodictable,knownasthepnictogens. 23

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24

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15 ]Intheirworktheyfoundthats-wavesuperconductivityissuppressedbymagneticimpurities.Specically,thecriticaltemperatureTcissuppressedaccordingtothefollowingrelationlnTc 21 2+ 2Tc 2 simpliestoTc 16 ]Inans-wavesuperconductor,thesuperconductinggapisthesamesignalongthewholeFermisurface(seeFig. 2-1 ).Forthecriticaltemperaturetobesuppressed,theimpurityhastobreakaCooperpair,whichconsistsofanup-andadown-spinelectron.Magneticimpuritiesscatteranelectronaswellasippingitsspin,whichalwaysbreakstheCooperpair.Non-magneticimpuritiessimplyscatterelectronsfromonepartoftheFermisurfacetoanother.Inans-wavesuperconductor,sinceall 25

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Left(Right):Schematicofthesuperconductinggapinans-wave(d-wave)superconductor,plottedalongtheFermisurface.Red(blue)indicatesthephaseofthegapaspositive(negative). partsoftheFermisurfacehavethesamesignofthegap,Cooperpairsarenotbroken,andTcisnotsuppressed.Inad-wavesuperconductorlikethecuprates,however,impuritiescanscatterelectronstoapartoftheFermisurfacethathastheoppositesigngap.Since(isotropic)impuritiesscattertoallpartsoftheFermisurfaceequally,theyessentiallyaveragethegapfunctionovertheFermisurface.Thus,non-magneticimpuritiesdonotaffectthethermodynamicpropertiesofs-wavesuperconductors,butdosoind-waveones.ThishasbeenconrmedexperimentallybystudyingtheresistivetransitioninLa1.85Sr0.15Cu1xAxO4,whereAisoneofFe,Co,Ni,Zn,Ga,andAl.[ 17 ].Thecriticaltemperatureisquiterapidlysuppressedinallcases,withthecriticalconcentrationvaryingfrom2to4%.However,nosystematicvariationisseenwiththestrengthofimpuritymagnetism;naively,onewouldassumeCo/FetobeamagneticscattererandZntobenonmagnetic,butnodifferenceisseen.ThelackofanotabledifferenceintheTcsuppressionthusagreeswiththed-wavesymmetryofthecuprates,asdiscussedabove.Thehistoryofcupratesinvolvesacertainamountofconfusion,asdisordereffectsobscuredanumberofotherwiseeasilyinterpretedexperiments.Alargebodyofworkhasbeendevotedtounderstandingdisorderincupratesuperconductors,andas 26

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18 ]ThistypeordisordermayberesponsibleforlocalnanoscaleelectronicinhomogeneityinthesuperconductingstateofBi-2212,whichwasindicatedbySTMexperiments,[ 19 22 ]whichshownanoscalemodulation,ontheorderofthecorrelationlength,ofthesuperconductinggapnearthedopantatoms.Nunneretal.studiedtheeffectofimpuritiesmodeledaspotentialscatterers,aswellaspairingscatterers.[ 23 24 ]Theirworksuggeststhattheobservedgapmodulationsarecausedbyatomic-scalevariationsinthepairinginteraction,possiblycausedbythedopantatomsthemselves.Thisissupportedbyastrongcorrelationofthelocalgapmagnitudewiththepositionofthedopantatoms.Asmentionedabove,chemicalsubstitution(aswellasdefectscreatedbyirradiation),suppressesthecriticaltemperatureinthecuprates.[ 17 25 27 ]TheshapeofthecurveroughlycorrespondstothatofthediscussiononAbrikosov-Gor'kov(AG)above;however,theexperimentallyobservedinitialslopeofthesuppressionissignicantlysmallerthantheoreticallypredicted.Tolpygo,forexample,measuredthesuppressionoftheresistiveTcinYBCOlmsandfoundasuppressionratetwotothreetimessmallerthanexpected.[ 26 ]Rullier-Albenqueetal.alsoreportedasmallerinitialslope,aswellasalinearsuppressionofTcinelectron-irradiationstudiesofoptimallydopedYBCO,allthewaytoTc=0.SeveralauthorshaveproposedtheoreticalexplanationsforthedeviationsfromAGresult.Franzetal.nd,withinnumericalmeaneldmethods,includingself-consistentsuppressionofthesuperconductinggap,thattheorderparameteraroundtheimpuritysitehasstrongdeviationsfromtheAGresult.OtherauthorsextendedtheAGresulttoincludealonger-ranged,oranisotropicimpurities.[ 28 31 ]Inparticular,Graseretal.[ 32 ]concludedthatstrongcorrelationsorstrongcouplingcorrectionsareneededtoproperlydescribetheTcvs.behaviorinthecuprates. 27

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33 35 ]ItwasconrmedthatdisorderiscausingthelocalmomentformationbystudyingthecorrelationbetweentheNMRbroadeningandimpurityconcentration.Byextractingthecorrelationlengthfrom7ONMR,itisfoundthatthemagnetizationisindeedlocalizedaroundthedefects.TheoreticalcalculationsbyMaieretal.haveshownthatthemagneticsusceptibilityshowsCurie-Weissbehavioruponimpuritydoping,alsoindicatingthepresenceofmagneticmoments.[ 36 ]Mostofthetheoreticalworksintheeld,includingthosementionedabove,arebasedinthenormalstate.However,quasiparticlesareaffectedatallfrequencieseveninthesuperconductingstate.Gargetal.consideredtheeffectsofdisorderonthedensityofstatesinthepresenceofimpurities,andincludedstrongcorrelationsthroughtheGutzwillerapproximation.[ 37 ]Theyreportthattheeffectoftheimpuritiesissuppressedduetothecorrelations.Similarly,Andersenetal.includedcorrelationswithinasimpleHartree-Fockscheme,andalsondweakenedeffectsofdisorderonthedensityofstates.[ 38 ]Additionally,theyreportbreakdownofuniversaltransportinthesesystems.[ 39 ]Thisisbynomeansanexhaustivelistofimpurityeffectincuprates,andwerefertoBalatskyetal.andAllouletal.[ 40 41 ]foramorecompleteone.Inthischapter,wewillusetheDynamicalClusterApproximation(DCA)tostudytheeffectofasinglescattererinasquareCulattice.WeconrmthatwithintheDCAweobservelocalmomentformationandTcsuppression,anddiscussthebehaviorofthesequantitiesforvariousimpuritystrengths.Thisistherstcalculationofimpurityeffectswherecorrelations,superconductivity,anddisorderaretreatedonthesamefooting. 28

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2.2.1HistoryofQuantumClusterAlgorithmsInthestudyofmany-bodysystems,oneoftenusesanexpansionofthenon-interacting(free)systemforsmallinteractionparameters.However,astheinteractionstrengthincreases,thisapproachbecomeslessandlessvalid.Ameasureoftherelativeimportanceoftheinteractionsisrs,whichinitssimplestformistheratioofthemany-bodyinteractionstrength(oftendenotedbyU)andtheparticle'skineticenergy,rsU K 42 ]andtheCoherent-PotentialApproximation(CPA),[ 43 ]whichtreatasinglesiteexactly,embeddedinanon-interactingbackground.Averysuccessfulextensionoftheseisknownasdynamicalmean-eldtheory(DMFT),whichisaself-consistentprocedureinwhichthemeaneldisdescribedbyquantitiescalculated 29

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SchematicofDMFT/DCAalgorithm.Frequency(andmomentum)indiceshavebeensuppressed,dependingonwhichmethodisillustrated(seetext). onasingle,interactingsite.[ 44 ]Althoughverysuccessful,byonlyconsideringasinglesitetheDMFTneglectsnonlocaluctuations.Thismeansthat,tonameasimpleexample,DMFTcandescribeferromagnetism,butisincapableoftreatingantiferromagnetism.Ingeneral,nonlocalcorrelationsarerequiredtotreatanyphenomenoninvolvinganonlocalorderparameter.Anyquantitycalculated(e.g.theself-energy)ismanifestlymomentum-independent(althoughitiscertainlyfrequencydependent).TherehavebeenseveralattemptstoexpanduponDMFTandincludenon-localcorrelations.Thisprovedtobemoredifcultthanexpected,asthemethodssufferedfromcausalityviolationsandlackoftranslationalinvariance.Afewapproachesweresuccessful,notablycellulardynamicalmean-eldtheory(CDMFT)[ 45 ]andthedynamicalclusterapproximation(DCA).[ 46 47 ]Boththeseapproachesincludenonlocalcorrelationsbyembeddingaclusterintothemean-eldbackground,asopposedtoasinglesite.Quantitiescalculatedcannowhavesomemomentum-dependence,althoughtheresolutiondependsontheclustersizeandthusisstilllimitedbythenitesize. 30

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2-2 ,wherewecalculatetheself-energy(i!n),where!nisafermionicMatsubarafrequency. 1. Webeginwithaguessfor(i!n). 2. WecalculatetheinteractingGreen'sfunction,givenbyG(i!n)=1 3. WecalculatetheundressedGreen'sfunctionofthelatticeG,whichisformedbyexcludingtheself-energyontheimpuritysiteG1(i!n)=G1(i!n)+(i!n) 4. WesolvetheimpurityproblemassociatedwithG(i!n)usingsomemethodsuchasexactdiagonalizationonaniteclusterorQuantumMonteCarlo.[ 48 ]Thisresultsinanupdatedimpurity-modelGreen'sfunctionGimp(i!n) Weconstructanewself-energy(i!n)=G1(i!n)G1imp(i!n) 2-3 ).Thus,theGreen'sfunctionG(i!n)isreplacedbyacoarse-grainedversionGc(K,i!n)=Nc 31

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Coarse-grainingprocedureshownforNc=4.EachcelliscenteredonaclustermomentumK;theaveragingisdonewithineachzoneoverthemomentum~k(reproducedfromREF) TheGreen'sfunctionGcisnowusedtodenotetheclusterGreen'sfunction(andsimilarlyfortheself-energyc),and~kdenotesthemomentainthecoarse-grainingzonebelongingtoK,wherewehaveimplicitlyusedoneoftheDCAassumptions,namelythat(K+~k,i!n)c(K,i!n).Theself-energyobtainedafteriteratingthealgorithmisnowmomentum-dependent,andthuscontainsnonlocalcorrelations. 36 ],whomodeledZnsubstitutionofCuasanon-siteenergyinatight-bindingHamiltonian,H=H0+Hint+XVn~rimp, 32

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Figure2-4. Inversepair-eldsusceptibilityfromMaieretal.(Fig.1),calculatedona4-sitecluster.SolidlinesaretstothefunctionP1d/(TTc),whichisusedtodeterminethecriticaltemperature.Inset:criticaltemperatureasafunctionofimpurityconcentrationx. 33

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2-4 showstheresultfromMaieretal.[ 36 ]TheyndthatthecriticaltemperatureTcislinearlysuppressedwithimpurityconcentration.Theirresultsmissedanimportantpoint,however.Ad-wavesuperconductor(generally)needsatleastfoursitesduetothemomentumdependenceofthegap.Thesimplestfunctionwithd-wavesymmetryiscos(kx)cos(ky).Toexpressthisfullsymmetry,oneneeds(inmomentumspace)theminimalsetofpoints(,0),(0,),(,0)and(0,).Byremovingasinglesite,orequivalentlyoneofthefourmomenta,d-wavesuperconductivityisstronglydisrupted.Theaveragingprocedurethenlinearlysuppressesthesuperconductivityduetothelinear,small-concentrationapproximationdiscussedabove.Wehaveextendedthestudytoa16-sitecluster,wheresuperconductivitycanstillexistinthesitesawayfromtheimpurity. 49 ]cluster.Thedopingwasxedat10%,andweletU=4t,whichissmallenoughthattheQMCnegativesignproblemissmall.AsshownbyMaieretal.,[ 50 ]theestimatesforTcwithintheDCAformalismarefairlyrobustagainstnitesizeeffects.Furthermore,wecheckedthatthe!=0spin-spincorrelationfunctiondoesnotchangeappreciablybeyondasinglelatticespacingfromtheimpurity;thisindicatesthatourclusterislargeenoughtocapturethesingle-impurityeffects.First,weshallfocusonthesamequantityasshownabove,namelythecriticaltemperature.Asshowningure 2-5 ,forsmallimpuritypotentialsthecriticaltemperatureisessentiallyunchanged(notethattheerrorbarsinthegurearedeterminedfromtheextrapolationproceduretondTc,whicharelargerthanthosefromstatistical 34

