MassimoCapone,1,2MicheleFabrizio,3,4ClaudioCastellani,1andErioTosatti3,4
SMC,CNR-INFMandDipartimentodiFisica,Universit`a“LaSapienza”,P.leAldoMoro2,I-00185,Roma,Italy2
ISC-CNR,ViadeiTaurini19,I-00185Roma,Italy3
InternationalSchoolforAdvancedStudies(SISSA),andCNR/INFMDEMOCRITOSNationalSimulationCenter,ViaBeirut2-4,I-34014Trieste,Italy4
TheAbdusSalamInternationalCenterforTheoreticalPhysics(ICTP),P.O.Box586,I-34014Trieste,Italy(Dated:September4,2008)
ThetrivalentalkalifulleridessolidsofgenericcompositionA3C60,whereC60isthefullerenemoleculeandA=K,Rb,andCs,areawellestablishedfamilyofmolecularsuperconductors.Thesuperconductiveelectronpairingisofregulars-wavesymmetryandisaccountedforbyconventionalcouplingofelectronstophonons,inparticularbywellunderstoodJahnTellerintramolecularC60vibrations.Asourceofrenewedinterestinthesesystemsarealarmingindicationsofstrongelectron-electronrepulsionphenomena,whichemergedespeciallyincompoundswheretheC60-C60distanceisexpanded,byeitheralargecationsizeorbyotherchemicalorphysicalmeans.Severalexamplesarenowknownwherethiskindofexpansion,whileleadingtoahighsuperconductingtemperatureatfirst,graduallyorsuddenlycausesadeclineofsuperconductivityanditseventualdisappearanceinfavorofaMottinsulatingstate.Thiskindofinsulatingstateisthehallmarkofstrongelectroncorrelationsincuprateandorganicsuperconductors,anditsappearancesuggeststhatfulleridesmightalsobemembersatlargeofthatfamily.
Ourapproachtothefulleridesistheoretical,andbasedonthesolutionofaHubbardtypemodel,whereelectronshopbetweenmolecularsites.Wetakeadvantageofthefactthat,unlikemodelsforthestronglycorrelatedcuprates,stillunderdebate,inaHubbardmodeloffulleridesalltheimportantelectroncorrelationsoccurwithinthemolecularsite,efficientlysolubleintheDynamicalMeanFieldTheory(DMFT)approximation.DMFTsolutionsconfirmthatsuperconductivityinthismodelfulleride,althoughofs-wavesymmetryratherthand-wave,sharesmanyofthepropertiesthatarecharacteristicofhighTccuprates.Thecalculationsareheavy;andwhileourworkingmodelisseveralyearsold,thenewresultswepresentinthisColloquiumpertaintothemostinterestingcaseofthreeelectronsperC60molecule,appropriatetoA3C60,andhaveonlybeenpossiblerecentlythankstoastrongercomputationaleffort.Wehavecalculatedthezerotemperaturephasediagramasafunctionoftheratioofintra-molecularrepulsionparameterUovertheelectronbandwidthW,theincreaseofU/Wrepresentingthemaineffectoflatticeexpansion.Thephasediagramisclosetothatofactualmaterials,withadomeshapedsuperconductingorderparameterregionprecedingtheMotttransitionforincreasingcellvolume.Unconventionalpropertiesofexpandedfulleridesuperconductorspredictedbythismodelinclude:(i)anenergypseudogapinthenormalphase;(ii)againofelectronkineticenergyandofconductingDrudeweightattheonsetofsuperconductivity,asinhighTccuprates;(iii)aspinsusceptibilityandaspecificheatbehaviorthatisnotdrasticallydifferentfromaregularphononsuperconductor,despitestrongcorrelations;(iv)theemergenceofmorethanoneenergyscalegoverningtherenormalizedsingleparticledispersion,electronicentropyandthespecificheatjump.Thesepredictions,whichifconfirmedshouldestablishfullerides,especiallytheexpandedones,asmembersofthewiderfamilyofstronglycorrelatedsuperconductors,arediscussedinthelightofexistingandforeseeableexperiments.
1
arXiv:0809.0910v1 [cond-mat.supr-con] 4 Sep 2008PACSnumbers:71.30+h,71.10.Pm,71.10.Fd
Contents
I.introduction
II.ElectronInteractionsinFulleridesIII.ModelandInteractions
A.DynamicalMean-FieldTheoryB.T=0PhaseDiagramIV.UnderstandingStronglyCorrelated
23557SuperconductivityfromDMFTandtheImpurityModel
A.AndersonImpurityWithaRigidBath
B.AndersonImpuritywithaSelf-ConsistentBathin
DMFTV.DiscussionofExperimentalProbesVI.Conclusions
Acknowledgments
91112131516
References
16
I.INTRODUCTION
DiscoveredbyKamerlinghOnnesnearlyacenturyago(Onnes,1911),andfirstexplainedmicroscopi-callybackin1957intermsofelectronpairingbyBardeen,CooperandSchrieffer(BCS)(Bardeenetal.,1957a,b),superconductivityisstillsurprisinglya’lapage.Ononehand,superconductivityisbeingconstantlydis-coveredinaneverincreasingvarietyofsolidstatecom-pounds.Ontheotherhand,itappearsmoreandmoredifficulttousebasicallythesamestandardtheory,es-sentiallyBCSanditsextensions(e.g.Parks,1969,chap-ters10and11)toaccountforallofthem.Inthisstan-dard,conventionaltheory,superconductivityarisesfromthecondensationofelectronpairs,thetwoelectronsusu-allyboundinapairstateofs-wavesymmetryandheldtogetherbyexchangeoflatticephonons.TheCoulombrepulsionbetweenthetwoelectronsopposespairforma-tion,butitdoesnotsuppresssuperconductivitybecausescreeningmakesitweakenough.
Thesurprisinglyfavorableeffectofrepulsiveelectroncorrelationsonsuperconductivityfoundinsomesys-tems,particularlyinhigh-Tcsuperconductingcuprates,whereelectron-electronrepulsionisdominant,remainsastandingpuzzle.Animmenseamountofexperimen-talandtheoreticalworkhasaccumulatedoverthelasttwodecadesintheattempttounderstandthesephenom-ena,seee.g.thereviewbyBennemannandKetterson(2008).Actually,cupratesarebutthemostspectacularmembersofawiderclassofstronglycorrelatedsupercon-ductorsincludingheavyfermionandorganicmolecularcompounds(BennemannandKetterson,2008),systemsforwhichthereisnoreallycomprehensivetheoryeither.Amongotherfactors,theoreticaleffortshavebeenham-peredbythegeneralinter-sitenatureofelectroninter-actionsandcorrelationsinmanyofthesesystems,afactthatposeslargetechnicaldifficulties.Inthislight,iden-tifyingasuperconductorfamilywherecorrelationsareatthesametimestrong,simple,andon-siteiswelcome.Amorecrucialelementisoneofperspective.Ithasbeenawidespreadprejudicetodistinguishbetweensu-perconductorswhere(asinBCStheory)pairingofelec-tronstakesplaceinthes-wavechannelandismediatedbyphonons,fromthosewherethemechanismmaybeelectronicandnotphononic,andwherepairinginsteadtakesplaceinthed-wavechannel.Whereasitiswidelyheldthatstrongrepulsivecorrelationsareessentialtosu-perconductivityinthelatter(Anderson,1987),theyarenotconsideredcrucialintheformer.TheconventionalBCSscenarioanditsextensions,namelytheMigdal-Eliashbergtheory(Eliashberg,1960;Migdal,1958;Parks,1969)–acontrolledapproximationvalidwhenthetyp-icalphononfrequencyismuchsmallerthantheFermi-energy–aremoreorlessautomaticallyaccepted,andusedtoaccountforthesuperconductingproperties.In
2
thistheory,electron-electronrepulsionmerelyrenormal-izestheelectron-phononparameters,loweringthecriticaltemperatureTcratherthanenhancingit.
