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Thermodynamic and Transport Properties of Unconventional Superconductors and Multiferroics

Permanent Link: http://ufdc.ufl.edu/UFE0042391/00001

Material Information

Title: Thermodynamic and Transport Properties of Unconventional Superconductors and Multiferroics
Physical Description: 1 online resource (155 p.)
Language: english
Creator: Boyd, Gregory
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: cuprate, multiferroic, pnictide, 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: Often, the phase diagram for a given material can be quite complex, presenting evidence for multiple orders and it is the task of the condensed matter community to describe and quantify knowledge of these properties. Significant insight can often be gained by comparing model calculations of basic thermodynamic and transport properties of a material with experiment. Here we consider two classes of novel materials whose rich phase diagrams are actively under investigation: unconventional superconductors and multiferroics. In 2006, H.Hosono discovered a new class of iron-based superconducting materials which are not conventional superconductors. After the initial discovery, there is a range of questions of immediate interest; foremost among them is what is the structure and symmetry of the superconducting state, a question which took roughly a decade to answer for the cuprate superconductors. We present calculations that help reveal the structure of the superconducting gap using angle dependent specific heat measurements. We then calculate the electronic Raman scattering intensity for several polarizations of light and different models of disorder, providing information about the anisotropy and location of nodes in the superconducting gap. Understanding the influence of disorder is considered crucial because currently conflicting experimental results may be due to differences in sample quality. Recently, there has also been interest in multiferroics: materials with simultaneous non-zero polarization and magnetic order. We present calculations of fundamental thermodynamic properties, mean field behavior for the simplest ferromagnetic-ferroelectric, characterize topological defects, and use the perturbative renormalization group to help understand the critical point, as a beginning towards understanding the multitude multiferroic materials with increasingly complex magnetic and polar order.
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 Gregory Boyd.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Hirschfeld, Peter J.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-06-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042391:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042391/00001

Material Information

Title: Thermodynamic and Transport Properties of Unconventional Superconductors and Multiferroics
Physical Description: 1 online resource (155 p.)
Language: english
Creator: Boyd, Gregory
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: cuprate, multiferroic, pnictide, 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: Often, the phase diagram for a given material can be quite complex, presenting evidence for multiple orders and it is the task of the condensed matter community to describe and quantify knowledge of these properties. Significant insight can often be gained by comparing model calculations of basic thermodynamic and transport properties of a material with experiment. Here we consider two classes of novel materials whose rich phase diagrams are actively under investigation: unconventional superconductors and multiferroics. In 2006, H.Hosono discovered a new class of iron-based superconducting materials which are not conventional superconductors. After the initial discovery, there is a range of questions of immediate interest; foremost among them is what is the structure and symmetry of the superconducting state, a question which took roughly a decade to answer for the cuprate superconductors. We present calculations that help reveal the structure of the superconducting gap using angle dependent specific heat measurements. We then calculate the electronic Raman scattering intensity for several polarizations of light and different models of disorder, providing information about the anisotropy and location of nodes in the superconducting gap. Understanding the influence of disorder is considered crucial because currently conflicting experimental results may be due to differences in sample quality. Recently, there has also been interest in multiferroics: materials with simultaneous non-zero polarization and magnetic order. We present calculations of fundamental thermodynamic properties, mean field behavior for the simplest ferromagnetic-ferroelectric, characterize topological defects, and use the perturbative renormalization group to help understand the critical point, as a beginning towards understanding the multitude multiferroic materials with increasingly complex magnetic and polar order.
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 Gregory Boyd.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Hirschfeld, Peter J.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-06-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042391:00001


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THERMODYNAMICANDTRANSPORTPROPERTIESOFUNCONVENTIONALSUPERCONDUCTORSANDMULTIFERROICSByG.R.BOYDADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2010

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c2010G.R.Boyd 2

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Themysteryseemedtohimalmostcrystallinenow;hewasmortiedtohavededicatedahundreddaystoit.-J.L.Borges,DeathandtheCompass,inFicciones 3

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ACKNOWLEDGMENTS P.J.HirschfeldhadthepatiencetoadvisemefornearlyallofmytimeinGainesville,andIowemuchtohim.Thankyoufortheyearsofyourtime.IwouldliketothankD.MaslovandK.Ingersentfortheirlucidandinformativelectures.LexKemperandChrisPankowhaveeldedmanyofmyquestions,andIappreciatetheirinputovertheyears.TheworkdoneinconjunctionwithS.Graserwasparticularlyproductive,somethingforwhichIthankhim.ProfessorsDorsey,Stewart,Matchev,andPhillpotarealsomembersofmycommittee.P.KumarandIhavehadmanyconversationsaboutmultiferroicsandmightwriteapapertogetheronedayalso.IamgratefultohaveM.FischerasafriendsinceourtimeinCargese.IwouldliketothankU.Paris-SudatOrsayfortheirhospitalityduringourvisit.Jevousremercie.InthetimebeforeFlorida,IfeelparticularlyindebtedtoGerryGuralnik,HerbFried,BradMarston,andMikeKosterlitzevensomanyyearslater.B.OvrutandV.Balasubramanianhavealsohelpedgetmehereintheirownways,andthisjourneybegunwiththededicatedeffortsofArtTimmons,MarlaWeiss,andEdBerger.Toalltherestofmyformerandcurrentcolleagues,thankyou. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTIONTOSUPERCONDUCTIVITY .................. 13 1.1ConventionalSuperconductivity ........................ 13 1.2TheoreticalDescription ............................ 14 1.3ExperimentsinConventionalSuperconductors ............... 19 1.3.1SpecicHeat .............................. 19 1.3.2Tunneling ................................ 20 1.3.3NuclearMagneticResonance ..................... 23 1.3.4OpticalConductivity .......................... 25 1.3.5ThermalConductivity .......................... 26 1.4UnconventionalSuperconductivity ...................... 28 1.4.1UnconventionalPairing ......................... 28 1.4.2Migdal-EliashbergTheory ....................... 32 2SUPERCONDUCTIVITYINCUPRATESANDPNICTIDES ........... 35 2.1Cuprates .................................... 35 2.1.1BasicCupratePhysics ......................... 35 2.1.2Tunneling ................................ 39 2.1.3SpecicHeat .............................. 43 2.1.4AngleResolvedPhotoemissionSpectroscopy ............ 44 2.1.5NuclearMagneticResonance ..................... 49 2.1.6Neutrons ................................. 51 2.1.7PenetrationDepth ........................... 52 2.1.8ElectricandThermalConductivity ................... 53 2.1.9Nernst,Kerr,SR ............................ 56 2.1.10RemarksonHighTcExperiments ................... 59 2.2Pnictides .................................... 60 2.2.1OpticalConductivity .......................... 63 2.2.2ARPES ................................. 65 2.2.3NMR ................................... 67 2.2.4Neutrons ................................. 69 2.2.5PenetrationDepth ........................... 71 2.2.6HeatTransport ............................. 72 5

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2.3TheoreticalSuggestions ............................ 74 2.3.1SpinFluctuations ............................ 74 2.3.2ResonatingValenceBondsandSlave-Bosons ............ 75 2.3.3QuantumCriticality ........................... 78 2.3.4CompetingOrder ............................ 80 2.3.5BEC-BCScrossover .......................... 81 3ANGLE-DEPENDENTTHERMODYNAMICS ................... 83 3.1DensityofStatesandLowTemperatureSpecicHeatintheVortexState 83 3.2TemperatureVariationoftheSpecicHeatinanAppliedField ....... 89 4RAMANSCATTERING ............................... 95 4.1Theory ...................................... 96 4.2AddingImpurities ................................ 101 4.3ModelSuperconductingGapsfortheRamanResponse .......... 104 4.4ModelingExperimentalRamanData ..................... 108 5MULTIFERROICS .................................. 115 5.1History ...................................... 115 5.2FerroelectricBackground ........................... 115 5.3MagneticBackground ............................. 118 5.4MultiferroicIntroduction ............................ 120 5.5Thermodynamics ................................ 123 5.5.1MaxwellRelations ........................... 123 5.5.2PhaseDiagram ............................. 124 5.5.3AdiabaticProcesses .......................... 125 5.6FreeEnergyFunctional ............................ 126 5.6.1OrderParameter ............................ 127 5.6.2Susceptibility .............................. 128 5.6.3SpecicHeat .............................. 130 5.7InhomogeneousEffects ............................ 131 5.8SpeccHeatwithGaussianFluctuations ................... 135 5.9FutureWorkonDynamics ........................... 138 5.10Conclusions ................................... 139 6CONCLUSION .................................... 141 REFERENCES ....................................... 143 BIOGRAPHICALSKETCH ................................ 155 6

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LISTOFTABLES Table page 1-1D4hSelectedSingletRepresentations ....................... 29 1-2D4hSelectedTripletRepresentations ........................ 30 7

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LISTOFFIGURES Figure page 1-1TinSpecicHeat ................................... 20 1-2LeadDensityofStates ................................ 22 1-3AluminiumNMR ................................... 24 1-4ConductivityofAl ................................... 26 1-5ThermalConductivityinAl .............................. 27 1-6StrongCouplingtunnellinginPb .......................... 33 2-1CuprateStructure .................................. 35 2-2CartoonofPhaseDiagram ............................. 36 2-3BSCCODOS ..................................... 40 2-4STMBSCCO ..................................... 41 2-5LocalDOSZn .................................... 42 2-6specicheatofYBCO ................................ 43 2-7ARPESFS ...................................... 46 2-8ARPESphasediagram ............................... 46 2-9FermiArc ....................................... 47 2-10ARPESsuperconductingspectralpeak ...................... 48 2-11Pseudogap'sangulardependence ......................... 48 2-12NMRYBCO ...................................... 49 2-13NMRcartoon ..................................... 50 2-14NeutronExcitationspectrum ............................ 51 2-15YBCOpenetrationdepth ............................... 52 2-16Dopingdependenceofcuprategap ........................ 55 2-17LinearTresistivity .................................. 56 2-18NernstmeasurementinLSCO ........................... 57 2-19KerrEffect ....................................... 58 8

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2-20PolarNeutronMeasurement ............................ 59 2-21122Structure ..................................... 61 2-22122Phasediagram ................................. 62 2-23SDWdrudeweight .................................. 65 2-24Ba-122FermiSurface ................................ 66 2-25Nd-1111GapfromARPES ............................. 67 2-261/T1in1111 ..................................... 68 2-27InelasticNeutronscatteringinPnictides ...................... 70 2-28PnictidePenetrationDepth ............................. 72 2-29c-axisHeattransport ................................. 73 2-30Spinuctuationdiagrams .............................. 75 2-31RVBandQCPphasediagrams ........................... 78 3-1Angle-DependentDOS ............................... 87 3-2SpecicHeatvsT .................................. 88 3-3InversionofSpecicHeat .............................. 90 3-4HigheldSpecicHeat ............................... 91 3-511-pnictidespecicmeasurements ......................... 93 4-1TMatrix ........................................ 102 4-2A1gGap ....................................... 105 4-3Swithimpurities .................................. 106 4-4DwaveResponse .................................. 107 4-5ExperimentalRaman ................................. 108 4-6Co-122BrillouinZone ................................ 109 4-7ExperimentalRaman ................................. 110 4-8IntrabandScattering ................................. 111 4-9InterbandScattering ................................. 112 4-10RamanIntesityCo-122 ............................... 113 9

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5-1Cr2O3 ......................................... 120 5-2OrderParametersvsT ................................ 128 5-3Susceptibilities .................................... 129 5-4Magnetoelectricsusceptibility ............................ 129 5-5SpecicHeatforCoupledOrderParameters ................... 131 5-6FluctuationContributiontoCv ............................ 136 5-7OneLoopFeynmanDiagrams ........................... 137 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyTHERMODYNAMICANDTRANSPORTPROPERTIESOFUNCONVENTIONALSUPERCONDUCTORSANDMULTIFERROICSByG.R.BoydDecember2010Chair:P.J.HirschfeldMajor:Physics Often,thephasediagramforagivenmaterialcanbequitecomplex,presentingevidenceformultipleordersanditisthetaskofthecondensedmattercommunitytodescribeandquantifyknowledgeoftheseproperties.Signicantinsightcanoftenbegainedbycomparingmodelcalculationsofbasicthermodynamicandtransportpropertiesofamaterialwithexperiment.Hereweconsidertwoclassesofnovelmaterialswhoserichphasediagramsareactivelyunderinvestigation:unconventionalsuperconductorsandmultiferroics.In2006,H.Hosonodiscoveredanewclassofiron-basedsuperconductingmaterialswhicharenotconventionalsuperconductors.Aftertheinitialdiscovery,thereisarangeofquestionsofimmediateinterest;foremostamongthemiswhatisthestructureandsymmetryofthesuperconductingstate,aquestionwhichtookroughlyadecadetoanswerforthecupratesuperconductors.Wepresentcalculationsthathelprevealthestructureofthesuperconductinggapusingangledependentspecicheatmeasurements.WethencalculatetheelectronicRamanscatteringintensityforseveralpolarizationsoflightanddifferentmodelsofdisorder,providinginformationabouttheanisotropyandlocationofnodesinthesuperconductinggap.Understandingtheinuenceofdisorderisconsideredcrucialbecausecurrentlyconictingexperimentalresultsmaybeduetodifferencesinsamplequality. Recently,therehasalsobeeninterestinmultiferroics:materialswithsimultaneousnon-zeropolarizationandmagneticorder.Wepresentcalculationsoffundamental 11

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thermodynamicproperties,meaneldbehaviorforthesimplestferromagnetic-ferroelectric,characterizetopologicaldefects,andusetheperturbativerenormalizationgrouptohelpunderstandthecriticalpoint,asabeginningtowardsunderstandingthemultitudemultiferroicmaterialswithincreasinglycomplexmagneticandpolarorder. Thersttwochaptersreviewconventionalsuperconductivityanditsunconventionalcounterpartfoundinthecupratesandpnictides.Theoriginalworkconstitutingthebodyofthisdissertationappearsinchaptersthreethroughve.Chaptervecontainsabriefintroductiontothetopicswhicharerelevantformultiferroicsbeforepresentingtheoriginalwork.Portionsofthisthesisarebasedontheauthor'spublicationsandarecitedwhenrelevant. 12

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CHAPTER1INTRODUCTIONTOSUPERCONDUCTIVITY 1.1ConventionalSuperconductivity Sincetheprimaryfocusofthisworkisonunconventionalsuperconductivity,itisinstructivetoreviewaspectsofwhatwemeanbyconventionalsuperconductivityanddenewhatismeantbyasuperconductor.Superconductivityreferstotheobservationthatinmanymaterials,belowacertaincriticaltemperatureTctheelectricalresistancedropstozero,butthispragmaticpointofviewcanmisssubtlephysics.Perfectdiamagnetismisalsoasignatureofsuperconductivity.Deningtheonsetofsuperconductivityusingtheresistivetransitionalonewouldnotidentifyasuperconductorwheresuperconductingislandsformwithnormalnon-zeroresistivityregionsbetweenislands.Diamagnetismisnotsufcienteither,becausetype-Iandtype-IIsuperconductorsdifferinhowtheyrespondtoamagneticeld.Inatype-IImaterial,aboveaminimumeldHc1,theuxcanpiercethematerialatisolatedpointsalthoughthemajorityofitisstillsuperconducting. ThemicroscopicpictureofasuperconductoristhatinthesuperconductingstatetheFermiseabecomesunstableandelectronsformCooperpairs.Trueidenticationofasuperconductorwoulddemonstratetheexistenceofphasecoherentpairedelectrons,andwoulddistinguishitfromthecasewherethesepairsarenotphasecoherentacrossthesample.Cooperpairingisonlyoneofseveralwaystheelectronseacanbecomeunstableandformadifferentgroundstate.Othergroundstatesincludechargedensitywavesorvariousmagneticstates.Inconventionalsuperconductors,itwasdiscovered,notablybyFrolich[ 1 ],thattheeffectiveelectron-electroninteractionduetotheexchangeofphononswasattractive.ThiswasanimpetusforCoopertoproposetheideathatelectronsformtwo-particleboundstates[ 2 ]inthiseffectivepotential.Belowacertainenergyscale(roughlytheDebyefrequency)theFermiliquidisdestroyedbyformingboundpairsofelectrons.Itisworthnotingthatphonon-mediated 13

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superconductivityisnottheonlypossibility,andthepairinginteractioninmanyunconventionalsuperconductorsisstillanopenquestion.Itisevenaquestionofwhethertheformationofpairsneedstobemediatedbytheexchangeofbosonsatall. Theboundpairofelectronswillhavesomebindingenergy,2.Toexciteapairintofreeelectronsagain,breakingitapart,oneneedstoaddmorethanaminimumenergywhichdenesthegap.Thegapdoesnothavetobeasimpleconstant.Therearemeasurableconsequencesofthischangeinthespectrum.Intunnelingexperimentsinapuresystemweshouldseenoexcitationsbelowthisscaleinthedensityofstates,andinspecicheatmeasurementswewouldnolongerseetheelectron+phononnormalstatespecicheatoftheformC=T+bT3butratheractivatedbehavioroftheformC/e()]TJ /F6 7.97 Tf 6.59 .01 Td[(=(kBT))atlowtemperatures.Inthisthesis,wewillinvestigatewaystomeasurethequasiparticlespectrumanduseittohelpdeterminethestructureofthegap. 1.2TheoreticalDescription Bardeen,Cooper,andSchriefferprovidedacompletedescriptionofthesuperconductingstate,hereinreferredtoasBCStheory,forthersttimein1957[ 3 ].WewillonlygiveanoverviewofBCStheory,butfortunatelythereareanumberofexcellentsources[ 4 ].Weusethevariationalwavefunction, j >=Yk(uk+vkcyk"cy)]TJ /F4 7.97 Tf 6.59 0 Td[(k#)j0>(1) tominimizetheexpectationvalueofamodelHamiltonian,withrespecttothevariationalparametersukandvkH=Xkkcykck+XklVklcyk"cy)]TJ /F4 7.97 Tf 6.59 0 Td[(k#c)]TJ /F4 7.97 Tf 6.58 0 Td[(l#cl". (1) Normalizationrequires jukj2+jvkj2=1,(1) 14

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soarefreetochooseuandvtobesineandcosine.EvaluatingthemodelHamiltonian'sexpectationvalueisstraightforwardbutlengthyalgebra,leadingto:=Xkk2cos2k+XklVklsinkcosksinlcosl. (1) Alittletrigonometrylater,wedenethegap,k,suchthat tan2k=XlVklsin2l 2kk k.(1) Thepairamplitudegk12isanonvanishinganomalousexpectationvaluerepresentingthepairsofelectrons.Thisisrelatedtothepairingpotentialatzerotemperatureforaspinsingletpair,by(k)=)]TJ /F12 11.955 Tf 11.29 8.96 Td[(PkVkpgp.isreferredtoasthegaportheorderparameter.TheenergygainbycondensingintothesuperconductingstateisN(EF)2 2,whereNisthedensityofstatesattheFermilevel.Tobreakallpairsintonormalquasiparticlesagain,thisistheenergyoneneedstosupply.Forexample,tosuppresssuperconductivitywithanappliedmagneticeld,Hp,oneneeds202BNFH2p=NF2 2.ThislimitiscalledtheClogston-Chandrasekharlimit;itisnottheonlywaytodestroyasuperconductorwithamagneticeld,butitisonescenario. Thevariationalproblemhasthesolution: u2k=1 2(1+k Ek),v2k=1 2(1)]TJ /F3 11.955 Tf 14.33 8.09 Td[(k Ek).(1) Thesingleparticleexcitationenergiesare"k=p 2k+2k.Thenumber,thegap,representshalfthebindingenergyofthepair.BCStheoryisameaneldtheory,wherethemeaneldis< >.Theuctuationsoftheorderparametershouldbeexaminedtoensurethey'resmallenoughforameaneldtreatment.Thesmallcoherencelength,0=~vF ,ofcupratesuperconductorsmakeuctuationsrelevanttosomeoftheirobservedbehavior.Toseehowthemeaneldtheoryisconstructed,we 15

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takethepotentialtobelocal, H0=)]TJ /F5 11.955 Tf 10.49 8.09 Td[(V0 2XabZd3rya(r)yb(r)b(r)a(r).(1) ThisHamiltonianistreatedbyconsideringallpair-wiseexpectationvaluesaccordingtothestandardHartree-Fockapproach: Zd3rXaya(r)U(r)a(r)+Zd3r[(r)y"(r)y#(r)+(r)#(r)"(r)].(1) SothefullHamiltonianis: H=Zd3rya(r)H0a(r)+Zd3rXaya(r)U(r)a(r)+Zd3r[(r)y"(r)y#(r)+(r)#(r)"(r)],(1) whereH0isgivenby )]TJ /F15 11.955 Tf 15.58 8.09 Td[(~2 2m(r)]TJ /F5 11.955 Tf 27.8 8.09 Td[(ie ~c~A)2)]TJ /F3 11.955 Tf 11.96 0 Td[(.(1) Twoofthemethodsforperformingcalculationswillbeshowninbriefbutthereadershouldrefertostandardtextbooksformoredetail.TherstoftheseistousestandardeldtheorytechniquessuchasGreen'sfunctions,pathintegrals,etc.,oranothertechniquemoresuitedforinhomogeneousproblemsisthesolutionoftheBdGequations,namedaftertheircreatorsN.BogoliubovandP.G.deGennes. ThequadraticHamiltoniancanalsobediagonalizedwithaBogoliubov-Valatintransformation, y"(r)=Pn[yn"un(r))]TJ /F3 11.955 Tf 11.96 0 Td[(n#vn(r)]y#(r)=Pn[yn#un(r)+n"vn(r)](1) H=Eground+XEnynana.(1) Inthispicturewecanseethatexcitationsinthesuperconductingstatearelinearcombinationsofquasi-particlesandquasi-holeswithenergyEn.Theseexcitationsshouldbeexpressedaslinearcombinationsofelectron-holepairs.Toensurethat 16

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fa,ybg=absimultaneouslywithfya(r),b(r0)g=ab(r)]TJ /F5 11.955 Tf 11.96 0 Td[(r0).Werequirethat Xn[un(r)un(r0)+vn(r0)vn(r)]=(r)]TJ /F5 11.955 Tf 11.95 0 Td[(r0),(1) Xn[un(r)vn(r0))]TJ /F5 11.955 Tf 11.95 0 Td[(un(r0)vn(r)]=0.(1) Wecalculatetheequationsofmotion [He,(r")]=)]TJ /F8 11.955 Tf 9.3 0 Td[((H0+U(r))(r"))]TJ /F8 11.955 Tf 11.96 0 Td[((r)y(r#),(1) [He,(r#)]=)]TJ /F8 11.955 Tf 9.3 0 Td[((H0+U(r))(r#)+(r)y(r").(1) Afterdiagonalization,theHamiltoniansatises: [He,yna]=nyna,(1) [He,na]=)]TJ /F3 11.955 Tf 9.3 0 Td[(nna.(1) TheresultisknownastheBogoliubov-deGennesEquations [H0+U(r)]u(r)+(r)v(r)=u(r),(1) )]TJ /F8 11.955 Tf 11.95 0 Td[([H0+U(r)]v(r)+(r)u(r)=v(r).(1) Thesecoupleddifferentialequationsshouldbesolvedself-consistently.AssumingthequasiparticlesobeyFermi-Diracstatistics,wendthattheBdGsolutionsuandvdeneourorderparameterandpotentialselfconsistentlyby: (r)=V0Xn(1)]TJ /F5 11.955 Tf 11.96 0 Td[(fn")]TJ /F5 11.955 Tf 11.96 0 Td[(fn#)un(r),vn(r)(1) U(r)=V0Xn[jun(r)j2fn+jvn(r)j2(1)]TJ /F5 11.955 Tf 11.95 0 Td[(fn)].(1) Themagnetizationmustalsobecalculatedself-consistentlyifthemagneticresponseisconsidered. 17

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Weusuallyexpressequationsinreciprocalspace, (r)=1 p VPnPke)]TJ /F4 7.97 Tf 6.59 0 Td[(i!n+ik.rck.IntroducingNambunotation,^ k=0B@ck"cy)]TJ /F4 7.97 Tf 6.58 0 Td[(k#1CA,andFouriertransforming,Hcanberewritten[ 5 ]: H0=Xk yk)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(i!n^1+k^3+k^1 k,(1) fromwhichwecanreadoffthepropagatorinNambuspace: G(k,i!n)=)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(i!n^1+k^3+k^1)]TJ /F6 7.97 Tf 6.59 0 Td[(1=1 (i!n))]TJ /F8 11.955 Tf 11.96 0 Td[((2k+2k)0B@i!n+kkki!n)]TJ /F3 11.955 Tf 11.95 0 Td[(k1CA,(1) where, 1=0B@01101CA2=0B@0)]TJ /F5 11.955 Tf 9.3 0 Td[(ii01CA3=0B@100)]TJ /F8 11.955 Tf 9.3 0 Td[(11CA.(1) Thedensityofstates,DOS,iseasilycalculatedfromtheGreen'sfunction N(!)=)]TJ /F8 11.955 Tf 10.89 8.09 Td[(1 Xk1 2Tr[ImGR(k,!)].(1) Inans-wavesuperconductorwhere,k=0,thedensityofstatesis, N(!)=N(0)! p !2)]TJ /F8 11.955 Tf 11.96 0 Td[(2(!2)]TJ /F8 11.955 Tf 11.96 0 Td[(2).(1) Incontrast,thed-waveorderparameterk=0cos(2)haslowenergystatesintheDOS N(!)=N(0)Zd 2! p !2)]TJ /F8 11.955 Tf 11.96 0 Td[(20cos2(2)=N(0)2! 0K(! 0),!<(1) whereKisanellipticintegraloftherstkind,andfor!>,N(!)=N(0)2 K(0 !). TheapproachweoutlinedaboveisvalidwhenthepairingpotentialN(EF)V0<<1isweakenoughforperturbationtheorytobevalid.Insomeelementalsuperconductors,thisisvalid,butinotherslikelead,itisnot.Thereisastrongelectron-phononinteractionandthegapisenergydependent,aswewillshowattheendofthischapter. 18

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Fromthegappedquasiparticleenergyspectrum,wecanquicklycalculatephenomenologicaldescriptionsofmanyexperimentallyaccessiblequantities.Thisisthedictionarywewanttocomebacktointhecaseofunconventionalsuperconductors,sothatexperimentalworkguidesthetheoreticaldescriptionoftheunderlyingphysics. 1.3ExperimentsinConventionalSuperconductors 1.3.1SpecicHeat Inthissectionwewillreviewexperimentsonordinarysuperconductors.Amongtheseprobes,wewanttofocusonthespecicheatatconstantvolumemorethansomeoftheotherprobesbecausewewillrevisititlaterinthisthesis.Thespecicheatatconstantvolumecanbederivedfromtheentropyofthequasiparticlegas,whichisthatoffreefermionsS=)]TJ /F8 11.955 Tf 9.29 0 Td[(2kBXk[(fkln(fk)+(1)]TJ /F5 11.955 Tf 11.96 0 Td[(fk)ln(1)]TJ /F5 11.955 Tf 11.95 0 Td[(fk))] (1) throughCv=T@S @T.Keepinginmindthatthegapisafunctionoftemperature,(T),andapproximatingthesum(expandin,lowT),wecanreproducethelowtemperaturespecicheatCv=2N(EF) kBZdep 2+2 (ep 2+2+1)2((2+2(T)) T2)]TJ /F8 11.955 Tf 13.15 8.09 Td[((T) Td(T) dT). (1) Whichinthelowtemperaturelimitis:limT!0Cve kBT. (1) Figure 1-1 showstheactivatedbehaviorexpectedinthethermodynamicsofafullygappedsuperconductor,adecayingexponential.Laterwecontrastthiswiththepowerlawtemperaturedependencefoundinunconventionalsuperconductors,indicativeoflowenergyquasiparticles. 19

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Figure1-1. ThelowtemperaturelimitofthespecicheatinSncomparedwithBCStheory.Reproducedwithpermissionfrom[ 3 ].APSc1957. 1.3.2Tunneling Atunnelingexperimentputstwomaterialsclosetogether,withsomepotentialbarrierinbetween,likeaninsulatororavacuum,sothatparticleshavetotunnelacrossthebarrier.OneexampleisSTM,scanningtunnelingmicroscopy,whereasharpconductingtipisbroughtwithinseveralAofasample'ssurface,andabiasvoltageisappliedsothatcurrenttunnelsbetweenthematerialandthetip.Tunnelingisanonequilibriumtransportprocess,butsincethecurrentamplitudeissolow,thetimebetweentwotunnelingeventsismuchlongerthantypicalrelaxationrates,soitisappropriatetouseaquasi-equilibriumformalism.Wearespecicallyinterestedinthetunnelingbetweenanormalmetalandasuperconductor. WemodelthetunnellingHamiltonian[ 6 ],byH=Hn+Hs+Ht,whereHnandHsarethenormalandsuperconductorHamiltonians,andthetunnelingpartisgivenby: Ht=X(Tcy1c2+Tcy2c1)(1) where1and2refertothematerialsoneithersideofthejunctionandtheGreekindicesrefertosingleparticlestates,andTisthetunnellingmatrixelement.ThecurrentisjustdQ dt=e<_N>where_Nisthechangeinnumberwithrespecttotime,so 20