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CriticaltemperatureasafunctionofimpuritypotentialVfora16-siteclusteratimpurityconcentrationsx=3%andx=6%.Inset:blowupoftheregionofsmallimpuritypotential. error).ThisisfundamentallydifferentfromwhatAbrikosov-Gor'kovtheorypredicts.Asequation 2 shows,anyimpurityconcentrationisexpectedtodecreaseTc.Infact,wendaweakinitialincreaseinTc,ontheorderof4%abovethehomogenoussystem.Lookingatlargerimpuritypotentials,wendthatat3%impurityconcentrationd-wavesuperconductivitysurvives,andisunaffectedbyfurtherincreasesinimpuritystrength.Thisisconsistentwiththesimplepair-breakingpicturebypoint-likeimpurities,whereincreasingtheimpuritypotentialpastthebandwidth(4t)drivesthescatteringintotheunitarylimit,wherethescatteringratesaturates.[ 51 ]Ifweincreasetheimpurityconcentrationto6%,theimpurityscatteringresultsinamonotonicdropinTctozerowithincreasingVbeforetheunitarylimitisreached.Figure 2-6 showsthecriticaltemperatureasafunctionofimpurityconcentration.Foralltheconcentrationsconsidered,thecriticaltemperatureinitiallyremainsconstantorslightlyincreases.ThisisinmarkedcontrasttotheAbrikosov-Gor'kov(AG)curve. 35

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Criticaltemperatureasafunctionofimpurityconcentrationxfora16-sitecluster,atimpuritypotentialsV=t,V=4tandV=20t.TheAbrikosov-Gor'kov(AG)resultisattothecriticalconcentrationforV=20t. Sincethescatteringrateisunknown(seeEq. 2 ),itisnotpossibletofullydeterminetheAGcurve,sotheAGcurveshownhasbeenttedtothepointwhereV=20tentirelysuppressesthesuperconductivity.Thus,althoughthecomparisonisinnowayexact,wecanstillnoteastarkqualitativedifferenceintheinitialbehavior,whichforAGtheoryisastrongsuppressionofTcforanyniteimpurityconcentration.ThecriticalconcentrationwendisinagreementwiththeexperimentallyobservedcriticalconcentrationforCu-substitutionbybothmagneticandnonmagneticimpuritiesinLSCO.[ 17 ]Theinitialslope,however,doesnotquitematch.Thisislikelyduetoamismatchbetweentheparametersofthestudyandthephysicalparameters.Thedegreeofprotection,orweakeningoftheinitialslope,dependsontheparameterU,whichlikelydoesnotmatchthetruevalue.Inordertoexplaintheobservedbehavior,wecalculatethedynamicspinsusceptibilityS(~Q,!).Thepeakposition(in!)ofthespinsusceptibilityat~Q=(0,)isameasure 36

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Dynamicalspinsusceptibilityfora16-siteclusteratimpuritypotentialV=4tandimpurityconcentrationx=3%.Inset:EffectiveexchangecouplingJeasafunctionofimpuritypotentialV. ofthesystem'seffectiveexchangecoupling2Je.Figure 2-7 showsthedynamicspinsusceptibilityatT3Tc,forthesystemwithanimpurityconcentrationof3%andanimpuritystrengthV=4t.Fromthis,weextractthepositionofthepeakandplotthisasafunctionofimpuritypotential(seeinsetofFig. 2-7 ).WendthattheinitialriseofTciscorrelatedwithaninitialincreaseinJe.Instrongcouplingmodels,theinteractionparameterJecontrolsthestrengthofthespin-uctuationpairing,andcanthusbeexpectedtocorrelatewithTc.Aftertheinitialrise,Jeremainsroughlyconstant,whileTcdrops.ThisindicatesthatthedropinTcisduetopairbreakingbytheimpurity,insteadofweakeningofthepairsbydecreasingthecouplingstrength.UsingthemethodintroducedbyKrishnamurthyetal.,[ 52 ]wecanestimatethelocalmomentinducedbytheimpurity.Notingthatthemomentsquared,inthe 37

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Squaredlocalmagnetizationfora16-sitecluster,asafunctionoftemperatureforvariousimpuritypotentialsandconcentrations.Thelinesareguidestotheeye. low-temperaturelimit,isproportionaltoTtimesthemagneticsusceptibility,wearriveatm2induced/T(C1C0) 2-8 showsthattheimpurityformsalocalmomentatanimpuritystrengthofroughlyU=2.Thisiscoincidentwiththesuppressionofthecriticaltemperature,indicatingthatitisscatteringduetothelocalmomentsthatbreakstheCooperpairs. 38

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IllustrationofenhancementofJebytheseparationoftwoenergylevelsE0byasmallenergy. theimpurities.ThisiscomplementarytotheworkofMaskaetal.,[ 53 ]whondwithinthet-JmodelthatanisolatedimpuritycanlocallyenhancetheexchangecouplingJ.Note,however,thatthismechanismisspecictotheone-bandHubbardmodelandisnotgeneric.[ 54 ]Theinstantaneouspartofthepairingpotentialinthet-JmodelisproportionaltoJ,[ 55 ]thusthelocalpairingandtransitiontemperatureareenhanced.Intuitively,theincreaseinJcanbeunderstoodbyconsideringthe2ndorderexchangebetweentwospinsonsiteswithunequalenergies(seeFig. 2-9 ).Atthetimeofthiswriting,noactualincreaseinTcorcompleteinsensitivitytoweakdisorderhasbeenreported.Itisnotsurprising,however,thatourmodelcalculationoverestimatesthepairingenhancementeffectofdisorder,giventhecrudewayinwhichthedisorderaveraginghasbeenperformed.Nevertheless,theseresultsareanindicationthattheobservedslownessinthesuppressionofTc,whichhasbeenremarkeduponforquitesometime,[ 32 ]hasitsoriginsinlargepartduetocorrelationeffects. 39

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37 ]andAndersenetal.,[ 38 ]ourworksuggestsarobustnessofsuperconductivityinthepresenceofcorrelationstoimpuritiesinthechargechannel. 40

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12 ]thesformlargercrystalswithgoodsurfacesandarethereforemoresuitableformanykindsofexperiments.Thesuperconductivityappearsuponsubstitutionofthealkalineearthmetalbyanalkali,suchasK,orthesubstitutionofFebyanothertransitionmetal,suchasCo,whichiswhatwewillstudyhere.ThenominalelectroniccongurationofCoisquitesimilartoFe,andthereisindeedsomeevidencethatCo'sextraelectronmightbeentirelylocalized.[ 56 ]However,Cogenerallyhasdifferentmagneticbehaviorthaniron,andwillrepresentaneffectivescatteringpotentialinboththechargeandspinchannels.OnemightexpectthatthescatteringpotentialsampledbythequasiparticlesintheFeAsplaneduetoaCowouldbeconsiderablylargerthananout-of-planealkalidopantduetoitslocationintheplane.Thisargument,however,mustbeexaminedinmoredetailforthree-dimensionalsystems,whichwewillargueisamoreappropriatedescriptionofBaFe2As2.Asidefromthedifferentpositioninthecrystal,Coaffectsthe122pnictidesdifferentlyfromthealkalidopantsduetoitsinherentmagneticcharacter.Whereasalkalimetalsareparamagnetic,Coinitsmetallicformordersferromagnetically.TheFe-pnictidesarealsoinherentlymagnetic:atlowdoping,theymagneticallyorderintoaspin-densitywave(SDW)orstripe-likephase.Severalauthorshavesuggestedtheuctuationsarisingfromthisnearbymagneticstateasapairingmechanismforsuperconductivitythroughtheexchangeofspin-uctuations.[ 57 69 ]Thus,introducingadopantionwithitsownmagneticcharactercanbeexpectedtostronglyaffectboth 41

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1-5 ).AstheSDWstateissuppressed,superconductivitysetsin,witharegionofpossiblecoexistencenear5%doping.[ 1 70 74 ]WewillstudytheeffectoftheCodopantontheSDWstate,andconsideritsscatteringpotentialinboththechargeandspinchannels.Itisoftensuggestedinthecupratesthatthesuperconductivityathightemperaturesarisesbecauseofthelayerednatureofthematerials;theirnearlytwo-dimensionalcharacterenhancescorrelation.The1111classofpnictidesissimilarinthisregard.Theyarelayered,andtheFermisurfacesarecylinderswithweakcorrugations.[ 75 ]The122saredifferentinthisrespect;althoughtheFermisurfacesarestillcylindrical,[ 13 ]thereisasignicantamountofkzdispersion,leadingtolargercorrugationsthaninthe1111s.Thisisbackedupbyexperimentalobservationofloweranisotropyinresistivity,Londonpenetrationdepth,andcriticalelds.[ 73 76 77 ]Reportedvaluesofc=abareapproximately6forthedoped122s,comparedtonearly20forF-dopedNdFeAsO.Inthiswork,weshalladdresssomeoftherootcausesoftheincreasedkzdispersionandthree-dimensionality.Wehaveperformedrst-principlescalculationsofcobalt-dopinginBaFe2As2.WendthatCobreaksthespindegeneracybyinducinganetmomentinthesystem.Thescatteringpotentialasaresultofitsspinandchargedopingisfoundtobeofintermediatestrength,andlong-rangedinthemagneticchannel.Thescatteringpotentialsarehighlyanisotropic,reectingboththestructuralandmagneticanisotropyoftheunderlyingsystem.Wendthatthechargedopingremainsintheplane,andmainlycenteredonthedopantitself.Nevertheless,thereissomeeffectintheundopedplane;thebreakingofspinsymmetryisreectedinthelocaldensityofstatesintheundopedplane.Thelinesofmajorityspinareclearlydistinguishedfromthoseofminorityspin.TheCoionitselfshowsupasaresonanceinthelocaldensityofstatesat-800meVand+200meV.Finally,weshedsomelightontheobservedlackofanisotropycompared 42

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2mer2i+Vext(ri)+Xi6=je2 78 ]provedthat 1. ForanysystemofinteractingparticlesinanexternalpotentialVext(r),saidpotentialisuniquelydeterminedbythegroundstatedensityn0(r),uptoaconstant.SincethisfullydeterminestheHamiltonian,allpropertiesofthesystemarethusdeterminedbythegroundstatedensityn0(r). 2. AfunctionalE[n]intermsofthedensityn(r)canbedenedforanyexternalpotentialVext(r).Thedensitythatminimizesthisfunctionalisthegroundstatedensityn0(r)forthatparticularpotential.Thus,theproblemhasshiftedfromndingafullmany-bodywavefunctiontominimizingafunctionalofdensity.However,themethodforndingsaidfunctionalisnotdened,asidefromitsdenitionintermsofthemany-bodywavefunction.TheeldwasfurtheradvancedbyKohnandSham.[ 79 ]Theirapproachwastoreplacethedifcultsystem,withthegenerallydenedbuthard-to-determinefunctionalwithadifferent(auxiliary)systemthatcanbesolvedexactly.Theirguess,oransatz,is 43

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2Xihijr2jii=1 2XiZrjri(r)j2 2Zr,r0n(r)n(r0) 44

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2mer2+VKS(r) 45

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80 ]exchange-correlationfunctionals,asimplementedintheQuantum-ESPRESSOpackage.[ 81 ]Wechose40Ryand400Ryascutoffsfortheplanewavebasisanddensity,respectively.Forthecalculationswithoutlong-rangemagneticorder,weconsideredBaFe2As2inthehigh-temperaturetetragonalstructure(seeFig. 3-1 ).Thelatticeconstants,asreportedinRotteretal.,[ 82 ]area=b=3.9625Aandc=13.0168A.Weusedthelatticeconstantsfromthesameworkforthelow-temperatureorthorhombicstate,a=5.6146A,b=5.5742Aandc=12.9453A.Thecalculationswereperformedforadopingofx=1=16,whichiswithintheexperimentallydeterminedrangefortheSDWstate.[ 1 70 74 ]Foroursupercell,whichcontains16Featoms,thisisequivalenttoreplacingasingleFeionwithCo.SinceoursystemcontainstwoplanesofFeatoms,thiswillresultinaplanethatisnotdoped.Thephysicalsystem,however,willhaveallplanesdopedwiththesameconcentration.Toaccountforthis,wehaveperformedsimilarcalculationswithtwodopants,andhaveenlargedtheunitcelltokeepthedopingxed.Duetocomputationalconstraints,wehavenotrelaxedtheatomicpositionsforthelargercells,buthavecheckedthattheatomicdisplacementsdonothaveastrongeffectonthequantityreported. 46

