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DirectEvidenceforaMagneticf‐ElectronMediatedCooper PairingMechanismofHeavyFermionSuperconductivity inCeCoIn5 J.VanDykea,F.Masseeb,c,M.P.Allanb,c,d,J.C.Davisb,c,e,f,1,C.Petrovicb,andD.K.Morra,1 a. b. c. d. e. f. DepartmentofPhysics,UniversityofIllinoisatChicago,Chicago,IL60607,USA CMPMSDepartment,BrookhavenNationalLaboratory,Upton,NY11973,USA LASSP,DepartmentofPhysics,CornellUniversity,Ithaca,NY14853,USA DepartmentofPhysics,ETHZurich,CH‐8093Zurich,Switzerland SchoolofPhysicsandAstronomy,UniversityofStAndrews,FifeKY169SS,UK KavliInstituteatCornell,CornellUniversity,Ithaca,NY14853,USA Toidentifythemicroscopicmechanismofheavy‐fermionCooperpairingisan unresolvedchallengeinquantummatterstudies;itmayalsorelatecloselytofinding the pairing mechanism of high temperature superconductivity. Magnetically mediated Cooper pairing has long been the conjectured basis of heavy‐fermion superconductivitybutnodirectverificationofthishypothesiswasachievable.Here, we use a novel approach based on precision measurements of the heavy‐fermion band structure using quasiparticle interference (QPI) imaging, to reveal quantitatively the momentum‐space (k‐space) structure of the f‐electron magnetic interactions of CeCoIn5. Then, bysolving the superconductinggapequationson the twoheavy‐fermionbands , withthesemagneticinteractionsasmediatorsofthe Cooper pairing, we derive a series of quantitative predictions about the superconductivestate.Theagreementfoundbetweenthesediversepredictionsand the measured characteristics of superconducting CeCoIn5, then provides direct evidencethattheheavy‐fermionCooperpairingisindeedmediatedbythef‐electron magnetism. Keywords: heavy fermion superconductivity, quasi‐particle interference, f‐electron mediatedpairingmechanism 1 To whom correspondence may be addressed. E‐mail: [email protected]; [email protected] 1 HeavyFermionsandCooperPairing Superconductivityofheavy‐fermionsisofabidinginterest,bothinitsownright(1‐ 7) and because it could exemplify the unconventional Cooper pairing mechanism of the high temperature superconductors (8‐11). Heavy‐fermion compounds are intermetallics containing magnetic ions in the 4f or 5f electronic state within each unit cell. At high temperatures,eachf‐electronislocalizedatamagneticion(Fig.1A).Atlowtemperatures, interactions between the f‐electron spins (red arrows Fig. 1A) lead to the formation of a narrow but subtly curved f‐electron band near the chemical potential (red curve, Fig. 1B),whileKondoscreeninghybridizesthisbandwiththeconventionalc‐electronband ofthemetal(blackcurve,Fig.1B).Asaresult,twonew‘heavy‐fermion’bands , (Fig.1C) appearwithinafewmeVoftheFermienergy.Thiselectronicstructureiscontrolledbythe hybridization matrix element for inter‐conversion of conduction c‐electrons to f‐ electronsandvice‐versa,suchthat , 2 2 1 The momentum structure of the narrow bands of hybridized electronic states (Eq. 1, Fig 1C, blue curves at left) near the Fermi surface then directly reflect the form of magnetic interactionsencodedwithinthe‘parent’f‐electronband .Itistheseinteractionsthatare conjecturedtodrivetheCooperpairing(1‐5)andthustheopeningupofasuperconducting energygap(Fig.1C,yellowcurvesatright). TheoreticalstudiesofthemicroscopicmechanismsustainingtheCooperpairingof such heavy‐fermions typically consider the electronic fluid as a Fermi liquid but with strong antiferromagnetic spin‐fluctuations derived from the f‐electron magnetism. Moreover,ithasbeenlonghypothesizedthatitisthesespin‐fluctuationsthatgeneratethe attractiveCooperpairinginteractioninheavyfermionmaterials,specifically,inthe ‐wave channel(1‐7).Why,inthedecadessinceheavy‐fermionsuperconductivitywasdiscovered 2 (12), has this been so extremely difficult to prove? The crux is that identification of the pairing mechanism in a heavy‐fermion compound requires two specific pieces of information: the heavy band structures superconductinggapsΔ , , and the ‐space structure of the (whichencodetheessentialsofthepairingprocess).However, determination of the characteristic heavy‐fermion band structure requires precision , measurement of both above and below the Fermi energy (Fig. 1B) making it problematicforangleresolvedphotoemission.Moreover,duetotheextremeflatnessofthe heavybands| , / | → 0andamaximumsuperconductinggapoftypicallyonlyafew hundred eV, no techniques existed with sufficient combined energy resolution 100 eV and ‐space resolution to directly measure , and Δ , for any heavy‐ fermionsuperconductor.UnambiguousidentificationoftheCooperpairingmechanismhas thereforeprovenimpossible. Heavy‐FermionQuasiparticleInterference:ExperimentandTheory Very recently, however, this situation has changed (13‐15). Heavy‐fermion Bogoliubov quasiparticle interference (BQPI) imaging implemented with 75 eV at 250mK allowed detailed measurements of the ‐space energy gap structure Δ , for the archetypical heavy‐fermion superconductor CeCoIn5 (Ref.16). Its normal state properties are somewhat unconventional (17,18) and seem to reflect the presence of strongantiferromagneticspinfluctuations(19)(whetherthesefluctuationsareassociated with the existence of a true quantum critical point (20‐22) is presently unclear). The compound is electronically quasi two‐dimensional (23) and its 2.3K is among the highest of the heavy‐fermion superconductors (16). The Cooper pairs are spin singlets (24,25) andtherefore Δ , mustexhibitevenparity.Applicationoftherecentlydeveloped heavy‐fermionquasiparticleinterferenceimagingtechnique(13,15)toCeCoIn5revealsthe expecteddevelopmentwithfallingtemperatureoftheheavybands(26)inagreementwith angle resolved photoemission (27,28). Evidence for a spin fluctuation driven pairing mechanism is adduced by comparing the change in the magnetic exchange energy to the condensationenergy(29).Atlowertemperatures,thereisclearevidencethatCeCoIn5isan 3 unconventional superconductor with a nodal energy gap (30‐33) possibly with order parameter symmetry (15,34) (although this has not been verified directly using a phase‐sensitivemethod(35)).ThemicroscopicCooperpairingmechanismofCeCoIn5has, however,notbeenestablished(7,16,36,37). HeavyquasiparticleinterferenceimagingandBQPIstudiesofCeCoIn5at cannowyieldaccurateknowledgeofthek‐spacestructureofboth , 250mK, andΔ , (Ref.15) Using a dilution refrigerator based SI‐STM system operating down to an electron temperature of 75mK, we image the differential conductance resolutionandregister,andthendetermine , , with atomic ,thesquare‐rootofthepowerspectral densityFouriertransformofeachimage.Recently,itwasdemonstratedthatthisapproach can be used to identify elements of heavy‐fermion k‐space electronic structure (13,38) because elastic scattering of electrons from to generates density‐of‐states interferencepatternsoccurringasmaximaat 2 in , . TheonsetoftheheavybandsinCeCoIn5isthendetected(15)asasuddentransformation of the slowly changing structure of , which appears at at E≈‐4 meV, followed by a rapidevolutionofthemaximumintensityfeaturestowardsasmaller|q|‐radius,thenbyan abruptjumptoalarger|q|‐radius,andthenbyasecondrapiddiminutionofinterference pattern|q|‐radii.Thusheavy‐fermionQPIrevealsdirectlythemomentumstructureoftwo heavy bands in the energy range 4meV 12meV and the formation of a hybridization gap. Figures 2A,B are typical