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_N=i[H,N](1) Afterabitofalgebrawearriveatanexpressioninvolvingaconvolutionofthedensityofstates,inthesampleandthetip.Wehaveassumedafeaturelessdensityofstatesinthetip;otherwiseawaytodeconvolvethedataisessential.Inthiscase,onearrivesattheformula: I=eZd! 2XjTj2Tr[A1(,!)A2(,!+eV)][f(!+eV))]TJ /F5 11.955 Tf 11.96 0 Td[(f(!)](1) whereA(,!)isthespectralfunctionandrepresentsotherquantumnumbers(usuallymomentumwhichisnotagoodquantumnumberintheabsenceoftranslationalsymmetry,sowemighthavedifferentlabelsforinternalstates,forexample,inaquantumdot).ThequantityTdependsonthegeometryofthesystemandthedetailsoftheelectronicstatesinvolvedintunneling.AdetailedtreatmentandexplicitevaluationofthesematrixelementsshowsTisproportionaltothederivativeofthewavefunctioninthesampleandthederivativedependsontheorbitalstateoftheatomsinquestion[ 5 7 ].Itshouldalwaysbekeptinmindthat,evenwithatomicscaleresolution,wearelookingatthetunnelingvoltageandnotactuallyatoms.Thepatternsonegetsfromexperimentoftenhavewelllocalizedpeaksinthesignal,andit'stemptingtoidentifythemwithindividualatoms,butthetipmightbesamplingasmallregionlargerthanoneatom. UsingjustFermi'sgoldenrule,wecangetaverysimilarexpression: I=I(sampletotip))]TJ /F5 11.955 Tf 11.95 0 Td[(I(tiptosample),(1) =2e2 ~ZjTj2Ns(E)Nt(E+eV)[f(E)(1)]TJ /F5 11.955 Tf 11.86 0 Td[(f(E+eV)))]TJ /F5 11.955 Tf 11.85 0 Td[(f(E+eV)(1)]TJ /F5 11.955 Tf 11.85 0 Td[(f(E))],(1) =2e2 ~ZjTj2Ns(E)Nt(E+eV),[f(E))]TJ /F5 11.955 Tf 11.95 0 Td[(f(E+eV)](1) 21

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whichwesee,apartfromconstants,isthesameasabovesincetheimaginarypartoftheGreen'sfunctionintegratestothedensityofstates.UsuallydataisanalyzedusingthedifferentialconductancedI dV. Figure1-2. Tunnelingdatafromsuperconductinglead.Reproducedwithpermissionfrom[ 8 ].APSc1962.Thetwomostsalientfeaturesarethesuperconductinggap,belowwhichthereisnodensityofstates,andthewigglesinthecurvewhicharenotcapturedinweak-couplingBCStheory[ 9 ]. Ifweassumeastructurelesstip,andthetunnelingmatrixelementsdonotcomplicatetheanalysisthenthevoltagedependenceofthetunnelingconductanceessentialisathermallysmeareddensityofstatesofthesample, dI dVZd![@f(!)]TJ /F5 11.955 Tf 11.95 0 Td[(eV) @VN(r,!)](1) Thisisprobablythemostdirectmeasureofthedensityofstates.Wereproducetunnelingmeasurementsfromsuperconductinglead[ 8 ]inFigure 1-2 whichshowsthedensityofstates. 22

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1.3.3NuclearMagneticResonance Ingeneral,dependingonhowtheexperimentalprobecouplestotheHamiltoniantomeasurethesystem,theoccurrenceofsuperconductivitywilleithercauseapeakjustbelowthecriticaltemperature,Tc,oradrop.ThisisdiscussedinthepopulartextbyTinkham[ 4 ]andreferredtoascase1orcase2interactions.Thedifferenceisunderstoodasamanifestationofquasi-particlecoherencefactors.Thedifferenceincoherencefactorsistracedbacktowhethertheinteractiondependsonlyonk)]TJ /F5 11.955 Tf 12.21 0 Td[(k0orkand)]TJ /F5 11.955 Tf 9.3 0 Td[(ktransitions. TheNMRspin-latticerelaxationratecanbereliablycalculatedusingFermi'sgoldenrule,whichprovidesanicequalitativewaytointerpretdata.IntheoriginalpaperbyHebelandSlichter[ 10 ],theexpressionis:1 T1/XkXk0jVj2f(Ei)(1)]TJ /F5 11.955 Tf 11.96 0 Td[(f(Ef))N(Ef)N(Ei), (1) whereVisthematrixelementbetweeninitialandnalstates.Thethermalfactorsf(Ei)aremeasuresoftheinitialandnaloccupation.Soonecanlooselylookatthespin-latticerelaxationrateasameasureofthedensityofstatessquared,blurredbyadistributionfunction.Byreplacingthequasi-particleenergybythegappedforminans-wavesuperconductor,wewouldestimatethat1 T1/ZdEE p E2)]TJ /F8 11.955 Tf 11.95 0 Td[(2(E+~!) p (E+~!)2)]TJ /F8 11.955 Tf 11.95 0 Td[(2eE (eE+1)2, (1) whichexhibitsapeakintheNMRrelaxationrate,calledtheHebel-Slichterpeak.Thispeakisclearlyobservedinsuperconductingsamples.Figure 1-3 ,takenfromMasuda,YoshikaandRedeld[ 11 ],isplottedversusinversetemperature,sothepeakisadip,butotherwise,theexponentialtailandHebel-Slichterpeakareclearlyinagreementwiththeory.Thecompleteexpression[ 12 ]suitablefordetailedcalculationsforthespin-latticerelaxationrate,1 T1,spinechorate,1 T2,andKnightshiftareobtainedfromthedynamicspinsusceptibility(wherethesaregyromagneticratios,fistherelative 23

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Figure1-3. Thespinlatticerelaxationrateinsuperconductingaluminiumasafunctionof1 T,plottedalongsidetheBCSprediction.Reproducedwithpermissionfrom[ 11 ].APSc1962. abundanceoftheisotope,andAarehypernecouplings):ab=(gB)2Zdtei!t, (1)1 T1=lim!!02kBT 2~4XqjAqj200(!,q) !, (1)1 T2=2fA 4~6XqjAqj4[0(0,q)]2, (1)Ks=jA0j0(0,0) eN~2. (1) TheKnightshiftistheshiftinresonantfrequencyduetoalocalmagneticeldandisprimafacieevidenceforthespinstateofthepair,butinterpretationisnotcompletelytransparent.Atzerotemperatureinapuresingletpair,theKnightshiftgoestozero.Tripletpairscanhaveaspinpolarization,butasinglethasnonetmoment.Theexceptionstothisinterpretationneedtobeborneinmind.Orbitalmagnetismand 24

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spin-orbitcouplingcanalterthesenaiveconclusionsbyeithermaskingthesignalfromtheCooperpairsormixingsingletandtripletstates. 1.3.4OpticalConductivity Detailedexpressionsforthedynamicalconductivity,1)]TJ /F5 11.955 Tf 12.14 0 Td[(i2,havebeenderivedbyMattisandBardeen[ 13 ]whichlendthemselvestosimpleinterpretationofthephysicaloriginsoftheconductivity. 1 n=1 ~!2Z1d"(f("))]TJ /F5 11.955 Tf 11.95 0 Td[(f("+~!))("2+2+~!) ("2)]TJ /F8 11.955 Tf 11.96 0 Td[(2)1=2[("+~!)2)]TJ /F8 11.955 Tf 11.95 0 Td[(2]1=2+(1) 1 ~!2Z)]TJ /F6 7.97 Tf 6.59 0 Td[()]TJ /F14 7.97 Tf 6.59 0 Td[(~!d"(1)]TJ /F8 11.955 Tf 11.96 0 Td[(2f("+~!))("2+2+~!) (2)]TJ /F3 11.955 Tf 11.96 0 Td[("2)1=2[("+~!)2)]TJ /F8 11.955 Tf 11.95 0 Td[(2]1=2,(1) 2 n=1 ~!2Z)]TJ /F14 7.97 Tf 6.59 0 Td[(~!d"(1)]TJ /F8 11.955 Tf 11.96 0 Td[(2f("+~!))("2+2+~!) ("2)]TJ /F8 11.955 Tf 11.95 0 Td[(2)1=2[("+~!)2)]TJ /F8 11.955 Tf 11.96 0 Td[(2]1=2.(1) BelowTc,theresistivitydropstozero.Whatthisreallymeansisinniteconductivityatzerofrequency,thatisadeltafunctionat!=0termintheconductivity.Itisworthwhiletoobservethatintheabsenceofscatteringtheconductivityisinniteinthenormalstate,sothepresenceofanitelifetimeisimportanttoobtainsensibleresults.TherealpartoftheconductivityhastwotermsintheMattis-Bardeenresult,therstdescribingthescatteringofthermallyactivatedquasiparticles,theseconddescribingpairsbrokenbythephotons.Theimaginarypartoftheconductivitydescribestheresponseofthecondensateitself.WiththisinterpretationitisclearthatatzeroT,1(!)=0,thenwhen~!2itrisessmoothlytoconnectwiththenormalstatevalue. AtT=0,theintegralscanbecarriedoutbyhand.For~!>2,k=2)]TJ /F14 7.97 Tf 6.59 0 Td[(~! 2+~!, 1 n=(1+2 ~!)E(k))]TJ /F8 11.955 Tf 11.95 0 Td[(22 ~!K(k)(1) andtheimaginarypartoftheconductivityisgivenby,withk0=p (1)]TJ /F5 11.955 Tf 11.96 0 Td[(k2), 2 n=1 2(2 ~!+1)E(k0)+(2 ~!)]TJ /F8 11.955 Tf 11.96 0 Td[(1)K(k0)(1) 25

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Figure1-4. Therealpartoftheconductivityreproducedwithpermissionfrom[ 14 ]asafunctionoffrequency.APSc1959.Theminimaofeachcurveoccursat2asafunctionoftemperature. KandEareellipticintegralsoftherstandsecondkind.Thetemperaturedependenceshowstheexpectedpeak,behavingasln(2 ~!)forlowerfrequencies(if~!&0.5it'ssuppressed).AtnonzeroTasafunctionoffrequency,thereisaminimumin1at2(T)asshowninFigure 1-4 1.3.5ThermalConductivity Theelectronicthermalconductivityinans-wavesuperconductorcanbecalculatedusingtheBoltzmannequation.Inthenormalstate,theexpressionis, n=N(0)vf` 3Z10d""2 T)]TJ /F3 11.955 Tf 10.49 8.09 Td[(@f @"(1) Thecalculation,rstperformedforbyBardeen,Rickayzen,andTewordt[ 16 ],essentiallyjustgapsoutthelowenergyquasiparticles,sothat s n=R1dEE2 T@f @E R10d""2 T@f @"(1) TheresultsarecomparedwithameasurementonaluminiuminFigure 1-5 ,andareinexcellentagreement. 26

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Figure1-5. Thethermalconductivityofsuperconductingaluminiumreproducedwithpermissionfrom[ 15 ]APSc1962. Althoughthestatementthatasuperconductorisaperfectdiamagnetisoftenmade,theelectromagneticeldalwaysentersthesampleforsomedistancecalledthepenetrationdepth.Thepenetrationdepthcanbecalculatedfromthestaticlimitoftheelectromagneticresponse.Theresultofthatcalculationis[ 4 ], 1 2(T)=1 2(0)[1)]TJ /F8 11.955 Tf 11.96 0 Td[(2Z1dEE p E2)]TJ /F8 11.955 Tf 11.95 0 Td[(2)]TJ /F3 11.955 Tf 11.7 8.09 Td[(@f @E].(1) InSIunitse2 mns(T)=1 02(T)=K where1 2=nsisthesuperuiddensity.Forans-wavesuperconductorthisexpressionisexponentialatlowT.Theresultforananisotropicsuperconductoris 2(0) 2(T)=1 2Zd!Zdk 4tanh(! 2)Re2k (!2)]TJ /F8 11.955 Tf 11.95 0 Td[(2k)3=2.(1) SchawlowandDevlin[ 17 ]measuredthepenetrationdepth,andshowedthattowithintheexperimentalresolution,thetemperaturedependencecouldbetto(T)=1 q 1)]TJ /F11 5.978 Tf 9.37 3.26 Td[(T Tc4,butthismissestheasymptoticlowtemperatureexponentialformq Te)]TJ /F13 5.978 Tf 8.27 3.26 Td[( T.GorterandCasimir[ 18 ]advancedatwouidmodelforthesuperconductorinwhichthetwouidsarenormalquasiparticlesandthecondensate. 27

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1.4UnconventionalSuperconductivity 1.4.1UnconventionalPairing Thesymmetryofthesuperconductingorderparametercanbereadoffdirectlyfromtheexpectationvalueitrepresents.Thecrystalenvironmentrequiresthattheexpectationvalue,,transformsasarepresentationofthecrystalpointgroupGandfromthespinasSU(2).InordinarysuperconductivitytheU(1)gaugesymmetryofthenormalstateisbrokenandinunconventionalsuperconductorsadditionalpointgroupsymmetriesarebrokenaswell.Ifamaterialhasinversionsymmetrybothparityandtime-reversalaregoodquantumnumbersforthepairs.Statesmaythenbeclassiedaccordingtoevenparityspinsingletoroddparityspintripletpairs.Inthenoncentrosymmetriccasethereisanadmixtureofsingletandtripletpairs.Unconventionalsuperconductorsdonotpossessthefullsymmetryofthenormalstate.ItisstandardpracticenottousetherepresentationsofthecrystalgroupG,butrathertorefertothestandardsphericalharmonicsoftherotationgroupSO(3).Aconventionalsuperconductoristhereforecalledans-wavesuperconductor.Wetakeunconventionalsuperconductortomeanthattheorderparameterisnots-wave,i.e.thattheorderparameterhasalowersymmetrythanthatofthecrystal.Inthecuprateliterature,thed-waveorderparameterisoftenencounteredintwoways,cos(kx))]TJ /F8 11.955 Tf 11.97 0 Td[(cos(ky)andcos(2).TheformerisabasisfunctionofarepresentationofthetwodimensionalpointgroupD4h,andthelatterisad-waveangularfunctionwhichcanbeidentiedwithaTaylorexpansionofcos(kx))]TJ /F8 11.955 Tf 13.03 0 Td[(cos(ky),whereismeasuredfromthea-axisonacircularFermisurface.Therepresentationpossessingthefullsymmetryofthesquarelatticewouldbes-wave,e.g.aconstant.Extendeds-wavecos(kx)+cos(ky)wouldalsobecompatiblewiththefullsymmetryofthelattice,butisnamedextendedbecauseitdiffersfrom1.Thes-andd-wavedifferunderrotationsby 2.Thisoperationwillchangethesignoftheorderparameterforcos(kx))]TJ /F8 11.955 Tf 12.42 0 Td[(cos(ky),butnotforcos(kx)+cos(ky).Welistintable 1-2 somecommonspinsingletrepresentationsofthesquarelattice.Whatis 28

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meantbyconventionalthenistheA1grepresentation.Thegapisgiven,generally,byalinearcombinationofthebasisfunctionsofarepresentation.Foraone-dimensionalrepresentationthisistrivialbut,fromtable 1-2 ,thetwodimensionalE-representationismoreinteresting,=1kxkz+2kykz,whereisingeneralcomplex.ForamorecompletediscussionoftheseideaswereferthereadertoSigristandUeda[ 19 ]. Table1-1. D4hSelectedSingletRepresentations Irrep.BasisFunction A1g1,(k2x+k2y)s-wave,extendedsA2gkxky(k2x)]TJ /F5 11.955 Tf 11.96 0 Td[(k2y)g-waveB1g(k2x)]TJ /F5 11.955 Tf 11.96 0 Td[(k2y)dx2)]TJ /F4 7.97 Tf 6.58 0 Td[(y2-waveB2gkxkydxy-waveEgfkxkz,kykzg Theformalismfordescribingthed-wave,orforthatmatteranypairing,isasimpleextensionofBCStheorythataccountsformatrixelementswithmorepossibleanomalouspairings. =k=0B@k""k"#k#"k##1CA.(1) Here,canbecomplex.Clearlyforasingletsuperconductorthek""andk##componentshavetobezero.Likewise,weneedthek"#andk#"componentstoformthethreej"">,j##>,andj"#+#">statesofatriplet. OwingtotheantisymmetryofFermions,thes-wavesinglethasagap k=0B@0k"#k#"01CA=0B@0k)]TJ /F8 11.955 Tf 9.3 0 Td[(k01CAd0(k)i2(1) Inthetripletcasewehaveamorecomplicatedd-vectorsinceatripletcouldcorrespondtoanyofthreewavefunctions,ingeneral: (k)=~d(k).~i2=0B@)]TJ /F5 11.955 Tf 9.3 0 Td[(dx+idydzdzdx+idy1CA(1) 29

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Themostgeneralformofagapbecomes ab(k)=((k)1+~d(k).~)i2(1) Table1-2. D4hSelectedTripletRepresentations Irrep.BasisFunction A1u~d=^xkx+^ykyA2u~d=^xky)]TJ /F8 11.955 Tf 12.24 0 Td[(^ykxB1u~d=^xkx)]TJ /F8 11.955 Tf 12.25 0 Td[(^ykyB2u~d=^xky+^ykxEuf^zkx,^zkyg It'sasimplemattertoworkoutwhichcomponentcorrespondstowhichspinwavefunctionbyusingabasiswith(1,0)or(0,1)forupanddownspins,sothattheouterproductoftwoupspinsinatripletcontributethecomponent j"">0B@101CA0B@101CAy=0B@10001CA(1) So 1 p 2j"")-278(##>=0B@100)]TJ /F8 11.955 Tf 9.3 0 Td[(11CA,1 p 2j""+##>=0B@10011CA,1 p 2j"#)-277(#">=0B@01101CA(1) Usingwhatweknowaboutunconventionalgaps,ifweintroducemomentumdependence(k)=0g(k),thedensityofstatesisgiven,inthreedimensions,by N(E)=NFZd 4E p E2)]TJ /F8 11.955 Tf 11.95 0 Td[(20g(k)2(1) whereg(k)containsthek-dependenceofthesuperconductinggap.Generally,thecleandensityofstatesforlowEwilltaketheformofapowerlaw,N(E)/En.Thisimportantresultcanbeusedtointerpretthephenomenologyofmanyexperiments,seetable 1-3 .Forcuprates,thecleandensityofstatesislinearinE.Sincethespinlatticerelaxationratevariesasthedensityofstatessquared,thismeanswewouldexpectatemperature 30

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dependenceof1 T1/TRdEE2)]TJ /F2 11.955 Tf 5.48 -9.69 Td[()]TJ /F4 7.97 Tf 11.36 4.71 Td[(df dET3.Forboththespecicheat,C T,andthechangeinpenetrationdepth,,alineardensityofstates,N(E)E,resultsinalinearinTbehaviorintheregionwhereT=issmall.Toseethis,wemustrealizethatthenumberofstatesforagivenenergycomesfromaregionp 2+2k
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whichind-wavematerialsleadsto1 (E)/Eor1 EscalingifE<<0respectively.SimpleBoltzmannequationestimatessuggestthattransportcoefcientsaredeterminedlargelybyexpressionslike[ 21 ]: Zd!()]TJ /F3 11.955 Tf 11.12 8.09 Td[(@f @!)N(!)(!),(1) socatalogingtheassociatedresponseforanyparticulardisordermodeliscrucialtoaccurateunderstandingofexperiment. 1.4.2Migdal-EliashbergTheory Migdal-EliashbergtheoryisanaturaldevelopmentofBCStheorytoincludeamorerealistictreatmentofthephonons.TherewerealreadyanomaliesinthedatameasuredfromHgandPbcomparedtotheweakcouplingBCSapproach,soitwasnecessaryfromthepointofviewofexperiment.Mercury,werecall,wastheoriginalmaterialstudiedbyKamerlinghOnnesin1911.WehavealreadyshowndatainFigure 1-2 whichhasafewwigglespresentinthedensityofstates,whereBCStheorypredictsthesmoothform! p !2)]TJ /F6 7.97 Tf 6.59 0 Td[(2.Theresultofincludingamorerealisticelectron-phonontreatmentwastopredictatunnelingcurrentoftheapproximateform[ 22 ], I(V)=Zd!Re j!j p !2)]TJ /F8 11.955 Tf 11.96 0 Td[((!)2![f(!))]TJ /F5 11.955 Tf 11.96 0 Td[(f(!+V)](1) Thefrequencydependentgapwillaccountforthewiggles. IfollowtheaccountinJonesandMarch[ 23 ].InNambuspacetheselfenergyincludingbothelectron-phononandCoulombinteractionsis, (k,i!n)=)]TJ /F8 11.955 Tf 10.99 8.09 Td[(1 Xn,k03G(k0,i!m)3(Xjgkk0j2D(k)]TJ /F5 11.955 Tf 11.96 0 Td[(k0,i!n)]TJ /F5 11.955 Tf 11.95 0 Td[(i!m)+Vc(k)]TJ /F5 11.955 Tf 11.96 0 Td[(k0))(1) Aconvenientparameterizedformoftheselfenergyis =[1)]TJ /F5 11.955 Tf 11.95 0 Td[(Z(k,!)]!^1+(k!)3+(k,!)1(1) 32

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whichinturnresultsinaGreen'sfunction. G(k,!)=!Z(k!)^1+"(k,!)^3)]TJ /F3 11.955 Tf 11.95 0 Td[(2(k!)^1 !2Z2(k!))]TJ /F3 11.955 Tf 11.95 0 Td[("2(k,!))]TJ /F3 11.955 Tf 11.95 0 Td[(2(k!)(1) Here,"(k,!)=k+(k!)and(k!)=(k!) Z(k!). Figure1-6. TheBCSDOSisthedashedline.TheexperimentaldataonPbandthecalculationsarethesolidandlong-dashedlinesrespectively,showinghowEliashbergtheorycanaccountfortheenergydependenceinthetunnelingsignal.Reproducedwithpermissionfrom[ 9 ].APSc1963 Anumberofcommentsneedtobemade.Theseequationscanbesimplied,andsolvedself-consistently,resultinginexcellentagreementbetweentheoryandexperiment,inparticularreconcilingthediscrepancybetweentheweakcouplingtheoryandthedatainstronglycoupledsuperconductorslikemercuryandlead.Scalapino,Schrieffer,andWilkins[ 9 24 ]carrythisoutintwopapers,andcomparetoPbdata.TheentiremethodreliesonanexceptionalobservationbyMigdal,nowcalledMigdal'stheorem[ 25 ],thatthevertexcorrectionsfortheelectron-phononcouplingarealwayssmallerbyafactorq me Mi,theratioofthemassoftheelectrontothatoftheion.Thismeansthattheelectronphononinteractioncanalwaysbetakentobethebareinteraction,)-417(=(1+O(q me Mi))whereisthebareelectron-phononinteration[ 25 ].InthecaseofparticleholesymmetryandtheinnitebandwidththeGreen'sfunctioninEq 1 cantakethenon-interactingform.Oneofthemostimportantresultsof 33

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Migdal-Eliashbergtheoryisanunderstandingoftheisotopeeffect,whichexaminesthechangeinthecriticaltemperaturewithshiftingisotopemass: =)]TJ /F5 11.955 Tf 10.5 8.09 Td[(dlnTc dlnM.(1) Onemodernissuetheoristsmustcopewithistheabsenceofanequivalenttheoremforotherinteractionsbesidestheelectron-phononinteraction[ 26 27 ],limitingknowncontrolledtechniquesessentiallytoweakcouplingversionsofproposedinteractions.Furthermore,attemptsatpredictingTcforthehigh-Tccompoundswereunsuccessfulusingthisframework,suggestingnewphysicsatwork.IamnotawareofschemeswhichuseMigdal-Eliashbergtheoryforphononssimultaneouslywithotherinteractionsbeyondwhatisalreadycited. 34

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CHAPTER2SUPERCONDUCTIVITYINCUPRATESANDPNICTIDES 2.1Cuprates 2.1.1BasicCupratePhysics Afternearly30yearsofhightemperaturesuperconductivityresearchalready,therearealargenumberofreviews,forexample[ 28 35 ].Theentirefamilyofcopperoxidesuperconductorsconsistofoneormoreplanesofcopperoxideseparatedbylayersofrareandalkalineearthions.Thelatticesarealltypicallyoftheperovskiteform,orverysimilartoit.Theoriginalmotivationthatledtothediscoveryofhigh-TcsuperconductivitywasastudyofJahn-Tellerdistortions,sosmalloctahedraltiltsdifferingfromtheperfectperovskitestructurearenotuncommon.TheCuformasquarelatticewithoneObetweeneachcopperatom,andtheOformtheperovskitecage.Sometimestheapicaloxygenisnotpresent,butthepatternofsquareCuO2layersisubiquitous.ExamplesareshowninFigure 2-1 .Amongthecupratestherearedetailsspecictocertaincompounds.Forexample,YBCOhasCuOchainsalongsidetwoCuO2-layerswhichorderinseveralways,andforLSCOthereisadipinthesuperconductingdomeat1 8doping.BSCCOcleavesbetterthanothercuprates.NonethelessthereisaremarkablesimilaritythroughouttheentiresetofhighTccompounds. Figure2-1. Arepresentationofseveralcuprateswiththeirsymmetryclassications.(reproducedwithpermissionfromwww.tfm.phys.cam.ac.uk) 35

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Thecupratephasediagramissomewhatcontroversialeventoday.ThegenerallyagreeduponresultsareshowninFigure 2-2 ,butcarefulomissionswerebuiltinwhichwillbeaddressedalongtheexperimentaldatawhichsuggestsaparticularinterpretation.Thebasicphasesareanantiferromagnet,asuperconductor,ametalathighdoping,andtwostrangephasesreferredtoasthestrangemetalandpseudogap,whosepositionsarehardertodemarcate.Itissuggestedbysomeexperimentsthattheentranceintothepseudogapphasemakesalinecrossingthroughthesuperconductingdometherebyindicatingaquantumcriticalpoint.Ontheotherhand,accordingtoadifferentsetofexperimentsthelinepassesoverthedomeandsuchaquantumcriticalpointissuggestednottoexist. Figure2-2. Acartoonofthecupratephasediagram TheundopedcopperoxidesareunambiguouslyantiferromagneicMottinsulators.Bandstructurecalculations[ 36 ](LSDA=LocalSpinDensityApproximation)originallypredictedtheparentcompoundstobemetals,untilcorrelations(LSDA+U)wereincludedsuchthatthecorrectinsulatingphasewasobtained.TypicalparametersintheantiferromagneticphasegiveanexchangeJofaround1000K,andaNeeltemperaturearound300K.Theon-siteCoulombrepulsionisoforder8-10eV,whichislargerthanthebandwidth.Cupratesaregenerallychargetransferinsulators[ 37 ]withthechargetransfergapsignicantlysmallerthanU,around3-5eVtypically.Hoppings 36

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areestimatedtobeabout0.43eV[ 28 ]fortheHubbardmodelbutashighas1.3eVforthep-dorbitals.Asholesareaddedtothecopperoxideplanetheantiferromagnetismisrapidlydestroyed.Atlowtemperatureoutsidetheantiferromagneticphasethereisaregionsometimesreferredtoasthespinglassphase,wherefrozenshortrangemagneticorderispresent,thenasholeconcentrationincreasesweenterthesuperconductingdome.Thetermspinglassshouldbeusedloosely.Thephaselackslongrangeorder,andthereareupturnsintheresistivityatlowTinthisphase,buttheextenttowhichthenameimpliesglassphysicscanbemisleading.Thesuperconductivityisnowwidelyacceptedtohavead-wavegap,thoughthistookaboutadecadetoestablish.Itisconrmedexperimentallybyahostofprobesnow:STM(scanningtunnelingmicroscopy),ARPES(angleresolvedphotoemissionspectroscopy),tri-grainboundaryexperiments[ 38 39 ],thelistgoeson.Thehighesttransitiontemperatureasafunctionofdopingisreferredtoasoptimallydoped,andoccursaround16%doping.OnebitofdirtylaundryinthissubjectisthatdopingisusuallyunknownandTc-tstoawidelyappliedfunctionareoftenused:Tc(p)=Tmaxc(1)]TJ /F8 11.955 Tf 12.78 0 Td[(86(p)]TJ /F8 11.955 Tf 12.78 0 Td[(.16)2),wherepisholesperCuO2unit.Thesuperconductingdomeresidesbetweenabout5-25%doping.Sometimespisthesameastheconcentrationofadopant,x,inthechemicalformulasbutnotalways.Underdopedandoverdopedrefertodopinglessthanorgreaterthantheoptimalvalue(p=.16)respectively.Mostoftheanomalouspropertiesofthecupratesshowuptoagreaterextentintheunderdopedcompounds.Athighenoughholedoping,itisbelievedthatthenormalstateisagainaFermiliquid,borneoutbytransportorNMRdata.AlargeregionofthephasediagramoutsidethesuperconductingregionisclearlynotaFermiliquid,andisreferredtoasthestrangeorbadmetalphasecharacterizedbytheapproximatelylinearinTresistivity.Thepseudogapphaseappearsbelowacharacteristictemperaturetowardstheunderdopedsideofthesuperconductingdome.Whetherthesearecrossoversortruephasesisstillamatterofdebate,butitiscleartherearenotsharpthermodynamicsignaturesofaphasetransitionasonewould 37

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seeformagneticsecondordertransitions.Ontheotherhand,someprobesindicateasharptransitionlinebetweenpseudogapandbadmetalphases,sooneisagainforcedtoreconcileexperimentswhichwillbediscussedbelow.Theelectrondopedsideofthephasediagramissimilarinoverallstructurebutthereisanotableasymmetrybetweentheelectronandholedopedmaterials.Antiferromagnetismsurvivestolargerdopingontheelectron-dopedside,andthepseudogapregionisnotaspronounced.Superconductivityandantiferromagnetismaremuchclosertooneanotherontheelectrondopedside.Whileresistivitymeasurementsindicateananomalousmetalphaseontheholedopedside,theyprovideevidenceforaFermiliquidontheelectrondopedsideatalldopings. Thetypicalsizeofthegapat(0,)whereitismaximumisaround25-40meVatoptimaldoping.Thematerialsareextremetype-IIsuperconductors,sofromtheslopeofHc2atTcwecanestimatefromBCStheory015)]TJ /F8 11.955 Tf 12.43 0 Td[(30A.Alternatively,onecanuseFermivelocityandgapmeasurementstoestimatethecoherencelength,whichcomesouttoaround2nm.PenetrationdepthmeasurementsfromopticsandSRareoforder1500A.Thesearein-planeestimates.Itisalsotruethatcupratesareveryanisotropicmaterials,withveryhighresistivityandmuchlongerpenetrationdepthsalongthec-axiscomparedtotheab-plane. Agreatdealofattentionovertheyearshasbeenpaidtocertainmodels[ 40 ]whichareclaimedtocapturetheessentialphysicsofthecuprates.TheHubbardmodelisonesuchmodel.Itisarguedwhethera3-band(Cu+O)or1-band(Cu)modelisnecessary[ 28 36 ].ThesimplestformoftheHubbardHamiltonianis:HHubbard)]TJ /F5 11.955 Tf 11.96 0 Td[(tXcyicj+cicyj+UXni"ni#. Todatetheonlyknownsolutionsareinoneorinnitedimensions,thoughtherearemanyapproachestothemodelinotherdimensions.AtlargeU,theHubbardHamiltonianismappedtothet-JHamiltonian.ASchrieffer-Wolfftransformation[ 41 ]is 38

