PlanarTunnelingandAndreevReflec2on:Powerfulprobesofthesuperconduc2ngorderparameterLauraH.Greene

Department of Physics  Center for Emergent Superconduc8vity  Frederick Seitz Materials Research Laboratory  Center for Nanoscale Science and Technology 

University of Illinois at Urbana‐Champaign Urbana, IL 61801 USA lhgreene@illinois.edu

ICMRSummerSchoolonNovelSuperconductorsAugust2–15,2009 UCSB

Outline:

•  Promo:Grandstatement/DoE‐BESreport/newSCs•  Brokensymmetries(gauge,reflec@onand@me‐reversal)•  Tunnelingandorderparameter(OP)symmetry•  Andreevreflec2on(AR)•  TunnelingintoAndreevboundstates:Brokensymmetries

•  PointContactAndreevReflec@onSpectroscopy(PCARS)•  Blonder‐Tinkham‐Klapwijk(BTK)theoryandit’sext.tod‐wave•  Defini@onoftheissues(ARatHFSsandspectroscopyofHFs)•  CeCoIn5andrelatedHFs•  Describedatawitha

‐two‐fluidmodeland‐Fanoresonanceinanenergy‐dependentDoS

Lecture1(TunnelingspectroscopyonHTS):

Lecture2(Andreevreflec2onspectroscopyonHFs):

CollaboratorsWanKyuPark (Illinois)XinLu (Illinois)EricBauer (LANL)JohnL.Sarrao (LANL)JoeD.Thompson (LANL)ZackFisk (UCIrvine)

Acknowledgements:PhilAnderson,DonaldGinsberg*,TonyLegge],V.Lukic,DavidPines,HeikoStalzer,Dozensofundergraduates,NSF,andDoE.*in memory 

GroupSeminar,MondaySeptember15,2008

WanKyuPark

Outline

  TunnelingandAndreevreflec@on

  Blonder‐Tinkham‐Klapwijktheory

  Point‐contactSpectroscopy

  Examples:NbandMgB2

  PCSofheavyfermions:CeCoIn5andrelated)

  PCSofgraphite

Electron Tunneling Spectroscopy (last lecture)

‐4 ‐2 0 2 4 Normalized

 Con

ductan

ce

eV / Δ

 Current

• BiasdependenceoftunnelingconductancedirectlyprobesDOS.⇒ wellestablishedtoprobeSCgap.

• E.L.Wolf,Electron tunneling spectroscopy 

What will happen to an electron with E < Δ if ∃ no tunnel barrier?

• NoQPstatesavailable,nosinglepar@clesareallowedtoenterS.

• WillaNSsystembelessconduc@vethatasingleS?

insulator

e e

specularreflec@on

normalmetal

cf.Ataninterfacewithhugepoten@albarrierthatistransla@onallyinvariantalongthetransversedirec@on,incomingelectronsreflectspecularly.

• No!

Andreev Reflection (I)

• QMsca]eringoffSCpairpoten@alnearN/S

• Par@cle‐holeconversionprocessmul@‐par@cle(AR)vs.singlepar@cle(tunneling)

• Retro‐reflec@onvh = -ve

• WhiletryingtoexplaintherapidincreaseofthermalresistanceofSnintheintermediatestate,Andreevdiscoveredthatanaddi@onalsca]eringmustbeinvolved.A. F. Andreev, Sov. Phys. JETP 19, 1228 (1964)

N. V. Zavaritskii, Sov. Phys. JETP 11, 1207 (1960)

Andreev Reflection (II)

• Conservedquan@@es‐Energy(E)

‐Momentum(hk)(Δ<<EF)

‐Spin(S)

‐Chargeinc.Cooperpairs

• Sub‐gapconductanceisdoubled.• Andreevreflectedholecarriesinforma@ononthephaseofelectronstateandmacroscopicphaseofSC.phasechange=Φ +arccos(ε/Δ)

• Inverseprocess(S⇒ N):ARofaholeor

‐emissionofaCooperpair(“Andreevpairs”):proximityeffect

‐4 ‐2 0 2 4 0

1

2

Cond

uctance

eV/Δ

Current

Andreev Reflection (III)

• Ifametalelectrodehasunequalnumberofspinupandspin‐downelectronsasinferromagnetsorhalfmetals,Andreevreflec@onissuppressed.