examples of our CeCoIn5 heavy‐fermion QPI data,showingthecomparisonofmeasured , our precision model for the , andthepredicted , derivedfrom (SI Section 1.2). The Fermi surface deduced from these measurementsisshowninFig.2C,andconsistsofasmallhole‐likeFermisurfacearising fromtheheavy ‐bandandtwolargerelectron‐likeFermisurfaces , resultingfrom theheavy ‐band(15)(seeFig.S1).Equation1showshowtheprecisedispersions thetwoheavy‐fermionbandsarefixedby interaction energy ∙ , of ,whichisitselfgeneratedbytheHeisenberg between spins and at f‐electron sites and (Ref. 39) (Eq. S1, SI Section 1). Therefore, the real space ( ‐space) form of the magnetic interaction potential, , can be determined directly from the measured , (see 4 Eq.S2bSISection1).Carryingoutthisprocedurerevealsquantitativelytheformof for the f‐electron magnetic interactions of CeCoIn5 (Fig. 2D) and therefore that strong antiferromagneticinteractionsoccurbetweenadjacentf‐electronmoments(SISection1). SolutionofGapEquationswithMagneticf‐electronInteractionKernel These interactions are hypothesized to give rise to an effective electron pairing potential /2(seeEq.S14,SISection2.1),withtheoppositesignto because antiparallel spins at sites and (for 0) experience an attractive 0 pairing potential; we are assuming spin‐singlet pairing throughout. Fourier transformation of yields putativepairingpotential at 1,0 / ; 0, 1 as shown in Fig. 3A, revealing thereby that the isstronglyrepulsiveat / 1, 1 .Itisthisstrongrepulsionat / 1, 1 andattractive / whichisthe longanticipated(1‐11)requirementforunconventionalCooperpairingtobemediatedby antiferromagneticinteractions.Finally,insertingthishypothesizedpairingpotential , into the coupled superconducting gap equations for both heavy bands of CeCoIn5 yields Δ Δ 2Ω Δ Δ 2Ω tanh Ω 2 tanh Ω 2 Δ 2Ω Δ 2Ω tanh Ω 2 tanh Ω 2 2 HereΩ , , Δ , arethetwopairsofBogoliubovbands, and arethe coherence factors of the heavy fermion hybridization process, and the primed sum runs onlyoverthosemomentumstates whoseenergies energy, , liewithintheinteractioncutoff ,oftheFermienergy.OursolutionstoEq.2(SISection2)predictthatthe and bandsofCeCoIn5possesssuperconductinggapsΔ , ofnodal ‐symmetryasshown in Fig. 3B,C. This symmetry is dictated by the large repulsive pairing potential near 1, 1 / (see arrow in Fig. 3A) which requires that the superconducting gap 5 changessignbetweenFermisurfacepointsconnectedby ,asshowninFig.3B.Wepredict that the maximum gap value occurs on the ‐Fermi surface, with Δ , being well approximatedby Δ cos cos Δ Δ cos 2 cos Here Δ 0.49meV, predictionsfortheΔ 0.08, Δ 1.04meV represent the quantitative 600 eV.ThesolutionofEq.2 (without any further adjustable parameters) then also predicts of 0.66meV inEq.3derivedfromthehypothesisofEq.2with constrainingthemaximumpossibleenergygaptobeΔ reducedto cos 3 cos 3 3 0.61, , cos 2 cos 2 2.96K, which is 2.55Konceoneaccountsfortheexperimentallyobservedmeanfreepath 81nm(Eqs.S21inSISection2).Overall,thepredictedgapstructureΔ areinstrikingquantitativeagreementwiththemeasuredΔ , (Eq.3)and (Ref.15)and 2.3K (Ref.16)forCeCoIn5. Theseresultsraiseseveralinterestingquestions.First,while appearsaboveasa phenomenologicalparameter,thequestionnaturallyarisesofhowsuchacrossoverscale arisesfromtheinterplaybetweentheformofthespin‐excitationspectrum,thestrengthof the coupling between f‐electrons and spin‐fluctuations, and the flatness of the f‐electron bands.Toaddressthisquestion,itwillbenecessarytoextendtheabovemethodtoastrong coupling, Eliashberg‐type approach. Second, while our approach assumes that the formation of coherent (screened) Kondo lattice is concluded prior to the onset of superconductivity, it has been suggested (40) that singlet formation might continue into the superconducting state. New theoretical/experimental approaches will be required to determine if this type of composite pairing might be detectable in the temperature dependence of physical properties for ≲ . Notwithstanding these questions, our first focus is now to evaluate the predictive utility of the relatively simple approach that was proposedoriginally1‐4andhasbeenimplementedhere. 6 PhasesensitiveQPI Given this detailed new understanding of Δ , (Eq. 3 and Fig. 3B,C), a variety of othertestablepredictionsforsuperconductingcharacteristicsofCeCoIn5becomepossible. Inparticular,thephaseofthepredicted magnitude of points and , ‐symmetrygapsisdirectlyreflectedinthe . The scattering of Bogoliubov quasiparticles between momentum neartheFermisurfaceleadstoacontributionto , thatis directly proportional to the product Δ Δ . For time‐reversal invariant scalar‐potential scattering, this contribution enters , with a sign that is opposite to that fortime‐reversalviolatingmagneticscattering.Asaresult,changesin , generatedby altering the nature of the scattering potential provide direct information on the relative phase difference between Δ and Δ . This phenomenon has been beautifully demonstrated by considering magnetic field‐induced changes in the conductance ratio in Bi2Sr2CaCu2O8, a single band cuprate superconductor with AlthoughthepredictedsymmetryofΔ , inCeCoIn5issimilarly predictionsformagnetic‐fieldinducedchangesin , ‐symmetry (35). (Fig.3),thedetailed arequitecomplexbecauseofthe multiplesuperconductinggapsandintricatebandgeometry(SISection2.3).InFig.4Awe show our predicted values of the field induced QPI changes Δ , , 0 foratypicalenergy, , , , , 0.5meV,belowthegapmaximum(SISection2.3).For comparison,inFig.4BweshowthemeasuredΔ , , atthesameenergy(SISection 2.3). Not only is the agreement between them as to which regions of ‐space have enhanced or diminished scattering intensity evident, but they also demonstrate that Δ , , is positive (negative) for scattering vectors, connecting parts of the Fermisurfacewiththesamesign(differentsigns)ofthesuperconductinggap(seeFig.4C). Incontrast,theequivalentpredictionsforphasesensitiveΔ , , ifthegapsymmetry is nodal ‐wave, bear little discernible relationship to the experimental data (SI Section 2.3).Thus,theapplicationinCeCoIn5ofthephasesensitiveQPItechnique(35)revealsthe predictedeffectsofsignchangesin symmetrygapsofstructureΔ , . 7 SpinExcitations Lastly, by combining the f‐electron magnetic interactions predicted Δ , (Fig. 2D) and the (Fig. 3), we can investigate the spin dynamics of CeCoIn5 in the superconducting state (SI Section 3). One test for the nodal character of the predicted ‐symmetry superconducting gap is the temperature dependence of the spin‐lattice nuclearrelaxationrate1/ :itstheoreticallypredictedformisshowninFig.4D.Thisisin good agreement with that of the measured 1/ reproduced in Fig. 4E from Ref. 25. The theoretically predicted and experimentally observed power‐law dependence 1/ ~ with 2.5(straightlinesinFig.4Dand4E)showsanexponentthatisreducedfromthe expected 3 for a single‐band ‐wave superconductor. We demonstrate that this effectisactuallyduetofinedetailsoftheΔ , multi‐gapstructureonthe ‐and ‐Fermi surfaces(seeFig.3C).Finally,ourtheoreticallypredicteddynamicspinsusceptibilityinthe superconducting state, using the extracted , and computed Δ adjustable parameters, exhibits a ‘spin resonance’ peak at with no further 1, 1 / and at 0.6meV(SISection3)asshowninFig.4F.Such‘spinresonances’areadirectsignature of an unconventional superconducting order parameter: they arise