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appliedtothelargeUlimitoftheHubbardmodel.Thet-JHamiltonianis,withJ=4t2 Uis:H=)]TJ /F5 11.955 Tf 9.3 0 Td[(tXcyicj+cicyj+JX~Si~Sj)]TJ /F5 11.955 Tf 13.15 8.09 Td[(ninj 4. TheHilbertspacesaredifferenteventhoughthereisaformalrelationshipbetweenthesetwoexpressions:doubleoccupancyhasbeenprojectedoutinthet-Jmodel,whichhasadrasticeffectonthequasiparticlessinceanyinteractionaccountingforvirtualtransitionstodoublyoccupiedsitesisnotincluded. Startingwiththespinoperator,~Si=1 2Xcyia~abcib andusingthegrandorthogonalitytheoremforgrouprepresentationsXR2SU(2)D(R)abD(R)cd=abcd+)777(!ab)778(!cd=4 2acbd, wecanputthet-JmodelintotheHubbardHamiltonian'sform,howeveronemustbecarefultorecognizethedifferenceintheHilbertspacesforthemodels.Thet-Jmodelhasprojectedoutdoublyoccupiedstates. 2.1.2Tunneling Tunnelingexperimentsinthecuprateswerenotinitiallyeasybecauseofissuesinvolvingsamplequalityandcontrolofthetunnelingbarrier[ 7 ],butoncethesestruggleswereovercome,STMbecameaninvaluabletoolininvestigatingsomeofthesematerials.DuetoaweakbilayercouplinginBSCCO,itcleavesnicely,andiswellsuitedfortheseexperiments.Thisalsomeansthatthereareafewlayersofatomsbetweenthecopperoxideplaneandthesurface,whichraisesthequestionofhowthesurfacelayersalterthetunnelingsignal.BSCCOisalsouniqueindisplayingsupermodulation,aperiodicbutincommensuratechangeinthedensityoftheBiOlayerwithrelativelylongwavelength.Figure 2-3 fromHuangetal.[ 42 ],showsatypicalcupratedensityofstates,extractedfromdI dVcurves.Webearinmindthatinterpretingthemeasurementasthe 39

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densityofstatescouldbedifferentfromunderstandingitasaconvolutionofspectralfunctionsfromthesampleandtip.Insharpcontrastwiths-wavesuperconductors,thereisanon-zerodensityofstatesobservabledowntozerobias.Thisisanindicationofnodesinthesuperconductinggap.Generallythedensityofstatescanbewelltfromtheformexpectedind-waveBCStheory,thoughmuchusefulinformationhasbeeninferredfromthescatteringratewhichwouldenterthisexpressionE p (E+i\(E))2)]TJ /F6 7.97 Tf 6.58 0 Td[((E,)2[ 43 ]. A B Figure2-3. DensityofStatesextractedfromtunnelingdatainBSCCOnexttoacleand-waveDOScalculation.Scatteringwouldbroadenthepeaks,createalowenergyimpurityband,andremovethesharpfeatures.Reproducedwithpermissionfrom[ 42 ].APSc1989. OneofthemostremarkableexperimentalresultsinthehighTcsuperconductorsistheexistenceofthepseudogapphase.AclearindicationofwhatismeantbypseudogapisshowninFigure 2-4 ,whereweseeacontinuousevolutionofthedifferentialconductanceevenaswecrossTc,whereonewouldexpectthesuperconductinggaptoclose.Agap,however,remainssothephaseiscalledthepseudogap.Extensivetemperaturedependencemeasurements[ 44 ]suggestthereisauniversallowenergyshapetopartofthepseudogap,forlowbias,andatemperaturedependenthigherbiaspart,whichisattributedtoanti-nodalexcitationsasthoseoccurathigherenergy.TheangulardependenceofthepseudogapisreportedbyARPES[ 45 ],whereitisseento 40

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roughlyfollowad-waveform.Theobservationoftwoenergyscalesinmicroscopyisreectedinotherexperiments,likeRamanscattering[ 46 ],howeverseealso[ 47 ]. Anotherfeaturewhichcaughttheattentionofthesuperconductingcommunitywasthedipathighenergycomparedtothesuperconductingcoherencepeak.(SeeFigure 2-4 at4.2K,whereitisprominent)Itisproposedthatthepeak-dip-humpstructurecanbeexplainedwithbandstructure,modiedMigdal-Eliashbergcouplingtoaphonon[ 48 ],oranotherbosonicmodelikethe41meVmodeobservedbyneutrons[ 49 ]. A B Figure2-4. DensityofStatesforBSCCOwithTc=83K.Reproducedwithpermissionfrom[ 50 ].APSc1998.AgapinthespectrumremainsaboveTc.ThephasediagramasseenbySTM.Reproducedwithpermissionfrom[ 7 ].APSc2007. Inthesuperconductingstate,tunnelingintoaZnimpurity,whichsubstitutesforaCuatomin-plane,revealadiscrepancybetweentheoryandexperiment.Ithasbeenaddressedbyanumberofauthorsbytryingtotakeintoaccountthematrixelementsfrequentlyignoredinthecalculationoftunnelingconductance.ThecenterofascatteringresonancealwayscoincideswiththesiteofasurfaceBiatom,belowwhichaZnis 41

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located.Theextendedtailsfromtheimpurityareorientedwiththegapmaximainsteadofthegapminima,andthelocalmaximainthedensityofstatesonspecicatomicsitesisatoddswiththesimplesttheory[ 51 ].Anumberofexplanationsinvolvingmatrixelementstunnelingthroughthe4.5Alayerhavebeenproffered[ 52 ],whichcanbeusedtoobtainreasonableagreement.Ifthisiscorrectnofundamentallynewphysicsisneeded,butthisignoresotherexplanationsincludinganonlocalKondo-couplinginsteadofpoint-likeimpurity[ 53 ]whichalsoresolvesomeoftheissues.FestkorperphysikistSchmutzphysik(attributedtoPauli). AExperiment BTheory Figure2-5. (left)LogarithmicplotoftheintensityaroundaZnimpurityinBSCCO.ReproducedwithpermissionfromPanetal.[ 54 ].cNaturePublishingGroup2000.(right)Theoreticalpredictionaroundunitaryscattererind-wavesuperconductor.Thatintheexperimentamaximumintensityisfoundatthecenterincontrasttothetheoreticalresult.Reproducedwithpermissionfrom[ 55 ].APSc1996. FromtheseparationofthecoherencepeaksandTc,theBCSratio20 kBTc=4.3ford-waveweakcouplingtheory,isfarexceededinthecuprates.WithinMigdal-Eliashberg 42

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theory,thisvaluecandepartfromtheweakcouplingresult.Themagnitudeofthegapdeducedfromthisscaleisalsofoundtovarystronglyfrompositiontopositiononthesamplesurface,andtheoriginofthestronglyinhomogeneousstructureisakeyquestioninthesecompounds.McElroyetal.[ 43 ]captureddatawhichsuggesttheoxygenscorrelatewiththelocalgapamplitude.Theinhomogeneityiscorrelatedwiththewidthofthesuperconductingtransition.Furthermore,asafunctionofdopingtheaveragespectralgapvaluedecreaseswithincreasingdoping,rangingfromabout100to20meV.Typicallytheratio20 kBTcexceedsthed-waveBCSratio4.3. 2.1.3SpecicHeat AExperiment BTheory Figure2-6. (left)T-dependenceofthecoefcientinCv=T.(right)Pseudogapscalefromspecicheatmeasurements.Reproducedwithpermissionfrom[ 56 ]. InFermiliquidtheorytheelectroniccontributiontotheelectronicspecicheatshouldbelinearintemperature,C=T.Lorametal.[ 56 ]havedoneextensiveworkonthespecicheatinthehigh-Tcsuperconductors.Figure 2-28 showsthedopingandtemperaturedependenceofthespecicheatcoefcientinYBCO.WecanalsoinferfromthisdatathebehavioroftheentropybyintegratingRdTCv(T) T.Asdopingfallsbelow 43

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p=0.19thespecicheatjump(Tc)decreasesrapidly,andtheentropyextrapolatestounphysicalvalues.Thegeneraltrendisreectedinmanyexperiments.Foroverdopedsamples,favorablecomparisonscanbemadewithd-waveBCStheory,butasweunderdopepuzzlingbehaviorappears.ThereisagapopeninginthesystemaboveTc,alossofentropy,andallthesymptomsofpseudogapformation. 2.1.4AngleResolvedPhotoemissionSpectroscopy AngleResolvedPhotoemissionSpectroscopy,ARPES,isarelativenewcomertosuperconductivityresearch,andhasbeenmainlyappliedtoresearchaftertheconventionalsuperconductorswerestudied.Onceenergyresolutionbecameneenough(currentclaimedresolutionforsynchrotronbasedARPESis0.1to1meV)ARPESbecameausefulbridgebetweentheoryandexperimentbecauseitiscommonlythoughtthatitdirectlymeasuresthespectralfunction.Toderivethis,however,isnotstraightforward[ 57 ].Photoemissioncurrentsaredirectlyproportionaltotheintensityofincidentlight,forfrequenciesabovetheemissionthreshold.Weareinterested,intheexpectationvalueofthecurrentoperator,undertheperturbationHI=)]TJ /F8 11.955 Tf 10.68 8.08 Td[(1 cZd~r~A(r,t).~j(r) whereAistheelectromagneticvectorpotential.TheARPESintensityisformallyathree-currentcorrelationfunction=1 ~2c2Zdtdt0drdr0A(r,t)A(r0,t0) Duetothecomplicatednatureofthisexpectationvalue,thereareaseriesofapproximationsmadetobringthisintoasimplerform.Theseapproximationsarereferredtoastheindependent-particleapproximation,thethree-stepapproximation,andthesuddenapproximation.InthissimpliedmodelthebasicmeasuredquantityisproportionaltothespectralfunctionconvolvedwiththeFermidistributionwithinthelimitsofresolution 44

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ofthedevice:I(kjj,!)jhnalj~A.kjinitialij2Zdk0Zd!0A(k0,!0)f(!0)R(!,!0) HereAisthespectralfunctionwhile~Areferstothevectorpotential,fistheFermi-Diracdistribution,andRistheenergyresolutionoftheapparatus.Inthepresenceofaniteselfenergy,thespectralfunctiongenericallytakestheform:A(k,!)=1 00k(!) (!)]TJ /F3 11.955 Tf 11.95 0 Td[(k)]TJ /F8 11.955 Tf 11.95 0 Td[(0k(!))2+(00k(!))2 ItisperhapsmoreusefultothinkofaFermiliquidwithniteselfenergy.AnequivalentrepresentationofthespectralfunctionisthenA(k,!)=Z )]TJ /F4 7.97 Tf 6.78 -1.79 Td[(k (!)]TJ /F3 11.955 Tf 11.96 0 Td[(k)2+)]TJ /F6 7.97 Tf 18.73 4.11 Td[(2k+Ainc(k,!) whereZ=(1)]TJ /F10 7.97 Tf 13.37 4.71 Td[(@ @!))]TJ /F6 7.97 Tf 6.58 0 Td[(1(inatranslationallyinvariantsituation),andAinc(k,!)representstheincoherentbackground.ThenextimportantfacttokeepinmindisthatARPESisasurfaceprobe.Themomentumk?isnotconservedbecausetranslationalinvarianceislostacrossthesurface.Ifthesurfacedoesnotcleavewellorsignicantsurfacereconstructionoccurs,theseeffectswillplayaroleinunderstandingtheobservation.Thatsaid,ifthoseeffectsareunderstoodorunimportant,ARPESprovidesveryvaluableinformation:electronicdispersions,lifetimes,theFermisurface,thesuperconductinggap.ForareviewinthecupratesseeDamascellietal.[ 45 ]. Foranon-interactingFermigasthespectralfunctionoftheelectronisadeltafunction[ 5 ].WithinteractionsweexpectagoodFermiliquidtohaveasharpquasiparticlepeakcorrespondingtoabroadeneddeltafunctionwithsomekindofincoherentbackgroundsignal.ARPESinthecupratesdoesidentifyaquasiparticlepeak[ 45 ],althoughitisquitebroadalongwithasignicantincoherentpart.Whatisalsointerestingisthattheweightinthequasiparticlepeakstronglydecreaseswithdecreasingdoping. 45

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Figure2-7. TheFermisurfacefromARPESinBSCCO.Reproducedwithpermissionfrom[ 58 ].APSc2002. ARPESmeasurementsunambiguouslyconrmedtheFermisurface[ 58 ],showninFigure 2-7 ,andtheunconventionalsymmetryofthesuperconductinggap.Theimportanceofthesemeasurementsshouldnotbeunderstatedbutwewillfocusonmorecurrenttopics.ARPEShasfoundahighlyanisotropicsuppressionofspectralweightthatpersistsfaraboveTc. Figure2-8. ThedemarcationofthepseudogaptemperatureaccordingtoARPESfromLeeetal.[ 59 ]APSc2006. 46

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Figure2-9. Laser-ARPESdataonBSCCO,showingtheFermiarcphenomena,andacontroversialclaimtoseetheclosureofthearc.Reproducedwithpermissionfrom[ 60 ].cNaturePublishingGroup2009. AnopenquestionhereistheappearanceofFermiarcsinsteadofaFermisurface[ 61 ]whichseemstoviolateLuttinger'stheorem[ 62 ].JustaboveTc,theFermisurfaceappearsasanarc,showninFigure 2-9 ,whichgrowslinearlyintemperatureuptoTwhenitbecomesthefullFermisurfaceweexpect[ 63 ].InhighmagneticeldsquantumoscillationexperimentssuggestthatthesearcsareactuallyFermipocketswithlittletonospectralweightintheARPESintensityonthebacksurface,givingtheappearanceofarcswheretrulyaFermisurfaceexists.ThesuggestionisthattheeldmayhavereconstructedtheFermisurface,eitherbystabilizingperiodicorderorsomeothereffect.Figure 2-9 illustratesthepitfallswell:thereisclearlyanicespectralweightalongthearc,andsomesmallersubsidiarypeak.Thiscouldbeashadowband,orasurfaceeffect,especiallysincetheintensityisatandfaint,sohowisonetointerprettheresult?Roughly,thereisanarc,androughlywearemeasuringthespectralfunction,butinaninstancelikethiswebecomeconcernedwiththemoresubtleaspectsintheoriginalideathatthethreecurrentcorrelationfunctioncanbethoughtofasthespectralfunction. AtTc,asuperconductingquasiparticlepeakappearsat(,0).Itdisplaysapeak-dip-humpstructurereminiscentoftheSTMresult.ThepseudogapisobservedasathequasiparticlepeakrecedesfromtheFermienergybelowT.ThereisapolarizedARPESmeasurement[ 64 ]tosuggestthatthisstateisatime-reversalbreakingstate, 47

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Figure2-10. (,0)quasiparticlepeakforasamplewithTc=91K,showingthepeakdiphumpstructure.Reproducedwithpermissionfrom[ 45 ].APSc2003. Figure2-11. AngulardependenceofthepseudogapinLSCOandattothed-waveform.Reproducedwithpermissionfrom[ 45 ].APSc2003. 48

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butthisisstillunderdebate.Whatisunusualaboutthepseudogapisthatitisnotsimplyanisotropicsuppressionofthespectralweight,butratheritseemstofollowad-waveform,lendingcredibilitytotheideathatd-wavesuperconductinguctuationsexistaboveTc:forexamplesuperconductinguctuationsareknowntobesufcienttoproducefermiarcs[ 65 ].Ofcourse,thisisalsostillunderdebatebecausead-densitywaveorangle-dependentquasiparticleweightZ(k,!)inexotictheoriescanalsocapturethisfeature[ 66 ]. 2.1.5NuclearMagneticResonance Figure2-12. TheKnightshiftin89YNMRinYBCO.Atx=1theKnightshiftisnearlytemperatureindependent,evolvingtostronglytemperaturedependentasxischanged,allaboveTc.Reproducedwithpermissionfrom[ 67 ].APSc1989. Ifthecupratesweregoodmetals,thesusceptibilitywouldbeconstantaboveTc.InunderdopedYBCOasTislowered,thesusceptibilityfallsforTwellaboveTc.Thismeasureprovidedsomeoftherstevidenceforthepseudogap.Onceinthesuperconductingstate,ad-waveBCSmodelforthespinlatticerelaxationratepredictsarathersmallHebel-Slichterpeakwhichisnotobservedinexperiment,butcorrectlycapturesthelowTasymptoticbehavior.TheabsenceofaHebel-Slichterpeakisgenerallyexplainedawaywithimpurityscattering,butitisacuriouslyabsentfeaturenonetheless.ThelowtemperaturespinlatticerelaxationratescalesasT3,asone 49

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expects,reectingalineardensityofstates.Multiplebands,disorder,andothernodalgapsareissuesinthepnictideswheresimilar1 T1behaviorisobserved. Figure2-13. Aschematic(inFrench)ofhowNMRspectroscopyworks,courtesyofJ.Bobroff.Thearrowsontheleftrepresentmagneticmomentsinducedaboutanimpurity.Thislocalmagnetisminuencesthenearbynuclei,whosesignalisdepictedontheright.TheNMRspectraisahistogramofthelocalmagneticeldsthroughoutthesample. InFigure 2-12 ,weshowtheKnightshiftinYBCOsamples.TheKnightshiftunexpectedlydevelopsastrongtemperaturedependencewithunderdoping.Thissuppressionoccursasifagapwereopeningintheexcitationspectrumofthesystem,signalingthepseudogap.InanormalmetalweexpectthePaulisusceptibility/2BN(EF).Thisistrueuntilaboutroomtemperature,300Korso.Aswelowerthetemperaturefromthere,thesusceptibilityhaslostapproximately80%ofitsvaluebythetimeTcisreached,indicatingaremovalofdensityofstates.Thesamepaper[ 67 ]demonstratesthataKorringalaw,i.e.that1 T1TK2sisaconstant,holdswhichisanindicationofFermiliquidbehavior. NMRwidthsshowadistributionofmagnetichyperneeffectsorequivalentlyadistributionofT1relaxationtimes,duetotheinhomogeneityinthelocalmagneticenvironment,oftenduetodefects[ 34 ].AdepictionofatypicalspectrumcanbeexplainedusingFigure 2-13 ,wheresatellitemomentsaddtothedistributionoflinewidths.ThisimpurityphysicscanbeusedtounderstandtheevolutionofNMRspectrainthecupratesasafunctionofdisorder. 50

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2.1.6Neutrons ForarecentreviewofneutronscatteringinthecupratesseeTranquada[ 49 ].Neutronmeasurementsofthespinexcitationsintheinsulatingparentcompoundsarewelldescribedbylinearspinwavetheory.Withincreasingholedopingtheantiferromagneticpeakat(,)developsincommensurability.Thereseemstobeauniversalpattern,showninFigure 2-14 ,totheacousticspinexcitations,whichtakeanhourglassshapeinQ-energyspace.Attheresonanceenergyaround41meV,thereareantiferromagnetic(,)excitations.Asyouraisetheenergytherearetwobranchesofexcitationsawayfrom(,),andsimilarlyfortwobranchesastheenergyislowered[ 68 ].Theincommensurabilityseemsdirectlytiedtodoping,atleastatlowholedoping.Inelectrondopedcupratestomyknowledgetheexcitationsremaincommensurate[ 69 ].Theseobservationsaresuggestiveofantiferromagneticornearlyantiferromagneticexcitationsinthesuperconductingstate.Theresonancemode'sexplanationisanopenquestion. Figure2-14. Q-Energydepictionofspinexcitationsincuprates.Reproducedwithpermissionfrom[ 49 ]. Atoptimaldopingaspin-gap,ofapproximately8meV,developsintheexcitationspectrumincontrasttolowerdopedsampleswherethespingapisreduced[ 49 ].Thegappedspectralweightgetspushedtohigherenergies. 51

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Oneofthemostinterestingfeaturesrevealedbyneutronsisastatewhichshowsupat1=8thdopinginLBCOandLSCO,whichhasbeenstudiedintensively.Thereisdeniteobservationthatthesuperconductivityiscompletelysuppressedatp=.125inthosetwomaterials,andthatsimultaneouslyaperiodicmodulationofspinandcharge,resemblingstripes,showsup.Theappearanceofquasi-one-dimensionalphysicsandspinchargeseparationinthesemesoscalestructuressuggeststhatthisorderissomehowintertwinedwiththesuperconductivity[ 70 ].Thisperspectiveissupportedbynumericalexperimentsonthet-JandHubbardmodelswheretheholesnaturallysegregateandcanformstripelikestructures.ThereisaplateauinYBCO'ssuperconductingdomewhereinLSCOitisstronglysuppressed,anditissuggestedthatthestripeorderissomehowuctuatingornonstatic. Asapartingremark,weobservethatthefullymomentumandenergyintegratedstrengthofmagneticscatteringdecreaseswithholedopinguntilitsdeathatoptimaldoping.Thistantalizingbitofinformationsuggeststhatthereisacoexistingmagneticglassorinhomogeneouslocalmagnetismresponsibleforthehostofstrangeunderdopedproperties 2.1.7PenetrationDepth Figure2-15. Temperaturedependentpenetrationdepth,showinglinearbehaviorforlowTincontrasttotheexponentials-wavebehavior.Reproducedwithpermissionfrom[ 71 ].APSc1993. 52

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InitiallythehighTccompoundsweredifculttogrowinlargecleansinglecrystalssamplequalitywasamajorissue.Oncesamplesbecameadequatereliableexperimentscouldelucidatetheunderlyingnatureofthesuperconductinggroundstate.Hardyetal.[ 71 ]usedacavityperturbationtechniquetomeasurethepenetrationdepthinoptimallydopedhighqualityYBCOcrystals,clearlyexhibitinglineartemperaturedependence,showninFigure 2-15 .ThisisindicativeofpointnodesinatwodimensionalgapoveracircularFermisurfaceorlinenodesonacylindrical(3D)Fermisurface.PreviousworkonthinlmsoftenreportedT2behavior,whichcanbeattributedtoimpuritiesinad-wavesuperconductor.Recallthatatthetime,establishingthed-wavesymmetryofthegapwasamajorgoal.Absolutevaluesofthepenetrationdeptharehardtoestablishbecauseexperimentsareusuallymeasuringachangeinthesuperuiddensity.Nonetheless,aconsistentsubtractionmustbeused,whichprovidesthebestideaofwhatthepenetrationdepthshouldbe.SRcanprovideabsolutevalues,howeverthesedependondetailsofthemodelingofthevortexstate.YBCOtypicallycomesoutwith1400-1600Apenetrationdepths,andBSCCOisabouttwicethesizeat2100A.LSCOisevenlargerwith4000A. 2.1.8ElectricandThermalConductivity Inthesuperconductingstate,ifweassumethatad-waveBCSsuperconductorisanaccuratedescription,somethingremarkablehappensintransport.P.A.Lee[ 72 ]showedthatinthelowtemperaturezerofrequencylimit,theinterceptoftheconductivitiesapproachaconstant,universalvalue,becausetheresultisveryinsensitivetodisorder.Ifwetakethesimpleresultthat (T)/N(T)v2F(T)(2) thenaccountforthedensityofstatesofad-wavesuperconductor,N(E)/E,includingthatforweakscatterers/1 Eitseemsliketheconductivityisindependentofenergy.Whilethisisaverynaveargument,itspursonfurtherinvestigation.Intheabsence 53

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ofvertexcorrections,thesamewouldbetruefortheelectricalandspinconductivities,buttheyaremorestronglyrenormalizedbyvertexcorrectionsthanthethermalcurrentwhichremainstoagoodapproximation,independentofthescatteringrate[ 73 ].Deningv2=@k @k,theuniversalvaluesintheabsenceofvertexandFermiliquidcorrectionsareshowninequation 2 .Thesevalueswouldneedtobescaledbythenumberofplanesperunitcelltocomparewithmeasuredvalues. 0=e2 ~2vF v2,0 T=k2B ~3vF+v2 vFv2,s=s2 ~2vF+v2 vFv2(2) Neglectingvertexcorrections,weprovidetherecipeforobtainingtheconductivities.Generallyonemustcalculatethebubble, Imij()=Xk1 Xi!ng2~vi~vjTr[G(k,i!n)G(k,i!n+im)](2) Theparametergreferstothechargeorspinforthosecurrentsrespectively;isaPaulimatrix:0forchargecurrent,1and3forspinorheatcurrents.Forthermaltransportg=!+ 2,althoughtheprecisedenitionofthermalcurrentcanposesubtleissuesbecausewearereferringtoalocaltemperaturegradient,seeforexampleCatelaniandAleiner[ 74 ].FromEq 2 ,=)]TJ /F4 7.97 Tf 10.49 4.7 Td[(Im forthechargeandspincurrentswhileforthethermalcurrent, T=Im T2.Atthelowesttemperatures,weexpectimpurityscatteringtoplayanimportantrole.Astemperatureisraised,inelasticprocesseswillalsocontributetothetransportprocesses,soaccuratemodelsoftheconductivitieswillnecessarilyincludeatreatmentofbotheffects. OneoftheusefulaspectsoftheuniversalvaluesisprovidinganotherwaytomeasuretheFermivelocityandsuperconductinggap,tocomparewithexperimentslikeARPESandSTM.vF v2isabout14inYCBOandmorelike20inBSCCO[ 75 ].Thedopingdependenceofthegap,usingaFermivelocityfromasecondmeasurement(typicalvaluesare105m=s),canalsobeobtainedasshowninFigure 2-16 54

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Figure2-16. Dopingdependenceofthegapaccordingtothermalconductivitymeasurements.Reproducedwithpermissionfrom[ 76 ].APSc2003. Thenormalstatetransportpropertiesaremuchmorestrange.Weventurenoguessatexplainingthem.InthephenomenologyoftheFermiliquid,theresistivityscalesatT2withtemperature.Theresistivityisapproximatelylinear[ 77 ]forhighTinunderdopedmaterialswithdeviationsatlowT,linearinTnearoptimaldoping,andthencrossedoverfromTtoT2intheoverdopedregionforallhole-dopedmaterials[ 78 ].Quantumcriticaltheories[ 79 ]cangiveexponentsdifferentfromT2,butcannotreproducetheseresults.Aconvincingexplanationofthisresultisamajoropenquestion. ThedynamicconductivitydoesnotfollowtheDrudeform0 1+i!,butdatacanttoaDrude-likeformRe()=dc 1+(!)yor0 1+i!(!)[ 80 ].Quantitativeornot,theutilityoftheexpressionisinprovidingreasonablemeasuresofthescatteringratewhichcanhelpidentifythemicroscopicinteractionswhichcontroltransportproperties.Oneofthemoststrikingresultsfromtransportmeasurementsisaclueaboutthenatureofscatteringinthenormalstate.AssoonasTcisreached,thescatteringratedropsprecipitously,indicatingthatgappingtheelectronsalsoremovedscatteringphasespace.Thisisstrongevidenceforelectronicpairinginthesuperconductingstate.Theresidual 55

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Figure2-17. Theanomalouslinear-in-Tresistivityoverawiderangeoftemperaturesinthecuprates.Reproducedwithpermissionfrom[ 78 ] resistivityperplaneisameasureofthescatteringrateoftheconductionelectronsbytheimpuritypotential.Itcanbeobtainedbyextrapolatingthehigh-T(T)toT=0. Thefrequencyintegratedconductivityinunderdopedcupratesisfoundtobeproportionaltox,thedoping,indicativethattheholescontributetotheconductivitydirectly.Thesamespectralweightcanbecomparedtowhatisexpectedfromsingle-particlepicturebandcalculations.Bothab-initioandexperimentalmethodsareopentosomecriticisms,butasageneraltrend,cuprateswillprovetobestronglycorrelatedbythismeasure,aswellasmaterialslikevanadiumoxide,butincontrasttosimplemetals.Aclearsignaloftheformationofthepseudogapisseeninc-axisconductivitywhileitisdifculttoidentifyinthea-bplanedata[ 30 ]. 2.1.9Nernst,Kerr,SR TheNernst-Ettinghauseneffectistheconnectionbetweenatransverseelectriceldcreatedbytheathermalgradientinthepresenceofamagneticeldorathermalgradientinducedbyanelectriceld.eN(H,T)E jrTj 56

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Specically,NernstreferstothedetectedelectriceldduetoanappliedthermalgradientandEttinghauseniswhenacurrentdensityproducesathermalgradient,allinaperpendicularmagneticeldQEjrTj JH. TheusualexplanationinvokestheexistenceofvorticesaboveTc.Severalauthorshaveworkedonthetheoryofthistransportprocess[ 81 82 ].Inanormalmetalthesignalissupposedtovanishintheabsenceofelectron-holeasymmetry[ 83 84 ].Ontheotherhand,theNernsteffectshavealsobeenobservedinsemimetalslikeBi[ 84 ]whereitislarge,andingraphene[ 85 ],wherethesignalisattributedtothreephysicaleffects:increasingthescatteringtime,increasingthecyclotronfrequencyandreducingtheFermienergy. Figure2-18. NernstSignalinnV/(oK-T)forLSCO.Reproducedwithpermissionfrom[ 86 ].APSc2006. IfwearetoassociatetheNernstsignalwithuctuatingsuperconductingvorticesaboveTc,wheretheCooperpairsarenotlongrangephasecoherent,thenthedata[ 86 ]showninFigure 2-18 clearlyfavorstheinterpretationthatthereisnoquantumcritical 57