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11 83 ]Figure 3-2 showsthedensityofstatesforundopedBaFe2As2intheSDWandPM(paramagnetic)states.NeartheFermilevel,thereisalargedropofspectralweightintheSDWstateascomparedtothePMstate,whichmaybeassociatedwiththeopeningofanSDWgap.InordertofurthercontrasttheelectronicstructureoftheSDWandPMstates,ingures 3-3 and 3-4 ,weshowtheelectronicbandstructure.Foreasiercomparison,wehaveusedthetetragonalcellandthecorrespondinghigh-symmetrypointsinallbandstructureplots.OnecanseethatthebandstructureoftheSDWstateisquitedifferentfromthatofthePMstate.ThePMbandstructureisinagreementwiththatreportedinpreviousstudies,[ 13 84 86 ]withholepocketsaroundthepointandelectronpocketsaroundM.Thepocketsareofsimilarsize,whichleadstoalargepotentialfornesting.IntheSDWstate,asexpected,thisnestinghasentirelydisappeared.Finally,thePMstatehasbandscrossingtheFermilevelfromRtoA,whichhavebeenremovedbythetransitionintotheSDWstate. 3-5 ).Thissupercelltypeofapproachtodopingisbynomeanstheonlyone.Othermethodsincludearigidbandshift,whichsimplyassumesthatthechemicalpotentialshiftsduetotheextraelectronorhole.ThismethodworkswellwhenthedopantsitsfarawayfromtheatomsthatprovidethemajorityoftheFermileveldensityofstates,andtheextraelectrondopesthepartsofthecrystalwherethoseatomssit.Anotheristhevirtualcrystalapproximation,whichtreatsthedopingbymixingthepseudopotentialsofthedopantatomwiththatoftheatomitreplaces,in 47

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3-5 .Therstconguration(A)containsonlyasingledopant,andthesecondandthirdcontaintwo.Thesecond(B)hastwodopantsonunlike-spinsites,andthethirdhastwodopantsonlike-spinsites.ForcongurationA,allionswereallowedtorelax(withintheSDWstate)afterCosubstitution.Themajorityofdisplacementsoccurrednearthedopantsite,wheretheCoatompushesawaytheFeionsoflikespin.Also,thenearestAsatomsareattractedtotheCodopant,causingthedistancetodecreaseby0.03AcomparedtothenormalFe-Asdistance.LimitingourconsiderationstocongurationAfornow,asmallshiftinenergyofthedensityofstatesneartheFermileveloccursduetoCodoping(seeFig. 3-6 ).Asidefromthisshift,thedopedDOSissimilartotheundopedDOS,suggestingthatarigidbandshiftapproachtoCo-dopingofBaFe2As2isatleastqualitativelyvalid.However,acloserlookrevealsothereffectsthatarenotcapturedbythisapproach.Inparticular,Cohaseffectsonthespinstructureofthesystemwhicharenotcapturedbytherigidbandshift.Figure 3-7 showsthebandstructurealonghigh-symmetrylines.Itisclearfromthegurethatthedegeneracybetweenmajorityandminorityspinsisbroken.Thedopantchangesthenetspinofthesystemfrom0to0.46percell.Someoftheinducedlocal 48

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87 ]Wehavecalculatedthedensityofstatesprojectedontoatomicorbitalsofagivenspecies,whichareshowninFig. 3-9 .ThegureshowsthedensityofstatesonCoandFeionsfarfromtheimpurity.ThereareclearresonancesintheCoDOSat-800and+200meV.ThelatterenergyisclosetothevoltagebiasusedtoimagetheCoionsonthesurfaceofCo-dopedBaFe2As2inrecentSTMexperimentsofChuangetal.[ 88 ]OurpresentcalculationalsopredictsthattheCoatomsshouldshowupasminimainthelocaltunnellingcurrentrelativetothenearbyFeatomsat-200and+800meV;experimentally,theCodopantsareobservablefromconductancemapsat150meV.Next,weconsiderhowtheCodopantactsasascattererinthesystem.Inparticular,wewouldliketoaddressthevalueofthepotentialaswouldapplyinatight-bindingmodel;thisissimilartotheworkbyWangetal.[ 89 ]Toaddressthisinthesimplestform,wecalculatethepotentialdifferencewherewedidnotrelaxtheatomicpositions.InFig. 3-10 weshowlinecutsofthepotentialchangeintheFeplane,scaledtotheaverageFe-FedistancehaFeFei=2.7972A.Weuseacoordinatesystemwherethe(x,y)axesarealignedalongtheFe-Febonds.Wendthatinthechargechannel,thepotentialinducedduetothedopantislimitedto1/4oftheFe-Febondlength,ofroughly1A.Theoscillationsinthepotentialstrengthareduetotheelectroncloud 49

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90 ]andhasalsobeenobservedinheavyfermionsystems.Furthermore,theseeffectscanbecapturedinsimplemean-eldtreatmentsofthecorrelations.Densityfunctionaltheorydoesnotincludestrongcorrelations;infact,theCoulombinteractionistreatedonlyatthemean-eldlevel.Furthercorrectionscomefromthecorrelation-exchangefunctionals.SinceX-raymeasurementsindicatethatthepnictidesarenotstronglycorrelated,[ 91 ]however,DFTshouldbeabletocapturethepotentialstrengthinthiscasefairlywell.Forasingleimpurityinahost,themagneticeffectsdecayawayfromtheimpuritysiteonalengthscalecorrespondingtothemagneticcorrelationlengthinthecorrelatedhost.Thus,itisinterestingtondthatthemagneticpotentialdoesnotdecayappreciablyonthelengthscalesweareabletoobserveinourunitcell.Becausethemagneticpotential,whichcorrespondstotheantiferromagneticcorrelationlengthinsimpletheories[ 90 ],isapparentlylong-ranged,itmayexplaintherapidsuppressionofthespin-densitywavebyCo.[ 92 ]Tofurtherstudythesizeofthemagneticcorrelationlength,werepeatedthecalculationsinthePMstate.However,wendthatasingleCodopantalwaysreturnsthesystemtotheSDWstateacrossthewholeunitcell,bothinthetetragonalandorthorhombicallydistortedlatticestructures.Thisagainconrmsthatthelengthscaleofthemagneticpotentialislargerthanourcalculationunitcellsize. 50

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93 ]isanappropriatedescriptionofthesuperconductivity,thenscatteringatthesevectors,knownasinterbandscattering,breakstheCooperpairs,andthussuppressesthesuperconductivity.AscanbeseenfromFig. 3-10 ,theinducedspinpotentialispeakedneartheFe-Fedistance.Thus,thepotentialinFourierspacewillbepeakednear=,whichistheinterbandscatteringvector.However,thecalculationwasdoneintheSDWstate,andthustheappropriatewavevectorslieintheBrillouinzoneasfoldedbythemagneticorder.Nevertheless,wewouldliketoutilizetheinformationaboutthescatteringpotentialinvariouschannelsforinputtophenomenologicalmodels.Todoso,weprojectthepotentialontothelocalatomicFestates,withthecaveatthatthefollowinganalysisisonlyfullyvalidinthesystemwiththeSDWpresent.However,wedonotexpectthenonmagneticpotentialstovarystronglyforaCointheparamagneticstate.Thepotentialsthatareusefulfortight-bindingmodels,torstorder,arethematrixelementsofthepotentialwiththeappropriateFe3dorbital(denotedbyUmc),Umc(~R)=Xhml=2(~r~R)jV(~r)jml=2(~r~R)i, 3-1 .Sincetherearetwotypesoflike-spinandunlike-spinneighbours,thetableliststheaverage(althoughtheindividualvaluesarequiteclose).Theinter-orbitalelementsofthepotentialnotshowninthetable,butarefoundtobesmall. 51

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2or1 2.ThevaluesfoundareshowninTable 3-1 .Notethatthesignofthescatteringinthespinchannelwilldependonwhichspin-sublatticetheCodopantislocated.TheCoionchangestheon-siteenergyoftheimpuritysite,anditalsoinducessignicantpotentialsonthenearestneighbors.DuetotheorthorhombicityofthecrystalandthepresenceoftheSDW,theCodopantattractsbothchargeandspinalonglinesofoppositespin,andrepelbothalongtheperpendiculardirection;thisleadstothesuppressionoftheSDWstatewithincreasedCodoping.WenextconsiderhowtheCoiondopesthesystem.Figure 3-11 showsthelinearlyintegratedchargedensity(z)=Rdxdyn(~r).Mostofthedopingislocalizedaroundthedopantplane.TherearrangementoftheAsatomsintheundopedplanecausethechangesinthatplane;thesehoweverintegratetozero.Theextraelectronintroducedbythedopantremainslocalizedinthedopantplane.Infact,thedopedelectronislocalizedonthedopantatom,inagreementwiththereportbyWadatietal.[ 56 ]Figure 3-12 showsthechangeinchargedensityalongtheFe-FeandFe-Asdirectionsintheplane.Althoughthereissomeredistributionofchargeonthenearbyatoms,thisismainlyduetothemovementoftheFeionsandthenearbyAsatoms.Notethatthechangeinchargedensityveryclosetothedopantiszerobecausetheextraelectronlivesinthedorbitals.Althoughthechargedopingremainslocalizedintheplane,theeffectivespinpotentialandbreakingoftheup-downspinsymmetryinthesystemmayhavelonger-rangedeffects.Wefocusontheeffectonthedensityofstatesinasmallwindowaroundthe 52

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3-13 A,weshowacutsof(~r)throughboththeundopedanddopedFeplaneforcongurationA.Inboththedopedandundopedplanes,aclearmodulationofthedensityofstatescanbeseen.Furthermore,themodulationiscommensuratewiththespin-densitywave,i.e.(~r)isenhancedalongthelinesofmajorityspin.Notethatinanundopedsystem,alllineswouldlookequal.Thus,althoughthechargedopingislocalizedinthedopantplane,theCoimpurityproduceseffectsinthelocalDOSoftheundopedplane.Figures 3-13 Band 3-13 Cshowcutsof(~r)throughoneoftheplanesofcongurationsBandC.NotethattheatomicpositionsofcongurationsBandCarenotoptimized;wehave,however,compared(~r)foroptimizedandnon-optimizedatomicpositionsincongurationA,andndnoappreciabledifference.SincetherearetwoCodopants,therearetwopossibilitiesfortheirrelativepositionsinthemagneticstructure:inlike-spinandunlike-spincongurations.Wehaveplacedthetwodopantsasfarapartaspossible,andconsidertheireffectsonthelocalDOS.Fromthegure,itisclearthattheLDOSstripesseenincongurationAarenotaspronouncedineithercongurationsBorC.Thereisstillsomevisibledifferenceinthelinesofmajorityandminorityspin,butthecontrastissmallerthanincongurationA.Notethatalthoughthesearemagneticeffects,theyappearinthetotal(spin-summed)localDOS;assuch,theymaybevisibleinstandardSTMexperiments.Furthermore,thedopantdistributioninthesamplesisnotuniform,asfoundinsomeNMRexperiments.[ 72 ]Thus,allthreecongurationsconsideredabovecanbeconsideredviableexperimentalpossibilities.CodopingperturbstheSDWwithsignicanteffectsfarawayfromthedopantlocation.Withoutgoingintofurtherdetailregardingthemulti-dopantinterferenceeffects,theComightaffectthesystemonscales 53

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88 ] 77 ]Figure 3-14 showsthedensityofstatesintegratedoverasmallrangearoundtheFermilevel((~r))intheacplane.Tocomparetherelativedensityofstatesavailablebetweentheplanes,wehavescaledtheplotsbythedensityofstatesintheFeAsplane.ThegureshowsthatBaFe2As2hasalargerFermileveldensityofstatesavailablebetweentheFeAslayersthanLaOFeAs.Thisistrueinthestateswithandwithoutlong-rangemagneticorder,suggestingthatitisthecrystalstructureratherthanthemagneticorderthatisthecauseforthedecreasedanisotropy.Withthatinmind,wenotethatthedistancebetweentheFeAslayersisroughly3AlessinBaFe2As2thanitisinLaOFeAs,whichpartiallyaccountsforthedecreasedtwo-dimensionalityinthe122-typematerials[ 94 ]ascomparedtothe1111-typematerials,whoseFermisurfacesareessentiallycylinders.Additionally,inBaFe2As2,theAsatomsare180outofphase,withtheAsatomsoftwoFeAslayersalternatelypointingtowardsandawayfromeachother.Thisdecreasestheinter-layerdistanceevenfurther,andaddstothepresenceofadditionalstatesbetweenthelayers.InLaOFeAs,theAsatomsareinphase,andthedistanceremainsconstant.UponCodoping,thedensityofstatesavailablebetweentheplanesdecreases(rightmostpanelofFig. 3-14 ),suggestingthatthedopingincreasestheanisotropy.Theoreticalmodelsforthepnictidesuperconductorshavebeengenerallylimitedtotwodimensions,whichtheexceptionofaworkbyGraseretal.[ 67 ]Theseresults,combinedwiththeexperimentalobservations,[ 73 76 77 ]suggestthatathree-dimensionalmodelisappropriate. 54