from spin‐flip transitions that involve states and with opposite signs of the superconducting gap. The experimental data on spin excitations in superconducting CeCoIn5 from inelastic neutron scattering (reproduced from Ref. 37) are shown in Fig. 4G, and exhibit a strong resonancelocatedat 0.6meV.Thequantitativeagreementbetweenthepredictedand measuredenergyofthespinresonance,aswellastheformofthespinexcitationspectrum bothbelowandaboveTcisremarkable.Moreover,thevalueof usedinEq.2isnowseen tobequiteconsistentwiththeenergyscaleofthespinfluctuationspectrum(SISectionS4). Conclusions Tosummarize:the ‐spaceand ‐spacestructureofmagneticinteractionsbetween f‐electrons, and , in the heavy‐fermion state of CeCoIn5 are determined quantitatively(Fig.2D)fromourmeasuredheavy‐banddispersions , (SISection1).The coupledsuperconductinggapequations(Eq.2)arethensolvedusingthehypothesisthatit is these magnetic interactions that mediate Cooper pairing with /2. This 8 allows a series of quantitative predictions regarding the physical properties of the superconductingstateofCeCoIn5.Theseincludethesuperconductivecriticaltemperature , the ‐space structure of the two energy gaps Δ , (Fig. 3), the phase‐sensitive Bogoliubov QPI signature arising from the predicted symmetry of the superconductinggaps(Fig.4A),thetemperaturedependenceofthespin‐latticerelaxation rate 1/ (Fig. 4D), and the existence and structure of a magnetic spin‐resonance near 1, 1 / (Fig. 4F). The demonstrated quantitative agreement between all these predictions (themselves based upon measured input parameters) and the disparate experimental characteristics of superconducting CeCoIn5 (15,16,25,29‐31,34,36,37) provide direct evidence that its Cooper pairing is indeed mediated by the residual f‐ electronmagnetism(Eq.2,3). Acknowledgements We acknowledge and thank M. Aprili, P. Coleman, M. H. Fischer, R. Flint, P. Fulde, M. Hamidian,E.‐A.Kim,D.H.Lee,A.P.Mackenzie,M.R.Norman,J.P.ReidandJ.Thompsonfor helpful discussions and communications. Supported by US DOE under contract number DEAC02‐98CH10886(J.C.D.andC.P.)andunderAwardNo.DE‐FG02‐05ER46225(D.K.M., J.V.). AuthorContributions:J.V.andD.K.M.performedthetheoreticalcalculations,M.P.A.and F.M. performed the measurements and analyzed the data; C.P. synthesized the samples; D.K.MandJ.C.D.supervisedtheinvestigationandwrotethepaperwithcontributionsfrom J.V.,F.M.,M.P.A.,andC.P.Themanuscriptreflectsthecontributionsofallauthors. 9 FIGURE LEGENDS Figure1Effectsoff‐electronMagnetisminaHeavy‐FermionMaterial The magnetic sub‐system of CeCoIn5 consists of almost localized magnetic f‐ A. electrons(redarrows)withaweakhoppingmatrixelementyieldingaverynarrow bandwithstronginteractionsbetweenthef‐electronspins. B. The heavy f‐electron band (1A) is shown schematically in red and the light c‐ electronbandinblack. C. Schematicoftheresultofhybridizingthec‐andf‐electronsinBintonewcomposite electronicstatesreferredtoas‘heavyfermions’(blue).Intherightpartofthepanel, the opening of a superconducting energy gap is shown schematically by the back bending of the bands near the chemical potential. The microscopic interactions drivingCooperpairingofthesestatesandthusofheavy‐fermionsuperconductivity havenotbeenidentifiedunambiguouslyforanyheavyfermioncompound. Figure2Heavy‐FermionBand‐structureDeterminationforCeCoIn5 A. Typicalexampleofmeasured B. Typical example of predicted , withintheheavybands. , for our parameterization of the heavy band structure. They are in good detailed agreement as are the equivalent pairs of measuredandpredicted C. , (SISection1). TheFermisurfaceofourheavybandstructuremodel(SISection1). D. ‐space structure of the magnetic interaction strength, S2b.Thisformof , as obtained from Eq. reflectstheexistenceofstrongantiferromagneticcorrelations betweenadjacentlocalizedmoments. Figure 3 Predicted Gap Structure for CeCoIn5 if f‐electron Magnetism Mediates A. CooperPairing /2,ofCeCoIn5.Thearrow Magneticallymediatedpairingpotential, represents the momentum 1, 1 / where the pairing potential is large andrepulsive. 10 B. Angular dependence of the predicted superconducting gaps Δ bands (note that the angles and , in the and ‐ are measured around , and 0,0 , respectively). The thickness and color of the Fermi surface encode the superconducting energy gap size and sign, respectively. These results were obtainedat 0usingaDebyefrequencyof themomentum 1, 1 / 0.66meV.Thearrowrepresents whichconnectsmomentumstatesontheFermi surfaceswherethesuperconductinggapspossessdifferentsigns. C. PredictedvaluesofthesuperconductingenergygapsasafunctionofFermisurface angle.Thelargestgapislocatedonthe ‐Fermisurface,whereasasmallgapexists onthecentral –Fermisurfaceandtheouter ‐Fermisurface. Figure 4 Comparison of Predicted Phenomenology of CeCoIn5 if f‐electron A MagnetismMediatesCooperPairing,withExperimentalData Predicted phase sensitive quasi‐particle interference (PQPI) scattering pattern for the predicted Δ 0.5meV.Δ , with , , symmetry: Δ , , , , , , 0 for isnegative(red)forscatteringvectorsconnectingparts of the Fermi surface with opposite signs in the order parameter, such as panelC),whilesign‐preservingscatteringleadstoapositiveΔ as , , (see (blue)such (seepanelC).SeeSIsection2.3forfulldetails. B Measured PQPI Δ , , and , , , , , , 0 for 0.5meV. The , , 0 are measured in the identical field of view using identical measurement parameters at enhancementandsuppressionfor 0 and , 3 . Δ , , exhibits the same asinA.Thegoodcorrespondence,especially for the relevant scattering vectors between regions whose gaps are predicted to haveoppositesigns,betweentheoreticallypredictedΔ , , andmeasurements thereofisaphasesensitiveverificationofa ‐wavegapsymmetryinCeCoIn5.SeeSI section2.3foradditionalenergiesanddetails. 11 C Equalenergycontour(EEC)for pointswith Ω , 0.5meVusedinpanelsA,B(i.e.,momentum )inthesuperconductingstate.Scatteringprocessesyielding thedominantcontributionto , , ,with connectingpointsontheEEC withoppositephases(thesamephase)ofthesuperconductinggapareshown(the phases are indicated by +/‐). For simplicity, we show ′ Umklapp‐vector to 2π, 2π , the . Note that the coordinate system is rotated by 45o with respecttopanelsAandB. D Predictedpowerlawfortemperaturedependenceofnuclearrelaxationrate1/ for thesuperconductingstateofCeCoIn5baseduponcombiningthef‐electronmagnetic interactions E andthepredictedΔ , . Measuredtemperaturedependenceof1/ forthesuperconductingstateofCeCoIn5 takenfromRef.25. F Predicted imaginary part of the dynamical spin susceptibility 1, 1 / ispredictedbelow at andthepredictedΔ , .Astrongresonance 0.6meV. Measured imaginary part of the dynamical spin susceptibility 1, 1 / at in the superconducting state of CeCoIn5 based upon combining thef‐electronmagneticinteractions G , , at inthesuperconductingstateofCeCoIn5(fromRef.37).Thereisa strongquantitativecorrespondencetothemodelprediction. 12 Figure 1 13 Figure 2 14 Figure 3 15 Figure 4 16 References 1MiyakeK,Schmitt‐RinkS,Varma,CM(1986)Spin‐fluctuation‐mediatedeven‐paritypairing inheavy‐fermionsuperconductors.Phys.Rev.B34(9):6554‐6556. 2 Beal‐Monod MT, Bourbonnais C, Emery VJ (1986) Possible superconductivity in nearly antiferromagneticitinerantfermionsystems.Phys.Rev.B34(11):7716. 3 Scalapino DJ, Loh E, Hirsch JE (1986) d‐wave pairing near a spin‐density‐wave instability. 