• MeasuringconductanceofFM/Sjunc@ongivesinforma@ononspinpolariza@on,P.

Andreev Reflection (IV) • Proximityeffects:S/N,S/N/S(Josephsonjunc@on)

• Subharmonicgapstructure:S/N/S(MAR,KBT&Octavio1982‐3)

• Reentrantbehavior:mesoscopicS/N(vs.conjugatedmirror)

• Reflec@onlesstunneling:S/I/DN(enhancedARprob,ZBCP.)• Andreevboundstates:nodalsurfacesofp‐andd‐waveSC• Andreevinterferometer

• Andreevbilliard• Crossed(ornonlocal)AR• KondoQDcoupledtoSC:interplaybetweenAR&Kondoeffect

Andreevreflec@onisaninteres@ngandfascina@ngphenomenon,havingvariousapplica@onstosuperconduc@ngdevices,ANDNORMALSTATEPROPERTIES!

Various Quasiparticle Reflections at Interfaces

superconductor

eh

Andreevretro‐reflec@on

normalmetal

insulator

e e

specularreflec@on

normalmetal

superconductor

e h

specularAndreevreflec@on

graphene

Beenakker,PRL(2006)

an@ferromagnet

ee

spin‐dependentQ‐reflec@on

normalmetal

kF kF+Q

Bobkovaetal.,PRL(2005)

Assumeµ(x)=µ, V(x)=0,Δ(x)=Δ.

Planewavesolu@ons

Elementary QP Excitations in a SC

Bogoliubov–deGennes(BdG)Equa@ons

FourtypesofQPwavesforagivenE

BCS Quasiparticle: Bogoliubon

Bogoliubon:QP–acoherentcombina@onofanelectron‐likeandhole‐likeexcita@ons

Blonder-Tinkham-Klapwijk Model PRB 25, 4515 (1982), cf. Klapwijk for history, J. Supercond. 17, 593 (2004)

WhatisthefateofanelectronapproachinganN/Sinterface?

      a:Andreevreflec@on      b:Normalreflec@on      c:Transmissionwithoutbranch‐crossing      d:Transmissionwithbranch‐crossing

Fourtrajectoriesarepossible.

Poten@albarrier

Boundary Condition Problem

Reflection and Transmission Probabilities

A =AR,B = NR,C =TMw/branchcrossing(BC),D =TMw/BC

A+B+C+D=1

• A(E)peaksatΔforZ>0.⇒doublepeaksindI/dVvs.Vcurve

• AtE =Δ,A =1,B = C = D =0,independentofZ.

Probabilities (Cont’d)

I-V & dI/dV-V Formulas

How to Calculate BTK Conductance? • Numericalintegra@onusingMATLAB®

• OriginalBTKkernel‐singularpointsoftheintegrand

‐differentformulasforE<Δ andE>Δ.

• Tofitexp.data,usethreefiwngparameters

Z:dimensionlessbarrierstrength(0:metallic,~5:tunnellimit)Δ: energygap(peakposi@on)

Γ: (Dynes)QPlife@mebroadeningfactor,Γ=h/2πτ

• HowtoincorporateΓ?‐replaceE→ E ‐ iΓina, b,&calculateA = aa*, B=bb*

‐verycomplicatedtodothis!• d‐waveBTKKerneldevelopedbyTanaka&Kashiwayagivesthesameresultsfors‐waveSCbutmuchmoreconvenienttouse.

Current vs. Voltage Characteristics (Z-dep.)

Conductance vs. Voltage Characteristics (Z-dep.)

WithincreasingZ,AR→ Tunneling

IsARobservableinheavyfermions?

Quasiparticle Lifetime

a : tunnelingb : relaxa@onc : recombina@on(τR) 

Consider,e.g.,tunnelingprocessesinaN/I/Sjunc@on.