Thetrivalentalkalifulleridessuperconductors,excel-lentlyreviewed,e.g.,byGunnarsson(1997,2004)andRamirez(1994),areamongthesystemswherethiscon-ventionallogicseeminglyapplies.FulleridesaresolidstatecompoundsofgenericcompositionA3C60,whereC60isthefullerenemolecule(Gunnarsson,2004)andA=K,Rb,andCsarealkalications.Thethreealkalisdonateatotalofn=3electronstoeachfullerene,halffill-ingitsthreefolddegeneratet1umolecularlevel.Electronhoppingbetweenfirstneighboringfullerenesgivesrisetoametal,whereconductionisrestrictedtothethreenar-rowt1u-derivedbands,withatotalenergybandwidthofnomorethan0.6eV(Erwin,1993;Satpathyetal.,1992).Metallicfulleridesaregenerallysuperconducting,withcriticaltemperaturesTcreaching∼40K,depend-ingonvariousfactors.Anempiricallyimportantfactorappearstobethecellvolume.Whenthefulleridelatticeischemicallyexpanded,byeitherincreasingcationsizeorbyinsertionofneutralmolecules,orelsephysicallyexpandedbyremovingpressure,Tcundergoesadefiniteandsystematicchange.Itrisesinitiallywithagoodcor-relationwiththeC60-C60distance(Gunnarsson,1997;Yildirimetal.,1995).FurtherexpansionhowevercausesTctodrop,endingeventually,throughafirstordertran-sition,inaninsulatingstate,asweshalldiscusslater.Awealthofevidenceindicatesthatsuperconductingpairinginfulleridesisphononicandthattherelevantphononsarethestiffintra-molecularHgvibrationsoftheC60molecule,Jahn-Tellercoupledtothet1uconductionelectrons(Gunnarsson,1997).FurthersupporttotheapparentBCSnatureofsuperconductivityinfulleridescomesfromspecific-heatjumpsthatscalelinearlywithTcinagreementwithBCStheory(BurkhartandMeingast,1996;Kortanetal.,1992;Ramirez,1994),aswellasaregular(i.e.,notexceedinglyhigh)normalphasemag-neticsusceptibility.(Kortanetal.,1992;Robertetal.,1998)Superconductingenergygapsarelessclearlyde-fined,(Gunnarsson,1997)thegapratio2∆/Tc3.4−4.2inK3C60andRb3C60(Gunnarsson,1997;≃Ramirez,1994),butnotfarfromtheBCSvalueof3.53.TheseelementssuggestviewingtheA3C60compoundsasweaklycorrelatedFermi-liquidconductors(Ramirez,1994),thoughwithunusuallynarrowelectronbands,withalargeeffectivemassroughlythreefreeelectronmasses(Robertetal.,1998).EventheobserveddecreaseofTcunderappliedpressureinK3C60andRb3C60isinqualitativeagreementwithanincreasingbandwidthanddecreasingdensityofstatesattheFermilevel,whichfur-thersupportsastandardBCSpicture.
Thesereassuring,conventionallookingfactsarehowevercontrastedbyanumberofconflictingele-mentsthatarestrongenoughtocastseriousdoubtsonthegeneralapplicabilityoftheBCSscenariotosuperconductorsinthisfamily.Theseelementsareespeciallyapparentinthemoreexpandedful-
3
leridesincluding(NH3)xNaK2C60(Ricc´oetal.,2003)andLi3C60(Durandetal.,2003),andinthealloysCs3−xKxC60andCs3−xRbxC60(Dahlkeetal.,2000).FortheseexpandedcompoundsTcdecreasesuponex-pansion,contrarytoBCStheory.TheelectrondensityofstatesextractedbyNMRKnightshiftisatthesametimeanincreasingfunctionoflatticeparameter,smoothlyconnectingwiththatoftheunalloyedcompoundsunderpressure(Dahlkeetal.,2000).WithinBCStheory,thatincreaseshouldleadtoariseofTcandnottoadropasobserved.Thesameunconventionalbehaviorisob-servedinCs3C60,thefulleridecompoundwiththehigh-estTc∼40Kattainedunderpressure(Palstraetal.,1995).AnovelA15superconductingphaseofCs3C60withexpandedstructurehasveryrecentlybeensyn-thesized(Ganinetal.,2008)correspondingtoabody-centeredcubicarrangementoffullerenes.Superconduc-tivityemergesunderpressurethroughafirstordernon-structuraltransitionat4Kbar.ThecriticaltemperatureTcfirstincreaseswithpressure,reachingadome-shapedmaximumof38Karound7Kbar,abovewhichTcdrops.Sincenostructuralchangesareobservedunderpressure,theappearanceofsuperconductivityaswellasthedome-shapedTcvs.pressurebehaviormustbeascribedsolelytothevolumecontraction(Ganinetal.,2008).Thisnon-monotonicbehaviorofTcwithpressurefindsnoapparentexplanationwithintheconventionaltheory.
Thebasicandstrikinganomalyofexpandedfulleridesoccursinthecompoundswiththelargestinter-moleculedistances.Inthesematerialsarelativelymodestad-ditionallatticeexpansion(andminorchangeofsym-metryduetointercalatedammonia)isenoughtodra-maticallyturnthemfrommetallicandsuperconduct-ingtoantiferromagneticandinsulating(Durandetal.,2003;IwasaandTakenobu,2003).1Withtemperature,antiferromagneticorderintheammoniatedcompoundNH3K3C60changestoparamagneticdisorderataN´eeltemperatureslightlyabove∼40K(Prassidesetal.,1999).EvenabovetheN´eeltemperature,themicrowaveconductivityinNH3K3C60remainsseveralordersofmag-nitudebelowthatofK3C60(Kitanoetal.,2002),testi-fyingtheMottinsulator(correlationdriven)natureoftheinsulatingphase.ElectronsinalatticegiverisetoaMottinsulatingstatewhenelectronelectronrepulsionstopstheirfreepropagationandthelatticeappearsasacollectionofmolecularions.Correlationsleadtoanen-ergygapintheirspectrum,eveniftheirnumberdensitypercellisodd,insteadofevenasinregularbandinsula-tors(Mott,1990).ProximityofaMottinsulatorphaseinfullerideshadlongbeenadvocatedindifferentcon-texts(BaskaranandTosatti,1991;Chakravartyetal.,1991;ChakravartyandKivelson,1991;Lofetal.,1992)
molecularion.InmolecularC360−
,thestrengthoftheseinteractionshasbeenevaluatedinthepast,andtheJTstrengthhasbeenestimatedtoprevailnarrowlyoverHund’sruleexchange(L¨udersetal.,2002).Thisnar-rowbalancefavorsalow-spingroundstate,witharel-ativelysmall“spingap”–theenergybetweenthelowspingroundstateandthelowesthigh-spinexcitedstate,expectedtobeoftheorderof0.1eV(Caponeetal.,2001;L¨udersetal.,2002).Inagreementwiththisex-pectation,localmomentsindicatethatinantiferromag-neticMottinsulatingNH3K3C60theC60(3−)sitesareinalow-spinstate,S=1/2(Prassidesetal.,1999),theirhighspinstateS=3/2lyingabout100meVhigherinenergy.Alow-spinqualifiestheoverexpandedful-leridesas“Mott-Jahn-Teller”insulators–thatis,MottinsulatorswhosesitesareinaJTstabilizedlow-spinstate(asopposedtoaHund’srulestabilizedhigh-spinstate)(FabrizioandTosatti,1997).Underhydrostaticpressure,NH3K3C60undergoesatransitiontoametallicstatewheresuperconductivityre-emergeswitharatherlargeTc(Prassidesetal.,1999).Itisimportanttonotethatthissuperconductingphasestillbelongstothe“ex-panded”family,assignaledbythefacttheTcherein-creasesfurtherwithincreasingpressure(reaching28Kat14.8kbar(Zhouetal.,1995))atvariancewithnon-expandedfullerides,whereTcdropsunderpressure.Alateral,butrelevantadditionalelementcomesfromtetravalentcompoundsA4C60,whichareinsulatorsornearinsulators.Bycomparisonwiththetrivalentful-lerides,theslightreductionofband-energygainperpar-ticlecausedbyaddingonemoreelectronpermoleculeandbyslightlychangingthecrystalstructureissuffi-cienttoturnthetrivalentmetalsintotetravalentinsu-latorseveninnon-expandedmaterials.Carefuldensity-functionalelectronicstructurecalculationsindicatedthatitisnotpossibletodescribethetetravalentcompoundssuchasK4C60asstaticallydistortedJahn-Tellerbandin-sulators(Caponeetal.,2000).AstaticJTdistortionandtheassociatedorthorhombicstateisactuallypresentonlyinCs4C60(DahlkeandRosseinsky,2002),andinRb4C60aboveacriticalpressure(HuqandStephens,2006),whileitnevershowsupinK4C60(HuqandStephens,2006)(withtheexceptionofmonolayers,seeWachowiaketal.(2005)).Thepersistenceofinsulatingornearinsulat-ingbehaviorandtherecoveryofmolecularsymmetryobservedinthehightemperaturephaseoftetravalentfulleridessuggeststhatthesecompoundstooareMott-Jahn-Tellerinsulators(Caponeetal.,2000;Kluppetal.,2006;KnupferandFink,1997),liketheoverexpanded
trivalentmaterials.ThedynamicJTeffectineachC4−
ionassociatedwithMottlocalizationofcarriersiscru-60cialinexplainingthelowspingroundstateandthespingapofA4C60,exactlyasfortheexpandedtrivalentcom-pounds.