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Phys.Rev.B86(3):035129 39SenthilT,VojtaM,SachdevS(2004)Weakmagnetismandnon‐Fermiliquidsnearheavy‐ fermioncriticalpoints.Phys.Rev.B69(3):035111. 40 Flint R, Nevidomskyy AH, Coleman P (2011) Composite Pairing in a mixed‐valent two‐ channelAndersonmodel.Phys.Rev.B84(6):064514. 19 SupplementaryInformation: DirectEvidenceforaMagneticf‐ElectronMediatedCooper PairingMechanismofHeavyFermionSuperconductivityin CeCoIn5 J.VanDyke,F.Massee,M.P.Allan,C.Petrovic,J.C.Davis,andD.K.Morr 0.Outline Before describing the theoretical elements of our approach in the following sections, we briefly summarize it here. Our microscopic model for the description of heavy fermion materials in the normal (hybridized) state is described in Sec.1.1. The hybridized band structureofCeCoIn5inthenormalstateisextractedfromscanningtunnelingmicroscopy (STM)dataviaheavyquasi‐particleinterferencescattering,asdescribedinSec.1.2[seeEq. S8andFig.S1].Since,withinthemodel,thedispersionoftheheavybandisdirectlylinked to the strength of the f‐electron magnetic interactions in real space, , [see Eq. S2b], we can determine the latter from the measured hybridized band structure [see Eq. S13]. Starting from the proposal that these magnetic interactions act as the superconducting pairing interaction, we solve the multi‐band BCS gap equations [see Eq. S20] and predict ‐symmetry (see Sec.2.1). the multiband superconducting gap structure, all with a Using the predicted SC gap structure, we then compute the Bogoliubov quasi particle interference (BQPI) pattern [see Sec.2.2, Eq. S25] , as well as the phase‐sensitive quasi particle interference (PQPI) pattern [see Sec.2.3, Eq. S32]. Finally, using the magnetic interactionsaswellasthepredictedsuperconductinggap,wepredicttheemergenceofa resonance peak in superconducting state [see Sec. S3, Eq. S45], and the temperature dependenceofthespinlatticerelaxationrate,1/ [seeEq.S47].Finallyweshowthatallof these predictions from our theory are in good quantitative agreement with the experimentaldata,asdiscussedinmoredetailbelow. 1.HeavyFermionsintheNormalState 1.1TheHamiltonian,Mean‐FieldEquationsandEnergyDispersions To understand the heavy‐fermion electronic structure of CeCoIn5, we use a model based upontheperiodicAndersonlatticeintheinfiniteUlimit.Tothisend,weemploytheslave bosonapproach(1‐4)forwhichtheHamiltonianisgivenby , . . , , , , , ∙ , . S1 , Here † † , , createsanelectronwithspin andmomentum inthelightconduction ‐ band (magnetic ‐band) whose two‐dimensional (2D) dispersion is . , is the hybridization matrix element between site in the ‐band and site ′in the ‐band. The † , are slave‐boson operators, that account for fluctuations between unoccupied and singly occupied f‐electron sites via the constraint∑ , , 1(1‐3). , is the magneticinteractionstrengthinthe ‐bandbetweenmomentlocations , ′,anddescribed bythe 1/2spinoperator (here,thesumrunsoveralldifferentpairs , ′).Inapath integralapproach,oneemploysafermionicSU(2)representationofthespinoperatorvia ∑ where , , , is a vector of Pauli‐matrices (3,5‐7). The magneticinteractiontermisthendecoupledusingaHubbard‐Stratonovichfield, and the constraint is enforced by means of a Lagrange multiplier approximation,onereplaces, static expectation value , bytheir(real)expectationvalue ,and , ′, , . In the static , ′, byits , ′ . Minimizing the effective action then leads to the self‐ consistentequations(8) , ′ , ′ , n ω ImG , ′, ω S2a , n ω ImG , ′, ω S2b n ω ImG where , ′ 1 , and , , / , , ω S2c E 0 . The effective hybridization representsthescreeningofamagneticmomentwhile , ′ describes the antiferromagnetic correlations between moments. For a translationally invariant system, , ′ ′ , ′ . Here, ′ are , ′ ′ ,and , theeffectivehoppingmatrixelementsfor ‐electronshoppingbetween( ‐electron)sites and ′givingrisetoadispersion, ,ofthe‘parent’ ‐electronband[seediscussionbelow Eq. S11]. In momentum space the renormalized quasi‐particle Greens functions and , ,describingthehybridizationprocess,aregivenby , , , , S3a Γ Γ Γ Γ 1 1 , , Γ S3b Γ S3c withΓbeingtheinverselifetimeofthequasi‐particlestates,andheavyfermioncoherence factors , givenby 2 1 1 2 , S4a 2 S4b 2 2 and,mostimportantly, , 2 2 S4c are the renormalized band dispersions reflecting the hybridization between the ‐ and ‐ bands(thisisEq.1ofthemaintext). Equivalenttothestaticapproximationisamean‐fielddecouplingontheHamiltonianlevel (with the same mean‐field equations as given above) leading to the effective mean‐field Hamiltonian c , c , , , , c , . . S5 , , in which the conventional metallic ‐band states are hybridized with the magnetic ‐ electron states via an effective hybridization matrix element . Diagonalizing the mean‐ fieldHamiltonianusingtheunitarytransformation , , , with , , , S6a S6b , givenabove,thenyields , , S7 , , , 1.2 Heavy Quasiparticle Interference (HQPI) in the Normal State of CeCoIn5 In the presence of impurities, the STM tunneling conductance / varies spatially, and theQPIspectrumcanbemeasuredbytakingthesquarerootofthepowerspectraldensity, , ,of / , into ‐space. Here, , is the magnitude of the conventional Fouriertransformof / , into ‐space,thelatterofwhichwedenoteby ̅ , ,i.e., | ̅ , |. In the Born approximation, we then obtain for each of the spin , projections ̅ , ≡ , ̂ , ̂ S8a , , 1 Im 2 , , S8b where S9a 0 ̂ S9b 0 and , , , , , , , , , S10 Here, isthedensityofstatesintheSTMtip,and, and aretheamplitudesfor electron tunneling from the STM tip into the light and heavy bands (10), respectively. Moreover, and arethescatteringpotentialforintrabandscatteringinthe ‐ and ‐electron bands, respectively, while is the scattering potential forinterbandscatteringbetweenthe ‐and ‐electronbands.Forthetheoreticalresultsof , shown in the main text and below, we used / 0.05, ⁄ 0.15and ⁄ 0.16. ThepresentbestfitofthetheoreticalheavyQPIspectrumEq.S8ausingthebandstructure ofEq.S4c,totheexperimental , dataasshowninFig.S1thenyields ‐electronand ‐ electronbandstructuresseparately 2 1 cos cos 4 2 cos cos 2 3 cos 2 cos 2 S11a 2 1 cos cos 4 with 0.85meV , 2 cos 50.0meV , cos 2 4 3 cos 2 cos 2 cos 2 13.36meV , 0.35meV , cos 2 0.8meV , 2 cos 3 cos 3 S11b 16.73meV , 151.51meV , 0.1meV , 0.09meV , 0.5meV.Here, isrelatedtothehoppingmatrixelement as follows: ′ ′ 1, 1 , ′ 2, 2 , and for ′ ′ for ′ 1,0 ′ for 2,0 ′ ′ introducedinEq.S2 or 0, 1 or 0, 2 3,0 , , or 0, 3 ′ for ′ . Thus, for , , etc.arethematrixelementsfornearest‐neighbor,next‐nearest‐neighbor,etc.hopping. Moreover,to achievethis good fit we find that thek‐space structure of the hybridization processisgivenby sin sin S12 with 3meV, and 7meV. In Figs. S1(a) and (b), we present a comparison of the experimentally measured and theoretically computed HQPI dispersions. We note that for this set of parameters, the ‐electron occupation per site 0.85which is in very goodagreementwiththatobtainedinrecentx‐rayabsorptionnear‐edgestructurestudies (9). Fig.S1.Comparisonbetweenthedispersionsofmaximaintheexperimentaldataandthatinthe theoreticallycomputed , forthe(0,1)(a)and(1,1)(b)directions.Theblackdotsmarkthe positionsofmaximaextractedfrommeasured , layers;thesolidlinesmarkthepositions ofmaximaextractedfromtheoreticallycomputed , cuts;thedashedlinesaretheexpected HQPI dispersions arising from2 . ‐scattering (see Ref.10). (c), (d) , along 0and