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55

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56

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CrystalstructureofBaFe2As2. Table3-1. ProjectionsofthescatteringpotentialontodorbitalsonboththecobaltdopantsiteandthenearestneighborFesites,inboththechargeandspinchannels.AllvaluesareineV. ImpuritySiteSame-SpinNeighborOpposite-SpinNeighbormOrbitalsUmcUmsUmcUmsUmcUms 57

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DOSforBaFe2As2intheundopedPMandundopedSDWstates.TheFermilevelsforbothsystemshavebeenalignedat0. Figure3-3. UndopedPMbandstructurealonghigh-symmetrylines 58

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UndopedSDWbandstructurealonghigh-symmetrylines.Duetosymmetryinup-anddown-spin,theindividualspinstatesaredegenerateandthusonlyoneisshown. Figure3-5. CongurationsofBa(Fe1-xCox)2As2forx=1 16,intheSDWstate.(A)40-atomunitcellwithasingleCodopant(B)80-atomunitcellwithtwoCodopantsofoppositespin(C)80-atomunitcellwithtwoCodopantsofsamespin.Baatomsarelightblue,Asatomsareyellow,grayandredballsdenoteFeatomsofupanddownspin,andtheCodopantsaredarkblue.NotethatcongurationsBandChave2Codopantseach,whilemaintainingthesameconcentration. 59

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DOSforBa(Fe1-xCox)2As2intheundopedanddopedcongurationA,SDWstates.TheFermilevelsforbothsystemshavebeenalignedat0. Figure3-7. SDWbandstructurealongwith6.25%Codoping,plottedalonghigh-symmetrylines.Black(green)indicatesthemajority(minority)spin. 60

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Localspinpolarizationinthedopantplane.Arelativelylargepolarizationisinducedaroundthedopantsite,inadditiontoachangeinpolarizationonnearbyFesitesoflikespin. Figure3-9. Projecteddensityofstates(PDOS)foratomicspeciesCo(dashed)andFe(solid).TheFestatesbelongtoatomsintheunitcelllocatedasfaraspossiblefromtheCo. 61

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Top:Linecutsofthepotentialchangeinthecharge(left)andspin(right)channelsuponCodoping.Bottom:CutsthroughtheFeplaneofthesame. 62

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Top:linearintegratedchargedensityforthedopedandundopedsystems,innumberofelectrons.Bottom:differencebetweendopedandundopedlinearintegratedchargedensities Figure3-12. ChangeinchargedensityupondopingasafunctionofdistancefromthedopantalongboththeFe-FeandFe-Asdirections. 63

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PlanecutsofthelocalFermilevelDOS(seeEq. 3 ).Red(gray)ballsindicateFeioninthespindown(up)state.TheCodopantisindicatedblue,hasbeencircledwherevisible.FourunitcellsareshownforcongurationA,andtwounitcellsareshownforcongurationsBandC. 64

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CutofthelocalFermilevelDOS(seeEq. 3 )throughtheverticalplaneFeAsplaneforLaOFeAsinthePMstate,BaFe2As2inthePMstateBaFe2As2intheSDWstateandBa(Fe1-xCox)2As2(x=1 16)incongurationA.Colorsarescaledfrom0%to2%ofthemaximumlocalDOSinthehorizontalFeAsplane.ForBaFe2As2(undopedandincongurationA),red(gray)ballsindicatespindown(up)Featoms. 65

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9 ]basedoniron,theeldisstillinthephasewheretheexactsymmetryisunclear.Nevertheless,thewiderangeofbehaviorsseenfromvariousexperimentsonthedifferentmaterialsislargerthanexpected.[ 95 ]Herewepose,andattempttoaddresswithinaweakcouplinguctuationexchangetheory,[ 57 58 ]thefollowingquestions:isitpossiblethatthesuperconductingstateoftheferropnictidesissounusuallysensitivetoaspectsofelectronicstructurebecausethereareparametersthatcantunethepairingsymmetryororderparameterstructure?And,ifso,whichparametersarethese?Basedonrst-principlescalculations,[ 75 94 96 ]angle-resolvedphotoemission(ARPES)andquantumoscillationsexperiments,[ 97 102 ]theFermisurfaceoftheFe-pnictidesisbelievedtoconsistsofafewsmallholeandelectronpockets.ThisFermisurfaceisshowningure 4-1 ,wherethepredominantFe3dorbitalcharacterofthevarioussheets,takenfromtheDFTcalculationsofCaoetal.[ 75 ]fortheLaFeAsOmaterial,havebeenindicatedbycolor.FollowingtheconventionofGraseretal.[ 57 ]andelsewhere,weshallrefertotheholepocketsaroundthe(0,0)pointasthesheets,andtheelectronpocketsaroundtheX(,0)point(intheunfolded,one-ironzone)asthesheets.QuiterapidlyafterthediscoveryofsuperconductivityinLaOFeAs,Mazinetal.[ 103 ]andDongetal.[ 104 ]proposedthatthenestingoftheFermisurfacewouldleadtoapeakinthemagneticandchargesusceptibilitiesnear(,0)(intheunfoldedzone), 66

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4.2 ).Indeed,ARPESexperiments,whilenotsensitivetothesignofthegap,haveconsistentlymeasuredaconstantgaparoundthevarioussheets,andseemtoindicateanisotropicgapstructuremomentumspace.[ 97 101 ]Neutronexperimentsobservearesonanceintheinelasticneutronscatteringsignal,whichisastrongindicationofasignchangeofthesuperconductinggap,butcouldbeconsistentwithanisotropicsstate.[ 105 110 ]However,notallexperimentsshowalackoflow-lyingexcitations.Adiversegroupofexperiments,includingRamanscattering,[ 111 ]NMR,[ 112 117 ]andthermalconductivity[ 118 124 ]indicatetheexistenceoflow-energyquasiparticleexcitations.Additionally,bothintheLaFePO[ 125 126 ]andinP-dopedBaFe2As2,[ 118 ]alinear-Tdependenceofthelow-temperaturepenetrationdepthhasbeenobserved;inBa(Fe1-xCox)2As2,variesroughlyasT2overalargeportionofthephasediagram.[ 74 ]Foranisotropicgap,onewouldexpectexponentiallyactivatedbehaviorduetothelackofquasiparticleexcitationsbelowthesmallestgapscale.TheobviouswayofinterpretingtheseexperimentsistoassumetheexistenceofgapnodesonpartsoftheFermisurface,leadingtothelow-energyexcitationsseenintheaboveexperiments.However,caremustbetakentonotethatdisordercanalsocreatelow-lyingexcitationsinisotropicsign-changingsuperconductors,albeitonlyundercertainspecicconditionsregardingintra-andinter-bandimpurityscattering.[ 127 131 ]However,thereisnoknownmechanismfordisordertoproducelinear-Tdependenceinthepenetrationdepth.Thus,toclarifytheseexperiments,itisextremelyimportanttoestablishwhetherthelow-energyexcitationsareduetogapnodes,orinducedbydisorder;andtoestablishunderwhatcircumstancesonecanexpectanisotropicoranisotropicgap.IntheframeworkofuctuationexchangetheoriesofthepairingstatebasedonrealisticFermisurfaces,itisindeedobservedthatthepreferentialstatesareof 67

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57 58 64 ]Inadditiontothessymmetryclass,alloftheabovecalculationsndnearbystatesoflowersymmetry,mostnotableonewithdx2y2symmetry.However,atransitionfromanodelessstoanodaldstatewouldgiverisetothermodynamicsignalswhichhavenotbeenobservedexperimentally.Thus,mosttheoreticalworkshavefocussedontheA1gstatewiththepossibilityofaccidentalnodes,whicharepresentduetodetailsofthesystem,ratherthanimposedbysymmetry.Itremainstoanswerthequestionposedabove:whatparticulardetailsarethedrivingforcebehindthenodalornodelessbehaviorintheuctuation-exchangetheories?Afewauthorshavealreadyshedsomelightonthesubject.Maieretal.[ 107 ]notedthatwithinamodelwithintra-andinter-orbitalCoulombrepulsion,thepairingofelectronshappensprimarilywithinthesameorbital.Furthermore,thenodesaredrivenbyintra-orbitalpairscatteringbetweenthetwosheets,aneffectwhichcannotbecapturedbysimplertwo-bandapproaches.Kurokiandco-workersperformedanextensivestudywheretheyconsideredtheeffectoftheatomicstructureontheelectronicproperties,andthroughthatthepairingsymmetry.[ 132 ]TheyobservedthattheheightofthepnictogenabovetheFeplanehastheeffectofraisingandloweringabandneartheMpoint(intheunfoldedzone),whichpossiblyaddsanotherFermisurfacesheet.Thisnewhole-likepocket,whichadditionallyalsoappearsuponholedoping,wasfoundtostabilizeamoreisotropicsstate.Usingmodelswhichcontainbandinteractions(asopposedtotheorbitalinteractionmodelsintheaboveworks),Vildosolaetal.[ 133 ]andCalderonetal.[ 134 ]havediscussedtheeffectofthepnictogenheightontheelectronicstructureandpairing.aInparticular,theynotedthatthisparametercanmodifytheorbitalcontentaswellasraise 68

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135 ]wheretherenormalizationwasneededtoaccountforanunphysicalshiftofthedispersionduetoself-energyeffects.Theyobservenodesinthegapfunctionfortheelectrondopedmaterial,andnodelessbehaviorfortheholedopedsystem.[ 136 ]Wangetal.[ 69 ]furtherconsideredtheeffectoftheFermisurfacearoundM,whichweshallrefertoasthesheet,andemphasizedtheimportanceoftheorbitalmatrixelementsindeterminingthestructureofthegapinmomentumspace.Usingfunctionalrenormalizationgroupcalculations,theyndthattheycanchangetheanisotropyofthegapbytuningthetight-bindingmodel,whichaffectsthematrixelements.Thomaleetal.,[ 68 ]usingsimilarcalculations,alsoarguethatwhentheFermisurfaceisabsent,anodalsuperconductinggapispreferred.Theyreportthattheformationofnodesisduetointer-orbitalpairhoppingfromthetothesheetswhichdrivestheformationofweakaccidentalnodes.Toproperlymodeltheinteractions,wehavestartedfromthefulltwo-bodyCoulombinteraction,andhaveprojectedthisontotheFe3dorbitals, 58 ]andarerelatedtothoseusedbyGraseretal.[ 57 ] 69

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4-1 .Weshallusetheunfolded,1-Feunitcellthroughout.Forthisholedoping,thereisthepreviouslymentionedextraholeFermisurfacearoundthe(,)point,whichweshallrefertoasthesheet.Below,weshalluseanotationwhereweordertheFeorbitalsas(dxz,dyz,dxy,dx2y2,d3z2r2).Weshalldemonstratethatmatrixelementswhichrelatetheorbitalandbandstatesplayanimportantroleinthepairingsymmetry;wedenotethembya`(k)=h`ji(k).ThedominantorbitalweightsontheFermisurfaceareillustratedinthegure.InEq.( 4 ),wehaveseparatedtheinteractionintointra-andinter-orbitalCoulombrepulsion(UandU0),aswellastheHund'srulecouplingJandanalterm,denotedpairhoppingJ0.However,ifthereisnointeractioninthesystemthatbreaksthespin-rotationsymmetryofthesystem,forexamplespin-orbitcouplingorrenormalization,[ 137 ]thereareonlytwoindependentinteractions.Thus,ourinteractionsarerelatedbythespin-rotationalinvariancerelations:U0=U2J 4.4 .TheadvantageoftheRPAspin-uctuationexchangemethodisthepossibilityofanalyzingtheindividualorbitalcontributionstothepairformation.Thebasicpictureweputforwardisfairlysimple.Theintra-orbitalscatteringofdxzanddyzpairsbetweentheandFermisurfacesbyspinuctuations,whicharepeakedat(,0),leadtoasign-changinggapbetweenthetwotypesofsheets.However,intra-orbitalscatteringbetweenthedxysectionsofthesheetscompeteswiththeformationoftheisotropicA1gstate.Additionally,duetotheirreduciblevertex,thereisanintra-bandCoulomb 70