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pointunderthesuperconductingdomeasthesignalextendsallthewayaroundthesuperconductingregion. Figure2-19. OnsetofKerrsignalinYBCO.Reproducedwithpermissionfrom[ 87 ].APSc2008. Incontrastwiththissetofexperiments,isworkdonebyKapitulnik[ 87 ]inYBCO.TheKerreffectistherotationofthepolarizationoflightbyscatteringfromatime-reversalsymmetrybreakingperturbation.ThesettingfortheKerreffectisusuallyferromagnetsoranyuniaxialmaterialwhichhasatendencytorotatetheincidentlight,soitissurprisinginanapparentlynon-magneticphasetoobserveanyKerrrotationatall.Mineev[ 88 ]hasshownhowtimereversalsymmetrybreakingcanbecalculatedfromtheelectromagnetickernelforunconventionalsuperconductors.ThedatashowninFigure 2-19 clearlyintersectthesuperconductingdomeforYBCO,favoringtheinterpretationthattimereversalsymmetryisbrokeninthepseudogapstateandsuggestingthatthesuperconductingdomeobscuresaquantumcriticalpoint. Thisissupportedbypolarizedneutronmeasurements[ 89 ]whichclaimtoobserveamagneticorderthatpreservestranslationalsymmetry,andSRexperiments[ 90 91 ].Briey,SRworksthefollowingway.Thetimeevolutionofthemuon-spinpolarizationisdependentonthelocalmagneticelddistribution,andismeasuredbydetecting 58

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Figure2-20. Onsettemperatureofmagneticorderobservedbyneutrons.Reproducedwithpermissionfrom[ 89 ].APSc2006. themuon-decaypositronsbecausetheirdecayispreferentiallyalignedwiththesample'smagneticmoment.TheSRmagneticmomentisconsistentwiththeneutronexperiments.However,thevolumefractionofthesampleinwhichthesemomentsexist,about3%,inwhichthesemomentsexistdonotsupporttheoriesthatascribethepseudogaptoastatecharacterizedbyloop-currentorder[ 92 ],asclaimedbytheneutronpaper,ratherthatdiluteimpuritiesandcorrelationsinducethiseffect.Furthermore,themuonstudiesndthatthedopingdependenceofthesignaldoesnottracktheonsetofthepseudogap.Thereisacounterclaim[ 93 ]thatthemuonchargedisturbstheobservationofamagneticmoment,butthiscontradictsSRwhichobservethemagneticmomentinthespin-glassphase[ 94 ]consistentwithNMRexperimentsinthespin-glassphase. 2.1.10RemarksonHighTcExperiments Ineachoftheexperimentsdiscussedthereisanopenquestion.Anoverviewofthetheoreticalproposalsisgivenattheendofthischapter.Thepseudogap'snatureisyettoberevealed,butnotforlackofeffort.TheanomalousresistivityisoutsideofthescopeofFermiliquidphenomenology.Inthetunnelingmicroscopythepatternaroundasingleimpurityisnotwhatcomesoutofthemoststraightforwardtheory, 59

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soacriticalexaminationofthetunnelingmatrixelementswhichsupposedlyaccountforthedifferencesisimportantifwearetohaveacompleteunderstanding.Someoftheworknotshownhere,notablybythegroupofJ.C.Davis,showsmesoscopicinhomogeneity,andoftentweed-likepatternsinthelocalconductanceswhichisattributedtolocalelectronicnematicorder.InARPES,thebiggestopenquestioninmyopinionisthenatureoftheFermiarcsvs.Fermipocketsobservedbyquantumoscillationexperiments.It'shardtoseethequantumoscillationdataandnotbelievethatthereisunderlyingFermiliquidphysics,butthemagneticeldistheseexperimentsisstrongenoughtosuggestthatzeroeldgroundstateisdifferentfromthestateinseveralTesla,forexampleaeldinducedspindensitywave.ThecaveatsinARPESanalysisalsomeansthattherecouldbeaverysignicantlossofspectralweightonthebacksideofthepocketsduetosurfaceeffectsorincoherence.Theneutrondata'sgrossfeaturescanbecapturedwithinanRPApicture,butfailstoreproducethepseudogapseenatlowdoping.Thenormalstateresistivityisawideopenquestion.Relatedtoit,istheproposalofamarginalFermiliquidwhichwouldgiveastrictlylinearresistivity,butthispicturelacksamicroscopicfoundation.QuantumcriticaltheoriesgivearesistivitythatvariesasT4=3,soperhapsnewavorsofquantumcriticalitywillcapturethisscaling.QuantumcriticalityisalsopromisingfortheneutronscatteringandKerreffect,butithastroublewithNernstmeasurementsandtheexplanationofdiamagnetismaboveTc. 2.2Pnictides In2006Kamiharaetal.[ 95 ],inthegroupofHideoHosono,discoveredsuperconductivityinanewclassofiron-basedcompounds,nowreferredtobyanyofthefollowingnames:pnictides,iron-arsenides,iron-basedsuperconductors.Theoriginalcompound,LaOFeP,onlyexhibitedTcofaround5K,soitwastwoyearsbeforeasimilarcompound,LaFeAsO1)]TJ /F4 7.97 Tf 6.59 0 Td[(xFx,whoseTcwas26K[ 96 ],causedexcitementinthephysicscommunity.Lowercriticaltemperaturesareassumedtobeduetoelectronphononcouplingasintheelementalsuperconductors.Intheironarsenidesithasreachedoforder55K[ 97 ],which 60

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isanindicationofunconventionalsuperconductingmechanisms.Anumberofusefulreviewsexist,forexampleJohnston[ 98 ]orSadovskii[ 99 ]. Figure2-21. IllustrationofBaFe2As2structure. Theiron-arsenidefamilyofmaterialsissubdividedintoseveralclassesreferredtoasthe1111's,122's,111's,and11'sreferringtotheirchemicalformulae.ForexampleLaFeAsOisa1111,whileBaFe2As2isa122,LiFeAsa111,andFeSeisa11.Itisperhapsobviousthatthebeginningofunderstandinganewmaterialislistingtheelementscomposingthecompoundandthecrystalstructure.ThisquestionisusuallyresolvedwithX-rayorelasticneutronscattering.Oneoftheancillaryquestionsthatcanbeaddressediswhetherthereareanystructuraltransitionsinthematerial.Thetwodominantstructuresinthepnictidesaretetragonal(I4/mmm)andorthorhombicstructures.Theparentcompoundsshowamagneticphase,andwithdoping,justasinthecuprates,superconductivityappears.OneexceptionisLiFeAswhichis 61

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anonmagneticstoichiometricsuperconductorwithaTcof18K.Dependingonthespeciccompoundtheremayormaynotbearegionofsignicantoverlapbetweenthesuperconductingandmagneticphases.Thetetragonalstructureisthehightemperaturephase,andatlowertemperatureandlowdopingsthereisatetragonal-orthorhombicstructuraltransitionasinthecuprates,showninFigure 2-26 .OneoftheotherremarkablestructuralfeatureswhichissimilartothecupratesisthepresenceofasquarelatticeofFeandhighanisotropyintransportmeasurements,whichimmediatelyleadstothespeculationthatonlytheseFeAsplanelayerscontainthephysicsasinthecuprates.Tocritiquethisidea,weshouldcheckrstwhetherthebandstructurehasc-axisdispersion,andtrytocomparethephasebreakinglengthcomparedtotheinter-layerspacing.Bandstructurecalculationsandexperimentsmeasuringthedispersioncanhelpanswertherstquestion.Tomyknowledge,thesecondquestionhasnotbeenaddressed,butthereisanemergingconsensusthattheso-called122compoundsthreedimensionalityevolvesasthebandstructureisalteredbydoping,andconsideringthesematerialstobeideallytwodimensionalmaynotbejustied. Figure2-22. Ba(Fe1)]TJ /F4 7.97 Tf 6.59 0 Td[(xCox)As2phasediagramfromacombinationofspecicheat,resistivity,andothermeasurements.Tisthestructuralphasetransition,Tisthemagneticphasetransition,andthedomerepresentssuperconductivity.Reproducedwithpermissionfrom[ 100 ].APSc2009. Theanisotropyinthe1111compoundsisfargreaterthaninthe122s,andthecorrelationsinthe11materialsseemtobemoreimportantthanintheotherfamily 62

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members.Iintendtodemonstratethatwhileitmaybepossibletoapplythesamephysicstodifferentpnictidematerials,aswevarythecompositionthroughoutthisbroadclassofcompounds,experimentsreportdifferentseeminglyconictingresultsregardingthesuperconductingstate.Thediversityofresultsobtainedappearstobemuchgreaterthaninthecuprates,evenaccountingfortheearlystageofmaterialsgrowthdevelopment.Onecanspeculatealongthefollowinglines:eachinstanceofapnictidemightliveinadifferentregionofparameterspace(t,UforaHubbardmodel,inadditiontoimpurityscattering)therebydisplayingdifferentpropertiesunderonecommonHamiltonian,ortherecouldbesignicantvariationinsamplequalityandtheeffectofimpuritiesmightresultinthediversityofobservations. 2.2.1OpticalConductivity Modelbuildingisadelicatetaskthatrequiresexperimentalinput.Thepnictideshavesimilaritieswiththecupratesatrstglance:squareplanarlayers,non-phononinducedsuperconductivity,amagneticphaseintheparentcompound.Soanimportantquestionistoaddresstheroleofcorrelationsinpnictides.Thisquestioncanbeaddressedbyacombinationofbandstructure,ARPES,andopticalconductivitymeasurements.Thegeneraldynamicconductivitywilldependonimpurityscatteringandinelasticscattering,soimportantinelasticprocessesandarealisticmodelfordisorderarenecessarytoachieveanunderstandingoftheconductivityoverabroadfrequencyrange.TheintegratedDrudeweightforagivenmaterialcanbecomparedwiththesamequantityasobtainedfromtheorytoprovideaqualitativeestimateoftheimportanceofcorrelations.Coupledwithdensityofstatesinformation,italsohelpsdeterminewhetherthematerialsareinsulatingorconductinginthemagneticstate.Thesemeasurementsfavortheinterpretationthatthepnictidesareitinerantsystemswithmoderatetoweakcorrelationsaswewillshow. Bysummingtheareaunderneaththeopticalconductivity(f-sumrule),wecanconstructthespectralweightandplasmafrequency(thespectralweightoffreecarriers, 63

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onlytheDrudepart). K(1)2 Z10d!1(!)(2) Z!cuto0d!1(!)=0!2p 2(2) Inpracticethisisdonewithacutoffforboththeoryandexperiment,butitaccuratelysegregatesstrongly,moderately,andweaklycorrelatedmaterials[ 101 ].TheextremesofthismeasurearethatforastronglycorrelatedinsulatortheratioKexp Kband!0,whileforgoodconductorsKexp Kband=1.ForLOFPKexp Kband.5whileforLOFAKexp Kband.3)]TJ /F8 11.955 Tf 12.21 0 Td[(.4,implyingthatpnictidesaremoderatelybutnotstronglycorrelated[ 101 ].Thisperspectivehasbeencorroboratedbyx-rayabsorptionandresonantinelasticx-rayscatteringstudies[ 102 ]whichindicatedweakcorrelationsinthepnictides.SeveralmodelsyieldU2eVandJ50meV[ 103 ]forthepnictides(bandwidtharound4eV),tobecomparedwithU10eVincuprates.Otherinformationwhichcanbegleamedfromopticalmeasurementsarecarrierconcentration,whichinpnictidesaremeasuredtobeafew1021=cm3,andestimated(inplane)scatteringrates,at300K400-900cm)]TJ /F6 7.97 Tf 6.58 0 Td[(1,at10Ktensofcm)]TJ /F6 7.97 Tf 6.59 0 Td[(1. Opticalmeasurements[ 105 ]intheundoped122sshowaDrudepeak,implyingthepresenceofitinerantcarrierseveninthespindensitywavestate,incontrasttotheinsulatingcuprateantiferromagnetism.Italsocontrastswithsomemeasurementsofthe11compounds,withnoobservedDrudepeakintheFeTesystem,whichtendstobemorecorrelatedthantheotherpnictides.TheStonerfactorispeakedinthevicinityof(,),consistentwiththepicturethatnearnestingdrivestheformationofthisgroundstate,butit'shardtoruleoutwhetherthecarriersparticipatinginconductionarethesameasthepartofthematerialresponsibleforthespinstructure.Startingatlowdoping,intheSDWstate,theDrudeweightsteadilyincreaseswithdoping.[ 104 ]. 64

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A B Figure2-23. (left)OpticalspectroscopyinundopedBaFe2As2,i.e.inthespindensitywavestate.(right)EvolutionoftheOpticalspectruminBa(Fe1)]TJ /F4 7.97 Tf 6.58 0 Td[(xCox)2As2withdoping.Reproducedwithpermissionfrom[ 104 ].APPSc2010. 2.2.2ARPES AngleresolvedphotoemissioninthepnictideshasobservedboththeFermisurfaceandthegap.Itisimportanttounderstandhowsurfaceeffectsmightmanifestthemselves,however.Inparticular,sincetheperpendicularcomponentofthephotonmomentumisnotusuallyresolved,thedatawillreectonesliceinthek?plane.Thereisasmallbutimportantchangeinthebandstructureatthesurfaceaccordingtodensityfunctionalcalculations[ 106 ],andinthesuperconductingstate,thesurfacescatteringcouldbeplayinganimportantroleinaveragingoutnergapstructures.ThesuccessofARPEShereistheoverallconrmationofthecorrespondencebetweenthecalculatedandobservedFermisurfaceshapes[ 107 ]:twoholepocketsaroundthe)]TJ /F1 11.955 Tf -437.08 -23.9 Td[(point~k=(0,0)andtwoelectronandtwoelectronsheetsbytheMpoint~k=(,).ThistwodimensionaltopologyseemstobecommoninallthepnictidesasshowninFigure 2-24 .Thethirddirectioncouldhavesignicantvariationamongdifferentpnictides.Thepnictidesarecompensatedsemi-metals,meaningthattheholepocketandelectron 65

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pocketdipthroughtheFermienergyinsuchawaythatthellingoftheelectronpocketspillsintotheholepocket[ 98 ].Themultibandsemimetalnatureofthepnictidescanbeusedtoexplaintheunusualnormalstatesusceptibility[ 108 ],whichincreaseswithtemperature(exceptinthe11-pnictideswhichincreaseordecrease)contrarytoonebandFermiliquidexpectations. A B Figure2-24. (A)Calculated(DFT)FermiSurfaceofBarium-122at10%Co-doping.(reproducedwithpermissionfromtheauthor)(B)ARPESNdFeAsO1)]TJ /F4 7.97 Tf 6.58 0 Td[(xFx.Reproducedwithpermissionfrom[ 107 ] TheconsensusfromARPESonthenatureofthesuperconductinggapisthatthesuperconductivityisfullygappedintheferropnictides,allowingforonlyaweakvariationaroundtheFermisurface[ 109 ].ArepresentativescanisshowninFigure 2-25 .Smallvariationsinthegaparesuggestedfromthedata,buttheerrorbarsarecomparabletothevariation.Thisclaimedbehaviorstandsinsharpcontrasttothelowtemperaturepowerlawsseeninbulkprobeswhichsuggestgapnodes,sothesurfacesensitivityofangleresolvedphotoemissionispossiblyimportantintheinterpretation. 66

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Figure2-25. VariationofthesuperconductinggapinNdFeAsO0.9F0.1aroundthe)]TJ /F1 11.955 Tf 10.1 0 Td[(holepocket.Reproducedwithpermissionfrom[ 107 ] 2.2.3NMR Cooperpairsarecreatedbybindingtwospin1 2electrons.Intheabsenceofspin-orbitcoupling,thetotalspinremainsagoodquantumnumber,sowecanmakeacleandistinctionbetweenaspin0andspin1Cooperpair.TheKnightshiftistherstindicationofthespinofaCooperpair,asdiscussedpreviously.TheKnightshiftgoestozeroasT!0inaspinsingletsuperconductorbecauseanappliedeldcannotpolarizethecondensate.AnonzeroKnightshiftatT=0ismoredifculttointerpretbecauseoftheeffectsofpossiblespinorbitcouplingandtherelativeorientationofthecomponentsandtheappliedeld.Both31Pand75AsNMRextrapolatedtoT=0indicatesingletCooperpairsinthepnictides[ 110 111 ]. Nuclearmagneticresonanceinthesuperconductingstateoftheiron-arsenidesobservedaT3spinlatticerelaxationrate[ 111 113 ],whichisconsistentwithnodallinesinthegap.Forafullygappeddensityofstates,onewouldexpectexponentiallowtemperaturebehaviornomatterhowmanybandsexist.Thenextissueishowimpuritiesmodifythosenotions.Ifdisorderispair-breakinginansgap,meaningthephaseofthesuperconductinggapontwoFermisheetshasoppositesign,thenalowenergy 67

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impuritybandcouldcontributetosomeoftheseprobes.Innodalsuperconductorsthelowenergyimpuritystateswillalsomodifypowerlaws.Thestrongestconclusionwecandrawisthepossibilityoflinenodesinthesuperconductinggap. A B Figure2-26. ClearevidenceforT3spin-latticerelaxationrateinthepnictideLa(O1)]TJ /F4 7.97 Tf 6.59 0 Td[(xFx)FeAs.Reproducedwithpermissionfrom[ 112 ].(B)KnightShift,57Fe,inthesamecompoundprovidingevidenceforthesingletnatureofCooperpairs.Reproducedwithpermissionfrom[ 111 ].APSc2008. Theq-averagedstaticsusceptibility,Pq(q,0),whichoccursin1 T1T,isequivalenttothelocalsusceptibilityinrealspace.InNakaietal.[ 112 ],thereisnodivergencewithmagneticorderingasyouwouldexpectinasecondorderphasetransition.Theseauthorstthespindynamicstophenomenologicalformsderivingfromself-consistentspinuctuationtheory.TheconclusionsofNakaietal.supportothermeasurementsinsayingthatwehaveanitinerantsystemwhichhasaspindensitywaveinstability,incontrasttotheantiferromagneticMottinsulatorinthecuprates.Thisisoneofthe 68

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majordifferencesbetweenthecuprateandpnictidefamilies.Oneoftheotherimportantcontrastsisthewidevarietyofbehaviorinthepnictides.InLaFeOAs,thepictureofitineracyissupportedbydata,butinthe11's,thereseemstobeaspin-glassregion,andalocalmomentdescriptionofthemagnetismcouldbeappropriateforlowdoping.Thisisalsoconsistentwiththenotionthatthe11'saremorecorrelatedthanthe1111's. 2.2.4Neutrons Thenormalstateneutrondatainthemagneticallyorderedstateforlowfrequencycanalwaysbetbyspin-waveanalysis[ 98 ],sincethatistheappropriatelongwavelengthdescriptionofthoseexcitations,sothereareprimarilyseveralfeaturestoexaminefromneutrondata.Themeasuredorderedmomentsare0.35Binthe1111sand0.87Binthe122s[ 114 ].OneoftheoriginalproposalsforamodelofthemagnetisminthesematerialswasaJ1-J2Heisenbergmodel[ 115 ],undertheassumptionofanewstronglycorrelatedmaterialbeforeexperimenthadweighedinonthedebate.Themainproblemswiththismodelarethatthesystemhasitinerantcarriersandthattherequiredsizeofthemagneticmomentwastoolargecomparedtoexperiment,soitwasarguedthatcorrelationeffectsreducethemoment[ 103 ].Anotherissueisthatspinwavedampingisobservedinneutronscattering,soonecaneitheraddsomeinelasticprocesstodampthespinwaves,likeaspin-phononcoupling,ordecayintotheStonercontinuumshouldbefavored. Zhaoetal.[ 116 ]reportedneutrondataandatusingathree-dimensionalHeisenbergmodelforthespinstructureinCaFe2As2.ThemagneticmomenttendstoberightaroundhalfaBohrmagnetoninthe1111s,andalittlelargerinthe122s.ThesetsareshowninFigure 2-27 .Noglobaltusingspinwavetheorycanbeobtained[ 117 ].AnitinerantpictureisfavoredbyDialloetal.[ 118 ],whousethesamemodeltottheirverysimilarneutrondata,butreportsubstantialdampingduetoaStonercontinuum.Thispictureisalsosupportedbynon-zeroDrudeweightintheopticaldata,densityfunctionalcalculations[ 119 ],andStoner-enhancementofthesusceptibilityintheregionofthe 69

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Figure2-27. FitstoinelasticNeutrondatafortwocutsthroughtheBrillouin(a,b)zoneandtheintegratedintensityinpart(c).[ 116 ]cNaturePublishingGroup2009. 70

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antiferromagneticorderingvector.Amixedinterpretation,neithercompletelylocalizednorcompletelyitinerant,asinKeetal.[ 120 ]forexample,seemstobethewayoutofthesecontradictoryresults. Onenteringthesuperconductingstatethereisaspinresonance,onlybelowTc,believedtoexistsincescatteringconnectstworegionsoftheFermisurfacehavesuperconductinggapsofoppositesign.Itisthoughttobeaneffectfromthecoherencefactors[ 121 ],andisobservedind-wavesuperconductorsaswell.Theenergyoftheresonanceisoforder10meV,althoughthewavevectorwhereitoccursseemstodependonthematerial,seeJohnston[ 98 ]forachart.TheseobservationslendsupporttoasignchangeinthegapacrosstheFermisurface,whichisbeyondasimples-wavepicture. 2.2.5PenetrationDepth PenetrationdepthmeasurementshavebeentbothtoexponentialT-dependence[ 122 ]andlow-Tpowerlaws[ 123 ].Itispossiblethatthesedifferencesreectgenuinelydifferentgroundstatesindifferentmaterials,soweshouldtakecaretokeeptrackofthedopingandmaterialonwhichthemeasurementsweretaken.Anotherlessonfromthecupratesinthiscontextistowaituntilrelativelycleansinglecrystalmeasurementscanbetaken.Fortunatelytheeldisalreadyatthispointandreliabledataappearstobeavailable. Again,inthesimpleonebandpicture,thelinearresultisthesameasad-waveorderparameter,fromwhichweinferlinenodesinthesuperconductinggap,butthisignoresthemultibandnatureofthepnictides.Systematicstudiesofthepenetrationdepthintheferropnictidesversusdoping[ 124 126 ]revealsthegeneraltrendthatatoptimaldopingthesematerialsseemtolacknodes(whichisnottosaythegapsareisotropic).Asweoverdope,thepowerlawbehaviorindicativeofverysmallornodalsuperconductinggapsshowsup,withTnhavingnbetweenabout1and3Intheunderdopedspindensitywavecoexistenceregionresultsarelessclear.Aninteresting 71

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APrFeAsO BBa(Fe1)]TJ /F23 6.974 Tf 6.22 0 Td[(xNix)2As2 Figure2-28. (left)PrFeAsOexhibitingexponentialfullygappedbehavior.Reproducedwithpermissionfrom[ 122 ](right)Lineartemperaturedependenceofthec-axispenetrationdepthinBa(Fe1)]TJ /F4 7.97 Tf 6.59 0 Td[(xNix)2As2.Reproducedwithpermissionfrom[ 123 ].APSc2009,2010. featureofsomeearlydata,wasalowTupturninthechangeinpenetrationdepth.Thisisascribedtotheexistentoflocalmomentsintheheavymetal,forexampleNdinthe1111[ 127 ]. 2.2.6HeatTransport Heattransportmeasurementsinthesuperconductingstateswithnodalgaps,shouldyieldalineartermintheT!0limitofthethermalconductivity.Thisisgenerallytakentobeastrongindicationfornodes.Ind-wavesuperconductorsthislineartermisuniversal,inthesensethatitsmagnitude,toleadingorder,isindependentofdisorder[ 72 73 ].Theuniversalthermalconductivitywon'tbepresentifthereiscompetingorder,ifthescatteringrateismomentumdependent,orforananisotropics-wavegap[ 128 ].Fora-bplane Tthereisanegligibleresiduallineartermforalldopings[ 129 ]inBa(Fe1)]TJ /F4 7.97 Tf 6.58 0 Td[(xCox)2As2,evidencethattherearenonodesinthegap. HoweverFigure 2-29 showsc-axisthermalconductivityinBa(Fe1)]TJ /F4 7.97 Tf 6.59 0 Td[(xCox)2As2[ 130 ].Thepresenceofzeroenergyquasiparticlesisindicativeofnodesinthesuperconductinggap.Thisisoneofthekeyindicatorsforthreedimensionality,becausethiseffectisnot 72

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Figure2-29. Heattransportalongthec-axisinBa(Fe1)]TJ /F4 7.97 Tf 6.59 0 Td[(xCox)2As2,whichshowsanonzerothermalconductivityatT=0fromtheextrapolation.Thisindicatesthepresenceofzeroenergyquasiparticles.Reproducedwithpermissionfrom[ 130 ].APSc2010. observedbythesamegroupinthea-bplanetransportmeasurements.UntanglingtheeffectsofdopingontheFermisurfaceandtheevolvingsuperconductinggapwillbechallenging,butnecessarytocometoquantitativegripsonthisdata. Wecanputthisinperspectivewiththeseeminglyconictingpenetrationdepthstudies.Itisquiteeasytogetlostwhentryingtotrackwhichcompoundsforwhichdopingsdoordonotindicatethepresenceoflowenergypower-laws.Onereasonforthedifferencesisthatevenwithinasinglecompound,thestructuresseemtobedopingdependent.ThemorphologicalchangesinboththeFermisurfaceandsuperconductinggapresultinavarietyofconclusionsregardinggapstructure.Fromthatperspective,allthedataseemstomakesense:someunconventional,possiblynodalsignchangingextendeds-wavestatewhichissensitivetosmallchangesinelectronicstructure[ 106 131 ]. 73

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2.3TheoreticalSuggestions Theoryhastheirksometaskoftryingtocreateaframeworkwhichsimultaneouslyexplainsalloftheexperiments.Itcanbesubdividedbyphaseandmaterialbutthetaskisstilldaunting.Wefocusonthesuperconductivity.EarlydensityfunctionalcalculationsfoundthattheelectronphononcouplingwastooweakinthepnictidestoexplaintheTc'swithEliashbergtheory,soalternativesespeciallyelectronicmechanismswereproposed.Themagnetismcanbeunderstood,inparticulartheneutrondata,fromanitinerantpicture,andincuprates,pnictides,heavyFermions,andorganicsuperconductorsmagneticphasesabound,sospinuctuationsareanaturalcandidate.Zerotemperaturechangesinthegroundstatewithpressure,eld,ordopingalsosuggestlookingatquantumcriticalscenarios.Thelowsuperuidstiffnessand'cheapvortices'incuprates,alongwithdiamagnetismaboveTc[ 132 ],suggestaphase-disorderedbutpairedmaterial.Thereareyetmoreproposals.Anypersonassessingtheeldmustbecomeacquaintedwitheachoftheseapproachesandtheirawsifalegitimateattemptatunderstandingistobemade. 2.3.1SpinFluctuations Thetermspinuctuations,referstoaneffectiveinteractionbetweentwoelectronsmediatedbyuctuatingspinpolarizationoftheelectronicmedium.Animmediatedifcultyisthattheobjectswhicharepolarizing,theelectrons,areinteractingwiththemselves.Theelectronphononinteractionhastwoseparatescalesonhandbecausethevibratingionsaremuchheavierthantheelectrons.NosuchclearseparationofscaleoccurshereatthelevelofthebareHamiltonian,soitisdifculttounderstandwhattodowhentheobjectsbeingpairedmediatetheirownpairing.Diagrammatically,theapproachhereistostartwithperturbationtheoryinthelowUlimitoftheHubbardmodel[ 133 ],shownforthesingletchannelinFigure 2-30 )]TJ /F9 7.97 Tf 6.77 -1.79 Td[("#=U 1)]TJ /F8 11.955 Tf 11.95 0 Td[((U0(k)]TJ /F5 11.955 Tf 11.95 0 Td[(k0,!))2+U20(k+k0,!) 1)]TJ /F8 11.955 Tf 11.95 0 Td[((U0(k+k0,!))2 74

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Figure2-30. DiagramscontributingtothesingletchannelofspinuctuationseffectivepairinginteractionduetoUni"ni# SummingallthediagramsgivesrisetoaninteractionoftheformVkk0~.~!)]TJ /F6 7.97 Tf 26.96 4.7 Td[(3 2Vkk03 2U20 1)]TJ /F6 7.97 Tf 6.59 0 Td[((U0)inthesingletchannel.ForreferencetheomitteddiagramscontributingtothetripletchannelsumtoVt=)]TJ /F4 7.97 Tf 6.58 0 Td[(U20 1)]TJ /F6 7.97 Tf 6.58 0 Td[((U0)2.Foragaptobeconsistentwithagiveninteractionitmustsatisfytheequation:k=)]TJ /F12 11.955 Tf 11.3 11.36 Td[(XpVsf(k,p)p 2Eptanh(Ep 2T) Theminussigninthegapequationiscrucial.Iftheinteractionisrepulsive(>0),thentheonlywaytosolvethegapequationistohavethegapatmomentump,oppositeinsigntothegapatmomentumk.Inthed-wavecase,whereforcupratesthenestingvectoratwhichthesusceptibilitypeaksis(,),k=0(cos(kx))]TJ /F8 11.955 Tf 12.48 0 Td[(cos(ky))satisesk=)]TJ /F8 11.955 Tf 9.3 0 Td[(k+(,). 2.3.2ResonatingValenceBondsandSlave-Bosons TheundopedcupratesareMottinsulators.TheholedopedregionofthephasediagramisadopedMottinsulator,andthereisstillnocompletetheoryforthiskindofmaterial.Anideawhichhasbeenaroundsincethediscoveryofthehigh-TccompoundsisthatthereisanintermediatephasebetweentheantiferromagnetandFermiliquid,wheredopingdestroystheantiferromagneticorderandrandomizedsingletsdecoratethelattice.ExperimentallyitisclearthattheNeelstateisdestroyedwithsufcient 75