To obtain these fits for , , and , we first note that for energies sufficiently removed from the Fermi energy, one has , and the QPI spectrum is thus determined predominantlybythelightband.FittingtheexperimentalQPIspectrumfortheseenergies first thus allows us to obtain . For energies close to the Fermi energy, the available experimental QPI data provide a sufficient number of constraints to independently determine both and through a comparison with the theoretically predicted QPI spectrum for the resulting and , . In Figs. S1(c) and (d), we also present , along 0 ,respectively,wherewehaveindicatedthesingleFermisurfacecrossingofthe ‐band,andthetwoFermisurfacecrossingsofthe ‐band,denotedby and . Within the context of Eq. S2b the dispersion of the heavy ‐band is directly linked to the strength of the magnetic interaction, , . Next we use the quantitative results in Eq. S11 andEq.S12toobtainthenormalstateGreensfunctionsinEq.S3andtosubsequentlysolve Eq. S2b for the magnetic interactions, , in real space. By using this 6.44meV for 1,0 or 0, 1 , approach, we obtain 20.30meV for 1, 1 , 6.04meV for 2,0 or 0, 2 , 9.65meVfor 2, 2 , and 2.58meVfor 3,0 or 0, 3 . Here, a positive (negative) value of represents antiferromagnetic (ferromagnetic) coupling between spins is shown in Fig. 2D of the main located at , . The resulting real space structure of text,leadingtoantiferromagneticcorrelationsbetweenadjacentlocalizedmoments The momentum space structure of f‐electron magnetism in CeCoIn5, i.e., the Fourier transformof ,isthengivenby 2 cos cos 4 cos 2 4 cos cos 2 cos 2 cos 3 2 cos 2 cos 3 cos 2 S13 Ourobjectiveistodetermineifthismagneticinteractionbecomesthepairinginteractionin the heavy fermion superconducting state. Since the STM experiments from which we extractedthedispersionswereperformedattemperaturesbelow ,thisapproach[solving Eq.S2bfor usingthenormalstateGreensfunctionsofEq.S3b]neglectsthefeedback effect of superconductivity on the magnetic interaction, . While the inclusion of this , feedback effect in the calculation of and of the superconducting gaps,Δ , is computationally very demanding, its physical consequences are minor, as discussed in SectionS2. 2.HeavyFermionsintheSuperconductingStateofCeCoIn5 2.1PredictingSuperconductingEnergyGapsand Pursuing the idea that it is the magnetic ‐electron interaction potential Eq. S13 that specifically gives rise to pairing in the CeCoIn5 SC state, we next predict the superconducting energy gaps and the value of the critical temperature for this material. Such a quantitatively realistic model of the mechanism of superconductivity valid for a specific heavy fermion compound has only now become possible because of accurate determinationoftheheavyfermionbandstructure(10)inCeCoIn5. While the introduction of the Hubbard‐Stratonovich field , leads to an effective decouplingofthemagneticinteractiontermintheparticle‐holechannel,thedecouplingof thesameinteractiontermintheparticle‐particlechannelleads,intheory,totheemergence of superconductivity. Assuming that the superconducting pairing interaction arises from ∙ in Eq. S1 the spin‐flip component of the ‐electron magnetic interaction∑ , , givenby 1 2 ,↑ ,↓ ,↓ ,↑ S14 , , we apply the unitary transformation, Eq. S6 to and then decouple in the particle‐ particle channel. Here, we neglect superconducting pairing terms of the form 〈 ,↑ ,↓ 〉whicharestronglysuppressedduetothemomentummismatchofthe ‐and ‐ band Fermi surfaces (see Fig. 2 of the main text). The resulting superconducting pairing Hamiltonianthentakestheform Δ ,↓ Δ ,↑ ,↓ ,↑ . . S15 where the primed sum is restricted to all states within the Debye energy of the Fermi energy,i.e.,tothosestateswith , ThesuperconductinggapsΔ , aredeterminedviathegap‐equations 〈 Δ 〈 Δ ,↓ 〉 ,↑ 〈 ,↓ 〉 ,↑ ,↑ 〈 ,↑ ,↓ 〉 ,↓ 〉 S16a S16b /2is the effective where is the number of sites in the system, and superconductingpairingpotential,asshowninFig.3Aofthemaintext. These are the essential equations from which the predictions about the superconducting electronicstructureandCooperpairingmechanismarederived,andaresummarizedasEq. 2 of the main text. The total Hamiltonian (with in Eq. S5) can then be diagonalized using separate Bogoliubov transformations for the ‐ and ‐bands given by ,↑ ,↓ ,↑ S17 ,↓ yielding Ω Ω S18 with Ω , , Δ , S19 Applyingthesameunitarytransformationtothegapequationyields Δ Δ Δ 2Ω Δ 2Ω tanh Ω 2 tanh Ω 2 Δ 2Ω Δ 2Ω tanh Ω 2 S20a tanh Ω 2 S20b Noticethatthecoherencefactors , onther.h.s.ofEq.S20arisefromtheprojectionof the ‐electron pairing interaction (Eq. S14) onto the ‐ and ‐bands. Below, we solve Eq. , S20aandEq.S20btoobtainΔ forallmomentumstateswithin theFermienergy.To accountforfinitelifetimeeffectsofthe ‐and ‐electronstates(forexample,arisingfrom defects,interactionswithphonons,etc.),wewritethegapequationsEqs.S16aandS16bin analternativebutequivalentformgivenby Δ Δ Δ Im 2Ω , tanh Δ , tanh 2 Im 2Ω Im 2Ω , S21a 2 Δ Δ Im 2Ω tanh , tanh 2 S21b 2 where , , , , 1 Ω Γ S22a 1 Ω Γ S22b withΓbeing the inverse lifetime of the quasi‐particle states. We here assume that the microscopicmechanismgivingrisetoanon‐zeroΓdoesnotleadtoarenormalizationof the pairing vertex. This is the case, for example, whenΓarises from scattering of non‐ magneticdefectswithamomentumindependentscatteringpotential,offromscatteringof magneticdefects.Inallothercases,arenormalizationofthepairingvertexispossible,with the strength of this renormalization (and its effect on ) dependent on the microscopic formofthescatteringpotential(11).Finally,wenotethatinthelimitΓ → 0Eq.S21aand Eq. S21b become the standard gap equations Eq. S20a and Eq. S20b. In what follows, we assumethatanymechanismleadingtodecoherenceandthusanon‐zeroΓissuppressed bytheopeningoftheSCgapsuchthatfor 0wesetΓ 0 . Oursolutionsofthegapequations,Eq.S20aandEq.S20b,predictthatthe and bandsof CeCoIn5 possess superconducting gapsΔ , of nodal ‐symmetry as shown in Fig. 3B,Cofthemaintextandthatthemaximumgapvalueoccursonthe Δ , ‐Fermisurface,with beingwellapproximatedby Δ Δ 2 cos cos cos 2 cos 2 cos 3 cos 3 S23a Δ Δ 2 cos cos S23b Here, Δ 0.492meV , quantitative predictions for Δ 0.607 , , 0.082 , Δ obtained for 1.040meV represent the 0.66meV , yielding a maximum 600 eV. Note that a value of 0.66meVis superconducting energy gap of Δ consistentwiththeenergyscaleofthespinfluctuationspectrum(SOMSectionS4). Theemergenceofasuperconductinggapwith ‐symmetry,andinparticularitsnodal structure, can be simply understood by considering the structure of the pairing /2,inrealspace.Forexample,anantiferromagneticinteraction interaction, betweennearestneighbors( 0 andaferromagneticinteractionbetweennext‐nearest 0 ,translatesintoanattractivepairingpotentialalongthebonddirection neighbors,( ( , 0 ,andarepulsivepairingpotential( , 0 alongthediagonaldirectionofthe underlying lattice thus yielding a nodal structure of the superconducting gaps. The emergenceofthe ‐symmetrycanalsobeunderstoodinmomentumspace:thelarge repulsive pairing potential near 1,1 / (see arrow in Fig. 3A) requires, as follows from the BCS gap equation, Eq. S20, that the superconducting gap changes sign between Fermisurfacepointsconnectedby ,i.e.,for asshowninFig.3B. It is worth noting that the maximum superconducting gap on the ‐Fermi surface, Δ |Δ 95μeV is significantly smaller than the maximum superconducting gap | 600μeVwhich occurs on the Fermi surface. The reason for this is twofold. First,thestatesonthe ‐Fermisurfaceconsistprimarilyof ‐electronstates,whichreduces theireffectivecoupling(viathecoherencefactors intheBCSgapequationEq.S20b)to thesuperconductingpairingpotential, ,whichoriginatesfromthemagneticinteraction in the ‐electron band. Second, the magnitude of the pairing potential, , is smaller for smallmomenta,andthewave‐vectorsconnectingstatesonthe ‐Fermisurfacearesmaller than those connecting states on the ‐Fermi surface. Thus, the magnitude of the pairing potential, , is smaller for pairing within the ‐band, leading to a value ofΔ that is significantlysmallerthanthatofΔ .Wenotethatthereisnodirectsignatureofanon‐ zerogaponthe ‐bandinourtunnelingexperimentssinceavalueof Δ 95μeVis to close to the experimental detection limit (12) at 250 mK. Moreover, the phase shift betweenthesuperconductinggapsonthe ‐and ‐Fermisurfacesoriginatesfromthefact thatthepairingpotentialformomenta connectingpointsonthe ‐and ‐Fermisurfaces ispredominantlyrepulsive.ThesolutionoftheBCSgapequationEq.S20thusrequiresthat thesuperconductinggapsatthesepointspossessdifferentsigns. With 0.66meVfixed,wesolvethegapequationforΓ 0 withnofurtheradjustable 2.96K . With increasing Γ parameters, obtaining a critical temperature of (corresponding to a decreasing lifetime of the quasi‐particles) the critical temperature is suppressedasexpectedduetotheensuingdecoherence.ForΓ 0.05meV(corresponding to the experimentally observed (13) mean‐free path of approximately 81nm) we obtainfromEq.S21aandEq.S21bacriticaltemperatureof 2.55Kingoodagreement with the experimentally observed critical temperature 2.3K. Thus, within our approach,weobtain 2Δ 5.46 S24 Δ 0.6meVand 2.55K.Wenotethattheratio2Δ / possesses byusingΔ onlyaweakdependenceon andcanthusbeconsideredageneralresultofourtheory. This weak dependence of2Δ / on (which is in contrast to the conventional BCS result) arises from the curvature of the energy bands near the Fermi surface and the resultingenergydependenceofthedensityofstates. (or the experimentally determined ratio We note that the above ratio of2Δ / 2Δ / 6.05) by itself is not sufficient to determine whether CeCoIn5 is a weak, intermediateorstrongcouplingsuperconductor.Thereasonisthatinmulti‐bandsystems with multiple superconducting gaps, the ratio2Δ / withΔ being the maximum magnitudeofthesuperconductinggapinallbands,cansignificantlyexceedtheresultfora weak‐coupling, single band superconductor (14,15). Here, we recall that for a weak‐ coupling one‐band superconductor with ‐symmetry, one finds2Δ / 4.3, Ref.(16), in contrast to the BCS result of2Δ / 3.53for a single‐band s‐wave superconductor.Eventakingthisratioof4.3asthelowerboundforCeCoIn5wouldimply thatitisatthemostamoderatelycoupledsuperconductor. Furthermore,wenotethatwhiledHvAexperimentshaveobservedboth2Dand3Dbands in CeCoIn5, the combination of theoretical results presented in the main text and the subsequentsectionsandtheirgoodagreementwithexperimentalfindingsclearlysuggests thatthebandsrelevantfortheemergenceofsuperconductivityare2Dinnature. In order to study the feedback effect of superconductivity on the effective pairing interaction /2, we solve Eq. S2b in the superconducting state, using the superconducting Green’s functions given in Eq. S29. However, since the latter depend on thesuperconductinggapthemselves,itisnownecessarytosimultaneouslysolveboththe BCS gap equations, Eq. S20, and Eq. S2b self‐consistently. This calculation has to be repeated for every temperature below and is thus computationally very demanding. However,executingthisapproachfor 0showsthattheresultsforthesuperconducting gaps are basically identical to those obtained without including a feedback of superconductivityon iftheDebyeenergyisslightlydecreased.Thus,toreducethe computational demands of our approach, we have neglected the feedback effect in the calculationofthesuperconductinggaps. Finally, we note that while both the bandwidth Δ , ,arisefromthemagneticinteractions, andΔ of and the superconducting gaps, ,therespectiveenergyscales, 11meV 0.6meVare quite different. The reason for this lies in the functional relationshipsbetween and ononehand,andΔ , ontheotherhand.Toexemplify this, consider the simple case where only 0, while case, one has 8 and for vanishing hybridization, 0for all other terms. In this → 0, one obtains 2 / from Eq.S2b. On the other hand, we find that the scaling between the maximum superconducting gap and the maximum value, of at 1,1 π/ is approximatelygivenby 1 ~exp Δ thusexplainingthedifferenceinthescalesof andΔ . 2.2Bogoliubovquasi‐particleinterference(BQPI)inthe superconductingstateofCeCoIn5 After having obtained the momentum dependence of the superconducting gaps below , we can now predict the form of the quasi‐particle interference spectrum [the so‐called Bogoliubov quasi‐particle interference (BQPI) spectrum] in the superconducting state. In | ̅ , |for each of the spin the Born approximation, the BQPI spectrum , projectionsisgivenby ̅ , , ≡ ̂ , ̂ S25a , 1 , Im 1 , , S25b wherenow 0 0 0 0 0 0 0 0 , S26a 0 ̂ and 0 0 0 0 0 0 0 0 0 0 0 , S26b , , , , , , , , , , , , , Here,thenormalGreen'sfunctions , , , , , , , , , , , , , , , , , S27 〈 , , 0 〉, S28a 〈 ,↑ ,↓ 0 〉, S28b andanomalousGreen'sfunctions , , reflecting both the hybridization between the ‐ and ‐electron bands as well as the emergenceofthesuperconductingorderparameterinthe ‐and ‐bands,aregivenby Γ , , Γ Γ Ω S29a Ω Γ Γ , , Γ Γ Ω Ω Γ S29b , , Γ Γ Γ Ω Γ Ω S29c , Δ Γ Δ Ω Γ Ω S29d , Δ Γ Δ Ω Γ Ω S29e , Δ Γ Δ Ω Γ Ω S29f whereΓ isthelifetimeofthec‐andf‐electronstates.Notethatinthemomentumsumin Eq.S25b,Δ energy. , 0onlyforthosemomentumstateswithintheDebyeenergyoftheFermi Fig.S2.(a‐e)Theoretical , forpurepotentialscattering(nomagneticfield, 0),(f‐ j)experimental , for 0.(k‐o)Theoretical , (in‐field,B≠0)foracombination ofmagneticandpotentialscattering,asdescribedinthetext,(p‐t)experimental , for 1.7 wasusedgivingthebestcorrespondence 0.Forthein‐fieldsimulations, to the experimental data. Modified theoretical (u‐y) and experimental (z1‐z5) PQPI intensityΔ , , ,asdescribedinthetextandFig.S3.Modifiedtheoretical(z6‐z10)PQPI intensityΔ , , for a nodal ‐wave superconductor as described in the text. All theoretical , havebeenplottedinarepeatedzoneschemeusingastructurefactorof the form 1 1 to suppress high‐q scattering generally observedinexperiment.Therighthalfofeachpanelislow‐passfilteredtomimicthefinite experimentalresolution. ThetheoreticalBQPIdataforaseriesofenergiesinsidetheSCgapareshowninFig.S2(a)‐ (e), while the experimental BQPI results are shown in Fig. S2(f)‐(j). The good agreement between the two data sets further supports the theoretically predicted form of the superconductinggaps. 