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4.2 webrieyreviewspin-uctuationexchangemethodology,andintroducetheeffectivepairscatteringverticesthatdeterminethegapstructure.Wediscusstheeffectofdopingforaparticularsetofspin-rotationallyinvariant(SRI)interactionparametersinsection 4.3 .WendthatthestrongestpairingsoccurfornodelessgapsintheA1gchannel,andfocusontheroleofthesheetonthepairingstructure.Wefurtherdiscussoriginofthetendencytopairinlikeorbitals,andelucidatetheeffectoftheJparameter.Insection 4.4 ,webreakspin-rotationinvarianceanddiscusstheeffectofJandJ0separately.Thenextsectionsofthischapterdiscusstheeffectsoftheorbitalcharacterandthepresenceofasurfaceonthepairingstate.Next,wederivesomeapproximateformsforthepairingvertex,anddiscusswhenthesearevalid. 138 ]sowewillbrieyreviewithereandextendittothecaseofmultipleorbitals,asisappropriateforthepnictides.Figure 4-2 illustratesthesetofdiagramsinthespin-uctuationmediatedpairinginteractioninthesingletchannel,asderivedbyBerkandSchrieffer.[ 6 ]Thebubblediagramseriesontherightwilldivergeasthesystemgoesintothemagneticstate.TheseriesofbubblescanbesummedusingtheRandomPhaseApproximation(RPA). 71

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2U20 139 ]Theinteractionstrengthischaracterizedbyadimensionlessparameter,justasthe!lninEliashbergtheory,givenbySF=Z10d!h=VS(q,!)i 4-3 .Forthismodel,whichmaybeappliedtothecuprates.theinteractionVS(q)ispositive,i.e.repulsive.ThiscanbereconciledwiththenotionoftheinteractionformingCooperpairsbyexaminingtheBCSequation:k=Xk0VS(kk0)k0 72

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4-1 ),consistsofsmallsheetsaround,andsmallsheetsaroundX(intheunfoldedzone).Second,itwasfoundthatthesusceptibilityispeakedaround(,0).Thisleadstoaninteractionthatispeakedat(,0),andthusasignchangeofthesuperconductinggapbetweenthesheetsaroundandthesheetsaroundX.ThisargumentwasproposedearlyonbyMazinetal.,[ 103 ]anditpredictsastateknownass.Thisstatehasanisotropic,ornearlyisotropic,gaponboththeandsheetsoftheFermisurface(seeFig. 4-1 ),withoppositesign.TheargumentbyMazinetal.makesakeyassumption:thattheinteractionisconstantaroundtheFermisurfacesduetotheirsmallsize.Thisiscompoundedbythefactthatrst-principlescalculationsindicatethattheorbitalcharactervariesstronglyasonegoesaroundtheFermisurfaces.Thus,wehaveconsideredtheinteractionsinorbitalspace,asopposedtobandspace,andwillnowreformulatetheabovetheoryformultipleorbitals.[ 140 ]WerstdenetheorbitalGreen'sfunctionGps(k,!)=Xap(k)as(k) 4-15 )(0)pqst(q,!)=1

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4-4 andisgivenby[ 141 142 ]`1`2`3`4(k,k0,!)=3 2Usspin(kk0,!)Us+1 2Us1 2Uccharge(kk0,!)Uc+1 2Uc`1`2`3`4 4 ).TheformsoftheinteractionmatricesUsandUcaregiveninsection 4.7 .AsillustratedinFig. 4-4 thereareintra-orbital,inter-orbital,andmixed-orbitalpairscatteringprocesses.ThecontributionsofeachtothetotalpairscatteringvertexijinEq.( 4 )arequitedifferent.Inparticular,asdiscussedbelow,theorbitalmatrixelementsforkandkstatesontheFermisurfacefavorpairswhichareformedfromelectronsinthesameorbitalstate.Wethereforendthat,inspiteofthefactthatthemixed-orbitalscatteringcanbesignicant,itscontributiontothepairinginteractionisnegligible.ConningourconsiderationstothevicinityoftheFermisurfaces,wecanprojecttheinteractionvertexontotheFermisurface.ForapairscatteringfrommomentumkonFermisurfaceitok0onFermisurfacejij(k,k0)=Xpqsta`2i(k)a`3i(k)<[pqst(k,k0,0)]a`1j(k0)a`4j(k0) 143 ][g(k)]=PijHiHjdkjjdk0jj (2)2PiHidkjj 74

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4-4 ). 144 ] 4-5 showsthegapfunctionsg(k)foratypicalsetofspin-rotationinvariantinteractionparametersU=1.3,U=0.9,J=J0=0.2andtwodifferentllingsn=6.03andn=5.95.Forlightelectrondoping,thereisnoFermisurfacepocketaroundandwendresultssimilartothoseobtainedinGraseretal.;[ 57 ]thegapisanisotropic,withnodesonthesheets.Thenodesarisepartiallybecausethesystemisfrustratedby12pairscattering,inparticularfromthedxysectionsofthesheets,asdiscussedin 75

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107 ]Additionally,thesignchangehelpsuppresstheeffectoftheintra-bandCoulombrepulsionduetotheirreduciblevertex.Asthesystemisholedoped,thepocketappearsandthenodesarelifted.Figure 4-6 illustratestheliftingofthenodesaroundthesheetsforxedinteractionparameters.Uponholedoping,thefrustrationcausedbythe12pairscatteringisnotreduced,butiscompensatedbyintra-orbitalscatteringbetweenthesheet,whichisentirelydxycharacter,andthesheets.[ 132 ]Ascanbeseenfromthepairingeigenvalue,thesheetalsoincreasestheoverallpairingstrength;thiseffectwouldcorrespondtoanincreaseofthecriticaltemperature.Toelucidatethiseffect,wehaveplottedtheorbital-projectedpairingvertices,asdenedinEq. 4 ,alonghigh-symmetrylinesintheBrillouinzoneforbothdopings(seeFig. 4-7 ).Thelargestfeatureintheplotisaresonanceintheintra-orbitalscatteringchannel2222neartheXpoint.Notethatthereisanequivalentpeakin1111ifonerotatestheBrillouinzoneby=2.Thispeakarisesfromtheresonantscatteringofdyzpairsbetweentheandsheets.Forthehole-dopedsystem,asimilarpeakarisesin3333,whichisduetotheintra-orbitalscatteringofdxypairsbetweentheandsheets.Thisscatteringcausesastabilizationofthenodelessstatebyovercomingtheintra-orbitaldxyscatteringat(,).Additionally,thereisariseinthemixed-pairscattering(2233)uponholedoping.However,asnotedpreviously,thisissuppressedbytherequirementthatbothparticlescomefromthesameband,whichisreectedinthematrixelementsinEq. 4 ;thisleadstothepairingbeingdominatedbyintra-andinter-orbitalscattering.Toillustratethesuppressionofthemixed-pairscattering,wehaveplottedthepairingvertexafterconvolutionwiththeorbitalmatrixelementsinFig. 4-8 (seeEq. 4 ).Theblackcircleintheplotsdenotesthemomentumofonememberofa(k,k)pair;theplotshowsthestrengthofthepairinginteractionij(k,k0)associatedwithscatteringofthispairtoapairwithmomenta(k0,k0).Keepinginmindtheorbitalcharacterofthevarious 76

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107 ]Whiletherearealsointer-orbitalscatteringprocesses,theseareweakerandthusdon'tcontributeasstrongly.ThiscanbeseeeitherfromFig. 4-7 ,orbynoting(fromsection 4.7 )thattheinter-orbitalchanneldepends,torstorder,onJ0,whichismuchsmallerthanU(whichcontrolstheintra-orbitalchannel).ItisclearfromthegurethatthereisstrongscatteringwithinthedxysectionsoftheFermisurface,whichbothfrustratesthesystem(12scattering)andstabilizesthenodelessstate(scattering).Sincethescatteringisstronger,thenodelessstateprevails.OnelastthingtonoteisthatthepairingstrengthishighlyanisotropicalongtheFermisheets,whichviolatesoneofthekeyassumptionsunderlyingtheargumentfortheisotropicsstate.WenowreturntotheeffectoftheinteractionparameterJonthepairingsymmetry.Figure 4-9 showsthegapfunctiong(k)forthehole-dopedsystem,wherethepocketispresent.OneseesthattheeffectofastrongHund'srulecouplingJistoliftthenodes.Weshallshowthatthisoccursfortworeasons.First,theintra-orbitalpairingverticesaaaaarecontrolledbyUandJ,asderivedinsection 4.7 .Themainpairingisdeterminedbytheintra-orbitalscattering,andthusincreasingJenhancestheoverallpairingduetospin-uctuations.Oneofthecausesfornodesistheirreduciblepartofthevertex,namelythestaticintra-bandCoulombrepulsionwhichisproportionaltoUc+Us.Increasingthestrengthofthespin-uctuationsrelativetotheintra-bandCoulombrepulsionreducestheimportanceofthestaticpart,leadingtodecreasedfrustrationandtheliftingofthenodes.Secondly,wendthatforsmallJ,thereisanattractivepotentialinthedxz-dxyinter-orbitalchannel1331.Thisattractionwilltendtowardsthesamesignonthesheetsandthedxysectionsofthesheets,causingnodesalongthesheets.TheincreaseofJremovesthisattraction,againcausingtheremovalofaneffectwhichfavorsnodes.Again,thediscussionabovealsooccursinaframerotatedby=2,switchingthedxzanddyzorbitals.Onanalnote,wendthatuponboth 77

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57 ] 4.7 ,toleadingorder,therelevantparametersgoingintotheinter-orbitalscatteringvertexareU0andJ0.Asdiscussedintheprevioussection,forspin-rotationinvariantinteractions,whereU0=U2J,thepairingisdominatedbytheintra-orbitalchannels.However,asnotedbyZhangetal.,[ 137 ]therenormalizationoftheinteractionscanbreakthespin-rotationinvarianceoftheinteractionsthatentertheuctuation-exchangeapproximation.Spin-rotationinvarianceappliestothebareCoulombmatrixelements,assumingnospin-orbitcouplingorsimilareffectsarepresent.However,theuctuation-exchangeapproximationdoesnotincorporatevertexcorrections.Indeed,Zhangetal.reportthatarenormalizationoftheinteractionsyieldsanapproximaterelationbetweentheinteractionsU+JU0+J0.Ifthisoccurs,theimportanceoftheinter-orbitalinteractionsmaybeenhanced.Furthermore,itisinstructivetoobservetheeffectsofJandJ0onthegapfunctionseparately.Here,forthehole-dopedsystemwherethepocketispresent,weholdUandU0xedandexaminetherolesJandJ0play.Figure 4-11 showstherelevantorbitalscatteringverticesandthegapfunctiong(k)alongthe1sheetforvariouscases.Comparing(b)and(c)to(a),wenotethatincreasingbothJandJ0removethenodes,althoughatdifferentrates.Thelargestfeatureinthescatteringverticesis,again,theintra-orbitalscattering2222.ThisfeatureisfurtherenhancedaseitherJorJ0areturnedon;thisincreasestheoverallpairingstrengthinbothcasesbyincreasingthepairingonthedyzsectionsoftheandsheets.Thisisfurtherreectedintheeigenvalue,whichincreasesfrom(a)=0.10to(b)=0.41and(c)=0.23forincreasingJandJ0,respectively.Notethatasmentionedabove,thegapfunctionisnormalizedto1overthewholeFermisurface,andshouldbe 78

