Download Direct Evidence for a Magnetic f-Electron

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
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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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