2.3Phase‐sensitiveQuasi‐ParticleInterference(PQPI) Bogoliubov quasi‐particle interference scattering is a powerful tool to extract the gap symmetry of a superconductor, but is not sensitive to the phase of the order parameter. Phase‐sensitive quasi‐particle interference scattering on the other hand can in principle give information about the phase of the order parameter, and thereby enable one to and a nodal ‐wave superconducting gap distinguish between a sign‐changing (17,18).TheunderlyingideaofPQPIisthatthereisadifferentcontributionfrompotential andmagneticscatteringatzeroexternalmagneticfieldandfinitemagneticfieldinaquasi‐ particle interference scattering experiment. Since the magnetic scattering component is sensitive to a sign changing of the order parameter in a different way than the potential scattering component, comparison of data sets taken in a field and in zero field should enable one to extract the sign of the order parameter. Here we will first describe our theoretical expectations (section 2.3.1) based on the BQPI discussed in the previous section, and then compare these to our experimentally measured PQPI spectra (Section 2.3.2). 2.3.1TheoreticalsimulationofPQPI Inthepresenceofamagneticfield,theQPIspectrumiscontrolledbyasuperpositionofa term due to non‐magnetic scattering, ̅ , , which is also present for 0, and a magnetic‐fieldinducedcontribution, ̅ , , .ThelatteriscomputedfromEq.S25by usingamagneticscatteringmatrix 0 0 0 0 0 0 0 0 S30 where and are the magnetic scattering potential for intraband scattering in the ‐ and ‐electron bands, respectively, while is the magneticscatteringpotentialforinterbandscatteringbetweenthe ‐and ‐electronbands. Thus,onehas ̅ , , 0 ̅ , ̅ , , S31a ̅ , , 0 ̅ , S31b Whiletheprecisemagneticfielddependenceof ̅ , , iscurrentlyunknown,agood theoreticaldescriptionoftheexperimentalBQPIdatainamagneticfieldof achieved by setting for simplicity / / and / / 3 canbe , and by choosing 1.7 .AcomparisonofthetheoreticalBQPIlayersinamagneticfieldof | ̅ , , 3T,i.e., , , 0 0 |ofEq.S31a,showninFigs.S2(k)‐(o)withthe experimentalresultsshowninFigs.S2(p)–(t)demonstratesgoodagreementbetweenthe theoreticalandexperimentalresults. Wenextconsiderpredictionsforthephase‐sensitiveQPIspectrumofCeCoIn5,definedas | ̅ , , | | ̅ , , 0 | S32 Δ , , , , , ,0 TodevelopabetterunderstandingforthemicroscopicoriginofΔ , , ,wenotethat both the BQPI spectra for magnetic and non‐magnetic scattering asderivedfrom Eq. S26 can be written as a sum of two contributions: one, ̅ , , involving a combination of , ,involving a normal Green’s functions [Eq. S29a – Eq. S29c] only, and one, ̅ combinationofanomalousGreen’sfunctions[Eq.S29d–Eq.S29f]only.Aswedemonstrate below, ̅ , is a sensitive probe for the phase change of the superconducting order parameter, while ̅ , is not. We note that ̅ , enters the BQPI spectra for magneticandnon‐magneticscatteringwithoppositesign,suchthat ̅ , ̅ wherewehaveused ̅ , , ̅ , / / ̅ , and ̅ , S33a , S33b / / .Wethusobtain | ̅ , , | | ̅ , , 0 | Δ , , | ̅ , ̅ , ̅ , ̅ , | | S34 | ̅ , ̅ , In the above expression, only ̅ , is sensitive to the phase of the superconducting order parameter, while ̅ , is not (see below). The general forms of ̅ , , are rather complex (as follows from Eq. S25 – Eq. S29), since due tothe two‐band electronic structure of CeCoIn5, they contain contributions from intra‐ and inter‐band scattering. However,toexemplifythephase‐sensitivityof ̅ , itissufficienttoconsiderscattering processes involving only one of the bands. We therefore choose an energy , and two scattering vectors , for which the main contribution to ̅ , comes from scattering processesinvolvingmomentumpointsnearthe ‐FS,inwhichcase ̅ , ̅ , Im 1 , Δ Δ Ω Γ Γ Ω S35 Here , contains terms involving the tunneling amplitudes, , , as well as the coherence factors , arising from the hybridization of the light and heavy bands; it therefore does not contain phase sensitive information. In Fig. 4C of the main text we present the equal energy contour (EEC) for scattering processes involving SinceΔ possesses a , Ω 0.5meVtogether wth the that yield the dominant contribution to ̅ , , . ‐wave symmetry, scattering vector connects momentum points on the EEC whereΔ possesses different phases, while connects momentum pointswhereΔ possessesthesamephase.Asaresult,thesignof ̅ , ̅ , is differentfrom that of ̅ , , reflecting the relative changeof phase ofΔ along the , yieldsthat ̅ Fermisurface.Applyingthesameargumentto ̅ , possessesthe samesignas ̅ , andisthusnotsensitivetoaphasechangeofΔ . Based on the above discussion, it is clear thatΔ , , in Eq. S34 contains in general both phase‐sensitive and phase‐insensitive contributions, making it experimentally challenging to unambiguously identify a phase change of the superconducting order parameter. However, for 2 , one finds that ̅ , , and ̅ , , 0 in Eq. S34possessoppositesigns(notethat ̅ isreal),andoneobtains(withζ sgn ̅ , , ) Δ , , ζ ̅ , ̅ , ̅ , ̅ , ̅ , ̅ , ζ 2 2 ̅ , ̅ , 2ζ ̅ , S36 inwhichcasethePQPIspectrum,Δ , , ,isindeedasensitiveprobeforthephaseof thesuperconductinggaps. Acomparisonofthetheoreticallypredicted,Figs.S2(u)–(y),andexperimentallyobserved, Figs.S2(z1)–(z5),PQPIlayersfor 1.7 ,areinexcellentagreement.Asthisratio of / issufficientlycloseto 2,thePQPIspectradirectlyreflectthephaseoftheorder parameter,whichforCeCoIn5isthus .ThisisdemonstratedinFig.4AandBofthe main text where we present the theoretical and experimentalΔ , 0.5meV, B : Δ , 0.5meV, B 0(red) for the scattering vector connecting momentum pointsnearthe ‐FSwhereΔ possessesdifferentphases,whileΔ , 0.5meV 0(blue)forthescatteringvector connectingmomentumpointsnearthe ‐FSwhereΔ possessesthesamephase. A further test of the phase sensitivity ofΔ , , is by contrasting its form with that obtained for a nodal ‐wave superconductor in which the phase of the superconducting orderparameterisuniform.ThisimpliesthatinthecalculationofΔ , , ,wereplace Δ , by its magnitude Δ , in Eq. S25 – Eq. S29. The results for the same set of parametersasbefore[i.e.,thoseusedinFigs.S2(u)–(y)]areshowninFigs.S2(z6)–(z10). These results show a structure ofΔ , , that is qualitatively different from the experimental data shown in Figs. S2(z1) – (z5), clearly demonstrating the presence of a phase‐dependentsuperconductinggap.Thistogetherwiththenodalstructurenecessaryto explain the BQPI data, provides strong evidence for a ‐wave symmetry of the superconductingorderparameter. We note in passing that if the ratio / is such that ̅ , , and ̅ , , 0 possessthesamesign,oneobtainsfromEq.S34 Δ , , ζ ̅ , ̅ , ̅ , ̅ , ̅ , ̅ , ζ ̅ , ̅ , ̅ , S37 and the PQPI spectrum thus directly reflects the form of the magnetic BQPI spectrum, whichisnotadirectprobeforthephaseoftheSCgap. 2.3.2ExperimentalstudyofPQPI To study the field dependence of the scattering interference pattern, we image the differentialconductance , atzerofieldandB=3Twithatomicresolutionandregister. Inordertobeabletocomparethezerofieldandin‐fieldmaps,thetwomeasurementshave been taken at the exact same location, using the same tip and scanning parameters. The firststepinanalyzingthedataistouseadriftcorrectionalgorithm(19)toperfectlymatch andalignthemeasurementsase.g.thermaldriftwillbeslightlydifferenthowevercareful the measurements have been performed. Then, , , , the square root of the power spectral density of each image is determined. To increase signal‐to‐noise, we fourfold symmetrize , , , along the principle directions. The drift corrected , , and correspondingfour‐foldsymmetrized , ,areshowninFig.S3forfiveenergieswithin thesuperconductinggap.Toenableeasycomparison,pairsof , , ,and , , ,at identicalEareshownusinganidenticalcolorscale. Then, we subtract the zero field from the in‐field , , to obtainΔ , , . The experimentalΔ , 0.5meV, and its histogram are plotted in Fig. S4(b) and Fig. S4(a), respectively. To clearly visualize the enhanced and suppressed scattering vectors, the histogram has been divided into three colors: enhanced vectors (blue), suppressed vectors (red), and the background (white) to which the bulk of the pixels is attributed, resultinginFig.S4(c).ThepanelsshowninFig.4ofthemaintextandinFig.S2andFig. S4(d)are3‐pixelsboxcaraveragesofthethreevaluedimages. Fig.S3.ExperimentalPQPItakenat250mK(V=‐10meV,I=300pA).Ontheexactsamefield , , have been taken at 0 Tesla (a‐e) and at 3 Tesla (k‐o). The of view (19x19 nm2), correspondingsymmetrized , , areshownin(f‐g)and(p‐t)respectively(right‐top corneris(2π,0)). Fig. S4. Experimental PQPI analysis. The (a) histogram of the (b) difference of the symmetrizedinfieldandzero‐field , , isdividedintothreeparts,wherethebulkof the pixels (i.e. the background) is set to zero (white). The (c) three valued result is (d) boxcaraveragedoverthreepixels. 