Fromtheabovediscussiononemightbetemptedtoconcludethatstrongcorrelationsplayaroleonlyintetravalentandexpandedtrivalentcompounds,whileface-centeredcubic(f.c.c.)K3C60andRb3C60,wheresu-
4
perconductivitywasoriginallydiscovered,couldstillbeviewedasweaklycorrelatedsystems,andasBCStypesuperconductors.Wedonotbelieveinthisconclusion.Afinal,independentandstronglyunconventionalsignalisprovidedbyNMR.Infact,NMRdatashowdirectev-idenceofaspingapoforder0.1eV,appearingasananomalousactivatedincreaseofinverserelaxationtime.Mostlikelythisgapbetweenalowspingroundstateandahighspinexcitedstatereflectsthemultipletbe-haviorofthelocalizedCn60−
molecularion.Itshowsupubiquitouslyinallalkalidopedfullerides,includingsu-perconductingf.c.c.compounds(Brouetetal.,2002a,b;Thieretal.,1995).Theexistenceofthespingapsig-nifiesthatthemagneticresponseoffulleridesisveryfarfromFermi-liquidbehavior,whichhasnosuchfea-ture.Magnetically,thefulleridesbehaveasiflocalizedmolecularmultipletexcitationscoexistedwithdelocal-izedpropagatingquasiparticles.Asdiscussed,therecov-eryofmolecularphysicsischaracteristicofMottinsu-lators,suggestingthatthefingerprintofMottphysicsisstronglypresentalreadyinthenon-expandedsupercon-ductingf.c.c.compounds.Thissuggeststhatthef.c.c.compoundsaresomehowtheanalogoftheoverdopedcuprates,whereastheexpandedtrivalentmaterialsareanalogoustotheunderdopedcuprates.Theconclusionisthatbotharecrucially,evenifdifferently,influencedbyelectroncorrelations.
Webelievethattheaboveelementsarestrongenoughtocallforanewphysicalpictureforthewholefam-ilyofA3C60superconductors.ProximityoftheMottinsulatorstronglysuggeststhattheanomaliesofex-pandedfullerenesuperconductorsmostlikelyoriginatefromstrongrepulsiveelectroncorrelationinthenarrowt1ubands.TheprevalenceintheMottlocalizedstateofmolecularphysics,withitsorbitaldegeneracy,JTef-fectandintra-molecularexchangemustbetakenintoac-countalongwithitinerantelectronbandphysics.Uponexpandingthecellvolume,theintermolecularhoppingofelectronweakens,whereasalltheon-sitecorrelationterms–Coulombandexchangeelectron-electroninter-actionsaswellasmolecularJTeffect–arelikelytobe-comeincreasinglyrelevant.WeareledtoapicturewheretheMottinsulatorphysicsofweaklycoupledmolecularionsprogressivelyprevailsoverbandphysicsforincreas-inglatticeexpansion.Inparticular,superconductorsthatoperateinthisregimeareboundtodeviatefromthestandardMigdal-Eliashbergscenario,themoresoasthelatticespacingincreases.Toinvestigatethatregime,weneedtostartwithabroadertheoreticalschemefortriva-lentfullerides,capableofdescribingtheirbehaviorunderlatticeexpansionandneartheMotttransition.Whilethathasbeenthescopeofourworkforseveralyears,previousworkwasforpracticaltechnicalreasonslim-itedtotetravalentsystems(Caponeetal.,2002,2000,2001).ThestudyofA3C60systems,computationallymuchheavierduetothesimultaneousrelevanceofmag-neticandorbitalordering,hasonlynowbeencompleted,andweofferhereanoutlineofthemainresults.
5
III.MODELANDINTERACTIONS
Ourtheoreticalmodeloftrivalentfulleridesassumesalatticeofmolecularsites,eachrepresentingaC60molecule.TheC60t1uthreefolddegenerateLUMOcanforallpurposesbetreatedasanatomicplevel.Anaver-ageofthreeelectronspermoleculearedonatedbyalkaliatomsandpartiallyfilltheseorbitals,whichcanhostuptosixelectrons.Theelectronshopfromsitetositegiv-ingrisetohalf-filledbandsofwidthW∼0.6eV.OneachsitetheelectronsexperienceaHubbardrepulsionU∼1eV(correspondingtotheSlaterintegralF0∝U),
HU=
U
2
L·L
+
5
6
semicircular.Thisisareasonabledescriptionofarealis-ticthree-dimensionaldensityofstatesdevoidofacciden-talfeaturessuchasvan-Hovesingularities.Itismoreoverparticularlyconvenientsinceitleadstoaverysimpleandtransparentformoftheself-consistencycondition.TheBethelatticeisthez→∞limitofaCayleytreeofcoor-dinationz,scalingthenearest-neighborhoppingineach√oneofthezdirectionsast/
iωn−εka
,(4)
toberelatedtothelocalinteractingGreen’sfunction,Ga(iωn),notonlythroughtheDysonequationfortheAIM
−11G0(iωn)a=G−a(iωn)+Σa(iωn),
whichrequiresthefullsolutionoftheimpuritymodel,
butalsobytheadditionalselfconsistencyequation
−12G0a(iωn)=(iωn+µ)−tG(iωn)a.