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4 .Similarly,the1111channel(notshown)forcesthesignonand2tohaveopposingsigns.Thisleavesthesignofthedxysectionsonthesheetsfreetobedeterminedbytheotherorbitalverticesinvolvingadxycomponent.Asnotedabove,thepocketstabilizesanisotropicstatebycausingthedxyintra-orbitalscattering3333togrow,overcomingthescatteringandtheintra-bandCoulombrepulsion.Indeed,ascanbeseenfromFig. 4-11 (b),increasingJincreases3333neartheamomentumtransferof(,0),leadingtoaliftingofnodesandamoreisotropicgap.Thisisfurtherconrmedinsection 4.7 ,whereweseethatalargerJenhancestheresonancein3333,andthus3333.TurningnexttotheeffectofJ0,wenotethattheattractiveinteractioninthedxz-dxyinter-orbitalchannel(1331),whichwasunaffectedbytheturningonofJ,haschangedsignandisnowrepulsive.Asnotedinsection 4.3 ,thishastheeffectofliftingthenodesnear=0,.Comparingthegapfunctionin(c)to(b),weseethatalthoughthenodesareliftedinbothcases,thegapismoreanisotropicifjustJ0isturnedon.Thisiscoupledtoanoverallsmallereffectonthepairing,whichcanbeseenbothinthesmallerpairingeigenvalue,andinthesmaller3333and1331.Notethatintheapproximateformsderivedinsection 4.7 ,J0doesnotaffecttheintra-orbitalscattering;itonlyentersthroughtheoff-diagonalcomponentsof0,whicharesmall;thusJ0onlyweaklyenhancesthepairingincomparisontoJ.Finally,wediscusswhathappenswhenJ0isincreasedwithconstantJ,i.e.transitionb)!d).AsseenintheFigure,theanisotropyincreaseswithincreasingJ0inthiscase.ThemaineffectofincreasingJ0istoenhancethe2222intra-orbitalscattering,whichincreasesthegapnear==2asinthepreviouscases;however,theeffecton1331ismuchweakerthanina)!c). 79

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132 134 ]thereareothermethodsforbringingthisholebandabovetheFermisurface.Inparticular,theheightofthepnictogenabovetheFeplane,whichisknowntovaryinacrossthepnictidesuperconductors,canadjustthesizeofthepocket,whichadjuststhepairing.Additionally,insomematerialsthepnictogenheightissuchthatthepocketchangescharacterfromdxytod3z2r2.Tostudythisparticulareffect,wehaveperformedanadhocadjustmentofthehoppingintegralsinourmodel,whichwiththeproperchoiceofadjustmentcanmimicthiseffect.Wehaveadjustedthebandstructuresuchthattheonlymajorchangeisthechangeincharacterofthepocket,sotheonlyd3z2r2characterisfoundaroundtheMpoint,inagreementwiththebandstructurefoundbyCalderonetal.[ 134 ]Asarguedabove,thedominantcontributiontothepairingcomesfromintra-orbitalscattering;sincethenewpocketistheonlyplaceontheFermisurfaceonendsd3z2r2character,wecanexpecttorecovertheresultwherethesheetisabsententirely.Figure 4-12 showsthatthisisindeedthecase:thepairingeigenfunctionrevertstothenodalsolutionfoundfortheelectrondoped,andundoped(aspreviouslyreportedbyGraseretal.[ 57 ]). 58 ]whichoneimaginescanbeadjustedbypressure,chemicalpressure,andisovalentsubstitution.Hereweaddressanothermethod,onethatisparticularlyimportantforexperimentssuchasARPES:thepresenceofasurface.Asmentionedpreviously,todatetheARPESmeasurementsonthesematerialsallobserveanisotropic,ornearlyisotropicgap.Toprovidesomeinsightintothesemeasurements,wehaveperformedrst-principlescalculationsonaslabofBaFe2As2,consistingof6FeAslayers. 80

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81 ]Weusedultrasoftpseudopotentialstoallowforthestudyoflargersystemswithinaniteamountofcomputertime,theseallowedustouseanenergycut-offfortheplanewavebasisof40Ry,andadensitycut-offof400Ry.Tosimulateasurface,wekeptthemiddletwolayersxed,andlettheatomicpositionsofthesurfaceandintermediatelayersrelax.Wendadecreaseintheeffectivec-axislatticeconstantandAsheightby5%and13%,respectively.Tospecicallyaddressthequestionofthepresenceorabsenceoftheholepocket,wehavecalculatedtheelectronicbandstructurefortheentireslab(showninFig. 4-13 ).Notethattheseresultsareinthefoldedzone,andthatthisfoldingcausesthepockettoappearat(0,0)insteadof(,).Todeterminetheoriginofeach(k),weprojectedthewavefunctioncorrespondingtosaiddispersionpointontotheatomsofthesurfaceandbulkFeAslayers.Adispersionpointisconsideredtobelongtoaparticularatomiftheprojectionofitswavefunctionontheatomexceeds50%.Wehaveveriedthattheresultsdonotchangeappreciablyifthethresholdvalueisvaried.Onecanseefromthegurethatthepresenceofthesurfacecausestheholeband,whichisjustbelowtheFermilevelinthebulklayers,risesabovetheFermilevelinthesurface.Wehaveconrmedthatthepocketisstillofdxycharacter.Consideringthis,itispossiblethatthepresenceofthesurfacecausesthenodestoliftbythemechanismsdiscussedintheprevioussections,andthusresultinsurfacemeasurementsthatindicateanisotropicgap,evenwhenthebulkgaphasnodes. 4-4 ).SomeofthebasicverticesareshowninFig. 4-14 .Tolowestorder,thereareintra-orbital(a),inter-orbital(d)andmixed-orbital(b)and(c)pairscatteringprocesses.Forclarity,wehaveshownsomesecond-orderprocessesaswellin(e)-(g).Asdiscussedabove,themixed-orbitalprocessesshownin(b)and(c)willbe 81

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4-14 ,thattherst-orderintra-orbitalscatteringprocessesinvolveUandJ,whereastheinter-orbitalonesdependonU0andJ0.IntheRPAuctuationexchangeapproximation,thepairingduetothespin-uctuationsaregivenbyEq. 4 ,where 4 ),andcanthusbeexpressedinan(`1`2)basiswith 82

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Interactionmatrixinthereduced1(dxz),2(dyz),3(dxy)basis. 112233122113312332 11 and Asbefore,wehaveusedGreekletterstodenotethebandindices,andfistheFermifunction.Thetemperature,aseverywhereabove,hasbeenchosenas20meV.Combiningthelimitedweightofthed3z2r2anddx2y2andthesparsityoftheinteractionmatrices,wecannowreducetheinteractionmatricesto99;wehaveshownUsexplicitlybelow.Furthermore,ascanbeseenfromFig. 4-16 ,theoff-diagonalelementsofthebaresusceptibilityaresmallbecausetheyinvolvesingle-particlepropagatorsataxedmomentumbutwithdifferentorbitals.Thisallowsustosimplifyoursusceptibilityaswell.Thematrixbelowisshowninthereduced,3-orbitalbasis,andwehaveneglectedtheoff-diagonalcomponentsof0`1`2`3`4.Thecolumnslabelsofthematrixcorrespondto`1`2,andtherowlabelsto`3`4. 83

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20131 1(U0+J0)013+1 1(U0J0)013RPA1331=1 20131 1(U0+J0)0131 1(U0J0)013Inthesameway,fromtheupperleft33partoftheinteractionmatrixoneobtainsforexampleRPA1111=011U021U03J0203 84

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4(U0+J0)2013 2(U0)201331 4-14 ).Since01331isnegativenearX,thisresultsintheattractionnear(,0)mentionedabove.Thesignchangephysicallycorrespondstoaninteractionmediatedbydxz/dxyorbitalandspinuctuations.Theintra-orbitalpairscatteringinvolvesthe33subspace,andisthusmorecomplicated.However,inthediagonalsusceptibilityapproximation,thedeterminantthatentersthescatteringverticesisthesameasabove(seeEq. 4 ).Thus,thedenominatorisalsocontrolledbyUandJ.Additionally,Jcouplesthevariousintra-orbitalchannels,sothatforniteJ,1111reectsthepeaksinallthe0aaaa. 85

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58 ]asafunctionofdopingandpnictogenheight.UsingRPAspin-ucuationsasapairingmechanismthroughtheuctuationexchangeapproximation,wendthat,forourrangeofparameters,theleadingpairingstatehasA1gsymmetry.OurresultsshowthattheA1gsymmetrystatecanhavenodesdependingonthepresenceofthesheetandonthestrengthoftheHund'srulecoupling.Thesenodesareaccidentalnodes,i.e.notenforcedbysymmetryasinthecaseofd-wavesuperconductors,butratherdeterminedbytheexactdetailsoftheinteractionsandtheelectronicstructure.Wefurtherstudiedtheinteractionvertices,projectedontotheorbitalstates.Asdiscussedbyotherauthors,thedominantscatteringprocessesareintra-orbital.Thenodesintheundoped,andslightlyelectrondopedsystems,arecausedbytwoindependenteffects.First,intra-orbitalscatteringfrom1to2frustratestheisotropicstate.Secondly,theCoulombrepulsionfavorsanisotropy,andnodesinparticular.Theanistropyisfurtherenhancedbytheattractioninthedxz(dyz)todxychannel,whichdisappearsforniteHund'srulecoupling.Uponslightholedoping,aholepocketappearsaround(,).Forourmodel,basedonDFTcalculations,thepockethasdxycharacter.Intra-orbitalscatteringfromthe

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87

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Fermisheetsoftheve-bandmodelforn=6.03(top)andn=5.95(bottom)withcolorsindicatingmajorityorbitalcharacter(red=dxz,green=dyz,blue=dxy).NotetheFermisurfacesheetisaholepocketwhichappearsfor1%holedoping. 88

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Diagramscontributingtospin-uctuationinteractionsasnotedinthetext. Figure4-3. Schematicplotofthespin-uctuationmediatedinteractionforatwo-dimensionalsystemwithshort-rangeantiferromagneticinteraction. 89

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Top:pairingvertex`1,`2,`3,`4denedintermsoforbitalstates`iofincomingandoutgoingelectrons.Bottom:representativeexamplesofclassesoforbitalverticesreferredtointhetext:intra-,inter-andmixedorbitalvertices. 90

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Thegapeigenfunctionsg(k)foraspinrotationallyinvariantparametersetU=1.3,U0=0.9,J=J0=0,0.2. 91

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Thegapfunctiong(k)onthe1pocketforn=5.95,J=0.2(redsquares)andn=6.03,J=0.2(bluecircles).Heretheangleismeasuredfromthekx-axis. 92

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OrbitalpairingverticesalonghighsymmetrydirectionsinqspaceforU=1.3,andJ=0.2forn=5.95(bottom)andn=6.03(top),spinrotationinvarianceassumed.Solid(green)line:2222(intra);dashed(blue)2332(inter);dashed-dotted(red)2233(mixed).Notethattheverticalscalesinthetwopanelsaredifferent. 93

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Thetotalpairscatteringvertexij(k,k0)forn=5.95withparametersU=1.3andJ=0.2asafunctionofkwithk0settothepointontheFermisurfaceindicatedineachpanelbytheblackdot. 94

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Thegapfunctiong(k)onthe1pocketforn=5.95,J=0(blackcrosses)andn=5.95,J=0.2(redsquares).Heretheangleismeasuredfromthekx-axis. 95

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Theinter-orbitalpairscatteringvertex1331alonghighsymmetrydirectionsforn=5.95withparametersU=1.3andJ=0.0,0.2. 96

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Left:orbitalpairingvertices,andright:gapfunctiononthe1sheet,forU=1.3andU0=0.9andn=5.95.CasesshownareJ=J0=0(a,=0.10);J=0.2,J0=0(b,=0.41);J=0,J0=0.2(c,=0.23);J=0.2,J0=0.2(d,=0.64). 97

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Theeigenfunctionsforthehole-doped(x=1%)compoundwherethepocketcharacterhasbeenadjustedtobeofd3z2r2type.TheinteractionparametershavebeenchosenasU=1.3,J=0.2. 98

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TheDFTbandstructurecalculatedforaBaFe2As2slab(graypoints).TheredpointsshowthebulkcontributionsfromtheFeAslayers,whiletheblackpointsdenotethecorrespondingsurfacecontributions. 99

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Therstorder(a-d)andsomesecondorder(e-g)scatteringverticescorrespondingtointra-(a,e),inter-(d,f),andmixed-orbital(b,c,g)scatteringprocesses. Figure4-15. Noninteractingsusceptibility0`1,`2,`3,`4denedintermsoforbitalstates`iofincomingandoutgoingelectrons. 100

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Thenon-interactingsusceptibilities0`1`2`3`4forn=6.03. 101