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doping,evenifit'smorerobustontheelectrondopedside.P.W.Andersonproposedthisresonatingvalencebondspinliquidtosimultaneouslyaccommodatetheholekineticenergyandexchangeenergy.Tripletexcitationsoutofthisstatestillhaveaspingap,butinplanetheholesremaingaplessforchargetransport.Tobemoreprecise,thegroundstateisaphasecoherentsuperpositionofallsingletcongurationsonagivenlattice.ThenomenclaturecomesfromtheKelkulestructureforbenzenewherethedouble-bondresonatesbetweentwoequivalentpatterns.Inthesuperconductingstate,itissaidthatthesesingletscondenseintothesuperconductingpairs.Thepseudogaptemperaturescaleisassociatedwiththecondensationofsinglets.Thesuperconductingcarriersbecomephasecoherentatalowertemperature,formingthesuperconductingdome.Ifwewanttocomparethiscriticallytoexperiments,thenwewouldtakeissuewithexperimentswhichshowthepseudogapscalecuttingthedome.Eithersingletsformbeforethesuperconductingpairsortheycouldformsimultaneously,butitisdifculttoimaginesuperconductingpairsformingthatarenotsinglets.Anyregionofthesuperconductingdomeextendingbeyondthepseudogaplineisnotconsistentwiththissimpleidea. ThereisnoknownHamiltonianindimension2orhighertodateforwhichtheRVBstateisasolution.TheRVBproposalisavariationalansatz.Assuchitisofteninvestigatednumerically[ 134 ](variationalMonteCarlo),andthesecalculationshavenotbeenextendedtonitetemperature,accordingtoEdeggeretal.[ 135 ],representinganimportantdirectionforfuturework.Inlieuofreferencingthecommunities'workforthemorethanthirtyyears,whichseemsahopelesstask,werefertheinterestedreadertothereviewsbyEdeggeretal.[ 135 ]andLeeetal.[ 59 ].ToimplementtheRVBidea,oneneedstoproposeaHamiltonian,usuallythet-JoraversionoftheHubbardmodel,thenthevariationalwavefunctionjRVB>=^PN^PGjBCS> 76

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isused.Where^PN^PGaretheGutzwillerandparticlenumberoperators.^PGprojectsoutdoubleoccupancyand^PNxesN,sincethisisnotdenedfortheBCSwavefunction,jBCS>=Qk(uk+vkcyk"cy)]TJ /F4 7.97 Tf 6.59 0 Td[(k#)j0>.Thenumericaldetailsofsuchareprocedureareintheaforementionedcitations. Earlyonitwasrealizedhowtoformulateaslave-bosonversionofthetheory[ 136 137 ],whichhasbeenreviewedinLeeetal.[ 59 ],resultinginagaugetheoryofstronglycorrelatedelectronsystems.Thisapproachusuallygoesunderthenamerenormalizedmeaneldtheory.Iftheslavebosonmethod[ 138 139 ]isused,thereisaformulationbasedonafermionicrepresentationofthespinsandabosonicrepresentationoftheholesaswellasanoppositeformulationwithbosonicspinons,furthermoreisitpossibletoattachuxtotheseoperators.cyi=fyibiorfyibi Thelanguageassociatedwiththistransformationisthatelectronsfractionalizedintoaholon(b)andspinon(f),likespinchargeseparationinonedimensionalphysics.TheconnectionbetweentheRVBproposalandthisapproachisthattheslave-bosontransformationenforcestheno-double-occupancyconstraintforthet-Jmodel,providesasingletorderparameter,andthatthevariationalRVBwavefunctionwasfoundtobeaselfconsistentsolutiontothemeaneldequations.TheconstraintofsingleoccupancyintroducesaneffectivegaugeeldbyrequiringaconstraintviaaLagrangemultiplier.Xfyifi+bibi=1. Inoneapproachtotheconstructionofaslave-bosonHamiltonianwearriveat:H=)]TJ /F12 11.955 Tf 11.29 11.36 Td[(Xij(ijfyifj+c.c.))]TJ /F3 11.955 Tf 11.96 0 Td[(+Xifyifi+Xij[ij(fyi"fyj#)]TJ /F5 11.955 Tf 11.96 0 Td[(fyi#fyj")+c.c.] Anunresolvedissueisthatthesetheoriespredictthesuperconductingdomestartingatzerodoping.Anopenquestionishowtodestabilizetheantiferromagneticphase,make 77

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atransitiontothespingapregion,thenbeginthesuperconductingdomesinceatlowenoughTthesingletswouldalwaysformCooperpairs. Figure2-31. PhaseDiagramsaccordingtotheRVB,see[ 59 ](left)andQCP(right),see[ 79 ],proposals. 2.3.3QuantumCriticality Quantumcriticalityisamajorthemeinavarietyofcondensedmatterphysicsproblems[ 79 ].Thetermreferstoaphasetransition,atT=0withniteTconsequences,whichoccursasanexternalparameterotherthantemperatureistuned.Typicalexamplesofthecontrolparameterinpracticearedisorder,pressure,andmagneticeld.ThequantumcriticalpointinthestrictestsensereferstotransitionsatT=0,thoughweexpecteffectsatnitetemperaturefromcriticaluctuationsduetothiszeroTcriticalpoint.ThediscussionofquantumcriticalitydatesbacktotheseminalpaperbyJ.Hertz[ 140 ],whichledtoaparadigmthatgoesbythenameHertz-Moriya-Millistheory[ 141 ].ThisapproachtomagneticquantumcriticalityconstructsaLandau-Ginzburg-WilsonactionforthespinsectorofanonsiteHubbardinteraction,integratesoutthefermionswiththeHubbard-Stratonovichidentity,andstudiesthecriticalbehaviorusingstandardrenormalizationgrouptechniques.Wehavelearnedintheinterveningyears,thatthereareanumberofsubtlepointsinthisanalysis:itisnotalwayssafetointegrateoutthefermionsoranyotherslowmodes[ 142 ]becausenonanalyticcorrectionstothefreeenergyoccurathigherordersinvalidatingthisapproach.Onlyaposterioricanthevalidityofthisapproachbeexamined.Thisleadsnaturallytoconsideringtherenormalizationgroupforboththeorderparameterandthefermionssimultaneously 78

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[ 143 ].AnotherconsiderationistheeffectofBerry'sphasesintheaction,whichcanhaveanimportantimpact[ 144 ]onthequantumtoclassicalmapping.Yetanotherconceptimportantinclassifyingtheexistingapproachestoquantumcriticalityisthatoflocalquantumcriticality[ 145 ],whichwasstudiedintheKondo-latticemodelofheavyfermions.Atalocalquantumcriticalpoint,wehaveanewlocaldegreeoffreedom,i.e.localmoments,whicharecriticallycorrelatedintimebutnotspace.Manyexperimentsobserve! Tscalingintheuniformsusceptibility,whoseproposedexplanationislocalcriticality.Itisaquestionspecictoamaterial,whetherthelocalmomentsorthelongrangeorderdominatethephysics,butinthecasethatthelocalmomentsbecomecriticalbeforethemagneticorderparameterdoes.Differentlanguagetodescribethesameideaistoaskwhenthefelectronsdelocalizewithrespecttowhentheycouldorder.TheRKKYinteractioncancausethemtoorderwhentheyarelocalized,butbelowtheKondotemperaturethesemomentsarescreened,soitbecomesaquestionofwhicheffectwinsout,orderingbeforescreeningorscreeningthereforenoorder. Quantumcriticalityarisesinthecupratesbecauseofthenatureofthepseudogapandthenon-superconductingstate,aswehaveseeninKerr,neutron,andtransportmeasurements.Thecontroversyisthatfromonepointofview,thepseudogap'sonsettemperatureasafunctionofdopingdoesnotintersectthesuperconductingdome,andfromanotherpointofviewitdoes.Inthelatercase,thefunnelshapedregionofthisputativequantumcriticalpointdescribesthequantumcriticalstate.InthequantumcriticalscenariotheFermiliquidisthehighdopinglowTphase,andthepseudogaprepresentssomekindofasofyetunidentiedorder,andthestrangemetalisaquantumcriticalmetal. CharacterizingsomethingasaFermiliquidisthestatement,justiedperturbativelyandrecentlywithrenormalizationgroupapproaches,thatthereisadiabaticcontinuitybetweenthenoninteractingelectrongasandonewithinteractions.Therestrictiononthephasespaceofinteractingelectronsmeansthattheybehaveasanelectron 79

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gaswithrenormalizedparameters.AsweapproachtheFermisurface,assumingitexists,theexcitationsinaFermiliquid(thequasiparticles)areinnitelylonglived.AnumberofphenomenologicalconsequencesareassociatedwithFermiliquids.TheresistivityshouldscaleatT2,linearelectronicspecicheat,thesusceptibilityistemperatureindependent(toagoodapproximation),andcertainfamousratiosareconstant:Korringa,Wilson,Wiedemann-Franz.Inheavyfermion,cuprate,pnictide,andorganicmaterialstherearescoresofobservationswhichdonotfollowtheFermiliquidparadigm. 2.3.4CompetingOrder Anexplanationofthepseudogapisanopenquestioninthephysicsofhigh-temperaturesuperconductivity.Amongmanyscenarios,oneproposalisthatthepseudogapisduetoahiddenbrokensymmetry.Inthepseudogapphase,thedensityofstatesisdepletedatlowenergies,asifsomeofthedegreesoffreedomofthesystemweredevelopingagap.Toexplainthis,somehaveproposedanunconventionaldensitywave[ 146 ]intheparticle-holechannel.ToavoidasecondorderphasetransitionatthepseudogaponsettemperatureT*,thestatecannotbreakanycontinuoussymmetries,onlydiscreteones,orwewouldanticipateaGoldstonemodewhichhasyettoappearinexperiment.SuchadescriptionwouldpositaHamiltonianoftheformH=Xkkcykck+Xk,iW(k)cykck+Q+Xkkcyk"cy)]TJ /F4 7.97 Tf 6.58 0 Td[(k# whereQisthenestingvector.Foraconventionaldensitywave,Wconstant,butwecouldhaveunconventionalstateslikeaD-densitywavewithW(k)=cos(kx))]TJ /F8 11.955 Tf 12.2 0 Td[(cos(ky).Nochangeindensityoccursif=0,soitmaybedifculttodetectatransition.Thethermodynamicswouldresemblethatofanunconventionalsuperconductorbecauseoftheappearanceofagapinthespectrumentersmanycalculationsinroughlythesameway.Thestateisabletoviolatetimereversalsymmetry,wouldpossessgaplesslowenergyquasiparticles,andhasmanyinterestingmagneto-transport 80

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properties[ 147 ].TheexistenceofspontaneouslybrokentimereversalsymmetryhasbeenclaimedinKerrrotationandneutronexperiments,alongsidethetheoreticalproposalthatahiddenordercausestheseeffects.Inparticular,anotherexplanationoftheNernstdatacouldbeduetosuchadensitywave[ 148 149 ],butseealso[ 82 ].Coexistingchargeordercanreconciletheunderdopedthermalcurrent'sdeparturefromuniversality[ 150 ],althoughanexplanationinvolvingmagneticcorrelationsaroundimpuritiesisalsopossible[ 151 ].Firmlyunderstandingwhattheexactconsequencesofdifferentvarietiesofunconventionaldensitywaveorderareisstillanopenquestion. AnotherkindofspontaneousorderisatendencyfordopedMottinsulatorstophaseseparateintomesoscalestructures.Inhomogeneitiesinmanganitesarecommon,andespeciallyaround1 8thdopingforafewcuprates,X-rayandneutrondatarevealsstripeorderincuprates.Howthisconnectstosuperconductivityisadifcultopenquestion.ItmightbeacompetingphasecoincidentallyinthemiddleofthephasediagramandspecialtoafewcompoundsoritmightrevealuniversalaspectsofhighTcphysics.Itshouldbenotedthatwhenstripesoccur,Tcissuppressed,implyingacompetitionbetweenthetwophases. 2.3.5BEC-BCScrossover Superconductivitycanlooselybethoughtofastwodistinctphenomena.First,weformCooperpairsandthepairslaterBose-condense.Sincepairingdoesnotimplyasupercurrentwecouldformagapwithoutseeingasupercurrent.Whenlongrangephasecoherenceoccurs,thereisalsoasupercurrent.AruleofthumbtogaugewhentheCooperpairswouldcondenseiswhentheinterparticlespacingbecomesiscomparabletothecoherencelength.Forordinarys-wavesuperconductors,thecoherencelengthsextendoverhundredsofangstroms(1600AinAl),whereasinhigh-Tcmaterialstheytendtobeofordertens-ofangstroms(1.4nminYBCO).Thismeansthatthewavefunctionsins-wavecompoundstendtoalreadyoverlapandthereisnodifferencebetweentheBECandpairingtemperatures,whereasinhigh-Tcmaterials 81

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thepairscouldformthenwaituntillowTtoundergoBEC.Cupratesarealsomoretwodimensional,whereuctuationeffectsaremoresignicantthanhigherdimensions.Inthismodelthepseudogapphaseisaphasewithpartiallypairedelectronswithnolongrangephasecoherence,thusnosupercurrent.TheexperimentbyLuLietal.[ 152 ]usedtorquemagnetometrytomeasurediamagnetismaboveTcinLSCO,YBCO,andBSCCOwhichsupportthispicture.TheonsetofdiamagnetismiscompatiblewithNernstmeasurements. AquantitativeimplementationoftheBEC-BCScrossover[ 153 ]comesfromidentifyingthegapequationwiththescatteringlengthintraditionalBECproblems.StartingfromamodelHamiltonian, H= ()]TJ 11.17 8.09 Td[(r2 2m)]TJ /F3 11.955 Tf 11.96 0 Td[() )]TJ /F5 11.955 Tf 11.95 0 Td[(g # # ", aHubbard-Stratanovichtransformationcanbringtheactiontotheform S=1 gZddx264jj2)]TJ /F5 11.955 Tf 11.95 0 Td[(Trln0B@()]TJ /F3 11.955 Tf 9.29 0 Td[(@t+r2 2m+)()]TJ /F3 11.955 Tf 9.3 0 Td[(@t)]TJ /F9 7.97 Tf 13.44 4.71 Td[(r2 2m)]TJ /F3 11.955 Tf 11.95 0 Td[()1CA375. Thenthegapequationisrelatedtoboththecouplingconstantgandthescatteringratesinceweunderstandhowtoformbothagapequationandarelationbetweenthescatteringlengthaandthecouplingg. =1 2Tr(^G1),m 4a=)]TJ /F8 11.955 Tf 10.74 8.09 Td[(1 g+Xk1 2"k, resultsin)]TJ /F5 11.955 Tf 15.23 8.09 Td[(m 4a=Xk1)]TJ /F8 11.955 Tf 11.96 0 Td[(2f(Ek) 2Ek)]TJ /F8 11.955 Tf 18.55 8.09 Td[(1 2"k. ThisrelatesscatteringandinteractioneffectstothepairingandistheheartoftheBEC-BCScrossoveridea. 82

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CHAPTER3ANGLE-DEPENDENTTHERMODYNAMICS 3.1DensityofStatesandLowTemperatureSpecicHeatintheVortexState Thelowtemperaturephenomenologyofmanyprobesintothenatureofthesuperconductinggapdistinguishbetweenfullygappedandnodalsuperconductors,butdonotprovidephasesensitiveinformationaboutthelocationofnodes.Furthermore,exponentialbehaviorintemperaturecanmasqueradeasT4scalingoveraniterange,andtheeffectofimpuritiescanfurtherobscuretheresult.Itisthereforeofinterestwhenconfrontedwithanewsuperconductingmaterialtohavemorespecicprobes.Inanidealworld,experimentssensitivetothephaseoftheorderparametercouldbeperformedprovidingsomeofthesought-afterinformation,butinpracticetheseexperimentscanbequitechallenging,andprecisecontrolofthecrystalgrowthandinterfacequalityisnecessarybeforetheycanbeundertaken.Thisiswhyittookthebetterpartofadecadetocometoaconsensusabouttheorderparameterinthecuprates.Thischapterdiscussesthecompromisebetweenthesetwoextremes:asimplethermodynamicbulkprobewhichisnotphasesensitive,butprovidesmoredetailedinformationaboutthelocationandpresenceofnodesthantransportexperiments.Ameasurementofthespecicheatinthepresenceofamagneticeld,H,whichisrotatedwithrespecttothecrystalaxesofthesample[ 154 ],mapsoutoscillationsinthedensityofstatesrelativetothenodalpositions. AboveHc1,inans-wavesuperconductor,thelow-temperatureentropyisdominatedbythevortexcoreboundstates.Thelevelspacingisoforder20 EF,whichistypicallysmallfors-wavesuperconductors.Insuperconductorswithlinenodes,thevortexcorecontributiontothelow-energydensityofstatesissmallerthanthatfromthequasiparticlesoutsidethecoreregion.Insystemswithshortcoherencelength,suchascupratesandheavyfermionmaterials,theextendedquasiparticlestatesalsodominate 83

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theentropysincethelevelspacinginthevortexcoreislarge,andonlyfewsuchlevels(ifany)areoccupiedatlowtemperature. Thequasiparticlesoutsidethecoremovewiththesuperowaroundeachvortex,sotheeffectofanappliedmagneticeldcanbeaccountedforsemiclassicallybymovingintothisreferenceframe,thatis,Doppler-shiftingthequasiparticlespectrumaccordingtothelocalvalueofthesuperuidvelocity,vs(~r).ThisapproximationisvalidwhenHHc2,aslongasthevorticesarespacedfarenoughaparttoconsideressentiallyonevortexcell,neglectingvortex-interactions,andvs(~r)variesslowlyovermostofthevortexcellonthescaleofthesuperconductingcoherencelength.ThisapproximationhasbeenscrutinizedbyDahmetal.[ 155 ],whoperformedacarefulcomparisonofseveralcommonapproximationsfors-waveandd-wavesuperconductorsinthevortexstate,sowecanrelyonquantitativecertaintythatforthissituation,wecaptureallthequalitativephysics.Quantumeffects[ 156 157 ]whichinterveneatextremelylowtemperatureswillnotberelevanthere. Thesingle-particleGreen'sfunctioninthepresenceofasuperowvelocityeldvs(~r)isobtainedbyDopplershiftingthequasiparticlestateswithenergy!andmomentum(withunitssuchthat~=kB=1), G(k,vs(~r),!n)=)]TJ /F8 11.955 Tf 10.49 8.08 Td[((i!n)]TJ /F5 11.955 Tf 11.96 0 Td[(vs(~r)k)0+k1+k3 (i!n)]TJ /F5 11.955 Tf 11.95 0 Td[(vs(~r)k)2)]TJ /F3 11.955 Tf 11.95 0 Td[(2k)]TJ /F8 11.955 Tf 11.96 0 Td[(2k(3) where!nisthefermionicMatsubarafrequency,kisthebandenergymeasuredwithrespecttotheFermilevel,iarePaulimatricesinparticle-holespace.Theorderparameterkwilldependonthemodelweconsider,forexample0cos2ford-waveor0sin()forap-waveorderparameter.TheDopplershiftisgivenby vs(r)k=~kF 2mrsin( )sin()]TJ /F3 11.955 Tf 11.95 0 Td[()(3) where isthewindingangleofthesuperuidvelocityinrealspace,theazimuthalangleontheFermisurface,istheanglebetweenHandthea-axis,andristhe 84

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rescaleddistancefromthecenterofthevortex.Notethatweexplicitlyexcludefromconsiderationquasi-1DFermisurfaces.Wewillfurthermorerestrictourconsiderationtoanisotropicmaterials,andconsideressentiallytwodimensionalplanes. SincetheBCSquasiparticlesarenoninteracting,thedensityofstatescanbeusedtocomputetheentropyaccordingtothestandardrecipe, S=)]TJ /F8 11.955 Tf 9.3 0 Td[(2Z1d!N(!)[(1)]TJ /F5 11.955 Tf 11.96 0 Td[(f(!))log(1)]TJ /F5 11.955 Tf 11.96 0 Td[(f(!))+f(!)logf(!)](3) wheref(!)=(exp(!=T)+1))]TJ /F6 7.97 Tf 6.59 0 Td[(1istheFermifunction.ToobtainthetheheatcapacityatconstantvolumewedifferentiateatconstantvolumewithrespecttoT, CV(T,H)=T@S @TV(3) AtlowTandH,whenthegapvariesweaklywithtemperature,thetemperaturethederivativeactsonlyontheFermifunctioninEq.( 3 ),andthespecicheatisapproximatelygivenbytheform CV(T,H)1 2Z1d!N(!,H)!2 T2sech2! 2T.(3) whichcanserveasanasymptoticlow-Tchecktothefulltemperaturedependentresult. ThelocaldensityofstatesisN(!,r)=)]TJ /F8 11.955 Tf 14.03 8.09 Td[(1 2ImXkTrG(k,!)]TJ /F5 11.955 Tf 11.96 0 Td[(vs(r)k)'N0ReZ20d 2j!)]TJ /F5 11.955 Tf 11.95 0 Td[(vs(r)kFj p (!)]TJ /F5 11.955 Tf 11.95 0 Td[(vs(r)kF)2)-221(j()j2 HereN0isthenormalstatedensityofstates.ThenetDOSpervolumeisfoundbyspatiallyaveragingN(!,r)overaunitcellofthevortexlattice,whichwecrudelyapproximatewithacircle,containingoneuxquantum,0.Toaccountforanisotropyandthree-dimensionalmaterials,werescalethec-axistheunitcellareabyafactorc=ab,anduxquantizationdictatesthatthequasiparticlesnowexperiencethe 85

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effectiveeldH?=(ab=c)H.InthenewcoordinatestheradiusandtheareaofsuchacellareRH=p 0=H?andAH=R2Hrespectively.Introducingnon-dimensionalizedpolarcoordinates,r=RH(cos ,sin ),wendfortheaverageeld-dependentdensityofstates N(!,H,T)=1 Z10dZ20d N(!,r)(3) Therearetwofactorswhichcauseanangularvariation:theFermisurfaceanisotropy,andtheanglebetweenthesuperowandthequasiparticlemomentum,onwhichthesuperconductinggapkdepends.Itisthisangulardependencewhichisexploitedtodistinguishbetweendifferentsuperconductinggaps.Intheabsenceofimpurityscattering,thereafewimportantenergyenergyscalesintheproblem:thetemperatureT,thegapmagnitude0,andthemagneticenergyortypicalDopplershiftEH=vF RH,whichwecanexploittohelpinterpretresults. Whenthepnictidesrstachievedprominencearound2008,therewereearlyindicationsofunconventionalsuperconductivity,asdiscussedinChapter2ofthisthesis.Sincethelowesttemperaturespecicheatvariationsinloweldareproportionaltoaproductofthedensityofstateswithathermalfactor,arapidcalculationoftheangularvariationofthedensityofstatesatzeroenergywouldserveasausefulindicationthatthegapvariedaroundtheFermisurface,anditmightidentifythelocationofnodes.Themaincontributiontothedensityofstatescomesfromthenodalquasiparticles,soitisagoodapproximation[ 158 ]forT,EH0tosumoverthemomentaatthenodesalone(awayfromnodesquasiparticleenergiesarefullygapped).Theresultsofthiscalculation[ 160 ]areshowningure 3-1 .Thehopewasthatitwouldbepossibletoperformasimplebulkprobelookingforthedistributionofgapnodes. Onceexperiments[ 159 ]demonstratedavariationofthespecicheattherewasastrongindicatorforanodalgap,inthe11-pnictideusedfortheexperiment.Theevidenceforanodalgapwasrstthatthereisastrongangularvariationofthespecicheatineld,andsecondthattheangularvariationwasobservedtoinvert 86

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A B Figure3-1. Predictedelddependenceofthedensityofstates[ 158 ](left)foraselectionofunconventionalgapsinthe11iron-pnictides[ 159 ]. asthetemperatureandeldwerevaried,whichisaspecicallyattributedtonodalsuperconductingbehavior. Asecondeffectduetoappliedmagneticeldcanalsobeusedtodistinguishbetweennodalandfullygappedorderparameters.Ifwehaveoneuxquantumpervortexthenifthevortexunitcellistakentobeacircle,R2=hc 2e H 87

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Sincemoststateslieoutsidethevortexcore,ifweincludethepresenceofamagneticeldbyDoppler-shifting!!!)]TJ /F5 11.955 Tf 12.32 0 Td[(vs(r).k,andrecalculatethedensityofstates,thenatzerofrequency:N(E!0,H)/1 AreaZd2rImG=1 Area whichforacircularvortexcell,frompowercounting,canbeseentovaryas:1 R2ZRdrr~kf 2mr/1 R2R, resultinginVolovikscaling[ 161 ]:N(0,H)TTp H. Inafullgappedsuperconductor,weperformthesamecountingasaboveonlyweneedtoremovethelowenergydensityofstatesifwearebelowthegap-scale,i.e.thereiszerodensityofstatesbelow0sotheelddependenceisthecontributionfromthevortexcore,andislinearinH.Inthisway,scalinghelpsdistinguishtheessentialphysics. Figure3-2. Electronicspecicheatofad-waavesuperconductorasafunctionoftemperatureinzeroandnonzeroappliedeld.TheVolovikeffectmanifestsitselfintheoffsetatzeroT. 88

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ThemainmanifestationoftheVolovikeffectisseeninmeasurementofthelineartermintheelectronicspecicheat,CV=(H)T.InacleannodalsuperconductorinzeroeldCvmaybeshowntovaryasT2.DuetotheVolovikeffectthezerotemperaturevalueofCvwillhaveanoffsetduetotheeld,whichcanbeseeningure 3-2 .Forasinglebandsuperconductorwithanodalgap,thelinearcoefcientwillscaleasp H.Aswiththeotherexperimentswehavetoaskhowthiscleanbehaviorchangesinthedirtylimitwhendisordermodiestheseresults.WhentheimpuritybandwidthiscomparabletotheenergyassociatedwiththeDopplershift,magneticeldeffectsareasimportantasdisorder.Theinuenceofdisorderistochangethedensityofstatesinto[ 162 ]:N(H,!=0) N00 impH Hcln 2Hc H. Withtheseresultsinhand,wenowturntotheinterpretationoftheseexperimentswithmorecare.Thenodalapproximationisrelaxed,andthelowtemperatureconstraintisrelaxed.Sincetheexperimentsareperformedatnitetemperature,itisimportanttounderstandtheeffectthishasonthepreviousresults.Thepredictionsaboveforthelowtemperaturespecicheatshowinparticular,thatthespecicheathasminimawhenthemagneticeldpointsinthedirectionofthenodesintheorderparameter,andmaximafortheeldalongtheantinodes,butthisisnottrueingeneral. 3.2TemperatureVariationoftheSpecicHeatinanAppliedField CeCoIn5isaheavyfermionsuperconductorwhoseorderparameterwasonlyidentiedrecently.Theinitialmeasurementsoftheanisotropyofthespecicheat[ 163 ]andthethermalconductivity[ 164 ]appeartogiveresultsforthegapstructurewhicharerotated45withrespecttooneanother,dxyanddx2)]TJ /F4 7.97 Tf 6.59 0 Td[(y2respectively.InSr2RuO4,whichisnominallyap-wavesuperconductor,thespecicheatoscillationswereobservedtoinvert[ 165 ]asthetemperatureandeldwasvaried,thatistheminimaandmaximaasafunctionofanglechangedplaces.TheenigmaisthenhowtounderstandtheinversionofoscillationsandwhythethermalconductivityandspecicheatinCeCoIn5gave 89

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contradictoryresults.AresolutionwasprovidedbyVorontsovandVekhterin[ 166 ].Tounderstandthis,weperformedthefullcalculationofthetemperaturedependenceofthespecicheatforad-wavesuperconductorinanappliedeldinthesemiclassicalDoppler-shiftapproximation. Toincludethefulltemperaturedependence,weneedtoalsoaccountforthevariationofthesuperconductinggapwithtemperature.Tothisend,werelyonaweakcouplinginterpolationformuladuetoEinzel[ 167 ]: (T)=0cos(2)tanh Tc 0s 4 38 7(3)(1)]TJ /F5 11.955 Tf 14.92 8.09 Td[(T Tc)!,(3) where0=2.14Tcisthegapmaximum. A B Figure3-3. (left)C(T,H)=C(T,=0)vs.forasetofequallyspacedtemperatures,every.02Tc,from0to.18Tc,Eh=.2Tc.(Right)sameforasetofequallyspacedtemperaturesfrom0to.08Tcevery.02Tc,Eh=.05Tc. WedemonstratethatthisinversionisagenericfeaturebycalculatingtheangledependentspecicheatinunconventionalsuperconductorsusingboththeloweldDoppler-shift[ 168 ]andtheBrandt-Pesch-Tewordtapproximation[ 169 ]appropriateforeldsnearHc2.Therewillalwaysbeadifferencebetweennodalandantinodalquasiparticlessimplybecauseittakeszeroenergytoexciteanodalquasiparticle,butoncethetemperaturescaleisoforderthelargestgap,therearesomequasiparticlesexcitedfromallregionsoftheFermisurface,andthedifferencebecomesless 90

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pronounced.Asexpected,theanisotropyintheangle-dependentdensityofstatesiswashedoutathigherenergies.However,thedifferencebetweenN(!,H,T)forthetwodirectionsoftheeldreappearsatenergiesoforder0,theenergyscaleabsentintheversionsofthecalculationwithlinearizedgap,asaresultoflinearization. A B Figure3-4. (left)Resultsforthespecicheatasafunctionofeld,angle,andtemperatureintheBPTapproximationdemonstratingtheinversionofoscillations.Reproducedwithpermissionfrom[ 169 ].APSc2008.(right)Resultingall-Tall-HphasediagramfromtheBPTandDoppler-shiftapproaches[ 168 ]. TheDopplershiftapproachisonlyvalidintheloweldregimewhenitisappropriatetoconsideronevortexcell.Astheeffectofahigherappliedeldtakeshold,itisnecessarytouseadifferentformalism.PreviousworkusingtheBrandt,Pesch,Tewordt[ 170 ]quasiclassicalapproachwascompletedbyVorontsovandVekhter[ 171 ]andcomplementourlowHcalculations.Theresultoftheirworkisshowningure 3-4 .Wewillnotreviewthismethodindetail,onlyjustgiveanoverviewofhowitissetup. ThequasiclassicalGreen'sfunctionsobeythedifferentialequations[ 172 ], !n+ivF 2(r i+2 0A(r))f(i!n,,r)=(,r)g(i!n,,r)(3) !n)]TJ /F5 11.955 Tf 11.96 0 Td[(ivF 2(r i)]TJ /F8 11.955 Tf 13.48 8.09 Td[(2 0A(r))fy(i!n,,r)=(,r)g(i!n,,r)(3) 91