(5)
DMFTforagivenmodelthusamountstosolveit-erativelytheAIMuntiltheimpurityGreen’sfunction
satisfiesEq.(5).Inthiswork,wesolvedthethreefolddegenerateAIMbyexactdiagonalization.ThatrequirestruncatingthesumoverkinEqs.(3)and(5)toafi-niteandrelativelysmallnumberofbathsNb,sothattheHamiltoniancanbediagonalizedinthefiniteresultingHilbertspace.WegenerallyusedNb=4foreachor-bital.Mostoftheresultsweshallpresentareatzerotemperature,wherewecanusetheLanczosalgorithmtocalculatetheGreen’sfunctionwithoutfullydiagonalizingtheHamiltonian.ThefinitetemperatureresultsforthespecificheatanditsjumpatTcreportedinSec.Varetheexception.Theyareobtainedbymeansofthefinite-temperatureextensionofLanczos(Caponeetal.,2007),wherethethermalGreen’sfunctionisexpressedasasumoverthelow-lyingeigenvectors|nandeigenvaluesEnoftheimpuritymodel
Gaσ(iωn)=
1
Em−En−iωn
+
2
n|p†|maσ
n
consistencyconditionis
G−01(iωn)σA=(iωn+µ)−t2G(iωn)σB,
(10)
whichbecomesEq.(5)ifGσAWhenthesystem≡GσB,i.e.ifthesystem
isnonmagnetic.isantiferromagnetic,thenGσA≡G−σB.ThuswecaneliminatethesublatticeBfromEq.(10),andobtainthefollowingresultfortheself-consistencyequation:
G−01(iωn)σA=(iωn+µ)−t2G(iωn)−σA.
(11)
ThisequationisvalidforaBethelatticewithnearest
neighborhopping,acasewithperfectnestingthatisratherexceptionalinrealisticantiferromagnets.Awaytosimulateimperfectnestingtypicalofmorerealisticsit-uations,whilestilltakingadvantageoftheBethe-latticesimplifications,′istoaddanext-nearest-neighborhoppingt/zintheCayleytree(merelyadevicetoeliminatenest-ing,notmeanttosuggestnextnearestneighborhopping,smallinfullerides).Inthelimitz→∞oftheBethelat-tice,theself-consistencyequationbecomes
G−2
01(iωn)σA=(iωn+µ)−t2G(iωn)σB−t′G(iωn)σA
=(iωn+µ)−t2G(iωn)−σA−t′2G(iωn)(12)σA.
Forbothbroken-symmetryphases,ifthediagonalel-ementsoftheself-energymatrixatlowMatsubarafre-quenciesfollowtheconventionalFermi-liquidbehavior–whichwealwaysfindtobethecase,
Σ
1
diagonal(iωn)≃1−
7
N
a↓i
a=
x,y,z
p††
ia↑pi,
(14)
whereNisthenumberofmoleculesandp†iaσcreatesanelectrononmoleculei,withspinσandinorbitala=x,y,z.TheFermi-liquidscatteringamplitudeintheCooperchannel,measuringthestrengthoftheef-fectiveattraction,isA=−10J/3.Wenotethatow-ingtofairlystrongJTinteractions(Auerbachetal.,1994;Gunnarssonetal.,1995),ifHund’sexchangeJHwereneglected,thedimensionlessJTcouplingconstantoffullerenecontrollingsuperconductivitywouldbenu-mericallyverylarge,λ=ρ0ρ0(≃2.4eV−1)isthebaredensity|A|≃of0.6states−1.0,perwherespinandorbital.TurningonaweakCoulombrepulsionUontopofthatwillreducethepairingattractioninthisregime.PerturbativelyoneobtainsforsmallUthatA=−10J/3+U.SinceJisinsensitivetoexpansionwhileU/Wincreases,thisimpliesthatinthispicture,whereHund’sruleexchangeisneglected,Tcshouldal-waysdecreaseuponexpansion,apredictionwhichisatoddswithexperiments.
Infact,asanticipated,Hund’sruleexchangeisnotnegligible,anditseffectistointroduceasubstantialcancellationinJleadingtoalargelyreducedeffectivecouplingλeff≃10
2
Sincegaugesymmetryisbroken,weareallowedtoassumePSCreal
(a)(b)
FIG.2Superconductingsolution(triangles)andantiferro-magneticsolution(squares)zero-frequencyanomalousself-energies∆(0)asfunctionofU/W.Solidsymbolsareusedwhenthecorrespondingsymmetrybrokenphaseisstable,whileopensymbolswhenitismetastable,i.e.haslowerenergythentheotherphase.Thefirst-ordertransitionbe-tweenthetwophasesisindicatedbyaverticallineseparatingthesuperconductorfromtheantiferromagneticMottinsula-tor.Panel(b)correspondstothecaseinthepresenceofafrustratingnext-nearestneighborhoppingt′=0.3t,absentin(a).
andbeyond.Intheconventionalweaklycorrelatedpic-tureUwouldprovideintheelectronpairingproblemarepulsive“Coulombpseudopotential”whosebarevalueisµ∗=Uρ0barevaluesof≃λ3and(Parks,µ∗we1969).shouldSimplyconcludecomparingthats-wavetheseBCSsuperconductivityinfullerenesissimplyimpossible(withtheobviousprovisothatforsmallUanunretardedtreatmentofJTphononinteractionsisnotreallyjusti-fied).
ThefullDMFTsolutionofthemodelforJ=0.05WandincreasingU/WyieldsthephasediagraminFig.2.WhileconfirmingtheaboveexpectationsformoderateU,ithasasurpriseinreserveatlargerUvalues,wheretheMotttransitionisapproached.
Fig.3showsthezerofrequencyanomaloussingle-particleself-energiescalculatedwithinDMFTforthesu-perconductingandfortheantiferromagneticsolution.AtU=0,themodelisans-waveBCSsuperconductor,withanexponentiallysmallsuperconducting∆(0).Itistoosmalltobevisibleinthefigure,sincetheeffec-tiveexchange-reducedλ∼0.2isaswassaidveryweak.Beginningfromzero,theincreaseofUfirstrapidlyde-stroystheweakBCSsuperconductivity.Thesupercon-ducting∆(0)vanishesatroughlythemean-fieldvalueU=10/3J,andabovethisvalueofUthegroundstatebecomesanormalmetalasexpected.Uponfurtherin-creasingU/Wthemodelremainsanormalmetal–nosuperconductivity,noantiferromagnetism.However,theimportanceofelectroncorrelationsincreaseswithU,assignaledforexampleintheDMFTspectralfunction(notshown)bythegradualformationofincoherentHubbardbandsonbothsidesoftheFermilevel.ThemetalliccharacterpersistsuntilahypotheticallycontinuousMott
8
transitioneventuallyreachednearU/W∼1.5,whereZ=0andthemetalliccharacterisextinguished.
Beforethishappenshowever,s-wavesuperconductiv-ityre-entersfromthenormalmetalstate.Theanomalousselfenergy∆(0),proportionaltothesuperconductingorderparameterhas,asafunctionofU,abell-shapedbehavior–a“superconductingdome”asitiscalledincuprates–hittingalargemaximumbeforedroppingagain.There-entrantsuperconductivebehaviorisaclearrealizationofphonon-inducedstronglycorrelatedsuper-conductivity(SCS)(Caponeetal.,2002).ThesharplyrisingorderparameteredgewithincreasingU/WcaninourviewexplainthestrongriseofTcuponlatticeex-pansioninnon-expandedcompounds,previously(andwebelieveincorrectly)attributedtoaBCS-likeincreaseofdensityofstatesuponbandnarrowing.Pastthedomemaximum,anduponincreasingexpansion,theSCSsu-perconductingorderparameterdeclines,andwouldeven-tuallydroptozeroatthecontinuousmetalinsulatortransitionnearU/W∼1.5.Thiscontinuousdeclineofsuperconductivityispreemptedbyafirstordertransi-tiontoalowerenergyantiferromagneticMottinsulatingphase,withorderparameter
PAFM=
1
0.20.50.15Z∆0.4SCZ∆AFM∆SC0.1∆0.3AFM0.20.050.1000.40.81.2000.40.81.20.50.50.40.4P0.3SCP0.30.2AFM0.20.10.1000.40.81.2000.40.81.2U/WU/WFIG.3Superconductingsolution(SC,left)andantiferromag-neticsolution(AFM,right)anomalousself-energies(top)andorderparameters(bottom)asafunctionofU/W.Thetoppanelsalsoshow(greendiamonds)thespectralgapsobtainedmultiplying∆bythequasiparticleweightZofeachsolution.ThegapsareinunitsofthebandwidthW(theorderparam-etersarebydefinitiondimensionless).Dashedverticallinesmarkthefirst-orderphasetransitionbetweenthetwosolu-tions.