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1 ,thepnictidesuperconductorshaveinspiredalargeeffortinmanyareasofcondensedmatterphysicsandchemistry.Inthischapter,wewillfocusonsuperconductivityinthe122classofmaterials.Here,angle-resolvedphotoemissionelectronspectroscopy(ARPES)measurementsperformedonhighqualitysinglecrystalsofBa0.6K0.4Fe2As2[ 97 102 ]havebeenveryinuential,reportingaposition,shapeandsizeoftheFermisurfacepocketsthatqualitativelyagreewithrst-principlescalculations.[ 13 145 ]InadditiontomeasuringtheFermisurface,ARPESexperimentsalsoreporttwodistinct,andisotropic,valuesoftheorderparameteralongtheFermisurfacesheets.Theseresultsseemtoconrmthatthegapisisotropic,andwouldthusbeconsistentwiththepredictionbyMazinetal.andDongetal.ofasign-changingisotropics-waveorderparameter.[ 103 104 ]However,asdiscussedinchapter 4 ,theseresultsneedtotakeintoaccountthepresenceofsurfaces.[ 66 ]Thesgapstructureisfurtherconrmedbyneutronscatteringonthe122compounds,whichreportaresonancenear(,0)intheunfoldedzone,whichcorrespondstothewavevectorconnectingtheholeandelectronpockets.[ 108 110 ]However,despitetheresultsmentionedabove,thesymmetryofthegapinthe122sisstillcontroversial.Justasinthe1111s(seechapter 4 ),manyexperimentalresultsindicatetheavailabilityoflow-lyingexcitations,whichmayindicatetheexistenceofgapnodes.TheseincludeNMR,[ 112 117 ]superuiddensity,[ 146 149 149 150 ]thermalconductivity,[ 119 122 ]andRamanscattering.[ 111 ]Inthe1111materials,severalauthorshavesuggestedatransitionfromfullygappedtonodalstatesfrombothmulti-orbitalspin-uctuationexchangeapproximationsandfunctionalrenormalizationgroupcalculations(seechapter 4 ).[ 66 69 132 151 152 ]Theseworkssuggestasensitivityofthesuperconductinggaptobothinteraction 102

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3 ,itisimportanttoconsiderthefullthree-dimensionalstructureofthe122s,andtostudywhetherthishasanyeffectonthepairingstate.Inthischapter,wewilldiscussthedevelopmentoffullthree-dimensionalmodelfortheBaFe2As2material,andusethismodeltostudythepairingstateusingthesamespin-uctuationexchangemethoddiscussedinchapter 4 5-1 ).Secondly,theplanesaremorestronglycoupledduetothesmallerdistancebetweentheFeAsplanesandthepositionoftheAsatomsrelativetotheplanesbeingoutofphasefromoneplanetothenext.[ 153 ]Thirdly,unlikeinthe1111s,wherethecontributionduetothelanthanide-oxideplanesliesfarawayfromtheFermisurface,thereisaBabandrelativelycloseby(seeFig. 3-3 ).Thismeansthatreductiontoaone-ironmodelhastostartwithintegratingouttheBacontribution.Despitetheseissues,onecanperformthenecessarystepsandindeedformaneffectivebandstructurebasedonasingleironperunitcell,startingfromattotherst-principlesbandstructure.[ 154 ]WehavecalculatedthebandstructureforBaFe2As2usingrst-principlesdensityfunctionaltheory,usingultrasoftpseudopotentials,asimplementedwithintheQuantumESPRESSOpackage.[ 81 ]Weusedexperimentallylatticeconstantsaswellascrystalstructure,I4/mmm,witha=3.9625A,c=13.0168A,andzAs=0.3545.[ 82 ]Wehaveplottedthebandsalonghigh-symmetrylinesoftheconventional(tetragonal)unitcellforeasiercomparisonwithpreviousworkonboththe1111sand122s.To 103

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155 ]todisentanglethebandsandseparateouttherelevantonesbyformingamodelofWannierfunctions,andminimizingtheirspread.AscanbeseenfromFig. 5-2 ,theWannierfunctionbandsaccuratelyrepresenttheDFTbandstructure,withtheexceptionofabandthatismainlyBa-characternearthepoint.Thisreectsthedifcultymentionedabove;however,sincethestatesweareinterestedinliebelowtheenergyrangewherethisbandcomesintoplay,itshouldnotaffecttheresultsbelow.Finally,tofurthersimplifythecalculationofthesusceptibilityandpairing,wettheWannierbandstoasimple5-orbitaltight-bindingmodel,andunfoldtheunitcellsoitonlycontainsoneFeperunitcell.Thettingprocedureneglectsthesmaller,longer-rangedhoppings,whichhastheeffectofdiffusingthelocalizedbasis;however,nottoapointwherethedescriptionoflocalizedorbitalsbecomesunphysical.TheHamiltonianisgivenas 5.3 .Figure 5-4 showscutsoftheFermisurfaceforalightlyholedopedcompoundatkz=0andkz=.ThecolorsdenotethelargestorbitalcontributiontotheFermisurfaceatthatpoint.Itisinterestingtonotethattheshapeandorbitalcompositionofthekz=0sliceifsimilartothatofLaOFeAs(seechapter 4 ).Ontheotherhand,thekz=slicelookstopologicallyandcompositionallyquitedifferent.Oneofthemajorchangesistheappearanceofasignicantamountofdx2y2characterinthe2pocket,whichwasnotpresenteitheratkz=0orinLaOFeAs.Ifwefocusonthekz=0Fermisurfaceonly,wewouldexpectthattwo-dimensionalmodelsofthe122storeproducethebehaviorofthe1111sstudiedabove. 104

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154 ]Thedispersionstheyobtainagreewithours,howevertheymadeadifferentgaugechoicefortheorbitalphases. 5-2 and 5-1 tabulatethehoppingmatrixelementsderivedfromourtight-bindingt.Weordertheorbitalsas(dxz,dyz,dx2y2,dxy,d3z2r2).Theon-siteenergiesfortheorbitalsare1=2=0.0987,3=0.3595,4=0.2078,and5=0.7516,asmeasuredfromtheFermienergy.Here,aswellasinthetables,allenergiesareineV.Thedispersionsare11=22=2t11x=ycoskx+2t11y=xcosky+4t11xycoskxcosky2t11xx(cos2kxcos2ky)+4t11xxy=xyycos2kxcosky+4t11xyy=xxycos2kycoskx+4t11xxyycos(2kx)cos(2ky)+4t11xz(coskx+cosky)coskz4t11xxz(cos2kxcos2ky)coskz33=2t33x(coskx+cosky)+4t33xycoskxcosky+2t33xx(cos2kx+cos2ky)44=2t44x(coskx+cosky)+4t44xycoskxcosky+2t44xx(cos2kx+cos2ky)+4t44xxy(cos2kxcosky+cos2kycoskx)+4t44xxyycos2kxcos2ky+2t44zcoskz+4t44xz(coskx+cosky)coskz+8t44xyzcoskxcoskycoskz55=2t55x(coskx+cosky)+2t55xx(cos2kx+cos2ky)+4t55xxy(cos2kxcosky+cos2kycoskx)+4t55xxyycos2kxcos2ky+2t55zcoskz+4t55xz(coskx+cosky)coskz12=4t12xysinkxsinky+4t12xxy(sin2kxsinky+sin2kysinkx)+4t12xxyysin2kxsin2ky+8t12xyzsinkxsinkycoskz13=23=2it13xsinky=x+4it13xysinky=xcoskx=y4it13xxy(sin2ky=xcoskx=ycos2kx=ysinky=x)14=24=2it14xsinkx=y4it14xycosky=xsinkx=y4it14xxysin2kx=ycosky=x4it14xzsinkx=ycoskz8it14xyzcosky=xsinkx=ycoskz8it14xxyzsin2kx=ycosky=xcoskz

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4 )intwoways.First,weshallneglectthekzdispersionofthebandsandsimplycalculatethetwo-dimensionalsusceptibilitiesRPA(~q,!=0)usingeitherthekz=0andkz=planes,withtheinternalmomentumsum(seeEq. 4 )restrictedtotheappropriatetwo-dimensionalplane.Next,wecalculatethefullthree-dimensionalsusceptibilityusingthefull3Ddispersionderivedaboveandcomparetothetwo-dimensionalresults.Forclarity,themomentumtransferisalwaysdenotedbyq,andtheBrillouinzone(intheformercase)byk.Fortheintegrationweusea646420k-meshandwenoticeonlynegligiblenitesizeeffects,thatshowupasweakoscillationsatsmallq.ThecalculationoftheRPAsusceptibilityandpairingsymmetryisdescribedindetailinsection 4.2 ,andwewillfollowthesameprocedurebelow.Below,wepresentcalculationsofthesusceptibilityforzeroandniteHund'srulecoupling.Wewillmaintainspin-rotationinvariance,andthusonlytwofreeparametersremain:UandJ.Theparametersarechosentobeclosetothesuperconductinginstabilitytoremovethepossibilityofanotherstateovertakingtheonereported.NotethatalthoughtheratioofJtoUissmallerthanreportedinMiyakeetal.[ 154 ]fromrst-principlescalculations,buttheoverallscaleissmaller.However,asnotedbyBulutetal.theeffectiveinteractionsfortheRPAarealreadyrenormalized,andtendtobesmallerthanthebarevalues.[ 156 ] 106

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5-5 showstherealpartofthetwo-dimensional,zerofrequencysusceptibility(aswouldbemeasurede.g.byelasticneutronscattering)fortheholedopedcompound, 2Xsp(1)sspp(q,0)(5)Asdiscussedabove,the2Dsusceptibilityisobtainedbyintegrationovera2DFermisurfacecut.Weshowthesusceptibilitycalculatedforkz=0andkz=,fortwosetsofinteractionparameters.Oneimmediateobservationisthatthekz=0susceptibilityishardlypeaked,whereastheoneaskz=haspeakswithmaximaroughly10timeslargerthanthemaximaofkz=0(notethescalesintheplotsaredifferent).Oneofthepeaksinthekz=susceptibilityiscommensurate,i.e.centeredatq=(,0).Theenhancementofthepeaksisduetotheadditionalintra-orbitalnestingbetweenthedxysectionsoftheholeandelectronpockets;inthekz=0plane,neitherofthesheetshavealargedxycontribution.Comparingthetwosetsofinteractionparameters,wendthataniteHund'srulecoupling(andpairhopping)stronglyenhancethepeakatXrelativetothepeakatMinthekz=cut(notethatthevalueoftheinteractionparameterUmustbechosensuchthatthesusceptibilityremainsnite).Sincethesusceptibilityatkz=ismuchlarger,weexpectthatuponperformingthefull3Dcalculation,thecontributionfromkz=willdominatethesusceptibility,althoughlessstronglyduetotheeffectiveaveragingoverkz.Indeed,asexpected,themostsignicantpeakinthefull3Dsusceptibilityoccursatthecommensuratewavevectorq=(,0),forbothsetsofinteractionparameters.However,thepeakat(,)hastransformedfromastrongpeakattoabroadplateau-likestructure,withthelargestsignalinanincommensurateridgecirclingtheMpoint.ForniteHund'srulecoupling,thesubstructureinthefeatureatMdisappears,andabroadfeatureoccursinstead.AsimilarcommensurateresultwasobservedbyInosovetal.[ 157 ]inthenormalstateinelasticneutronscattering(INS) 107

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5-7 and 5-8 showequivalentquantitiestothoseabove,exceptforzerodoping.Inthetwo-dimensionalsusceptibility,astrongdependenceontheinteractionparametersdevelopsforthepeakat(,0)inthekz=cut,andadditionalincommensurateresponsesappearinthesameplane.Thethree-dimensionalsusceptibilitylosesallpeakstructuresinfavorofincommensurateridgesofresponseencirclingtheMpoint.ThecommensuratepeakatXthatwasseeninthehole-dopedcasehasshiftedalongtheX-Mline,andisofequalweightastheridge. 4.2 above.ForthenumericalcalculationofthegapfunctionweparametrizetheFermisurfacebyadensemeshof1200kvaluesdistributedoverthe5differentFermisurfacesheetsasshowninFig. 5-9 .FirstwestudythepairingfunctionataxedkzcutoftheBZ.Inordertosolvetheeigenvalueproblemwerstuseaneffectivepairinginteractionij(k,k0)calculatedfromthetwo-dimensionalandthree-dimensionalsusceptibilitiesderivedintheprevioussection.SincetheFermisurfacehasstrongvariationintheorbitalcontributionsgoingfromkz=0tokz=,weexpectstrongvariationinthepairingfunctionsforthevariousslicesaswell.Figure 5-10 showsthepairingfunctionsforU=0.65andJ=0,forthelightlyhole-dopedcompound.The 108

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66 144 ]andinchapter 4 ,originatesinafrustrationbetween(,0)and(,)scattering,theintra-bandCoulombinteraction,andintheorbitalcharacteroftheFermisurfacestates.Foroursecondsetofinteractionparameters,U=0.55andJ=0.25U,althoughtheoveralleigenvaluesarereduced,bothcutsnowhaveans-wavestateastheleadingeigenvalue(seeFig. 5-11 ).ReferringtothecorrespondingsusceptibilitywenotethatforthesecondsetofinteractionparametersthepeakatMisreducedwithrespecttotheoneatX;thisremovessomeofthefrustrationandthusalsosomeoftheanisotropyinthegapfunction.Asisevidentfromtheabove,thegapfunctionstructureinthevariouscutsisnotthesame.Ifthegapsymmetryinthekz=0andkz=planesweresimilar,thenanargumentcouldbemadethatthetwo-dimensionalmodelissufcient.However,ourresultimpliesthatoneneedstodoafull3Dcalculationtoreconcilethetworesults.Yet,itisinterestingtonotethatthemajorityofthepairingstrengthcomesfromthekz=cut.Thenextstepistocalculatethepairingfunctiononthefull3DFermisurface.Thisinvolvesthethree-dimensionalsusceptibilityreportedabove.Asitonlyweaklydependsonqz,weexpectthepairingtobemorehomogenousacrosstheBrillouinzone.Figure 5-12 showstheleadingpairingfunctionsfortwosetsofinteractionparameters,onewithJniteandonewithout,forthehole-dopedcompoundatkz=0andkz=.ForbothsetsofinteractionparameterswendanA1gstatewithhighanisotropy,andnear-nodes,onthesheets.Furthermore,nodesdevelopasafunctionofkzonthesheets.ThelackofcleardependenceontheparameterJmaybeduetotheeffective 109

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5-13 showstheleadingandsubleadingpairingfunctionsalongthesamekzcutsforU=0.9andJ=0.25U,fortheundopedcompound.Notethatthesheetisnotpresentforthisdoping,andthe1sheetisabsentatkz=.Wendthattheleadingeigenfunctionhereisd-wave,withasubleadings-wavesolution.