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Whichissolvedself-consistently,togetherwiththefollowingconditions:thenormalization, g(i!n,,r)=(1)]TJ /F5 11.955 Tf 11.96 0 Td[(f(i!n,,r)fy(i!n,,r))1=2(3) andtheeldandgapequations, (,r)=N02TX!n>0Z20d0 2V(0,)f(i!n,,r)(3) rra(r)=)]TJ /F8 11.955 Tf 9.29 0 Td[(22 TX!n>0Z20d0 2^k ig(i!n,,r)(3) wherea(r)istheinternaleld,A(r)=1 2Hr+a(r).TheBrandt,Pesch,Tewordtapproximationreplacesg(i!n,,r)byitsspatialaverage,andthefunctions,f(i!n,,r)areexpandedinanappropriatebasis,resultinginaself-consistentsetofequations.Thistechniquewasemployedtoobtainthehigheldresults,beyondthevalidityoftheDoppler-shiftapproximation. Ourmainndingisthatthesignoftheoscillationsofthespecicheatasafunctionoftheeldorientationdependsonthetemperatureandeldstrength.Weconrmedthatatlowtemperaturesandeldsthespecicheathasaminimumwhentheeldisalonganodaldirection.However,asHandTareincreased,minimaofthespecicheatbegintooccurfortheeldalongthegapmaxima,i.e.,aninversionoftheoscillationpatternoccurs.Thetechniqueshouldproveuseful,generally,inidentifyingpossiblyunconventionalsuperconductorsassoonassinglecrystalsbecomeavailable. Theinversionofspecicheatoscillationshasbeenobservedin[ 173 ]aswellasbyH.H.Wenetal.[ 159 ].InthepaperonFeSe0.45Te0.55[ 159 ],the2.6-2.7Kdataconvincinglydisplaysfour-foldoscillationswithaminimumwhichappearswhentheeldisalongthe)]TJ /F1 11.955 Tf 6.77 0 Td[(Mdirection.InaonebandmodelinthelowTlowHregion,thisidentiesthenodesasalongthesamedirection.Theanalysisismoresubtlehere,becauseofthefourbandsinthepnictides.ItshouldalsobekeptinmindthatFermisurfaceanisotropycancontributetothevariation.Onthismatter,itistheinversion 92

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A B Figure3-5. Datademonstratingtheinversionofspecicheatoscillationswithincreasingtemperatureina11-pnictide[ 159 ](left).TheheavyfermionsuperconductorCeCoIn5displaysasimilarinversionandoscillationpatternconsistentwithadx2)]TJ /F4 7.97 Tf 6.59 0 Td[(y2superconductinggap.Reproducedwithpermissionfrom[ 163 ].APSc2001. ofthespecicheatoscillationswithtemperaturewhichisaclearsignatureofanodalgap,whichisnotpossiblefromsimpleFermisurfaceanisotropy.Weneedtoestimatetheeldandtemperaturescalesintheproblemtoseeiftheseminimacorrespondtoantinodes.Thein-planemagneticeldintheseexperimentsis9T.Hc2isnotgiveninthepaper.MeasurementsbyTsurkanetal.[ 174 ],ndthatHc2is85Tinthea-bplaneforthesamecompoundatslightlydifferentdoping.ThismeanswecanestimateH Hc2tobeaboutatenth,whichissufcienttobeintheloweldregime.Tcis14.5Kinthespecic 93

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heatmeasurement,sothisisabout20%ofTc.Accordingtoourmodelphasediagram,theminimahereshouldcorrespondtomaximainthenodesbecausethatistheinvertedtemperatureregime. Thisanalysisisalsoresponsiblefortheresolutionoftheheavyfermionparadox[ 166 ].TheinterpretationisthatCeCoIn5hasadx2)]TJ /F4 7.97 Tf 6.59 0 Td[(y2superconductinggap,andthattheapparentcontradictionbetweenthetwoexperimentsissimplythatoneoftheexperimentswasbeingperformedintheinvertedregimewhennodaldirectionscorrespondtomaximainsteadofminima.ItwasconcludedthatCeCoIn5hasadx2)]TJ /F4 7.97 Tf 6.59 0 Td[(y2superconductinggap. 94

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CHAPTER4RAMANSCATTERING Thestructureofthesuperconductinggapisresponsibleforalargepartoftheobservedphenomenologyintransportandthermodynamicexperiments,soprimacyisgiventoitsidentication.ThischaptercontinuesinthesameveinasChapter3,infocusingonexperimentswhichelucidatethegapstructure,withoutbeingasdemandingasphasesensitivemeasurementssothattheymightbeperformedsoonerafterthediscoveryofanewsuperconductor.AswesawinChapter2,nuclearmagneticresonanceshowedaT3spinlatticerelaxationrate,1 T1,reminiscentofagapwithnodes.Angleresolvedphotoemissionmeasurementsonsinglecrystalsof122-typematerialsreportednearlyisotropicgapsaroundtheFermisurface.PenetrationdepthmeasurementshavebeentbothtoexponentialT-dependence,indicativeofafullygappedstate,andlow-Tpowerlaws,suggestiveofnodes.Itispossiblethatthesedifferencesreectgenuinelydifferentgroundstatesasthespeciccrystalanddopingisvaried.However,thecomplexinterplayofmultibandeffects,disorder,andunconventionalpairingleavesopenthepossibilitythatasinglemechanismforsuperconductivityexistsinalltheFe-pnictides,andthatdifferencesinmeasuredpropertiescanbeunderstoodbyaccountingfordisorder,bandstructure,anddopingchanges.Itislikelythataconsensuswillbereachedonlyaftercarefulmeasurementsareperformedusingvariousprobesonthesamematerial,forthesamedoping,andsamecrystalquality(impurityscatteringrate)systematicallymappingouttheparameterspaceforthepnictidefamily.Herewecontinuethethemeofgapidenticationandfocusontheroleofdisorderspecically. Ramanscatteringistheinelasticscatteringofpolarizedlightoffofasolid.Inelasticscatteringissensitivetotheexcitationsinasystem,butit'sthepolarizationdependenceofRamanscatteringwhichoffersadegreeoffreedomthatcanbeexploitedtotheexperimentalist'sadvantage.Justaswesawwiththeangle-dependentspecic 95

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heat,measurementoftheelectronicRamanscatteringinthesuperconductingstatecanprovideimportantinformationonthestructureoftheorderparameterthroughitssensitivitytosymmetry,throughthepolarization,andgapscales.Intheangle-dependentspecicheat,itwastheexternalmagneticeldwhichprovidedanextradegreeoffreedomfortheinvestigationofthegapstructure.WhetheragivenRamanpolarizationweightsagivengapstronglyorweaklydependsonthepolarizationofthelightviatheRamanvertexk.Thisisparticularlyimportantforsuperconductorswherethegapisstronglymomentumdependent,andwasexploitedsuccessfullyincupratematerialstohelpdeterminethed-wavesymmetryinthosesystems,whereasins-wavesystemstheresponsewouldlookthesameindependentofpolarization.Opticalprobes,aswesawins-wavesuperconductors,aresensitivetothepairbreakingscale20.Itcomesasnosurprise,then,thattheenergyofthepeaksintheRamanintensityaredirectlyrelatedtothemagnitudeofthegapsonthevariousFermisheetsintheunconventionalcasealso.Inaddition,thepresenceofnodesandthedimensionalityofthenodalmanifoldmaybedeterminedthoughnotalwaysuniquelybycomparisonwithlowenergypowerlawsintheRamanintensityindifferentpolarizationstates.Thesepowerlawscanbealteredbydisorder,sosystematicirradiationordopingstudiescancomparetheevolutionofthesepowerlawsincomparisonwiththeoreticalpredictionstodeterminetheoverallconsistencyofamodel.Dopingisnotthesameasdisorder.Ramanscatteringwithuseofdifferentpolarizationsmaythereforebeausefulmethodofacquiringmomentum-dependentinformationonthestructureofthesuperconductingorderparameter.Wediscussseveralcasesbelowwhichshouldallowextractionofthedominantgapsymmetry,thegapmagnitudes,andpossiblenodalstructureoftheorderparameterovertheFermisurface. 4.1Theory Theintensityofscatteredlightisproportionaltotheimaginarypartofthechannel-dependentRamansusceptibility[ 175 ],whichresemblesadensity-densitycorrelationfunction 96

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weightedbyavertex, ,(!)=Z0de)]TJ /F4 7.97 Tf 6.59 0 Td[(i!mhT~()~(0)iji!m!w+i.(4) Off-resonance,i.e.awayfromcollectivemodesordirectinterbandtransitions,thevertexisindependentoffrequency,andwecanwritetheeffectiveRamanchargeuctuationsinthepolarizationwithvertexas ~=Xk,Xn,mn,m(k)cyn,(k)cm,(k),(4) wheren,mdenotebandindices.n,m(k)denesthemomentum-andpolarization-dependentRamanvertices.Generally,thevertexisdeterminedbymatrixelementsbetweentheconductionbandandtheexcitedstatesthroughdensityandcurrentoperatorswhichmayincludebothintrabandandinterbandtransitions.Eveninsimplemetals,likeAl,thevertexisoftenunknownordifculttocalculate,soitisstandardpracticetomakeuseofcertainapproximations.Thepolarizationsoftheincomingandoutgoingphotonsimposeanoverallsymmetryduetothewayinwhichexcitationsarecreatedinthedirectionsdeterminedbytheelectriceld,soitbecomespossibletoclassifytheRamanverticesintobasisfunctionsoftheirreduciblepointgroupofthecrystal.Wewillusethisbasishere.Theintermediatestateswhichcontributetotheverticesarestilltoocomplicatedtobeamenabletoanalyticwork.FurtherprogresscomesfromrequiringthatthemomentumtransferredbythephotonbesmallcomparedtotheFermimomentum,goodinalmostallcases,andthefrequencyoflightneedstobelessthanthebandgap.TheintermediatestatespresentinthegeneralRamanvertexincludeboththestatescreatedfromtheinitialstateaswellasstatesseparatedfromtheconductionband.Thus,thenon-resonantapproximationisvalidiftheenergyoftheincidentandscatteredphotonsfallsinthisrange.Theintraband(n=m)contributiontotheRamanamplitudesnforacleansystemmaybeexpressedintermsoftheeffectivemassandtheincidentandscatteredpolarizationvectorsesi 97

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n(k)=m ~2X,ei@2n(k) @k@kes.(4) FornbandscrossingtheFermilevel,theintrabandRamanresponseisgivenby ~,~(!)=1 NXkXnn(k)2n(k,!),(4) intheabsenceofchargebackoweffects.Here n(k,!)=tanhEn(k) 2kBT4jn(k)j2=En(k) 4E2n(k))]TJ /F8 11.955 Tf 11.95 0 Td[((~!+i)2(4) istheTsunetofunctionforthenthbandandthebanddispersionisn(k),energygapn(k),andquasiparticleenergyEn(k)=p 2n(k)+2n(k). SinceRamanscatteringprobeschargeuctuationsinthelong-wavelengthlimit,theroleofthelong-rangeCoulombinteractionisquiteimportant.Inparticularitisimportanttoaccountforscreening.Thelong-rangeCoulombinteractioncanbetakenintoaccountbyincludingcouplingsoftheRamanchargedensity~totheisotropicdensityuctuations, scr~,~=~,~)]TJ /F3 11.955 Tf 13.15 8.09 Td[(~,,~ ,,(4) with ~,=,~=1 NXnXkn(k)n(k),(4) and ,=1 NXnXkn(k).(4) Equations( 4 )-( 4 )constituteclosedformexpressionsfortheintraband,non-resonantcontributiontotheRamanresponse. ItisclearfromEqs. 4 4 theRamanresponseisingeneralthesumofeachbandseparately,andthereforeanon-linearfeedbackoccurs.Correctionsenforcenumber-conservationofchargedensityuctuationsandgaugeinvariance,inparticular,thescreeningofthechargeuctuationshavebeenincluded.Dueto 98

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thetothedependenceonthedirectionofthepolarization,incidentphotonscancreateanisotropicchargeuctuationsandthosechargeuctuationsrelaxbytheusualscatteringmechanisms:electron-impurity,electron-phonon,electron-electroninteractions,orviabreakingofCooperpairs.Thesechargeuctuationstransformaccordingtotheelementsoftheirreduciblepointgroupofthecrystal.ForD4htetragonalsymmetry,in-planechargeuctuationstransformaccordingtothefullsymmetryofthelatticeA1g,meaning90degreerotationsdonotchangethesignasdotheB1gandB2grepresentations,whichchangesignunder90rotations.Ramanscatteringprobeslongwavelengthchargeuctuations,thereforetheB1gandB2gchargedensitiesthataveragetozerowithineachunitcellarenotscreenedviathelong-rangeCoulombinteraction.Asaconsequence,theRamanchargedensitiesforthesechannelsdonotcoupletotheisotropicchargedensitychannel,andthetermsinEq. 4 vanish.ThustheB1gandB2gRamanresponsesforamulti-bandsystemconsistsimplyofthesumofthebare-bubblecontributionsfromeachband.Incontrast,A1guctuationsneednotaveragetozeroovertheunitcell,andthereforetheycancoupletoisotropicchargeuctuations,givingthecorrectionsrepresentedbythesecondterminEq. 4 WhilethethebarebubbleexpressionfortheB1gandB2gchannelsaregenerallyaccurateforthecuprates,theA1gcontributionissignicantlymorecomplicatedduetotheissuesassociatedwiththescreeningoflongwavelengthuctuations.Insystemswithseveralpairinginstabilitieswhicharenearlydegenerateenergetically,itispossibletohavestrongexcitonicpeaksintheA1gpolarization.Thesecanbethoughtofastransitionsbetweentwonearlydegeneratesuperconductinggaps[ 176 ].SpuriousexcitonpeakscanoccurinRamancalculationsusingonlythebarebubblebecausegaugeinvarianceisnotpreserved.ByincludingthelongrangeCoulombeffectsitisthisrequirementofgaugeinvariancewhichoftenliftsthelowlyingcollectivemodestohigherenergy.AprimeexampleistheAnderson-Bogoliubovmode,whichisacousticforcharge 99

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neutralsuperuidslikeHelium,butisliftedtotheplasmaenergyoncevertexcorrectionsandlong-rangeCoulombinteractionsareincludedrestoringgaugeinvariance. InordertofocusongeneralfeaturesforRamanscatteringinthepnictides,wewillapproximatetheanisotropicFermisurfacesfrommorerealisticbandstructurecalculationsandtreatallFermisurfacesheetsascircles.Thisallowsforsimplesymmetry-basedasapproximationstotheRamanverticesandwecanexpandtheverticesasFermisurfaceharmonics,andhasbeendoneinthecuprates[ 177 ].Thishasimportantconsequencesforthenon-vanishingofthevertex.Expandingthepolarization-dependentverticesinFermisurfaceharmonicsforcircularFermisurfaces, n()A1g=an+bncos(4)n()B1g=cncos(2)n()B2g=dnsin(2), (4) withangle-independentbandprefactorsan,bn,cn,dnsettingtheoverallstrengthoftheRamanamplitudesforbandn. Isotropicdensityuctuationvanishforq!0,soitcanbeshownviaEq. 4 thattheA1gcontributionantotheRamanvertexdoesnotcontributetothescatteringcross-section.Thelowestordernon-vanishingcontributiontothevertexinthisexpansionisfoundtobecos(4). TakingtheimaginarypartofEq. 4 wethenobtainatT=0 Im~,~(!)=XnImn~,~(!)=XnNF,n !ReZd2n()jn()j2 p !2)]TJ /F8 11.955 Tf 11.96 0 Td[(4jn()j2. (4) FromthedenominatorintheRamanresponseinthecleanlimitweseethatinthecaseofanisotropicgap,k=,thereshouldalwaysbeapeakat2,whentheenergynecessarytobreakpairsissupplied.Inanunconventionalsuperconductor,depending 100

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onthepolarization,thisabsorptionpeakisreplacedbyapeakorotherstructureat20,twicetheextremumofthegapovertheFermisurface,nonethelessafeatureremainsatthissignicantenergyscale.Thepeakwillbebroadenedbyscattering,butstillprovidesameasureofthemagnitudeofthegapsinthesystemandmaybecomparedtothosedeterminedfromotherexperiments,usuallyARPESandtunneling. TheB1gandB2gverticesinEq. 4 havezerosink-spaceandthereforeweightthepartoftheBrillouinzoneawayfromthesezeros.Thisgivesrisetoanasymmetrybetweenthetwopolarizationchannels,asobservedinthed-wavecuprates,wherethesharp2peakoccurringintheB1gchannelonly,andalesspronouncedfeaturecorrespondingtoachangeinslopeoccursatthesamescaleintheB2gchannelwhichweightsthenodalregionsmoststrongly.Furthermore,theexistenceofnodesinagapcreateslowenergyquasiparticlesthatcauseanonzeroresponseforallfrequencies.ThisisinsharpcontrasttofullygappedsuperconductorswhoseT=0responseshowasharpgapat!=20edgewithnolowenergyquasiparticlesbelowthat.Takingadvantageofthispolarization-dependencecangiveinformationaboutthestructureofunconventionalsuperconductinggaps. 4.2AddingImpurities ThepresenceofimpuritiesinanunconventionalsuperconductormaygenerateanitedensityofquasiparticleexcitationsatzerotemperaturesinceitisnotsubjecttoAnderson'stheorem.Sincetheseimpurity-inducedquasiparticlesarebothgeneratedandscatteredbyimpurities,itisimportanttoincludetheeffectofdisorderinourcalculation.Toincludetheeffectofdisorderweself-consistentlysolvefortheselfenergiesbyincludingallscatteringsoffasingleimpurityandperformingadisorderaverage.Thisneglectscrossingdiagrams,responsibleforthephysicsassociatedwithweaklocalizationandinterferencephenomena.TheT-matrixisrepresenteddiagrammaticallyingure 4-1 .TheT-matrixcanbedenedas ^G=^G0+^G0^T^G0+(4) 101

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Figure4-1. Diagramsrepresentingtheself-consistentT-matrixapproximation(SCTMA).Thedashedlinesarescatteringoffanimpurity,andthesinglelineistheself-consistentGreensfunction. ThefullmatrixGreensfunctioninthepresenceofscatteringinthesuperconductingstateis^G(k,!)=~!^0+~k^3+~k^1 ~!2)]TJ /F8 11.955 Tf 11.83 0 Td[(~2k)]TJ /F8 11.955 Tf 13.01 2.66 Td[(~2k, where~!!)]TJ /F8 11.955 Tf 12.27 0 Td[(0,~kk+3,~kk+1,andthearethecomponentsofthedisorderself-energyproportionaltothePaulimatricesinparticle-hole(Nambu)space. UsingtheNambunotation,thesingle-particleselfenergyinasuperconductorcanbedecomposedas: ^(k,!)=X(k,!)^(4) where^arethePaulimatricesand^0istheunitmatrix.NotethatthebandindexisimplicitlycontainedinthekindexsincewerestrictpairingtoindividualFermisurfacesheets.Treatingimpurity-scatteringinT-matrixapproximationgivesrisetothefollowingself-energy ^(k,!)=ni^Tkk(!),(4) whereniistheimpurityconcentrationandTkk(!)isthediagonalelementoftheT-matrix ^Tkk0(!)=Vkk0^3+Xk00Vkk00^3^G(k00,!)^Tk00k0(!)(4) Wedene)-277(=nin N0 whereniisthedensityofimpurities,nofelectrons,andN0thedensityofstatesattheFermilevel.Foraconstantpotential,thecasewhichwewillconsiderinthispaper,this 102

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expressionbecomestheseries:^(!)=niV0^3Xn(Xk0^G(k0,!)V0^3)n LaterwewillrestrictVkk00tobeconstantforparticularsetsofmomenta,eithertoallowtransitionsbetweenalltheFermisheetsortorestricttransitionstoremainwithinFermisheets. Theself-energy~(k,!)hastobesolvedself-consistentlyincombinationwiththesingle-particleGreen'sfunction ^G(k,!))]TJ /F6 7.97 Tf 6.59 0 Td[(1=^G0(k,!))]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F8 11.955 Tf 13.35 2.66 Td[(^(!).(4) Aftersolvingfortheself-energies,weinsertthemintothegeneralexpressionfortheRamanresponse.Weshouldnote,inpassing,thewell-knownissuethatthereisnocalculationofthedisordered-averagedresponsefunctioninthegeneralcase.WeaklocalizationcorrectionsfromthediffusonandcooperonhavebeenstudiedbyAltshuler,Aronov,andcoworkers[ 178 ],aswellastheseT-matrixcorrectionswhichneglectthecrossingdiagrams.WearechoosingtoreplacethesingleparticleGreen'sfunctionbyitsdisorder-averagedversion,insteadofdisorder-averagingoverthefullresponsefunction.Weneglectvertexcorrections,whichmustinprinciplebeincludedifsuchaprocedureisfollowed.Inthesimplestmodelofanelectrongaswithimpurities,itisthevertexcorrectiontolowestorderwhichgivesthetransportscatteringtimecomparedtothebarelifetime.Vertexcorrections,however,wereshowntohaveasmalleffectontheRamanresponseinthesinglebandd-wavecase[ 177 ].Forisotropics-wavesuperconductors,vertexcorrectionsfroms-wavescatterersvanishidentically.Wedidnotincludethemhere,becausetheeffectisusuallysmallandmostnoticeablenearthegapedge[ 175 ]. Beginningwithaspectralrepresentation:G(k,i!n)=Zdx)]TJ /F8 11.955 Tf 9.29 0 Td[(1 ImG(k,x) i!n)]TJ /F5 11.955 Tf 11.95 0 Td[(x 103

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wearriveatazero-temperaturelong-wavelengthform,Im,()=XkZ0)]TJ /F6 7.97 Tf 6.59 0 Td[(dx1 2kTr[ImG(k,x)^3ImG(k,x+)^3]. IntermsofretardedandadvancedGreen'sfunctions,ImG=GR)]TJ /F4 7.97 Tf 6.58 0 Td[(GA 2i:Im,()= 1 4hN()ImZd2Z0dx[FRR(x)]TJ /F8 11.955 Tf 10.41 0 Td[(,x))]TJ /F5 11.955 Tf 10.41 0 Td[(FRA(x)]TJ /F8 11.955 Tf 10.41 0 Td[(,x))]TJ /F5 11.955 Tf 10.41 0 Td[(FAR(x)]TJ /F8 11.955 Tf 10.41 0 Td[(,x)+FAA(x)]TJ /F8 11.955 Tf 10.41 0 Td[(,x)]i(4) whereFa,b=Tr[^Ga(k,x)]TJ /F8 11.955 Tf 11.96 0 Td[()^3^Gb(k,x)^3]a,b=A,R WehavetakenN()=N0inthispaper.TheangularbracketsrepresentanaverageovertheanglearoundtheFermisurfaces.Whileforsimples-waveorevend-wavegapstherealpartoftheselfenergyhasadenitesign,thisisnotthecaseingeneral,socaremustbetakenwiththeimplicitbranchcutswiththeexpressioninEq. 4 .ThedifferencebetweenthezerotemperatureresponseandthenitetemperatureresponseistoreplacethehardcutoffatbyonesmoothedbyniteT. 4.3ModelSuperconductingGapsfortheRamanResponse Webeginbyconsideringmodelone-bandcleansystemswithgapsinspiredbyproposalsfortheFe-pnictidestoillustratewhatintuitionwecangainregardingtheRamanresponseforvariouspolarizations.Figure 4-2 illustratesalltheprototypicalbehaviorinRamanscatteringspectra.Basedonspin-uctuation[ 179 ],functionalrenormalizationgroup[ 180 ],andotherearlycalculations,ananisotropics-wavegapwithsignchangesbetweentheelectronandholepocketsbecameanearlyubiquitouspossibilityforthesuperconductinggap,despitethevarietyofmethodsusedtoderiveit.Wechoseasimpleparameterizationofsuchastate.ThegureshowstheRamanspectraforamodelgapoftheformk=0 1+r()]TJ /F8 11.955 Tf 9.3 0 Td[(1+rcos(2)).First,weobservethepositionsofthepeaks.Thesecorrespondtogapenergyscales,i.e.twicetheminimum 104

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A B C D Figure4-2. Ramanscatteringintensitiesfor:(A)Aweaklynodalgap.Thex-axisisthedistancearoundacircularFermisurface(B)Thesame,butwithoutnodes.(C)Anisotropics-wavegapwithnodes(D)Anisotropics-wavegapwithoutnodes andmaximumofkaroundtheFermisurface,justasinans-wavesuperconductorpairbreakingdoesnotoccuruntilascale20.AkeydifferencebetweenthesetwoexamplesisthatthereisnoRamanresponsebelow20inthefullygappedcase.Thenonzeroresponsedownto!=0isacluesignalingthepresenceofnodes.Thenbyexaminationofthepolarizationdependenceweseethatthereisasharppeakinonepolarizationwhichisnotpresentintheother,signalingtheanisotropyofthegapsink-space.Thechannelwhichshowsthepeakatthehighestfrequencygivesthepredominantsymmetryoftheenergygap,ingure 4-2 thiscorrespondstotheB1gchannel.ThisisthesensitivityoftheRamanvertextothepresenceofgapnodes.Ausefulruleof 105

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thumbistothinkofRamanasthevertex-weightedconvolutionoftwodensitiesofstates,N(!)N(!+).Ifthevertexweightsthenodeheavily,thenit'sconvolvingtwo,say,lineardensityofstates,sotheintensityisproportionalto!andsignicantweightisatlowfrequencies.Comparethistothevertexweightinganti-nodes,wherewewouldconvolvetwosquare-rootdivergencesatthegapedge.Comparatively,twodivergingdensityofstateswouldhaveamuchlargerpeak,andthisisreectedingure 4-2 ,thelargerpeakoccursfortheB1gpolarization.Inelasticprocesseswhichhavenotbeenincludedinthisapproachwillbecomeimportantatlargefrequency.InactualRamandata,adescriptionofinelasticscatteringwouldbenecessarytomodelthehighfrequencyintensity. A B Figure4-3. (left)QuasiparticledensityofstatesN(!)forisotropicsstate,normalizedtonormalstatedensityofstatesN0vs.!=0,where0isthevalueofthegaponholeandelectronsheets.N0isassumedconstantonallFermisheets.Shownarevariousinterbandimpurityscatteringrates)]TJ /F1 11.955 Tf 10.1 0 Td[(inunitsof0.(Right)RamanresponseIm(!)vs.!=forbothB1gandB2gpolarizationsforansstateforvariousinterbandimpurityscatteringrates)]TJ /F1 11.955 Tf 6.78 0 Td[(,inunitsof. NowwewishtoexaminethengerprintsoftheproposedgapsintheironarsenidesintheRamanintensities.OneoftheearliestproposalswasforanS-wavegap,whichtakesisotropicgapsoneachFermisurfaceandallowsforasignchangebetweenthem.TherearetwoelectronandtwoholepocketsintherealBrillouinzone,buttosimplifythediscussionweconsiderjusttwoFermisheets,oneholeandoneelectron.Interband 106

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impurityscatteringispair-breakingbecauseofthesigndifferencebetweenthegaps.Assuch,weexpectthecreationoflowenergyquasiparticleswhenisotropicimpuritiesarepresent.ThedensityofstatesandRamanintensityforthismodelisshowningure 4-3 .Inthecleancase,therearenoquasiparticleexcitationsbelow20,andasharppeakoncepair-breakingbeginsat!=20.Asisotropicdisorderisaddedzeroenergyquasiparticlesarecreated,andweseetheformationofanimpurityband.IfweusetheheuristicargumentthattheRamanintensityissomewhatsimilartoaconvolutionofthedensityofstateswithitselfatashiftedexternalfrequency,weunderstandthattheconsequenceoftheimpuritybandistocreateexcitationsbelowthegapedge.Thisisreectedingure 4-3 ,whereasmallnonzerobumpappearsbelowthepairbreakingscale,20.Theconvolutionoftheimpuritybandwithitselfleadstoweakweightbelow0asshownintheinsetofgure 4-3 andtoamoresignicantstepat0wheretheimpuritybandisconvolutedwiththegapedge.AllRamanintensitiesmustgotozeroatzerofrequency. A B Figure4-4. (left)DensityofstatesN(!)=0vs.energy!=0forad-wavesuperconductorforvariousvaluesofscatteringrate)]TJ /F3 11.955 Tf 6.78 0 Td[(=0inunitarylimit.(Right)EffectofdisorderonT=0Ramanresponseofdwavestatevs.energy!=0.Shownaretwopolarizations,B1gandB2g,forvariousvaluesofscatteringrate)]TJ /F3 11.955 Tf 6.78 0 Td[(=0inunitaritylimit. OneofthekeyfeaturesinthesimpleexampleextractedbycalculatingRamanintensityastheimpurityscatteringraterises,isthatthenodeswhichareallowedin 107