spin-1/2antiferromagneticinsulator.Besidesspinrota-tionalsymmetry,thiskindofstatealsobreaksorbitalrotationalsymmetry,signalingthatspinorderingmustbegenerallyaccompaniedbyorbitalordering.Inam-moniatedfulleridesthatagainisconsistentwithexperi-ment(IwasaandTakenobu,2003).Weconcludethat,inspiteofstrongsimplifyingassumptions,ourmodelseemsabletoreproduceveryimportantfeaturesofthephasediagramofexpandedfullerides.Inthefollowingweshalldiscussinmoredetailthestronglycorrelatedsupercon-ductingphaseneartheMotttransition,andalsoproposeexperimentsthatmightdistinguishitfromastandardBCSstate.
IV.UNDERSTANDINGSTRONGLYCORRELATEDSUPERCONDUCTIVITYFROMDMFTANDTHEIMPURITYMODEL
There-emergenceofphonon-drivensuperconductivityclosetotheMotttransition–StronglyCorrelatedSu-perconductivity(SCS)–wasdiscussedinCaponeetal.(2002)intermsofFermi-liquidtheory.Akeypointofthatphenomenonistherenormalizationoftheeffectivebandwidthandthusoftheeffectivemass,bothcon-trolled(inaBethelattice)bythequasiparticleweightZ,Eq.(13).Z(U)decreasesasafunctionofU/Wandvan-ishesatthecontinuousmetalinsulatortransitionpointU=Uc,wheretheeffectivemassm∗/m=1/Z(U)di-verges,andtheeffectivequasiparticlebandwidthW∗=ZWvanishes.Anestimateoftheinteractionbetweenchargedquasiparticlesrequirestheevolutionoffluctua-tionsthattakeplaceinchargespace.Becausecharge
9
fluctuationsaregraduallyfrozenawayneartheMotttransition,theeffectiverepulsionbetweenquasiparticlesisalsorenormalizeddowntosomesmallervalueU∗ suppresssuperconductivity,aswasnotedearlier.How-ever,ifUisclosetothecriticalmetal-insulatorvalueUc,thenZ∼(Ucturnsnegative−Uin)/Ucspite≪of1andalargethescatteringU.Thisisamplitudequalita-tivelythereasonfortheSCSre-entranceofsupercon-ductivity(thoughinthisregionofcoursetheactualpairscatteringamplitudemightdeviatefromthissimplefor-mula(Caponeetal.,2002)). Inaddition,theFermi-liquidargumentsuggestsanex-planationforthelargevalueofsuperconductingorderparameter,implyingalargeTc,intheSCSregime,seeFig.2,comparedtotheU=0BCSvalues.Infact,whenA∗particleattraction≃ZW,andA∗Zwillisatdroppingsomepointsharply,U=theU∗quasi-equalthecoherentquasiparticlebandwidthZW.Thatveryuncommonsituation,ofmetallicquasiparticleswithanpairattractionequaltotheirenergybandwidth,isknowntoyieldmaximumsuperconductivityforagivenattrac-tion.Asshownbystudiesofpurelyattractivemod-els(Micnasetal.,1990),themaximumsuperconductingtemperaturekBTcattainableinthatcaseisabout7%ofthepairattractionenergyitself.Inourmodeloftrivalentfullerides,thisestimateyieldskBTcwhichhasthecorrectmagnitudeofroughly∼0.07A∗40K∼for0.2J|J∼|,20meV–avalueinturnfullyconsistentwiththeob-servedspingap0.1eV∼5J.Whilethiscoincidenceofnumbersisprobablyfortuitous,itdoesindicatethator-dersofmagnitudeimpliedbyourmodelwithrealisticpa-rametersarequiteconsistentwithexperimentalfacts.Atfacevalue,italsosuggeststhat0.07|J|=0.07(JJTinfullerides.−JH)couldbethemaximumattainablekBTcWeconcludethatstrongcorrelationsplayacrucialroleinbringingthesuperconductinggapmagnitudetotheright 0.050.04W0.03/cs∆Z0.020.01000.20.40.6U/W0.811.21.4FIG.4QuasiparticlesuperconductingenergygapinunitsofthebandwidthW≃0.6eVandonalargerscalethanFig.3,computedthroughtheanomalousself-energy,panel(a)inFig.2,multipliedbythequasiparticleresidueZ(U).Noticethat,aboveU/W≃1.1,thesuperconductingsolutionismetastablesincetheantiferromagneticonehasloweren-ergy,seeFig.2.WenotethatthemaximumgapZ∆c≃0.045W≃27meV. rangeofvaluesascomparedwiththeexperimentalones,seeFig.4.SuchvaluesandlargecriticaltemperatureswouldneverbeattainedwithinconventionalBCStheoryusingavalueofλ≃0.16−0.2,includingasitshouldthelargecancellationofJTbyexchange.Theycouldinpointoffactbeattainedifthecancellationduetoex-changewere(incorrectly)neglected;butthenalatticeexpansionshouldalwaysleadtoadecreaseofTc,con-trarytoexperiment. Toappreciatefurthertheeffectofexchange-JTcan-cellation,itisinstructivetoconsider,aswasdoneforasimplifiedmodelbyCaponeetal.(2004),thebehav-iorwithU/Wofthesuperconductingself-energy∆(0)(proportionaltotheT=0gap,androughlyspeakingtoTc)startingwithpureJTandwithoutexchange,andthenproceedingtoturnonexchangeandgradualcan-cellation,seeFig.5.ForthebareJT,λ≃1(astrongcouplingvalue)thesuperconductingself-energydecreasesmonotonicallywithincreasingU,inagreementwiththeMigdal-EliashbergpredictionofanincreasingCoulombpseudo-potential.Aboveacriticalvalue,thesystemturnsdirectly,viaasecond-orderorweaklyfirst-orderphasetransition,toaMottinsulatingphase.ThisresultisfullyconsistentwithpreviouscalculationsbyHansletal.(Hanetal.,2003),wherethesametypeofHub-bardmodelwasstudiedwithinDMFTatfinitetemper-ature.Treatingexplicitlytheelectron-phononcoupling(includingthefullphonondynamics)withλ=0.6andneglectingexchangetheyobtainedasuperconductorwithmonotonicallydecreasinggap. Throughaprogressivereductionofλ(mimickingJTcancellationbyexchange)wefindthatanon-monotonicsuperconductingbehaviormakesitsappearanceasafunctionofU.Initiallythereisstillasinglesuper-conductingphaseforallU/Wvalues;buttwodiffer- 100,4J=0.02W0,3J=0.05WJ=0.1WJ=0.2W∆SC0,20,1000,40,81,21,6U/WFIG.5Anomalousself-energy∆(0)(relatedtothesupercon-ductinggapbyafactorZ−1)forthesuperconductingsolutionanddifferentvaluesofthecouplingparameterJ=0.02,0.05,0.1and0.2,whichcorrespondtoλ=0.09,0.21,0.42,0.85,respectively.NotethatforlargeJ,correspondingtoacasewheretheJahnTellercouplingisnotcanceledbyHund’sruleexchange,superconductivityisstrongestatU=0.Forincreasingcancellation(decreasingJ),twoseparatesuper-conductingpocketsemerge,aBCSpocketnearU=0,andaSCSpocketneartheMottinsulator.Whenthecancellationisnearlycomplete,SCSismanyordersofmagnitudesstrongerthanBCS.Thisisthesituationweproposeinourmodeloffullerides.SimilarphysicswasdescribedforasimplermodelinCaponeetal.(2004). entregionsnearzeroandnearUcbegintomaterialize.