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158 ]thec-axispenetrationdepthshowsalinear-Tbehavior,indicativeofgapnodesalongthecaxis.Notethatourworkhasnotfullyexploredtheparameterspaceavailable,aswasdoneinthepreviouschapters.Instead,wefocusedonthenovelaspectsoftheBaFe2As2modelwhichwerenotcapturedintheprevious1111-basedworks,inparticulartheeffectofthree-dimensionality.Furthermore,wehaveonlyconsideredthepossibilityofholedoping.Asdiscussedabove,thereareothervariableswhichaffecttheelectronicstructure,includingthepresenceofasurface,pressure,andpnictogenposition,thatcanaffectthepairingstate.Furtherworkisneededtofullycapturetheseeffectsforpossiblequantitativecomparisonstoexperiment. 111

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Figure5-1. SketchoftheBrillouinzoneoftheI4/mmmcrystalsymmetry(a)andofthelargeeffectiveBZcorrespondingtothe1Fe/unitcell(b).Thebluelineshowsthetwopathsinthe1Fe/unitcellBZthathavetobefoldedbythereciprocallatticevectorT=(,,)(redarrow)togivethecorrespondingpathinthe2Fe/unitcellBZoftheP4/nmmsymmetry. Table5-1. TheinterorbitalhoppingparametersusedfortheDFTtofthe5orbitalmodel. 112

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TheintraorbitalhoppingparametersusedfortheDFTtofthe5orbitalmodel.

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(b) Figure5-2. TheparamagneticDFTbandstructure(fullline)andaWanniert(crosses)ofthe10bandsinthevicinityoftheFermisurfaceontotheFe-3dorbitals(a).The5-orbitaltight-bindingt(coloredpoints)ofthe10-orbitalWanniert(blackpoints)withacolorcodingofthemainorbitalcontributions(b).Thecolorscorrespondtodxz(red),dyz(green),dxy(blue),dx2y2(orange),andd3z2r2(magenta).AllenergiesaremeasuredfromtheFermienergyEF=10.86eV

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Thepartialdensityofstatesofthe5-orbitaltight-bindingt,usingthesamecolorcodingasinFig. 5-2 b. 115

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(b) Figure5-4. ThemainorbitalcontributionstotheFermisurfacesatkz=0(a)andkz=(b)usingthesamecolorcodingasinFig. 5-2 b. 116

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117

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5-5 118

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5-5 119

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5-5 120

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Fermisurfacemeshforthecalculationofthepairingfunctions.Hereweused2410k-pointsforeveryFermisurfacesheetwith1(red),2(blue),1,2(green),and(yellow). 121

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122

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123

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124

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125

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6-1 ).QuantumoscillationsinP-dopedBaFe2As2,LaFePOandSrFe2As2indicatethattheelectronpocketshavealongermeanfreepath.[ 159 161 ]Similarly,Halleffectmeasurementsndthatthetransportisdominatedbytheelectronpockets,[ 162 164 ]andopticalmeasurementsandRamanscatteringreportalargerscatteringrateontheholepocketsaswell.[ 111 165 ]AmodelcalculationbyJaroszynskietal.ndsthatatleastanorderofmagnitudedifferenceinthemobilitiesisneededinordertoexplainthetemperaturedependenceoftheuppercriticaleldinF-dopedNdFeAsO.[ 166 ]Fangetal.[ 163 ]performedadetailedstudyandndthattherelaxationratefortheelectronsrstdecreasesuponcooling,andthendropsprecipitouslyuponenteringthespin-densitywavestate(SDW).Furthermore,upondoping,thedisparitybetween 126

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57 69 103 104 ]Wenowconsidertheeffectofspinuctuationsonthelifetimesinatight-bindingmodelforLaOFeAs.Todoso,wecalculatethelifetimecorrectionsduetothesecondorderself-energydiagramshowninFig. 6-2 ,wherewehaverenormalizedthebaresusceptibilitywiththerandomphaseapproximation.Inprincipleoneneedstocalculatetherealpartoftheself-energyaswell,whichrenormalizesthebandstructure.However,asnotedinsomedetailbyIkedaetal.,therenormalizationofthedispersionsduetoself-energyeffectscausesanunphysicalshiftofthebandstructurerelativetoexperiments.[ 135 ]Thisinsomesenseduetoanovercountingproblem;thebandstructureisbasedonrst-principlesdensityfunctionaltheory(DFT).DFTincludesanumberofeffectsbeyondthefree-electronapproximation,includingHartree-Fockcorrectionsandbeyond.Theself-energycalculatedbyIkedaetal.alsoincludesthesecorrections,thusdouble-countingtheircontributions.Indeed,onerecoverswhatappearstobethecorrectARPESbandstructurebysimplyneglectingthebandrenormalizations.Ontheotherhand,astudybyOrtenzietal.[ 167 ]ndthatthebandstructurefromDFTinfactneedsthesecorrectionstoagreewithARPESresults.Theexactoriginofthisdisagreementisunclear,howeveritisworthnotingthatthetwogroupsusesubstantiallydifferentinteractions:Iketaetal.useanorbitalbasisfortheinteractions,whereasOrtenzietal.useabandbasis.Thisfact,inadditiontothedifferencesinmethodologyandunderlyingDFTbandstructuremakesadirectcomparisondifcult.Inlightofthiscontradiction,wefocusjustonthelifetimecorrectionstotheself-energy.Withinaspinuctuationmodelwithorbitalinteractions,wendthat,inagreementwithexperiments,thescatteringrateontheelectronpocketsissignicantly 127

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57 66 107 ] 57 ]Tosimplifythecalculationslightly,wewilltransformtheHamiltonianthroughagaugetransformationsuchthattheeigenvectorsarereal.Initially,theHamiltonianisoftheformH=0BBBBBBBBBBB@1112 128

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0000e3=2 00 0e3=3000 00e3=31CCCCCCCCCCCA=0BBBBBBBBBBB@i0 0000i 00 01000 0011CCCCCCCCCCCA 6-2 ).Thisisa 129

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168 ],whichcaneasilybeextendedtomultipleorbitals:Im~ps(q,)=1 2Nk~Uwzpq~UuvrsXkIm~wzvu(qk,k)~aq(qk)~ar(qk) 4 ,andwethesumoverinternalorbitalindicesisimplied.WeshallworkatT=0,wherethisexpressionsimpliesto:Im~ps(q,)=1 4 ).Fortractability,weshallderivealltheinteractionlinesforthecaseoftwoorbitals.Alltheinteractionsinvolveeitherscatteringwithinasingleorbital,orbetweentwoorbitals;therefore,weshallrestrictthederivationtoscatteringinatwo-orbitalsubspace,withtheunderstandingthattheinteractionsderivedbelowwilloccurineverytwo-orbitalsubspace.Weshalluseanoccupationnumbernotationinthetwo-orbitalbasistoderivetheinteractionlines:j>=j1"1#2"2#> 130

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2U0j1001> 2(U0J)1 2Jj0110> 2J1 2(U0J)j0101> 2U0j0011> 1. 2(U0J)(U1)abab=0(U1)baab=1 2(U0J)(U2)aaaa=U(U2)bbaa=1 2U0(U2)abab=J0(U2)baab=1 2J(U3)aaaa=U(U3)bbaa=1 2J(U3)abab=J0(U3)baab=1 2U0 2cabcd+1 6sabcd~~ 131

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2cabcd+1 6sabcd,triplet2 6sabcd,singlet 2(U1)2+(U2)2c+1 6(U1)2+(U2)2+2(U3)2s 6-3 showstheinverselifetime(1 57 ]Intheundopedsystem,thedyzsectionofthe1sheetiswellnestedwiththedyzsectionof2,whichleadstostrongscatteringalongthosesectionsoftheFermisurface.Thisiscompoundedbythefactthatthepresenceofmatrixelementsfavorsintra-orbitalscattering;aneffectwhichisampliedwhenonlyUisnonzero.Whenthechemicalpotentialisloweredandthesheetappears,weshouldthusexpectthatthescatteringwillbecomemoreisotropicasthedxysectionsofthesheetscannowscattertothesheet,whichisentirelydxy;wendthatthis 132

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6-1 showstheaveragescatteringratesobtainedforeachoftheFermisurfaces.Incomparisonwiththegure,theaveragescatteringrateshowsanevenlargerdisparityintheratioofholetoelectronscatteringrate.ThisenhancementisbecausethebandswithlowFermivelocitieswillhaveahigheraveragescatteringrate(seeEq. 6 ).Notethatthisisacrudeestimateofascatteringratewhichmightenteratransportproperty,includingtheeffectsoftheFermivelocityanisotropy.However,itneglectsthedifferentverticesandvertexcorrectionsappropriateforthetransportcoefcientsmeasuredexperimentally.Thus,thisestimateisonlyintendedtoprovidearoughmeasureofthehole-electronscatteringanisotropy. 133

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4 ,wediscussedtheimportanceoftheintra-orbitaleffectswithrespecttothevariousinter-orbitalones.Thus,uponinclusionoftheotherinteractions,weexpectthattheseconclusionsdonotchangeappreciably.Thishastobechecked,however,anditiscurrentlyunderinvestigation. Doping12 Table6-1. AveragescatteringratesforeachFermisurfacesheetat!=10meV.TheinteractionparametersareU=1.55eV,V=J=0eV. 134

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Fermisurfacefor8%holedopedLaOFeAs.ThecolorsindicatethemajorityorbitalcontributionstotheFermisurface.Intheundopedandelectrondopedsystems,thepocketfallsbelowtheFermisurfaceanddisappears. 135

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Firstcontributiontothequasiparticlelifetime.TheRomanindicesdenotetheorbitalcharacter;theGreekindicesdenotethebandindices. Figure6-3. Inverselifetime1 136

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AlexanderKemper,betterknownasLex,wasbornintheNetherlandsin1981,inthetownofZaanstad.Hewouldbynomeansremainthere.Overthecourseofthenext29years,hemovedwithintheNetherlandstwice,backandforthacrosstheAtlantictothesmallislandofCuracaothreetimes,andnallytotheUnitedStates,wherehemadehiswayupfromMiamitoGainesville.Movingaroundalothadadvantagesanddisadvantages.Bylivinginmultiplecultures,onecanpickupthebestpartsofeach,andlearnalotaboutvariouspeoples.Ontheotherhand,itleavesonewithoutapermanenthome.However,GainesvillehasbecomeLex'shome,ashehasspentthemajorityofhisadultlifethere.HestartedseriouslystudyingphysicsintheFallof2003,whenhemovedtoGainesville.Afterreceivinghisbachelor'sdegreeinmathandphysics,heremainedattheUniversityofFloridaandenteredthephysicsdoctoralprogram.Now,nearlysixyearslater,hehasgraduated,andislookingforwardtotherestofhislifeasaphysicist. 147