3. Spin Excitations of CeCoIn5: Spin‐Resonance and the Spin‐Lattice RelaxationRate1/T1 In superconductors with an unconventional symmetry, strong magnetic interactions can lead to the emergence of a magnetic resonance peak in the superconducting state. This peak was observed (20) in CeCoIn5 in the imaginary part of the dynamical spin susceptibility at an energy ofE 0.6meV. To describe the emergence of the magnetic resonance peak in CeCoIn5, we compute the spin susceptibility in the random‐phase approximation(RPA).Thisapproachhaspreviouslybeenemployedtosuccessfullyexplain the existence of a resonance peak in the high‐temperature superconductors (21‐23) and heavyfermionmaterials(24,25). Since the by far largest contribution to the magnetic susceptibility, , arises from the magnetic ‐moments, and sincethe energy and momentum position ofthe resonance are unaffected by contributions from the light ‐bands, we neglect the latter and define in Matsubara ‐spacevia 〈 , ∙ 0 〉 1 1 〈 〈 0 〉 0 〉 〈 0 〉 2 2 ∓ , , , S38 where isthespin‐1/2operatordescribingthemomentsofthe ‐electrons. Theretarded,non‐interactingspinsusceptibilityinthesuperconductingstateforasingle , ,isthengivenby spindegreeoffreedom, , 1 2 , , , 1 , Ω Ω Ω Ω Ω Ω Ω Δ Δ 1 Ω Δ Δ Ω Ω 1 Ω Ω Ω Ω S39 where , if if S40 and 0 .Note that in the momentum sum in Eq. S39,Δ 0only for those momentumstateswithintheDebyeenergyoftheFermienergy. Startingfromabare(static)magneticinteractionbetweenthe ‐momentsinthespin‐flip channel(similartothefullinteractiongiveninEq.S14) 1 1 ̅ ,↑ ,↓ ,↓ ,↑ ,↑ ,↓ ,↓ ,↑ S41 2 , , , , with ̅ /2we obtain within the random phase approximation for the full transversesusceptibility 1 , S42 , ̅ 21 , To obtain , we first note that the magnetic interaction in Eq. S13 is the fully renormalizedinteractionsinceitwasextractedfromtheexperimentalQPIdata.Itcanbe withintherandomphaseapproximationinthenormalstatevia relatedto 2 ̅ ̅ Re , S43 Here, we relate ̅ and ̅ in the normal state since was computed from Eq. S2b using the normal state Greens functions. SinceRe , in the normal state is only weaklyfrequencydependentatsmallfrequencies,asshowninFig.S5afor π, π ,we , Re , 0 , 0 andthusobtainforthebareinteraction setRe ̅ ̅ , 0 S44 , 0 by , 0 in Eq. S44, and thus We note in passing that replacing consideringtherenormalizationoftheinteractioninthesuperconductingstate,hasonlya weakeffectontheformof . , inthenormalstateasafunctionoffrequency, ,at 0.Itexhibits Fig.S5.(a)Re onlyaweakdependenceon .(b)Re , inthenormalstateasafunctionoffrequency, , at 0, as obtained from Eq. S50. Shown are the results for three momenta near , . InsertingEq.S44intoEq.S42thenyields 1 , , 2 ̅ ̅ , , 0 , 0 S45 Theimaginarypartofthedynamicalsusceptibility,Eq.S45,exhibitsamagneticresonance peak at π, π andE 0.6meV, as shown in Fig. 4F of the main text, in very good agreementwiththeexperimentalfindings(20)reproducedinFig.4G. Moreover, we note that the position of the resonance peak, whose position in energy determinedvia ̅ Re , ,0 0 S46 isunaffectedbycontributionstothespinsusceptibilityfromthe ‐electronband,sincethe magneticinteractionoccursonlywithinthe ‐electronband. Finally,thespinlatticerelaxationrateisgivenby 1 1 lim 2 2Im , → 47 where is the hyperfine coupling constant, and the factor 2 on the r.h.s of Eq. S45 reflectsthetwospindegreesoffreedomcontributingtothetotaltransversesusceptibility. Since the microscopic form of is unknown, we will consider a direct hyperfine constantonly,inwhichcase ismomentumindependent,i.e., . Tocomputethetemperaturedependenceof1/ ,wecalculatethetemperaturedependent superconductinggap,Δ itsformincomputing , ,fromsuperconductinggapequation,Eq.S20,andthenuse , , inEq.S39.Toobtain , , inEq.S41,whichenters , thecalculationof1/ inEq.S45,itisnownecessarytocomputethebareinteraction, fortheentiremagneticBrillouin zonefor 0. Moreover,sincewecompute1/ overa largertemperaturerange,andtoavoidanyspuriouseffectsinthecalculationof atsmallmomenta ,wenowrelate ̅ ̅ and , inthesuperconductingstatevia , 0, 0 S48 suchthatthefullspin‐susceptibilityintheentireBrillouinzonefor ̅ , , , 0, 1 , , ̅ 2 , , , 0, isgivenby 0 0 S49 Theresultingtemperaturedependenceof1/ ,isshowninFig.4Dofthemaintext. The temperature dependence of1/ is in general determined by a superposition of contributionsfrominter‐andintra‐bandscattering.Asaresult,onerecoverstheexpected ‐waveresult1/ ~ onlyfor muchsmallerthanthesmallestsuperconductinggap (onthe ‐FS).Intheexperimentallyrelevantintermediatetemperatureregimewhere is larger than (or comparable to) the small superconducting gap on the ‐FS, but smaller than the large gap on the ‐FS, the temperature dependence of1/ is given by a superposition of different power‐laws. Nevertheless, we find that in this regime, the theoreticalresultsfor1/ canbewellfittedbyanapproximatepower‐law1/ ~ with 2.5,asshowninFig.4Dofthemaintext. 4. Debye Energy, Spin Fluctuation Spectrum and Renormalized Interactions While was introduced as a phenomenological parameter in section S2, the question naturally arises of whether it can be related to the energy scale of the spin fluctuation spectrum in the normal state; the latter can be considered the equivalent of the weak‐ coupling inastrongcouplingtheoryofsuperconductivity.Toinvestigatethisquestion, we consider the energy dependence ofIm , for , as shown in Fig. 4F of the maintext.Itexhibitsamaximumaround 0.2meV,andapproximately50%ofitstotal spectral weight is located below 1meV, consistent with an in the range of0.5to 1meV. Inaddition,wecanconsiderthefrequencydependenceofthefullRPAinteraction ,ω 1 2 50 , asobtainedusingthebareinteraction, inEq.S44. In Fig. S5b, we present the frequency dependence of the real part of , ω (as obtained obtained in Sec. S3) for several momenta near , . The from Eq. S50 using interaction decreases on the scale of 0.3 0.5meV, a scale that is also (at least qualitatively)consistentwiththatoftheDebyeenergy, .Thequalitativeconsistencyof wehadpredictedinsectionS2.1,withtheenergyscaleofthespinfluctuations,aswell asthatoftherenormalizedRPAinteractions,mightprovidethefirstcluetoitsmicroscopic origin. A more detailed investigation of , ω , its relation to the Debye energy, and the resulting frequency dependent hopping parameter, , requires the development of a theory that accounts for the dynamical aspects of both the slave boson and the magnetic correlationswhichisbeyondthescopeofthepresentstudy. 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