(NotethatUcsimultaneouslyshiftstohigherUasλdecreases).Whenthecancellationissostrongthatλisstillpositivebutsmall,thetwosuperconductingre-gionsbreakaparttoformtwoseparatepockets,leav-inganormalmetalphaseinbetween.IntheleftmostpocketnearU/W=0theanomalousself-energyhasaBCS-likeexponentialdependenceonλandindeedsu-perconductivityinthiscornerisBCS.Superconductivityintherightmostpocketnearthemetalinsulatortransi-tionbehavesquitedifferently.Heretheλdependenceofsuperconductivityismuchweaker,andthesuperconduc-tivegapmuchstronger,thanintheBCSpocket.Super-conductivityinthispocketcaninfactbecharacterizedasSCS(Caponeetal.,2002),duetonarrowquasipar-ticlepairingasdescribedabove.AsimilarbehaviortoFig.5,withtwoseparateBCSandSCSregimesemergingfromasingleinitialonewhentheeffectivepairingattrac-tionisprogressivelyweakenedbyexchange,wasderivedandillustratedinasimplertwofolddegeneratemodelinCaponeetal.(2004). TheSCSsuperconductingpocketneartheMotttran-sitionisexpectedtodifferfromtheBCSpocketeveninitsnormalstateproperties.Thenormalstateunderly-ingaBCSsuperconductorisFermi-liquid-like.Ontheotherhand,previousanalysissuggestthattheFermi-liquidpictureislikelytobreakdowninourmodelwhentheMotttransitionisapproached.ThekeyreasonforthebreakdownoftheFermi-liquidispreciselythat,when 11 Z→0,theattractionbetweenquasiparticlesmusteven-tuallyreachandexceedinmagnitudethequasiparticlebandwidthZW,asituationdifficulttosustain.3Pos-sibledeviationsfromtheFermi-liquidparadigmwereinfactoverlookedinRef.(Caponeetal.,2002)astheyarerelatedtothevery-lowenergybehaviorofthenormalphaseclosetotheMotttransition,notexploredinthatwork.Later,thenon-Fermi-liquidbehaviorwasdiscov-eredinthetwo-orbitalmodelwherethephysicsisverysimilar(Caponeetal.,2004). A.AndersonImpurityWithaRigidBath TheDMFTcalculationsdescribedaboveinvolvedtwosteps,onesolvingtheAndersonimpuritymodel(AIM),theothermakingthatselfconsistentwiththebath.Fol-lowingareasoningproposedbyFabrizioetal.(2003),onemaystartoffwiththefirststepalone,namelyanalyz-ingthebareAIMwithoutimposinganyself-consistencyconstraint.Theconductionbathcanbeassumedtohaveaflatdensityofstates,andthebath-impurityhy-bridizationtobestructureless,asituationwhichavoidsnumericaluncertaintiesandyieldsaccuratelow-energyproperties.ThiskindofanalysisappliedtotheAIM(3)shows(DeLeoandFabrizio,2005)thattwodiffer-entimpurityphasesarestabilizedaccordingtothera-tiobetweentheattractionJ,andtheKondotempera-tureTK(Hewson,1997).BelowthistemperatureandwhenJ=0,thespinofanimpuritycoupledtoaFermiseaisscreenedoutandabsorbedintheconductionsea(Hewson,1997).InthelatticecontextwithinDMFT,theKondoscalemeasuresmetalliccoherenceandcorre-spondstotherenormalizedquasiparticlebandwidthZW.ForfiniteJ=0butsmallerthanTK,Kondoscreen-ingremains,thusstillimplyingaFermi-liquidbehavior − inDMFT.Infullerides,theimpurityrepresentstheC360ion,carryingthreeorbitalsandthreespins.IntheKondophaseeachofthethreespinsisseparatelyscreenedbythebathandthusincorporatedintheFermisea.Conversely,whenJ>TKtheKondoscreeningislost,andthatwasshowntoimplyanonFermi-liquidphasecharacterizedbyapseudogapinthesingle-particlespectrumandbyseveralothersingularproperties(DeLeoandFabrizio,2005).Averyqualitativedescriptionofthisphaseisthat,unliketheKondophase,twospinsoutofthreepairoffantiferromagneticallyatanygiventime,leavingoutasinglespin1/2availableforKondoscreening.How-ever,sinceorbitaldegeneracyisunbroken,thisresidual anarrowresonanceintheFermi-liquidregion,andtoanequallynarrowspectraldensitydip(the“pseudogap”)inthepseudogapphase(DeLeoandFabrizio,2005).Sinceinthisphasetheimpuritystillcarriesaresidualspin-1/2,thereremainsafinitevalueofthespectralfunctionatthechemicalpotential,andthegapisnotcomplete.ThepseudogapwidensifJisincreased,andthecusp-likedipintheimpurityspectralfunctionsmoothlyturnsintoacusp-likepeak,thevalueatthechemicalpotentialstayingfixedandconstant.ThisbehaviorisshownbythedottedcurveinFig.6correspondingtoaverylargepseudogap(seetopinset),possessingaverytinypeakatthechemicalpotential.ThisindicatestheexistenceofyetanotherenergyscalebesidesT+andT−thatsetsthewidthofthecusppeak. B.AndersonImpuritywithaSelf-ConsistentBathinDMFT TherigidbathAIMbehavioranditscriticalpointbrieflyreviewedaboveprovideaguidetotheDMFTre-sultsoncetheimpurity-bathcouplingisself-consistentlydetermined.Firstofall,sinceTKcoincideswithinDMFTwiththerenormalizedZWwhichinturnvanisheswhenthecontinuousMotttransitionisapproached,theimpu-ritycriticalpointisinevitablymetbeforethemetalinsu-latortransitionasU/Wisincreased,atsomeUcp<∼Uc.Thisentailsseveralimportantconsequences: •ThelowthenormalcontinuousstatemaymetalbeainsulatorFermiliquidtransition,onlyfarsuchbe-asperhapsmaybethecaseinthenon-expandedfullerides.ExpandedcompoundsontheotherhandareexpectedtohaveanonFermi-liquidnormalstate,andeventuallyapseudogap,possiblydevel-opingbeforethefirstordertransitiontotheanti-ferromagneticinsulator. •ThetransitionSCSsuperconductingreflectstheleadingpocketinstabilitynearofthetheMottim-puritycriticalpoint.InotherwordsSCSsuper-conductivityisthewayinwhichthelatticemodelrespondstoimpuritycriticalityandavoidsit.•Theicalpoint,lowerenergyT−=scale0.HerevanishesT+astheonlyenergyscalecontrolling≃rightJatthecrit-thethusmagnituderemainsofthesuperconductingenergygap.AwayfromU=Ucp,T−perconducting=gap0,shouldandthedecreaseamplitudemonotonicallyofthesu-with(T+cut-offsthe−T−)/T+(Caponeetal.,2004),sinceT−localpairinginstability.Thereforethegapshouldbemaximumrightattheimpuritycrit-icalpoint(thetopofthedome). •EvenwelldefinedthoughBogoliubovthenormalquasiparticlesphaseisnon-Fermishouldliquid,existinsidetheSCSsuperconductingpocket. 