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theA1gsymmetryclassforthepairstatearenotprotected.Thiswouldnotbetrueforrepresentationswhichdonotpossessthefullsymmetryofthelattice,panddwaveinrotationgroupterminology.Showningure 4-4 aretheDOSandRamanintensityforasinglebandd-wavemodel.Theeffectofscatteringforasinglebandd-wavesuperconductor,ismostlytoroundoffsharpsingularitiesandsuppressthepeaks.Amoretellingsignisthelowestfrequencybehaviorwhichchangesfrom!3to![ 177 ].Thepresenceofsymmetryprotectednodesisimpliedwhennoamountofscatteringbyimpuritiesisabletoaveragethegapintoanisotropicstate.ThisclueisimportantinthepnictideswherefullygappedandlowtemperatureTnscalingisobservedinseveraldifferentprobes(T3forNMR,Tforpenetrationdepth,etc.). 4.4ModelingExperimentalRamanData Figure4-5. ExperimentaldataonBa(Fe0.939Co0.061)2As2.Reproducedwithpermissionfrom[ 181 ].APSc2009. Weturnnowtocreatingandexaminingmodelsforthepnictides.ThesearesignicantlymorecomplicatedthanthepreviousexamplesjustbecauseofthefactthattheyhaveseveralFermisheets.Anotherissueishowscatterersaffectthecompounds.Itisunclearatpresentwhethertheappropriatemodelforvariousdopantimpuritiesis 108

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closertotheBornorunitarylimit,howextendedthescatteringis,andhowinterandintrabandpotentialsdependontheimpurity.Wehaveconsideredtwoextremelimitstotrytoextractascleanlyaspossiblethequalitativepossibilitiesforimpurityscattering,understandingthatanyrealmaterialwillbeinbetweentheminsomenon-idealizedblendofinterband,intraband,andmoderatestrengthscatteringlimits.Inallcaseswehavechosenanextendeds-wavestatewhosegrossfeaturesreproducedataoncobaltdoped122pnictides.Theextendeds-wavestatewasanaturalchoicegivendataonpenetrationdepth,ARPES,andNMRshowingsingletexponentialandT2behavior,implyingthats-wavegaps,possiblywithnodes,werepresent.Theotherchoicewewillmakeistoputimpuritiesintheunitarylimit,simulatingthelargeshortrangeeffectivepotentialfoundforCodopantsinBa-122[ 106 ]withindensityfunctionaltheory. Figure4-6. DepictionoftheBrillouinzoneandRamanPolarizationsforBa(Fe0.939Co0.061)2As2Reproducedwithpermissionfrom[ 181 ].APSc2009. Let'srstfocusonbuildingacleanmodel.TheexperimentalresultisthatthereisarelativelysharppeakintheexperimentalB2gpolarizationwhichappearsbelowTc.TheB1gpolarizationisalmostlinearanddoesnotchangewithT,whiletheA1gA2gpolarizationsarenearlyfeatureless.Thereisanonzeroresponsedowntozerofrequencysowededucethatweareworkingwithanodalsuperconductinggap.TomodelthesignalmeasuredinMuschleretal.[ 181 ]thegapshouldbelargeinthepartsoftheBrillouinzoneweightedweightedbytheB2gvertex,andweneedanalmostfeaturelessB1gchannel.FromthepositionoftheFermisurfaces,weseethatB2gis 109

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samplingtheelectronpockets,whiletheB1gpolarizationsamplestheBrillouinZonediagonalwherenoFermisurfacesheetsarepresent,soweshouldbuildagapwithmaximaonthesheets.Thelowfrequencyanalysisofthedatashowsp behavior,correspondingtoanosculatingnode,asomewhatspecialaccidentalcase,sothenodeshouldalsoliveonthe-sheets.Themodelwhichcapturedthesefeatureswas: 1()=01+rcos(4) 1+rr=.75 (4) 2()=01)]TJ /F5 11.955 Tf 11.95 0 Td[(rcos(4) 1+rr=.75 (4) 1()=)]TJ /F8 11.955 Tf 9.3 0 Td[(01)]TJ /F5 11.955 Tf 11.95 0 Td[(rcos(2) 1+rr=1 (4) 2()=)]TJ /F8 11.955 Tf 9.3 0 Td[(01+rcos(2) 1+rr=1 (4) Figure4-7. PlotofthecleanRamanintensityforthegapandverticesdescribedinequations 4 4 Notethatwhilethegapononeofthesheetsapparentlyhasd-wavesymmetry,itisalwayssupplementedbyanadditionalgaprotatedby90ontheothersheet,preservingthefulltetragonalsymmetry.Itisalsoimportanttoexaminethenormalstateresponse.Thereislittlechangeinsomeofthespectraacrossthenormaltosuperconductingtransition,sowhentheverticeswerechosen,itisimportanttotakethis 110

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intoaccount.Theverticesmustrespectthesymmetryoftheproblembutlessweightwasgiventothe-FermisurfacesheetsasaconsequenceoftheobservationsfortheA1gpolarization.Thesetofverticesusedtomodelthedatawas: 11g=0.2g=0. (4) 21g=.25()]TJ /F8 11.955 Tf 9.3 0 Td[(2)sin()cos()2g=.25cos(2) (4) 11g=.5()]TJ /F8 11.955 Tf 9.3 0 Td[(2)sin()cos()2g=+1. (4) 21g=.5()]TJ /F8 11.955 Tf 9.3 0 Td[(2)sin()cos()2g=)]TJ /F8 11.955 Tf 9.3 0 Td[(1. (4) Theresultofthesechoicesintheabsenceofanyimpuritiesisshowningure 4-7 .Theresultisnormalized,sowewouldidentifythescale20withthe70cm)]TJ /F6 7.97 Tf 6.58 0 Td[(1peakintheB2gchannel,andwillshowthattherelativelyweakfeatureat20intheB1gchannelisengulfedbydisordereffects,orisotherwiseunobservablebecauseofscatterintheexperimentaldata.Thenextimportantquestionishowthesegapsevolveasafunctionofimpurityscattering;thiswillenableustocriticallyexamineourchoiceofmodelgapstructure. A B Figure4-8. (A)DensityofstatesN(!)=N0fortheextendeds-wavestatevs.!=0forunitaryintrabandscattereringrates)]TJ /F3 11.955 Tf 6.78 0 Td[(=0.Insertshowslow-energybehavior.(B)Ramanintensityfortheextendeds-wavestatevs.!=0forvariousunitaryintrabandscattereringrates)]TJ /F3 11.955 Tf 6.78 0 Td[(=0andpolarizationstatesB1gandB2ginthe2-Fezone.Insert:lowenergyregion. 111

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Intherstcasewehaverestrictedscatteringtopurelyintrabandscatteringintheunitarylimit,showningure 4-8 .Theintrabandscattering'smaineffect,beingrestrictedtoeachbandindividually,istoaveragethegapovereachFermisheet,creatingamoreandmoreisotropicstate.ThisisseenintheRamanintensityasthelowfrequencyedgepullsbackfromzerofrequency.Thereisnofurtherpairbreakinginthislimitbecausethereisnoscatteringbetweenpairswithamplitudesofdifferentsigns.Anderson'stheoremshouldholdinthiscase. A B Figure4-9. (A)Thedensityofstatesforanextendeds-wavestatevs.!=0forunitaryisotropicscattereringrates)]TJ /F3 11.955 Tf 6.78 0 Td[(=0.(B)Ramanintensityforanextendeds-wavestateforvariousunitaryisotropicscattereringrates)]TJ /F3 11.955 Tf 6.78 0 Td[(=0andpolarizationstatesB1gandB2ginthe2-Fezone.Insert:low-energyregion. Ingure 4-9 ,weconsiderunitaryscattererswithequalscatteringratesbetweenallFermisheets(isotropicscatterers).HereAnderson'stheoremwillnothold,sowehavetwocompetingeffects:thetendencytocreatealowenergyimpuritybandalongsidetheaveragingeffectfromscattering.Thedensityofstatesbearsacertainsimilaritywiththeone-bandd-wavemodel.Thepresenceofdisordercreatesalowenergyimpuritybandwhichllsupasthescatteringrateincreases.Intheunitarylimitandforthisbandstructurethistendencyalwaysdominatestheintrabandaveragingeffectonthesuperconductinggap,soalthoughthegapbecomesmoreisotropicasanecessaryconsequenceofscattering,lowenergyquasiparticlesremaininallthe 112

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casesweconsidered.Wedoseeachangeinthelowfrequencyslope,increasingasthescatteringrateincreases.AsonemighthopefromseeingthecuspintheB1gcleancase,thatfeaturebecomesalmostcompletelyextinguishedwithmorescattering. Figure4-10. RamanintensityfortheBa(Fe0.915Co0.085)2As2sampleshowingasmallgap.Reproducedwithpermissionfrom[ 181 ].APSc2009. Thephenomenologicalmodelcannothopetocapturethelargefrequencyresponsebecauseadditionalscatteringduetoinelasticprocessesisneeded.Furthermore,therecanbenowaytogetanexactmapofthestructureofthesuperconductingstatefromthisprobe.Nonetheless,thereareanumberofrmconclusionswhichcanbedrawnwhichconstitutesignicantresults.First,giventhedatafromMuschleretal.[ 181 ]thesuperconductinggapforthatdopinglikelycontainsnodes.Thisconclusionissupportedbythepresenceoflowenergyquasiparticles,andthelackofsymmetryintheresponsesfortheA1gandB1gpolarizations.UntilsystematicRamanstudiesofthedopingorscatteringrate,e.g.studieslikeirradiation[ 182 ],areperformedfortheentirepnictidefamilywecannotsaywithcertaintywhattheeffectofdisorderis,butwehaveamodelwhereincreasingthescatteringforintrabandscatterersapparentlycreatesgaps.ThisisconsistentwiththedataofMuschleretal.[ 181 ]onasecondsamplewithhigherCocontent,whereacleargapoforderseveralmeVwasobserved,showninFig. 4-10 .Ontheotherhand,theearlyindicationisthatirradiationreducestheexponentTninthelowTpenetrationdepth.Therearealsoindicationsfromthedopingdependenceofpenetrationdepth[ 183 ]thatoptimallydoped122sarefullygappedwhiletheoverdoped 113

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sideisnodalwhichhasbeeninterpretedintermsofinterbandscattering.Thiswouldpointmoretowardsasituationwhereitisbandstructurechangeswhichcausethegapgeometrytoevolve,butsharpconclusionsalsorequireunderstandingtheexacteffectofdopantswhichiscurrentlyunderinvestigation.Itwouldbeanovergeneralizationtosaythatthesethingsaretrueforalldopingsandallpnictides,butwecansaymaketheseconclusionsforthiscompound.Theideaofananisotropicextendeds-wavestatewhichevolveswithdopingisstartingtotakerootinthepnictidecommunity.ItexplainsthediscrepancybetweenARPES,NMR,penetrationdepth,andRamanexperiments.Whatislefttodoisunderstandwhichfactorsamongscatteringrates,bandstructurechanges,andtheunknownpairinginteractionaredominantincontrollingthedetailsofthesuperconductinggap. 114

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CHAPTER5MULTIFERROICS 5.1History Multiferroicscouplepolarization,magnetization,andtheelasticresponseinamaterial.Interestintheelectriccontrolofthemagneticpropertiesofasystemforapplicationsrequiresthemagneticandpolarordersbecoupledasstronglyaspossible.Therearetechnologicalimplicationsforamaterialforwhichpolarizationandmagnetizationcanbeswitchedbyexternalmagneticandelectricelds,sotherehasbeenalotofrecentinterestinmultiferroics.SmolenskiiandChupis[ 184 ]summarizedthesubjectuntiltheearly1980sandmorerecentlythearticlebyWangetal.[ 185 ]reviewsrecentadvances.Manyphysicalsystemscanbedescribedbytheinterplayofmorethanoneorderparameter,butitisthenewcontextofmultiferroicswhichleadsbacktothisclassicproblem. Inthischapter,afterabriefoverviewofferroelectricityandmagnetism,wewillconsiderthesimplestpossibledescriptionofthecouplingofmagneticandpolarorders,anddiscussthethermodynamicconsequencesindependentofanyspecicfreeenergy.Inthenextsection,wespecifythesimplestformforthefreeenergyandexaminethesusceptibilitiesatbothtransitiontemperatures.Wendthatintheabsenceofauctuation-inducedresponse,themagnetoelectricsusceptibilityisonlynonzerowhenbothPandMarenonzero.Finallyweexaminethexedpointsforthetheoryandtheeffectofinhomogeneities. 5.2FerroelectricBackground TheearliestknownferroelectricsareRochellesalt,NaKC4H4O44H2O,andKDP,KH2PO4,intheearlierpartofthe20thcentury.BaTiO3wasdiscoveredafterwards,andopenedupanewerainferroelectricresearch.Awonderfulresourceonferroelectricsuntilthe1977isthetextbyLinesandGlass[ 186 ].Morerecently,thevolumebyRabeetal.[ 187 ]isusefulfordevelopmentssince1977. 115

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Ofthe32crystalclasses,11arecentrosymmetric.Onlynoncentrosymmetriccrystalsareconsistentwithanonzeropolarizationwhichbreaksinversionsymmetry.Ofthe21remainingclasses,20exhibitapolarresponsewhenstressisapplied[ 186 ].AtrueferroelectrichasaswitchableP,consequentlythisisdifferentiatedfromonlyaspontaneousnonzeropolarizationwiththetermpyroelectric.Piezoelectricityisapolarresponsefromappliedstress.Piezoelectricityisthebasisforthedigitalquartzwatch,andADP,similartoKDP,wasatransducerinWWIIerasubmarines(soundinducesapolarization).Somememoryapplicationsmakeuseofferroelectricrandomaccessmemoryandfrequentlycameraashesmakeuseofferroelectricsasacapacitor. KDPandBaTiO3exhibittwodifferenttypesofferroelectricbehaviorwhichformsthebasisforunderstandingtheformationofamacroscopicpolarization.InKDP,wehavetheso-calledorder-disordertypeferroelectric,wheretheplacementofhydrogenwithinaphosphatetetrahedraisrandomathightemperatureandmacroscopicallyordersatlowtemperature.InBaTiO3,itisthedisplacementofoneoftheionswhichgivesrisetothemacroscopicpolarization,sothisiscalledadisplaciveferroelectric.Inthispicture,aphononbecomesunstable,theso-calledsoftmode,andachangeintheunitcellcreatesapolarization.Inthecontextofmultiferroics,wecanalsohaveimproperferroelectrics,wherepolarizationisnotindependent.Itis,instead,inducedbysomeotherorder,usuallymagnetic. Thetheoreticalunderstandingofthesematerialscomesfromseveralperspectives:Ginzburg-Landau-Devonshiretheory[ 188 ],ab-initiocomputations[ 189 ],andmicroscopicmodels[ 190 ].Thestrengthofamacroscopicapproachisthesimplicityandwiderangeofapplicability,whileitsdrawbackisthatnomicroscopicunderstandingisgained.ThefoundationforamicroscopicpictureofdisplacivetransitionswasgivenbyAnderson[ 190 ]andCochran[ 191 ].Thoseauthorsestablishedthatatransverseopticalphonon,withhelpoftheLyddane-Sachs-Tellerrelationship[ 192 ]!2L !2T=(0) (1),hasafrequencywhichvanishesas(T)]TJ /F5 11.955 Tf 12.3 0 Td[(Tc)neartheferroelectrictransition.Takingthepolarizationas 116

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theorderparameter,aLandauexpansionofthefreeenergyinpowersofPisthebasisoftheLandau-Devonshiretheory[ 188 ].Toestablishtheorder-disordertypeferroelectricmechanismexperimentally,avarietyofstructuralprobesareneededtostudyprotonmotion.InKDPthiswasaccomplishedbyinelasticandelasticneutronmeasurements[ 186 ].Alessrigorousbutusefulindicatorofrelaxorferroelectricityistolookatthesharpnessofthepeakinaresponsefunctionlikepermittivity,whichwillbebroadinarelaxorandsharpinadisplaciveferroelectric.Thedynamicsofanorder-disordertypeferroelectricalsohavetheeffectthatincreaseoffrequencydecreasesthepermittivitypeak,whichisnottrueofasecondorderstructurephasetransitionwithoneuniqueTc. Thecalculationofpolarizationfromrstprinciplesonlybecamepossibleinthe1993.Thebasicproblem,thatthechargedistributioncannotdetermineP[ 187 ],isasfollows:imagineNaClwhichhasnomacroscopicpolarization.Iftheunitcellcontainsaverticalpairofions,thepolarizationinthatunitcellpointsvertically.Redeningtheunitcellhorizontallycausesthepolarizationpercelltopointhorizontally.Thisarbitrarinessisunacceptable,andaformulationindependentoftheunitcellisnecessary.ThesolutionwasgivenbyKing-SmithandVanderbilt[ 193 ]whoprovidedawaytocalculatethepolarizationusingtheWannierfunctionstopologically,fromtheBerryphase. Ferroelectricsaresimilartosuperconductorsintheircriticalbehavior,inthatmeaneldtheoryusuallyworksverywell[ 186 ].Additionalevidenceforthis,isthewidthofdomainwalls,usuallyonlyafewlatticespacings,10-30A[ 194 ],incontrasttomagneticdomainwallswhicharemoreappropriatelycalledmesoscopicinsize,typicallym. Thelowestorderfreeenergydescribingasecondordertransitionfromaparaelectrictoaferroeelctricisgivenby FE=P2 2E0+bEP4.(5) Manyferroelectrictransitionsarerstorder,andthusmustbesupplementedbyatermP6.ThedetailsofthisapproachcanbefoundinLinesandGlass[ 186 ]. 117

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5.3MagneticBackground ForacomprehensivereviewofmagnetismthereistheseriesbyRadoandSuhl[ 195 ],aswellasanumberofusefultextbookssuchasWhite[ 196 ]orMattis[ 197 ].Magnetismarisesfromthespinsofelectrons.Twoextremepointsofviewarethatofcompletelylocalizedandcompletelyitinerantelectrons.Intheformercase,wecantakethepositionstobexedandwritethehamiltonianasH=)]TJ /F12 11.955 Tf 11.29 8.97 Td[(PijJij~Si.~Sj.InthelattercaseournotionsarebasedontheStonermodel,see[ 5 ],whereahighdensityofstatesleadstoadivergenceofthestaticsusceptibility.Intheintermediateregime,aremodelswhichcombinelocalanditinerantelectrons,suchastheAndersonmodelwherelocalizedelectronshybridizewithaconductionbandandwillformamomentdependingwhatthespecicparametersare.Alsointhiscategoryisantiferromagnetismduetosuperexchange,orferromagnetismduetodouble-exchange[ 198 199 ]. Multiferroicsareinsulators,mostlytransitionmetaloxides.Wewillfocusonthemacroscopicapproach.Thereareanumberofsmallermagneticinteractionswhichareimportantinanaccuratedescriptionofthephysicsofinsulatingmagnets.First,isthecrystaleldanisotropy.Thecrystalenvironmentdeterminesalocalpotentialwhichisnotisotropicandbiasesthealignmentofspins.Thisresultsinpreferredaxesorplanesforthespins,andcanbecapturedbyaddingtermssuchasaz(~S.^z)2totheHamiltonian.Thesignofazwouldfavorordisfavoralignmentalongthez-axis,resultingeitherinIsingorXYspins.AnotherimportantinteractionformultiferroicphysicsistheDzyaloshinskii-Moriya[ 200 201 ]interaction, Xij~Dij.(~Si~Sj)(5) whichisaconsequenceofthespin-orbitinteraction.Dijisproportionaltoxrijwherexisthevectorperpendiculartorijandfromrijtotheligand(oxygenusually).Itistypicallyweak,butincreasesinstrengthforincreasingatomicnumberZ,andresultsintheweak-ferromagnetismofmanycompounds. 118

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Theorderparameteristakentobethemagnetizationwhichiszeroabovethecriticaltemperatureandnonzerobelowit.Strictly,weshoulduserepresentationanalysis[ 202 ]toconstructabasiswhichtransformsasthedifferentirreduciblerepresentationscompatiblewiththelatticesymmetry,andfromthisconstructtheLandauexpansion,butsincetheinteractionrespectsthesymmetryofthelattice,asimpliedapproachwithonlyonefouriercomponentisadequate.Thegeneralapproachstartswiththeidenticationofthepropagationvectorkassociatedwiththemagneticstructure,andwhichspacegroupsymmetryoperationsleaveitinvariant.Thepropagationvectorisanexperimentalinputalongwiththespacegroupofthecrystalandthepositionsofthemagneticions.Thelittlegroupofk,Gk,determinetheirreps(irreduciblerepresentations)ofGk.Astandardgrouptheorytechnique[ 203 ]usingaprojectionoperator,projectsoutabasis,whichisnotingeneralunique.Thedifferentirrepsclassifythemagneticstructurescompatiblewithsymmetry,andtheorderparameter(s)arethebasisfunctions.Werestrictthisexpositiontoamagneticstructurewithsomewavevectorusuallydeterminedfromexperiment,whichistheminimumoftheexchangeinreciprocalspace. Nowwewanttodemonstratetheconnectionbetweentheseapproaches.Webeginwiththestandardexpressionforthefreeenergy, F=U)]TJ /F5 11.955 Tf 11.95 0 Td[(TS(5) Wecanexpresstheinternalenergyasthefouriertransformoftheexchange, XqJ(q)~Sq.~S)]TJ /F4 7.97 Tf 6.59 0 Td[(q(5) wherewetakeclassicalspins.Theentropyisaddedtotheenergyandexpandedinpowersof~Sq.Wewanttoconsiderasimplemagneticstructurewithjustonefouriercomponent,~SQ0.Itisworthnotingthatanexternaleldcouplestotheuniformmagnetizationnotthestaggeredmagnetizationsoinadditiontothefouriercomponentwhichisresponsibleformagneticorderinthisexample,~SQ0,wemustalsoincludethe~S0 119

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component. F(J(0)+q2J0(0))~S0.~S0+(J(Q0)+(q)]TJ /F5 11.955 Tf 11.96 0 Td[(Q0)2J0(Q0))~SQ0.~SQ0(5))]TJ /F5 11.955 Tf 9.3 0 Td[(T )]TJ /F3 11.955 Tf 9.3 0 Td[(Xq=0,Q0...~Sq.~S)]TJ /F4 7.97 Tf 6.59 0 Td[(q)]TJ /F3 11.955 Tf 11.95 0 Td[((~Sq.~S)]TJ /F4 7.97 Tf 6.59 0 Td[(q)2+...! ThisisageneralapproachforconstructingtheGinzburgLandaufreeenergyfromalocalHamiltonian.Inthecaseofaferromagnetthecomponentwekeeponly~S(0),whichafteraveragingisthemagneticmomentM, FM=M2 2M0+bMM4.(5) 5.4MultiferroicIntroduction Figure5-1. TemperaturedependentmagnetoelectricresponseofCr2O3[ 204 ]. Theearliestknownmagnetoelectric,isCr2O3whichexhibitsalinearmagnetoelectriceffect.Themagnetoelectriceffectiswhenamagnetization(polarization)isinducedbyanexternalelectric(magnetic)eld.Dzyaloshinskii[ 201 ]rstpointedoutthat,onsymmetrygrounds,itwaspossibletoinduceamagneticmomentwithanelectriceldorviceversa,possiblyintheantiferromagnetCr2O3.TheatomsinCr2O3belongtotherhombohedralgroup[ 205 ]D63d.Therearefourspinsintheunitcell,andthemagnetic 120

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structureofthegroundstateneedstorespectthesymmetryofthedirectproductofthelatticeandtransformationsactingonthespinsthemselves,D63dR,whereRinvertsthesignofthespins.Theaxialvectorssi,foreachofthefourspins,formareduciblerepresentationofthisgroup.Linearcombinationsofthesespinscanbeconstructedwhichformthebasisofanirreduciblerepresentation.Usingthesesymmetryarguments,heconcludedthatthefreeenergywouldincludeaterm,F=F0)]TJ /F3 11.955 Tf 11.96 0 Td[(kEzHz)]TJ /F3 11.955 Tf 11.96 0 Td[(?(ExHx+EyHy) Astrovetal.[ 206 ]conrmedthisprediction.Rado[ 207 ],offersanexplanationfortheobservedtemperaturedependenceinCr2O3[ 204 ],bypointingoutthatthecrosssusceptibility'stemperaturedependencecanbeunderstoodfrom/L,whereListhestaggeredmagnetization.Duetothesymmetryoftheantiferromagnet,thesusceptibilitycouplesquadraticallytoeld.InthefreeenergythetwoinvariantstolowestordermustcombineinsuchawaythatwhenLisparalleltotheexternaleld?vanishes,implyingF/L2B2)]TJ /F8 11.955 Tf 12.49 0 Td[((L.B)2.WeobservethatCurie-Weisssusceptibilitiesvaryas1 Twhereasthespontaneousmeaneldorderisgrowingasq T TN)]TJ /F8 11.955 Tf 11.95 0 Td[(1,andthecompetitionbetweenthesetwotrendsresultsinthetemperaturedependenceofk,whilein?wejustobservetheq T TN)]TJ /F8 11.955 Tf 11.96 0 Td[(1ofL. Fromatechnologicalpointofview,themutualcontrolofelectricandmagneticpropertiesisadesirablepropertyformicroelectronics.Duringthe1960sand1970sseveral(52arelistedin[ 184 ])multiferroicmaterialswerediscovered,suchasBiFeO3,BiMnO3,andtheboraciteNi3B2O13I.Recently,theinvestigationofrareearthmanganitemultiferroics,notablyTbMnO3[ 208 ],reignitedinterestinmultiferroicresearch.Abroadclassofthesemultiferroicsfollowthechemicalformulas,RMnO3andRMn2O5[ 209 ],whereRisarareearthion.Theappearanceofspiralorderintherare-earthmanganitesaccompaniesthepresenceofapolarizationinthese,whereasthespinelmultiferroicsHgCr2S4andCdCr2S4,arebothferromagneticandferroelectric.Inthemacroscopic 121

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approach,ifinversionsymmetryisabsent,thetermM.(rM)[ 210 ],isknowntoinducespiralorder.Inthismorerecentclassofspiral-magnetmultiferroics,theferroelectricityisreferredtoasimproper.Theprimaryorder,themagnetism,issaidtoinducetheelectricpolarizationwhichoccursasaconsequenceofthemagnetism.Thisistobecontrastedwithproperferroelectrics,whereaspontaneouspolarizationexistsofitsownaccord.Mostovoy[ 211 ]providedageneralargumentbasedonsymmetryandGinzburg-Landautheoryforwhythishappens.Mostovoystudiesafreeenergydescribedby Fm+Fe+Fem=Fm+P2 2e+P.(M(r.M))]TJ /F8 11.955 Tf 11.96 0 Td[((M.r)M)(5) Themagneticfreeenergyisnotgiven.Theauthorobservedthatintheabsenceofaspontaneouspolarization(akintoincludingP4term),thepolarizationisonlyinducedbythemagneticorderandisgivenby, P=)]TJ /F3 11.955 Tf 9.3 0 Td[(e(M(r.M))]TJ /F8 11.955 Tf 11.96 0 Td[((M.r)M)(5) Thisanalysisclariestheroleofmagneticstructureindeterminingapolarization,andinparticulardemonstratesagenericmechanismformagnetismtoallowpolarizationwithoutreferencetoalatticeormicroscopicdetails.Moredetailedgrouptheoreticalarguments[ 212 ]cansometimesdisagreewiththepredictedorientationsofthetwoorderparameters,especiallyformorecomplicatedinvariants,butthepointisthattrilinearcouplingsnowallowedbysymmetryaregenerallyresponsiblefortheinducedferroelectricity.AdescriptionfortheoriginofmagneticorderisfrequentlyattributedtothepresenceofaDzyaloshinskii-Moriya[ 200 205 ]interaction[ 213 214 ].ThoseauthorshaveshownusingamicroscopicmodelthatapolarizationP/rij(~Si~Sj)arisesfromsuperexchangebetweenthespins,whererijconnectsthespins.Magnetostrictioncanprovideanothermicroscopicmechanismcouplingtheelastic,magnetic,andpolarresponses.Thespin-latticeinteractioncanalsoinduceaninteractionP2M2.Themagnetostriction'smagnitudedoesnothavethesameinherentweaknessofthe 122

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Dzyaloshinskii-MoriyainteractionbecausetheexchangeJistypicallyalargerenergyscale. ToquotefromJ.Sethna[ 215 ],Firstyoumustdenethebrokensymmetry.Secondyoumustdeneanorderparameter.Thirdyouaretoldtoexaminetheelementaryexcitations.Fourth,youclassifythetopologicaldefects.Anothernovelaspectofmultiferroicphysicsisinthelowenergyexcitationsasaconsequenceofcouplingthelatticeandmagneticorder.Anew,hybrid,lowenergyexcitationcalledanelectromagnonwillalsobepresentintheexcitationspectrum,resultinginthecouplingofspin-wavesandpolarlatticevibrations.Theseexcitationsallowforspinwavestobeexcitedbyappliedelectricelds.Theyhavebeenrecentlyobserved[ 216 217 ]inRMnO3andRMn2O5compounds. Thetopologicaldefectswillbeintertwinedjustastheelectromagnonsintertwinethemagneticandpolarexcitations.GdFeO3,forexample,isaweakferromagnetandaferroelectric[ 218 ].MagnetostrictionthroughtheexchangeinteractionbetweentheGdandFespinscouplesthetwoorders.Thedomainwallsforelectricandmagneticorderswillrespondtooneanother[ 219 ],soitisinterestingtoinvestigatethenovelaspectsofhowamagneticdomainrespondstoanappliedelectriceld. 5.5Thermodynamics 5.5.1MaxwellRelations Tobeginconsiderthesimplestconsequencesofcouplingtwoorderparameters,usingtheHelmholtzF(P,V,M,T)andtheGibbsG(E,P,B,T)freeenergieswhicharefunctionsofthevolumeV,magnetizationM,temperatureTandElectriceldE,polarizationP,pressureP,andmagneticeldB.Inthoseensembles, dF=)]TJ /F5 11.955 Tf 9.3 0 Td[(SdT)]TJ /F5 11.955 Tf 11.95 0 Td[(PdV+EdP+BdM(5) dG=)]TJ /F5 11.955 Tf 9.3 0 Td[(SdT+VdP)]TJ /F25 11.955 Tf 11.95 0 Td[(PdE)]TJ /F25 11.955 Tf 11.96 0 Td[(MdB(5) 123