Quasiparticle vs. Thermal Smearing (Large Z)

Z=10,T=0

Smearingduetofinitelife@meoftransferredQP

Z=10,Γ =0

SmearingduetobroadeningofFermifunc@on

Quasiparticle vs. Thermal Smearing (Small Z)

Z=0.308,T=0

Smearingduetofinite“life@me”oftransferredQP

Z=0.35,Γ =0

SmearingduetobroadeningofFermifunc@on

Zero-bias Conductance vs. Temperature (Z-dep.)

• ZBCvs.Temp.w/inc.Z

AR→ Tunneling

• UsefultocharacterizethetypeofN/Sjunc@on

• Couldbeusedtoes@matelocalTc

S. Kashiwaya et al., PRB 53, 2667 (1996)  

ExtendedBTKtheory(tod‐wave)

d‐wave:c‐axisorlobedirec@on d‐wave:nodaldirec@ona=π/4

s‐wave

a=0

BTKModelfors‐wave and  extended to d‐wave.

YET AGAIN: d-wave BTK Model

an2nodaljunc2on

nodaljunc2on

Y.Tanakaetal.(PRL,1995)S.Kashiwayaetal.(PRB,1996)

ABS Tunneling Spectroscopy of High-Tc Cuprates

• ZBCPduetoABSsplitsundermagne@cfield(Dopplershiz).

Further Extensions of BTK Model • MismatchinFermisurfaceparameters

‐Fermivelocity⇒enhanceZeff

‐Effec@vemass,Fermiwavevector⇒renormalizedversionofBTK

‐Fermienergy:breakdownofAndreevapproxima@on(Δ <<EF)

⇒ Imperfectretro‐reflec@on

• Tunnelingconeeffect

What is Point-Contact Spectroscopy (PCS) ?

• Iftwobulkmetalsareincontactwitheachotherandthecontactsizeissmallerthanelectronicmeanfreepaths,quasipar@cleenergygain/lossmostlyoccursattheconstric@on.

• Nolineari@esincurrent‐voltagecharacteris@csreflectenergy‐dependentquasipar@clesca]eringsinthecontactregion.

‐eV/2 +eV/2

Junction Size Matters in PCS!

Contact Size d

Ballistic Diffusive Thermal

Wexler’sformula G.Wexler,Proc.Phys.Soc.London89,927(1966)

I.K.Yanson

Point Contact Techniques

SimilartotunnelingresultsbyMcMillanandRowell(1965)

Sov.Phys.JETP39,506(1974)

Needle‐anviltech.developedbyA.G.M.Jansenet al.

Example (I): Au/Nb Park&Greene,Rev. Sci. Instum.77,023905(2006)

Δ:Energygap

Γ:Quasipar@clesmearing

Z:Tunnelbarrierstrength

Example (II): Au/MgB2

Two types of conductance curves from MgB2/Au

Δ=1.97,6.90meVΓ=0.18,0.01meV

Z=0.47,0.25

ωπ=0.972

Δ=2.43,7.00meVΓ=0.41,0.45meV

Z=2.5,0.9

ωπ=0.90

Park&Greene,Rev. Sci. Instum.77,023905(2006)

 Blonder‐Tinkham‐Klapwijk(BTK)theorycanexplainthetransi@onalbehaviorfromAndreevreflec@ontoTunnelingusingasingleparameter,theeffec@vebarrierstrength(Z).

 BTKandextendedBTKtheoriesprovideausefulframeworktounderstandchargetransportphenomenainvarioustypesofN/Shetero‐structures.

 ToanalyzePCSdataonasuperconductorusingBTKtheory,threefiwngparameters:Z,Δ, Γ

  BTKtheoryhasbeensuccessfullyappliedtoanalyzeourPCSdataforNbandMgB2.