120.8DDDrude (Normal)superfluid(Super)0.6∆/W0.40.200.40.60.8U/W11.21.4FIG.7Calculatedweightsofthezero-frequencydelta-functioncontributiontotheopticalconductivityforthesu-perconductingphase(superfluidstiffness,squares)andthenormalphase(Drudeweight,circles).Thezero-frequencyanomalousself-energyisalsoplotted.NotethehighervaluesintheSCSsuperconductor,relativetothenonsuperconduct-ingmetalphase. Thelaststatementcomesfromthefactthat,attheim-puritycriticalpoint,superconductivityprovidesanewscreeningchannelwhichhelpsthesystemgetridofthefiniteresidualentropyatthecriticalpoint,thuseliminat-ingnon-Fermiliquidsingularities(DeLeoandFabrizio,2005;Schir`oetal.,2008). Theroleofsuperconductivityasanovelscreeningchannelclosetotheimpuritymodelcriticalpointshouldreflect,inthelatticemodel,intoagainofbandenergy(thetightbinding“kineticenergy”)attheonsetofsu-perconductivity.Owingtoasumruleconnectingkineticenergyandzero-frequencyopticalconductivity(some-timesreferredtoas“Drudeweight”)(Scalapinoetal.,1993)theonsetofSCSsuperconductivityclosetotheMotttransitionshouldleadtoaDrudeweightincrease.ThispredictioniswellborneoutbythefullDMFTso-lutionofourHamiltonian.InFig.7weplottheω=0(d.c.)opticalconductivityofourmodelsuperconductor(whereitcoincideswiththesuperfluidstiffness),definedby(Toschietal.,2005) Ds=−Ekin+χjj(q→0,Ω=0), (16) whereEkinisthekineticenergyandχjjthestaticlimitoftheparamagneticpartoftheelectromagnetickernel χjj= 2 0-0.001-0.002-0.003∆ E -0.004 ∆ Ekinpot-0.0050.40.60.8U/W11.21.4FIG.8Energeticbalance∆Epot=ESpot−EN underlyingsuperconductivity.potisthedifferencebetweenthepoten-tialenergiesofthesuperconductingwhile∆Ekin=ESkin−EN andthenormalsolution, ofthekinisthesamedifferencebetweenthekineticenergiestwosolutions. thisstatemeanttoprovideacartoonoftherealnormalphaseaboveTc.TheDrudeweightisgivenbyEq.(16)and(17)withF(ǫ,ωn)≡0.UponenteringtheSCSdomefromthelowU/Wside,thesuperconductingphaseini-tiallyloseskineticenergyoverthenormalstateasinordinaryBCStheory.However,uponincreasingU/Watandbeyondthedomemaximum,thelossrevertsovertoagain,andindeedmostoftheSCSsuperconductingpocketispredictedtohavealargerweightofthezero-frequencyopticalabsorptionthanthenonsuperconduct-ingstate.Thesamebehaviorisdisplayed(Fig.8)bytheenergybalancebetweenthetwosolutions.Onlyfarbe-lowtheMotttransitionthesuperconductorisstabilizedbyapotentialenergygain,asisthecaseinBCStheory.Inthepseudogapregime,correspondingtotheexpandedfulleridesneartheMotttransition,thestabilizationisassociatedwithakineticenergygain. Asimilarphenomenoniswellknownintheopticalconductivityofhigh-Tccopperoxides(Carboneetal.,2006).Ourcalculationsshowthatanincreaseinthezero-frequencyopticalconductivityorakineticenergygaindonotactuallyexcludeanelectron-phononpairingmechanism,butratherdemonstratesthekeyimportanceofstrongelectroniccorrelations.Thankstopairing,themotionofcarriersinthesuperconductingphaseisfacili-tatedwithrespecttothepseudogapnonsuperconductingmetal.Inthatanomalousmetal–anonFermiliquid–theinteractionconstraintsjamthefreepropagationofquasiparticles,causingakineticenergycost,partlyre-leasedwiththeonsetofsuperconductivity.Itwouldbeofextremeinterestiftheopticalconductivityincreasedemonstratedincupratescouldbeinvestigatedinful-lerides,bothregularandexpanded,sincethatwouldhelpdiscriminatebetweenconventionalBCSandSCS. 13 V.DISCUSSIONOFEXPERIMENTALPROBES Inthefollowingwediscusswhetherandhowthedom-inanceofstrongcorrelationweproposeforthesuper-conductingfulleridescanbereconciledwiththelistofexperimentalfactsgivenearlier,apparentlysupportingaconventionalBCSbehavior.Thefirstquantitywedis-cussisthespecificheatjumpatTc.WithinBCStheory,thespecificheatjump∆CVatTcisapproximately ∆CV 4 AsmentionedinSec.III.A,forthepresentthree-orbitalmodelithasnotbeenpracticaltoincludeinEq.(6)anumberofex-citedstatessufficienttomakethetruncationerrorintheGreen’sfunctionnegligible.ForthedatareportedinFig.9wehaveanaverageerrorof3-4%whichdoesnotallowustodetermineTcwithsufficientaccuracy.Yet,thejumpofthespecificheatrela-tivetoTcturnsouttobealmostindependentonthetruncationerrorandonthedetailsofthecalculations. 604 χ/χ2000.5U/W11.5FIG.10Normal-phaseuniformmagneticsusceptibilityχnormalizedtothenon-interactingvalueχ0.ForU=0,χ≃χ0,thedifferencebeingextremelysmallsinceJ≪W.TheplateaubetweenU/W=0.5and1signalstheeffectivecrossoverfromaFermiliquidwithS=3/2persite,toanonFermiliquidwithS=1/2persite. stantaround4γ0uptotheMotttransition,despiteadiverging1/Z(SeeFig.9).Thisshowsthattheenergyscalethatcontrolsthesuperconductinginstabilityiscon-stantneartheMotttransition,consistentwiththesingle-impuritypredictionthatthisscaleshouldbeidentifiedbyT+,aquantityoforderJ.Thebottomlineconclusionhereisthatanormallysizedspecific-heatjumpdoesnotimplyBCSsuperconductivityinfullerides. Anotherphysicalquantitywhichseeminglypointedto-wardsweakcorrelationsinfulleridesisthemagneticsus-ceptibilitymeasuredinthenormalphase,apparentlycon-sistentwithaweaklycorrelatedaFermiliquidandaStonerenhancementofaboutafactor2to3.Thisargu-mentagainbecomesinconclusiveoncetheFermi-liquidscenarioisabandoned.InFig.10weplotasfunctionofU/WtheuniformmagneticsusceptibilityinthenormalphasecalculatedbyDMFT.AfteraninitialStoneren-hancementatsmallU/W,thesusceptibilityflattensoutandremainsalmostconstantbeforearapidgrowthwhichtakesplacesextremelyclosetotheMotttransition.Intheplateauregionthesusceptibilityenhancementofafactorbetween2and3withrespecttoU=0,surpris-inglyclosetotheexperimentalenhancement,coveringthewholesuperconductingdomain.Physically,theori-ginofthissusceptibilityplateauforincreasingUisquiteinstructive.ItcorrespondstothegradualcrossoverofthemaximumspinavailableateachsitefromS=3/2intheFermiliquidatsmallU Finally,wewishtoaddresssignaturesofthestrongly 14 correlatedscenariowhichweexpectshouldshowupinimportantspectroscopiesincludingthetunnelingI-Vcharacteristicsandangleresolvedphotoemissionspec-troscopy(ARPES).Thisisinitiallyembarrassingontwoaccounts.First,ARPESspectroscopiesarek-vectorre-solved,whereasinDMFTwedonothaveaccesstoanyspatialstructure.Second,tunnelingspectroscopiesareextremelywellresolvednearzerovoltage,whereasourLanczosmethodyieldsamuchpoorerspectralfunctionresolutioninthisregion. Letusaddressingtunnelingfirst.AlthoughFig.6 