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Thedifferentiabilityofthefreeenergiesmeansthatcrossderivativesareequalanditfollowsthat, @M @EjB=@P @BjE(5) Whichisthestatementthatthecrosssusceptibility12issymmetricinEq. 5 .Bylookingattheothercrossderivatives,wend,suppressingthetensorindices: @2C @E2=T@2E @T2@2C @B2=T@2M @T2(5) and @C @B@E=T@212 @T2,(5) @212 @P2=@2(V) @B@E,@2E @P2=@2(V) @E2,@2M @P2=@2(V) @B2.(5) Hereisthedifferentialsusceptibility(thesubscriptEforelectricalandMformagnetic),isthecompressibility,VisthevolumeandCthespecicheatatconstantvolume. Theconvexityofthefreeenergy,forstability,alsoleadstotheinequality @2F @M2@2F @P2)]TJ /F8 11.955 Tf 11.95 0 Td[((@2F @M@P)20(5) whichinturnimpliesthat 212em.(5) ThisinequalitywasrstderivedbyBrownetal[ 220 ]. 5.5.2PhaseDiagram TheEhrenfestdenitionoftheorderofaphasetransitionneedstobeconsideredcarefullywherethereismorethanonemechanicaleld.ThethermodynamicphaseboundariesforarstordertransitionwithalatentheataredescribedbytheClausius-Clapeyronequation.Forasecondorderphasetransition,thecorrespondingequationistheEhrenfestequation.TheClausius-ClapeyronequationsforTM(E),whichcanbe 124

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obtainedfromdGi=)]TJ /F5 11.955 Tf 9.3 0 Td[(SidT+VdP)]TJ /F5 11.955 Tf 12.66 0 Td[(PdE)]TJ /F5 11.955 Tf 12.65 0 Td[(MdBbychoosingtwopointsoneithersideofthecriticallineintheT-Bplane, @TM @E=)]TJ /F25 11.955 Tf 10.49 8.09 Td[(P1)]TJ /F25 11.955 Tf 11.96 0 Td[(P2 S1)]TJ /F5 11.955 Tf 11.96 0 Td[(S2.(5) Forcompleteness,wepresentalltheEhrenfestequationsfortwomechanicalelds,inthiscontextEandB: @TE @E2=TEE C,@TM @E2=TME C(5) @TE @B2=TMM C,@TM @B2=TMM C.(5) Inaferromagnet(orferroelectric)inanyniteeldthereisnophasetransition.Andyet,ifthetransitionissecondorderataniteeld,asitcanbeforanantiferromagnet,theevolutionofTcshouldbedescribedbytheEhrenfestequationsabove.Whenthetwoorderparametersarecoupled,aswewillshowbelow,MiszeroatTEbutEisnon-zeroatTM.TheferromagnetictransitionatTM(B,E)issecondorderinE,unliketheuncoupledresult. 5.5.3AdiabaticProcesses Adiabaticprocessesareusedincooling,suchasdemagnetizationatlowtemperatures.Itisaquestionofefciencywhethermagneticorelectriccoolingisuseful.ThecoolingarisingfromanadiabaticprocesswhentheentropydependsontwoexternaleldsisdescribedbydS=0whichuponsubstitutingtheMaxwellrelationsbecomes: dP dTdE+dM dTdB+dV dTdP+Cv TdT=0(5) dT dE=)]TJ /F5 11.955 Tf 12.03 8.09 Td[(T CvdP dT(5) Wewilldeferanyexplicitcalculationoftheintegralwhichdependsontheexplicittemperaturedependenciesofthespecicheatandalsooftheorderparameter. 125

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5.6FreeEnergyFunctional Asystemwithmorethanonevectororderparameterhasbeenconsideredmanytimesbefore,forexample[ 221 224 ]andreferencestherein.Thefreeenergycanbewrittenas:F=FE+FM+Fint FE=P2 2E0+bEP4(5) FM=M2 2M0+bMM4(5) Fi=k(M.P)2(5) )]TJ /F6 7.97 Tf 6.59 0 Td[(1E0=aE0T TE0)]TJ /F8 11.955 Tf 11.95 0 Td[(1(5) )]TJ /F6 7.97 Tf 6.59 0 Td[(1M0=aM0T TM0)]TJ /F8 11.955 Tf 11.96 0 Td[(1(5) WeareconsideringaferromagnetdescribedbyFMandaferroelectricrepresentedbyFEwiththeinteractionbetweenthemgivenbyFi.ThecoefcientsofthefreeenergyhereareallconstantswiththeexceptionofthesusceptibilitieswhichareassumedtobedescribedbytheCurie-Weissformforlocalmoments.ForspecicitywetakeTE0>TMasthetransitiontemperaturesrespectivelyfortheelectricpolarizationandthemagnetization.Ingeneral,theinteractionbetweenthemagnetizationandelectricpolarizationmustbeascalarandcouldbeintheformk1M2P2+k2(M.P)2whichisrequiredbythetimereversalinvariance.ThistermdeterminestherelativeanglebetweenMandP.Oncetherelativeangleisdetermined,thefreeenergybecomesoftheforminEq. 5 withitscoefcientsrenormalized.Thebiquadraticformisalsowhatwecouldexpectfromthenextterminthemagnetostrictionpicturealreadymentioned. TheequationsofstatearetheconventionalthermodynamicequationsforaHelmholtzenergy: E=@F @P(5) B=@F @M(5) 126

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Forahomogeneoussystem,wederivethethermodynamicpropertiesusingfreeenergyabove.Inamultiferroic,thelinearresponseincludescrosssusceptibilities12,denedby M=MB+12EP=12B+EE(5) Thefreeenergyfunctionalintheconventionalformincludestheexternalelds)]TJ /F25 11.955 Tf 9.3 0 Td[(P.Eand)]TJ /F25 11.955 Tf 9.29 0 Td[(M.B,andthusthegroundstateisaminimumoff.Thedimensionlessfreeenergyf=F=FE(0),iswrittenintermsof:p=P=P0,m=M=M0,`=FM(0)=FE(0),=E FE0P0,.m`=B.M=FE0. f=F FE0=fe+fm+2km2p2=2(T TE0)]TJ /F8 11.955 Tf 9.3 0 Td[(1)p2+p4)]TJ /F5 11.955 Tf 9.3 0 Td[(p.+`(2(T TM0)]TJ /F8 11.955 Tf 9.3 0 Td[(1)m2+m4)]TJ /F5 11.955 Tf 9.3 0 Td[(m.)+2k(m2p2)(5) Theorderparametersinthescaledvariablesaresolutionsof, 4=(T TE0)]TJ /F8 11.955 Tf 11.96 0 Td[(1)p+p3+km2p(5) 4=(T TM0)]TJ /F8 11.955 Tf 11.96 0 Td[(1)m+m3+k `mp2(5) 5.6.1OrderParameter WeconsiderTE0>TM0.Thesolutionsare: p0=m0=0T>TE0>TMm20=0,p20(T)=(1)]TJ /F4 7.97 Tf 16.87 4.7 Td[(T TE0)TE0>T>TM.m20(T)=1)]TJ /F11 5.978 Tf 7.78 3.26 Td[(k ` 1)]TJ /F11 5.978 Tf 7.78 3.26 Td[(k2 `(1)]TJ /F4 7.97 Tf 15.9 4.7 Td[(T TM),p20(T)=1)]TJ /F4 7.97 Tf 6.59 0 Td[(k 1)]TJ /F11 5.978 Tf 7.78 3.26 Td[(k2 `(1)]TJ /F4 7.97 Tf 15.22 4.7 Td[(T TE)TE0>TM>TTE TE0=1)]TJ /F4 7.97 Tf 6.58 0 Td[(k 1)]TJ /F4 7.97 Tf 6.59 0 Td[(kTE0 TM0TM TM0=1)]TJ /F4 7.97 Tf 6.58 0 Td[(k=` 1)]TJ /F11 5.978 Tf 7.78 3.26 Td[(k `TM0 TE0(5) 127

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AsshowninFig. 5-2 ,themeaneldorderparameterPappearsatTE0continuously. Figure5-2. (ColorOnline)Thepolarizationandmagneticmomentversustemperature.PshowsajumpatTMwhenMcondenses. WehavechosenTE0=TM0>1.WendTE=TE0>1andTM=TM0<1.TheelectricaltransitiontakesplaceatTE0,whichisunchangedbytheinteraction,andTM0isre-normalizedbyinteractions.ThephysicalmagnetictransitiontakesplaceatTM,notTM0.AsshowninFig. 5-2 ,thereisakinkinp0atTM.Thiskinkisduetothefactthatthescaletemperatureforp0,forTTE0. 5.6.2Susceptibility Followingtheequationofstate,wecanderivetheelectric,magnetic,andthecrosssusceptibilitiesasdenedinEq. 5 .TheybothtakeaCurie-Weissform,alsohaveadiscontinuityattheothertransitiontemperature.Figures( 5-3 5-4 )showthesefeaturesforaspecicchoiceoftheparameters(k=.4,k `=.3).ThecrosssusceptibilitydivergesatTM.Asexpected,itvanishesaboveTM.Thisexcludesthepossibilityofauctuationinducedresponse,whichweneglectinourstudy.Theferroelectricsusceptibility,exhibitsadivergenceatitstransitiontemperatureandananomalyatthemagnetictransitiontemperature.Thesizeofthisjumpdependsontheenergeticsoftheferroelectricandmagneticfreeenergies,andoccursbecausetheelectricsusceptibilitybelowTM.Themagneticsusceptibilitydemonstratesparamagneticbehavioraboveit'stransitionanddivergesatTM.Itshowsacuspattheferroelectrictransitiontemperaturecoincidingwith 128

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thesmoothonsetofP.Theinverseelectricsusceptibility,)]TJ /F6 7.97 Tf 6.59 0 Td[(1E,iszeroatitstransitiontemperatureandjumpsatthemagnetictransitiontemperature.Theinversemagneticsusceptibility,)]TJ /F6 7.97 Tf 6.58 0 Td[(1M,iszeroatTM,andshowsachangeinslopeattheferroelectrictransitiontemperatureduetotheonsetofP.Thisresultsincusp-likebehaviorratherthanajump. Figure5-3. (ColorOnline)(Left)Theferroelectricsusceptibility,thebluesolidline,andthemagneticsusceptibility,thedashedredline.(Right)Theinversesusceptibilities,thebluesolidline,)]TJ /F6 7.97 Tf 6.59 0 Td[(1E,andthethindashedlinerepresentingtheinversemagneticsusceptibility,)]TJ /F6 7.97 Tf 6.59 0 Td[(1M.Anauxiliarydashedlineisalsopresenttounderscoretheslopechangein)]TJ /F6 7.97 Tf 6.58 0 Td[(1MatTE0. Figure5-4. (ColorOnline)Thecross-susceptibilitybecomesnonzeroonlybelowTMandhastheoppositesignoftheothersusceptibilities. 129

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Theformalexpressionsare, E(T>TE0)=1 4(T TE0)]TJ /F8 11.955 Tf 11.95 0 Td[(1),E(TMTM)=1 4`((T TM0)]TJ /F8 11.955 Tf 11.95 0 Td[(1)+k `p20)(5) M(T
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atTMandTE0.Thealgebraicresultsare: CV F0=0T>TE0,2T T2E0TM
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ThesetermsarelinearinPandthereforespontaneouslybringaboutabrokensymmetryvalueofP.Inotherwords,asnotedbyMostovoy,wehaveaneffectivelocalelectriceldEinthere,proportionaltothemagnetizationgradients. Eint=z3M(r.M)+z4[(M.r)M](5) WhichleadstoapolarizationP=eEint.Replacingthepolarizationwithitsinducedresponse,wendafreeenergyformagnetization: F=M2 2M)]TJ /F8 11.955 Tf 13.15 8.09 Td[(1 2Ejz3M(r.M)+z4[(M.r)M]j2(5) ThespatialproleofthegroundstateisdeterminedbythemomentumdependentM(q),theqforwhichitisamaximum.TheferromagneticgroundstatewouldbeunstableiftheeffectoftheLifshitzinvariantweretomovethegroundstatetoniteq.TheeffectivefreeenergyforM,resultingfromtheLifshitzinvariantisquarticinM,thereforeaferromagnetisstableaslongasbM&E(q)z2i.SinceEisdivergentatTE,aferromagneticgroundstateisunstablenearthattemperature.Itmaydependonthedetailswhethertheferromagneticgroundstatecanbeexpectedtobestableanywhere.Thethermodynamicuctuations,leadtoaqualitativechangeinalltemperaturedependentproperties.Thereisadownwardshiftinthecriticaltemperatureandachangeinasymptoticmeaneldbehavior,resultinginadifferentcriticalexponent. Anotherimportantfeatureofinhomogeneoustermsisthepossibilityofdefects.Apossiblewaytocauseimproperferroelectricityistohavedefectsinducetheotherorderparameter,soouranalysiswouldbeincompletewithoutconsideringhowthismightbeaccomplished.Inaferromagnetthereareseveraltypesofdomainwalls,namelytheso-calledBlochandNeelwalls,whichdifferinhowthemagnetizationrotatesfromonegroundstatetotheother.ThereisanimportantdistinctionbetweentheminthesematerialsduetotheLifshitzinvariant.Take,withoutlossofgenerality,theBlochwallto 132

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bedescribedbyMB=(cos((z)),sin((z)),0) andtheNeelwalltobeMN=(0,cos((z)),sin((z))) TheBlochwallisdivergencefree,furthermoretheexpressionM(rM)vanishesfortheBlochwall.ThisisnotsointheNeelcase.Sothegeneralexpressionb1P.M(r.M)+b2P.M(rM) whichisequivalenttotheLifshitzinvariant,whencrystalsymmetrysetsb1=b2inforexampleacubicenvironment,vanishesfortheBlochwall.ThismeansthatwhileaNeelwallhasthepossibilityofinducingpolarization,butitdoessoatacostinenergy. Itisquitenaturalinthislight,toaskhowinhomogeneousstructuresareinuencedbyanelectriceld,asDzyaloshinskii[ 225 ]didformagneto-electricmaterials.WefollowDzyaloshinskii'sapproachinthenewcontextincludingthesymmetricinteraction,cP2M2,todemonstratethatthelengthscaleassociatedwithaNeelwallistunablewithelectriceld.WeswitchtoageometrywhereM=(cos((x)),sin((x)),0) whichisstillcompletelygeneral.Therstchoicethatismadeistoapplyanelectriceldpreferentiallyalongthey-axis,andthenextistoincludeacrystaleldanisotropyalongy,)]TJ /F5 11.955 Tf 9.3 0 Td[(aM2y.InthissituationthefreeenergyisF=A(@x)2)]TJ /F5 11.955 Tf 11.95 0 Td[(asin2())]TJ /F8 11.955 Tf 11.95 0 Td[((b1sin2()+b2cos2())E0@x+cE20sin2() Wehavemadetheassumptionthatwe'reinthelinearresponseregimesothatP=eEandabsorbedthesusceptibilityintoourinteractionparameters.Dzyaloshinskiiconsideredthisactionwithc=0inhisearlierpaper[ 225 ].Theequationofmotionisa 133

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formofthewell-knownsine-Gordonequation,A@2x=)]TJ /F8 11.955 Tf 9.3 0 Td[((a)]TJ /F5 11.955 Tf 11.95 0 Td[(cE20)sin()cos() Theclassicalminimaoftheeffectivepotentialoccurat 2.Integratingourequationofmotionwiththisinmind)]TJ /F5 11.955 Tf 9.3 0 Td[(A(@x)2 2)]TJ /F8 11.955 Tf 13.16 8.09 Td[(1 2(a)]TJ /F5 11.955 Tf 11.96 0 Td[(cE20)(sin2())=K Kmustbe)]TJ /F6 7.97 Tf 10.49 4.7 Td[(1 2(a)]TJ /F5 11.955 Tf 11.82 0 Td[(cE20)fortheseboundaryconditions.Theequationcanbeintegratedintermsofelementarytechniqueswithhelpofthesubstitution,tan( 2),resultingin(x)=2arctan(tanh(r a)]TJ /F5 11.955 Tf 11.96 0 Td[(cE20 Ax 2)) Thelengthscalewhichcontrolsthedomainwallwidthis=2q A a)]TJ /F4 7.97 Tf 6.59 0 Td[(cE20.Thisimpliesthereisacriticalelectriceld,perhapsunrealisticallylarge,wherethemagneticstructureonamacroscopicscaleisessentiallythatofthedefect.WecomparethescaleinDzyaloshinskii'spaper,athresholdelectriceldfortheproliferationofdomainwallsEt=2p aA b,toourscalehereEc=p a c.Theanisotropicenergyscaleaiscommonnotationtobothpapers,andbreferstotheLifshitzterm'scouplingtypicallycausedbyspin-orbitterms.Aisofordertheferromagneticexchange. Thesedefectsareadhocinthesensethatanansatzwasusedtoproducethem.Amorefundamentalinvestigationwillnotrelyonthisassumption.ConsiderafreeenergywhichinducesPalongaparticularaxiscoupledviatheLifshitzinvarianttothemagnetization.F=Zd3x(rM)2 2+P.(M(r.M))]TJ /F8 11.955 Tf 11.95 0 Td[((M.r)M) Introduceapolarizationpotential,P=r,whichisdenedeverywhereoutsidepointdefects,normalizethefreeenergytothevalueofthemagneticmoment,^n=M M0,and 134

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integratebyparts:F M20=Zd3x(r^n)2 2)]TJ /F3 11.955 Tf 11.95 0 Td[(r.(^n(r.^n))]TJ /F8 11.955 Tf 11.96 0 Td[((^n.r)^n)+r.((^n(r.^n))]TJ /F8 11.955 Tf 11.96 0 Td[((^n.r)^n)) Thethirdtermisatotaldivergence,soitdoesnotinterveneinsolvingtheEuler-Lagrangeequations.Usuallythisisthrownawaywithoutregret,butinthepresenceofdefectsitcouldcontainimportanttopologicalinformation.WehavenotincludedhigherpowersofPorM.Astheyarenormalized,theywouldsimplyaddconstantstothefreeenergywhichonlyplayaroleinthetotalenergyofastate,notinthevariationalproblem.Withthisinmindweproceedbyneglectingthetotaldivergence.Theexpression1 2r.(^n(r.^n))]TJ /F8 11.955 Tf 9.6 0 Td[((^n.r)^n)istheGaussiancurvature,K,forasurfacewithnormal,^n=rh(x,y) jrhj.Thetermisfamiliarasthesaddle-splayenergyinliquidcrystalswhere^nisthedirector.F M20=Zd3x(r^n)2 2)]TJ /F3 11.955 Tf 11.95 0 Td[(2K ThisprovidesageometricunderstandingoftheLifshitzterm,asawaytobalancebetweenthersttermfavoringuniformmagnetization,andacurvaturewhichdependsonthepolarization'sctitiouspotential.The-terminducesGaussiancurvatureKtolowerthefreeenergy.Thefullconsequencesofthischaracterizationofthispointofviewareatopicforfuturework. 5.8SpeccHeatwithGaussianFluctuations Weturnbacktotheconsequencesofinhomogeneousterms.Theuctuationeffectsarisefromgradientterms,e(rP)2andm(rM)2,inthefreeenergy(Eq. 5 ).NearTE0thecorrespondingcorrelationlengthdiverges.ThesubleadinglengthscalederivedfrommstartstoplayanimportantroleinthetemperaturerangeTM
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negligible.ThesetermsarealsoirrelevantintheRGsense.Ind-dimensions,(x7!x0 b),ziscalesasb2)]TJ /F4 7.97 Tf 6.58 0 Td[(d. Theuctuationcontributiontothespecicheatwascomputedinthegaussianapproximation,andshowninFig. 5-6 Figure5-6. Themostsingularcontributiontothespecicheatfromgaussianuctuations,forarbitrarilychosencorrelationlengths Werstincludegradients,e(rP)2andm(rM)2inthefreeenergy.Then,weexpandaboutthesaddlepoint,m0+m,p0+p,andneglectcross-termssincethisismeanttobeanestimate.Therearetwoindependentcontributionsinthatapproximation,oneforthegaussianintegraloverthepolarizationandonefromthemagnetization.ThemostsingulartermsinthespecicheattaketheapproximateformCvuctT2G(T)F(T)d 2)]TJ /F6 7.97 Tf 6.58 0 Td[(2 WherethefunctionsGandFfollowapattern.LetOidenotetheithorderparameter,MorP.ThenG(T)=a=Tc+12O1dO1 dT+2kO2dO2 dTandF(T)=(T Tc)]TJ /F8 11.955 Tf 9.82 0 Td[(1)+6b1O21(T)+kO22(T).Thecorrelationlengthwilltaketheform,2i=i F(T).ThewidthofthecriticalregionaroundeitherTcinthreedimensions[ 226 ]isgivenbytheexpressionT1=2G=kB 4Cmf3i,whereCmfisthesizeofthejumpinthespecicheatfrommeaneldtheory.ItshouldbenotedthatthereisnoexactGinzburgcriterion[ 226 ].Itisasemi-quantitativemeasureofthecriticalregion.WeplotthebehaviorofthesedecoupleductuationsinFig. 5-6 .It 136

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willtakeameasurementtodeterminetowhatextentthesystemcanbetreatedwithinmeaneldtheory.Weexpectthedispersionofpolarphononstoberelativelyweak,andrelyingonexperienceinferroelectrics,thepolartransitioncouldevenberstorder. Figure5-7. Thediagramsatoneloop.a)two-pointfunctionb)vertexforoneorderparameterc)vertexcouplingtwoorderparameters.Dottedorsolidlinesrefertothepropagatorsforeachorderparameterseparately. Itisinstructivetoconsidertherenormalizationgroupowandanalyzethexedpointsofourfreeenergy.ThecriticalbehavioroftwocoupledorderparameterseachwithO(n)symmetryhasalreadybeencomputedinothercontexts[ 224 ],andtheresultscanbeprotablyreappliedinthisnewcontext.Thenecessarydiagramsareshowning. 5-7 .Theveowequations,resultingfromintegratingoutashellofmomentumfrom bto,aretwocopiesof: 0ri=b2[ri+4(ni+2)uiZ bddq ri+q2+2njkZ bddq rj+q2](5) 0ui=b[ui)]TJ /F8 11.955 Tf 11.96 0 Td[(4(ni+8)u2iZ bddq (ri+q2)2)]TJ /F8 11.955 Tf 11.96 0 Td[(4njk2Z bddq (rj+q2)2](5) andtheowforthecouplingofbothorderparameters: 0k=b[k)]TJ /F8 11.955 Tf 9.29 0 Td[(16k2Z bddq rm+q21 re+q2)]TJ /F8 11.955 Tf 9.3 0 Td[(4(nm+2)kumZ bddq (rm+q2)2)]TJ /F8 11.955 Tf 9.3 0 Td[(4(ne+2)kueZ bddq (re+q2)2](5) 137

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Therearesixxedpointsdeterminedbytheowofthequartictermsuiandk.Infourofthese,theorderparametersdecouple,k=0,andareunstableexceptforthepossibilitythatthesystembreaksintotwoindependentvectormodels.Theremainingtwonon-trivialxedpoints,theHeisenberg-Heisenbergandso-calledbiconicalxedpoints,interchangetheirstabilityasafunctionofhowmanycomponentstheorderparametershave.Fluctuationdrivenrstordertransitions[ 227 ]canoccurifthereisarunawayowintheRG(whenthereisnoxedpointaccessibleunderRGiterations),andareabsentinthismodel.Atthedouble-Heisenbergxedpoint,umc=uec=kc.Linearizing,theeigenvaluesareallnegativeforne+nm4and>0,butchangesitsstabilityifthecomponentsoftheorderparameterchange.Forthecasen=3,thatisforthreedimensionalvectorsPandMconsideredhere,thebiconicalxedpoint(umc6=uec)isstable,andthedoubleHeisenbergxedpointisunstable,meaningthatthesimplecoupled-freeenergywebeganwithalsocontrolsthecriticalbehaviorinthecasethatitisnotmean-eldlike.Ifwehavetwoeasyplanesinsteadandn=2,orverystrongcrystaleldanisotropicsuchthatn=1effectively,thenthedouble-Heisenbergxedpointisagainstable.Theanomalousdimensionasinallone-loopscalareldtheoriesvanishes,butwecananticipatecorrectionsathigherorder. 5.9FutureWorkonDynamics Oneoftheongoingquestionswewishtounderstandisthedispersionofthelowenergyexcitations.Thedispersioncanbesensitivetomicroscopicinteractions.Oneapproachtounderstandingthedynamicsisthroughaneffectiveequationofmotion.Forferroelectricorder,theequationofmotionisessentiallythatofaphonon:P=)]TJ /F5 11.955 Tf 9.3 0 Td[(fF P. 138

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Thisequationcouldbemodiedtoincludedampingwithatermproportionalto_P.Linearizing,P=P0+p,wherep0isthestaticcomponent,p=)]TJ /F8 11.955 Tf 9.3 0 Td[(4f(T Te)]TJ /F8 11.955 Tf 11.95 0 Td[(1)+cq2)(P0+p)+P30+3P20p wereproducewhatonewouldexpectforasimplemodelofopticalphonons.!2=4f[cq2+2(1)]TJ /F5 11.955 Tf 14.85 8.09 Td[(T Te)] TheinteractionM2P2willalteronlythelinearterm,inducingalargeropticalgapproportionaltoM20P0.TheBlochequationforthesemi-classicalprecessionofamagneticmomentinaneffectiveeldreads:_M=MF M. Afterlinearization,theBlochequationreproducestheknownferromagneticresult:)]TJ /F5 11.955 Tf 9.3 0 Td[(i!~m=)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F5 11.955 Tf 9.3 0 Td[(cq2(~m0~m) eveninthepresenceofferroelectricity.Thisresult'sutilityisthatinhomogeneouscouplingsaredistinguishedfromitinthattheyaltertheferromagneticspectrum.ZvezdinandMukhin[ 228 ]considerthedynamicsofanantiferromagnetcoupledviatheLifshitzinvarianttotheferroelectric,asdodeSousaandMoore[ 229 ].Themerepresenceoftheinhomogeneoustermaltersthespectrumoftheantiferromagneticmagnons.DeterminingtheeffectofLifshitzinvariantsonthespectrumaswellasthespectrumforothermultiferroicsisthesubjectoffuturework. 5.10Conclusions Wehavemadesomegeneralconsiderationsmostlybasedonthermodynamicsandsymmetryaboutmultiferroics.Ifthegoalwhenanalyzingamodelistounderstanditsgroundstates,excitations,andtopologicaldefects,thenwehaveworkedtowardsgaininganunderstandingofthephasediagramandtopologicaldefectswhileexcitations 139

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canbeleftforfuturework.WeshowedthatnonzeromagnetoelectricresponseonlyexistswhenbothPandMarenonzeroandthatthiseffectisboundedbytheelectricandmagneticsusceptibilities.Wedeterminedtheuniversalityclassofferroelectric-ferromagnetstobegovernedbythenumberofcomponentsintheorderparameter,inwhichcasewewouldobservecriticalbehaviorcontrolledbythedouble-Heisenbergxedpointorbiconicalxedpoint.Weexaminedtheeffectofinhomogeneousterms,inparticularontopologicaldefects.Theelectriceldwasshowntohaveaninuenceonthewidthofdomainwalls.Thereareanumberofinterestingdirectionsinwhichthisworkcouldbeextended:generalizationstomoreexoticmagneticorpolarorders,inclusionofelasticinteractions,numericalorexactsolutionsfornewdefects,andtheexaminationofdynamics. 140

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CHAPTER6CONCLUSION Logicandmathematicscanstandontheirown,butitiscomparisontotheexternalworldwhichcanfalsify,nevercompletelyverify,oursuppositionsinphysics.Wecanextendourmodelsinsofarastheyareconsistentwithwhatnaturepresents.Inthisthesis,theobservedpropertiesofsuperconductingcupratesandpnictideswerepresentedindetailinChapter2.Alongwiththiscamesomeofthemaintheoreticaltrendsregardingasynthesisofthisbodyofevidence.Noconsensushasbeenreachedyet,sothismaterialservesasmotivationfortheactivescientist.InChapter3,wepresentedamethodfordeterminingtheunderlyingmicroscopicstateofunconventionalsuperconductorsbasedonabulkthermodynamicprobe.Whileitisnotsensitivetoeverydetail,itcandiscriminatebetweenlargelydifferentsuperconductinggapstructures,andruleoutconventionalbehaviorwhenoscillationsofthespecicheat,includingtheinversionoftheseoscillations,occurbeyondeffectsfromFermisurfaceanisotropy.InChapter4wesummarizedmodelcalculationsandexperimentalworkonRamanscatteringintheiron-arsenidesuperconductors,includingtheeffectofimpurities.ThoseconsiderationsledtoaproposalforthesuperconductinggapinBa(Fe0.939Co0.061)2As2,whichcanprovideabasisforfurtherworkcomparingtheevolutionoftheRamanintensitytothemodelastheimpurityscatteringrateischanged.InChapter5,welookedatanaltogetherdifferenttopic,multiferroics.Wecalculatedthethermodynamicconsequencesofthecouplingoftwoorderparameterscouplingtodifferentexternalelds,includingafundamentalboundonthemagnetoelectricresponsewhichdoesnotdependonanyparticularchoiceoffreeenergy.Wethenlookedatasimplemodelforcoupledorderparameterswherethemagneticmomentbreakstime-reversalinvarianceandthepolarizationbreaksinversionsymmetry.Wecalculatedthestaticthermodynamicresponsesforthismodel,andstudiedtheinhomogeneousgroundstates,e.g.topologicaldefects,thatcanariseinthatsituation.Finallyweused 141

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therenormalizationgrouptounderstandthecriticalbehaviorofthemodel,allowingtheuniversalityclasstobeunderstood.Theutilityamacroscopicapproachprovidesistoconrmordenyournotionsofwhattheessentialphysicsisinagivensystem. 142

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