Conclusions for for BTK Model

4f or 5f electrons

1-1-5 Heavy-Fermion Compounds

CeMIn5 CeCoIn5 (Tc=2.3 K, gel=290 mJmol-1K-2)

PuMGa5 PuCoGa5 (Tc=18.5 K, gel=77 mJmol-1K-2)

4f

CeCoIn5 CeRhIn5 CeIrIn5 CeCoIn5

TheHeavyFermionSuperconductorCeCoIn5:PhasediagramofseriesCeMIn5(M=Co,Rh,In)

Pagliuso et al., Phys.Rev.B64(2001)100503(R)

 Anisotropictype‐IISC Heavy‐fermionliquid

meff=83m0T *~45K

 Non‐Fermiliquidρ~T 1.0±0.1,Cen/T~‐lnT, 1/T1T~T –3/4

Theheavy‐fermionSuperconductorCeCoIn5:Someinteres2ngproper2es

 QuantumPhaseTransi2onwithchemicalsubs@tu@on,hydrosta@cpressure,magne@cfield, (similartocuprates)

 FFLOPhaseTransi2on

V. A. Sidorov et al.,  PRL 89, 157004 (2002) 

TheHeavy‐FermionSuperconductorCeCoIn5

C. Petrovic et al., J. Phys.: Condens. Ma`er 13, L337 (2001) 

WhyitisourHFSofchoice(idealforPCS):•     Tc=2.3K(highforHFS)• Superconduc@vityincleanlimit(mfp = 810Å  >>ξ0)

CrystalStructureandFermiSurface:Quasi2‐dimensional

R.Se]aietal.,JPCM13,L627(2001)

BTKmodelhasworkedwellforawiderangeofmaterials,butaswewillsee,NOTforheavy‐fermionsuperconductor/normalmetal(HFS/N)interfaces

Recalleffec@vebarrier

strength:

TheFermivelocitymismatchissogreatattheHFS/NinterfacethatAndreevreflec2on(AR)shouldneveroccur(Z>5expecttheextremetunnelinglimit).

However,ARisrou@nelymeasuredattheN/HFSinterface,albeitsuppressedcomparedtoN/conven@onal‐S.

Andreevreflec2onattheN/HFSinterfacecannotbeexplainedbyexis2ngtheories

Defini@onoftheissues

1.UnderstandingchargetransportacrossHFinterface Exis@ngmodelscannotaccountfor

Andreevreflec@onattheHFS/Ninterface2.  SpectroscopicstudiesofCeCoIn5(OPsymmetry,mechanism,…)

The“Rose]astoneforheavyfermions”

CeCoIn5:Superconduc2ngOrderParameterSymmetry:Previouswork

• Evidencefortheexistenceoflinenodes:Powerlawdep:Cen/T~T,κ~T 3.37,1/T1~T 3+ε,λ~T 1.5

• Four‐foldsymmetryoffield‐angledepinthermalcond.:smallangleneutronsca]ering⇒dx2‐y2specificheat ⇒dxy

• Spectroscopicevidencewaslackingtodeterminetheloca@onsoflinenodes:(110)or(100)i.e.dxyordx2‐y2?

Gold@p‐electrochemicallyetchedCeCoIn5singlecrystal‐(001), (110) and (100)oriented‐etch‐cleanedusingH3PO4

Coarseapproach‐donebeforeinser@ngprobeFineapproach‐doneduringcooldown‐piezodrivenbycomputercontrolOpera@onrange‐Temperature:downto300mK‐Magne@cField:upto12T

OurExperiments:PointContactAndreevReflec@onSpectroscopy(PCARS)1)Can@lever‐Andreev‐Tunneling(CAT)RigW.K.Park,LHG,RSI(06).

BasicsofPCS:Tipproduc2on

OurexperimentsareintheSharvinLimit,andarereproducible.

Forourexperiment(RN=1‐4Ω)andnotT‐dep:

*Upperlimitof2a=46nm*lelatTc=is81nm(fromthermalconduc@vity),

andincreaseswithdecreasingT,to4‐5µmat400mK.

Thesharpgold@piselectrochemicallyetchedinhydrochloricacid

Conductance asymmetry begins at T* and saturates below Tc

(110) (001)

AndreevReflec@onConductanceofAu/CeCoIn5

=>AddingelectronstoCeCoIn5abovetheFermienergyismoredifficultthanremovingthem

(+) CeCoIn5; (-) Au

ConsistencyAlongThreeOrienta@ons ‐Conductancemagnitude (AR) ‐Conductancewidth (Δ) ‐Backgroundasymmetry (2‐fluid&DoSpeak?)

Notetheshapesoftheconductancecurves

•  Compare shapes for data with calculation. •  Flat vs. cusp-like •  Z is always finite. •  Nodes are along (110) direction.