referstotheimpurityC360− molecule,webelievethatsimilarfeatureswouldremainafterfullDMFTself-consistencyinthenormalphase–ifweonlyhadabet-terlow-frequencynumericalresolutionthanwepresentlyhave.Hence,wesuggestthattunnelingI-Vspectraofexpandedfulleridesbemeasuredandexamined,inor-dertobringouttheexpectedrichstructureofthekindsketchedinFig.6. Next,letusconsiderphotoelectronspectroscopy.Againaccordingtothesingle-impurityanaly-sis(DeLeoandFabrizio,2005),theimaginarypartofthesingle-particleself-energyshouldbefiniteandoforderT+almosteverywhereinthenon-Fermiliquidnor-malphaseaboveTc.ThishasthefollowingimplicationsforARPES: •thet1ufulleridebandsdispersingphotoemissioninthespectrumBrillouinshouldzoneshowwithnonzerobandwidth,governedbytheenergyscaleT+.ThevalueofT+decreaseswithincreasingU/W(increasingexpansion),fromWatU=0to(largerthan)JattheMotttransition;•therethesameshouldorderbeofamagnitudespectralpeakT+broadeningastheappar-ofentk-resolvedbanddispersion.InparticularthebroadeningshouldremainconstantapproachingtheFermisurface–unlikeaFermiliquidphasewherequasi-particlepeaksbecomenarrowerandnarrower. Themomentum-independenceoftheDMFTself-energyimpliesthat,inthisapproximation,thek-modulationoftheelectronicdispersionisassumedtore-mainunaffectedbyinteractions.Withinthisassumptionandwithoutrequiringtoohighfrequency-accuracy,wecancomputeatoyk-resolvedspectralfunctionaccord-ingto A(k,ω)=− 1 ω−εk−Σ(19) DMFT(ω) , whereεkisthenoninteractingdispersionandΣDMFT(ω)istheDMFTself-energycalculatedwithafinitenumberofbaths.Theeffectofthelocalself-energywillbetochangetheeffectivebandwidthandtogiverisetofinitelifetimeeffects,evenifthek-modulationofthedispersionisunrenormalized.ForourBethelattice, εε = -W/2ε= -W/4 = 0PESεε = W/4 = W/2IPES-0,1-0,05ω00,050,1-0,1-0,05ω00,050,1FIG.11SimulatedphotoemissionspectraelaboratedformtheDMFTresultinthenormalphase.TheleftpanelreferstoU/W=1.1(correspondingtounexpandedfullerides),whiletherightpanelisforU/W=1.3(correspondingtoexpandedfullerides). thereisnostraightforwarddefinitionofmomentum,andweremedythatbycomputingA(ε,ω),whichcorrespondstoEq.(19)withεkInFig.11weshow→theoreticalε. ARPESresultsforsomechoicesofεobtainedbyusingtheDMFTself-energyfortemperaturesaboveTc,bothforavalueofU/W=1.1whichliesclosetothemaximumofthesuperconductingdome,butstillonthelesscorrelatedside,correspond-ingtounexpanded(ormoderatelyexpanded)fullerides,andforavaluewhichliesinthedownwardbranchofthedome(U/W=1.3),correspondingtoaveryexpandedfulleride.Wenoteinbothcasestheexistenceofaninco-herentlow-energyfeaturedispersingwithareducedbutnonzeroelectronbandwidthof0.1W.Intheexpandedcasethepseudogapfeatureisclearlypresent. RecentphotoemissionspectraofK3C60(Goldoni,2007)indicateanoveralldispersionbandwidthofabout160meV,aboutaquarterofthebarecalculatedband-widthinthelocal-densityapproximation.Theexper-imentalspectralpeakdoesnotshowtheusualFermi-liquid-likenarrowingonapproachingtheFermilevel,afactwhichisinagreementwithourexpectationforanonFermiliquid(althoughanonexpandedfulleridelikeK3C60probablyliesatthebeginningoftheSCSdome,wheredeviationsfromFermiliquidarenotmassive).ExperimentallythepeakdoesnotappeartocrosstheFermilevel,andtheintensityinsteaddrops,suggestiveofapseudogap.Unfortunatelythespectrumshowsverystrongvibroniceffects,reflectingtheretardedstrongelec-tronJahnTellercoupling.Thisaspectisnotcoveredbyourunretardedapproximation,butitheavilyaffectsthelineshapeandhamperstheextractionofpurelyelec-tronicfeatures.Treatmentofthevibroniceffects,andaquantitativedescriptionofdispersion,willrequireaban-doninginthefutureourapproximationofinfinitelyfastphonondynamics,aswellasapossibleextensiontoclus-terextensionsofDMFTwhichallowfordifferentrenor-malizationsofdifferentmomenta(Hettleretal.,2000). 15 VI.CONCLUSIONS Summarizing,weaddressedtheapparentlycontradic-torypropertiesofexpandedtrivalentfulleridessupercon-ductorsandinsulators–andtosomeextentofthewholefamilyoffullerides–andpresentedatheoreticalscenarioemphasizingtheroleofstrongelectroncorrelations.Thatisespeciallydesignedandappropriateforthemoreex-pandedmembersofthefamily,suchas(NH3)xNaK2C60,Li3C60,Cs3−xKxC60andCs3−xRbxC60,andtherecentlydiscoveredA15Cs3C60thatarenearorpasttheMotttransition. Ourmodelexplainsthedome-shapedincreaseandsub-sequentdecreaseofTcuponexpansionofthelatticespac-inginfullerides;thecoexistenceofmetallicbehaviorandofMottinsulatorfeaturessuchasthelargeNMRspingapinallfullerides,andtheS=1/2spininthein-sulatingstate(identifiedasaMott-Jahn-Tellerinsula-tor).Itexplainswhythes-waveTccanbeashighas40KeventhoughtheCoulombinteractionstrengthisprohibitive,andwhyTcdoesnotautomaticallydecreaseuponincreaseofU/W.Italsoaccountsformorestan-dardobservations,suchasregularspecificheatjumpsandmoderatelyhighspinsusceptibilities,factsthatweresofarconstruedasevidenceforconventionalBCSsuper-conductivity. BesidesthoselistedinthepreviousSection,onecananticipateanumberofadditionalexperimentsthatcouldprovide“smokinggun”evidenceforstronglycorrelatedsuperconductivityinfullerides.ThetunnelingI-Vchar-acteristicsobservable,e.g.,byascanningtunnelingspec-troscopytipshould,inanexpandedfulleride,developthelowenergyfeaturestypicaloftheKondoimpurityspectralfunction.Theisotopeeffectuponcarbonsub-stitutionshouldalsobehaveveryunconventionally,andeventuallygetsmallerasthesuperconductingdomeispassedandtheMotttransitionisapproacheduponex-pansion.Inthisregime,asthequasiparticlebandwidthZWgraduallyfallsbelowthetypicalenergyh¯ωofanincreasingfractionoftheeightHgJahnTellermodes,theassociatedretardationeffectshouldinfactdisap-pear.Theexpandedfulleridesandrelatedmaterials,clearlynotenoughinvestigatedsofar,deserveinourviewthestrongestexperimentalattention.Theycom-bineelementsthatmakethemmembersatlargeofthehightemperaturesuperconductorfamily.TheycombineneighborhoodoftheMotttransitionandpredominanceofstrongelectroncorrelations,withconventionalelementssuchaselectron-phonons-wavepairing,thataretypicalofBCSsystems.OurstudyidentifiesapseudogapandotherfeaturesintheIVtunnelingspectrum,anincreaseofzero-frequencyopticalweightintheopticalresponseofthesuperconductingphase,andtheemergenceoftwoseparateenergyscalesinARPESasthemosturgentex-perimentalundertakingsthatcouldconfirmoffalsifyourclaims. 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