WKP et al., PRL 100, 177001 (2008)

kx

ky

Spectroscopic Evidence for dx2-y2 Symmetry

SC surface

Andreev Bound States (ABS)

Exp. Data

Calc.

BackgroundConductanceAsymmetryofAu/CeCoIn5

T*

Tc

Backgrounddevelopsanasymmetry*attheheavy‐fermionliquidcoherencetemperature,T*~ 45K.

Thisasymmetrygraduallyincreaseswithdecreasingtemperatureun@ltheonsetofsuperconduc@ngcoherence,Tc =2.3K.

*el‐hasymmetrydescribedbyNakatsuji,Pines&Fisk,PRL92,016401(2004)

• Co‐&Ir‐115:qualita2velysimilar

• Rh‐115:addi@onalstructureduetoAFM(also,inCd‐dopedCo‐115)

SC SC

non‐SC

BackgroundConductanceAsymmetryofAu/CeMIn5

Why is the conductance asymmetric?

RelevanceofProposedModels

• Compe@ngorder(Hu&Seo,PRB2006)

‐DoesnotexplainSTSdataonUD‐Bi2212,norourCeIrIn5data.

• Non‐Fermiliquidbehavior(Shaginyan,Phys.Le].A2005)

‐Asymmetryiss@llseeninfield‐inducedFermiliquidregime.

• LargeSeebeckeffectinHF+thermalregime(Itskovich‐Kulik‐Shekhter,Sov.JLTP1985):asymmetrypersistsinSCstates.

• Energy‐dependentQPsca]ering(Anders&Gloos,PhysicaB1997)‐Explainsbothreducedsignal&asymmetry,butunclearorigins.

• Stronglyenergy‐dependentDOS(Nowack&Klug,LTPhys.1992)

•  Asymmetry is reproducible; conductance is always smaller when HFs are biased positively for the two SC 115s.

Two‐fluidpictureofheavyfermions

T *  Tmin

Petrovicet al.(2001)

Shishidoet al.(2002)

• EmergingheavyfermionsinKondolawcesystemsbelowacoherencetemperature,T *(~45KinCeCoIn5).

• f (T):rela@veweightofheavy‐fermionliquid,increaseswithdecreasingTandsaturatedbelow2K.Nakatsuji,Pines,Fisk,PRL92,016401(2004).

• Thistwo‐fluidpictureappearsvalidinotherheavy‐fermionsystems.Curroet al.,PRB70,235117(2004).

• “Heavyelectronssuperconductbutlightelectronsdon’t.”Tanataret el.,PRL95,067002(2005).

Conductance Asymmetry vs. Two-Fluid Behavior

TN

• AsymmetryfollowsHFspectralweightqualita@vely.

• Satura@onordecreasebelowSCorAFMtransi@on.• NdRhIn5(non‐HFAFM)shownoasymmetry.

T

*?

(001)CeCoIn5

Moresupportfor2‐fluidmodelinCeCoIn5PCARTSforbothN/Sjunc@onsofAu/Nb&CeCoIn5/Nbarecomparable,wherethereisno2‐fluidmodelforSNbsoalltheCooperpairspar@cipateintheAR.

RecallforN/SAu/CeCoIn5isgreatlyreducedandwearguethat“oneofthe2fluidsdoesnotpar@cipateintheAR”

N/S

N=CeCoIn5

Two-channel Model Based on Lorentzian DOS

fiwngparameters   ωh=0.51   σ = 1, Γ = 5 meV   ε0 = -2.1 meV   Ζ = 0.28   ΓDynes = 95 µeV   Δ = 600 µeV  cf. 1-channel BTK fit   ΓDynes = 218 µeV   Δ = 460 µeV

SCheavyel. Non‐SClightel.

WKPet al.,PRL100,177001(2008)

fiwngparameters   ωh=0.51

  ΓDynes = 95 µeV   Δ = 600 µeV  cf. 1-channel BTK fit   ΓDynes = 218 µeV   Δ = 460 µeV

WKPet al.,PRB72,052509(2005)

• Qualityofthefitissensi@vetoωh.

• MuchsmallerΓDynesthanthatobtainedfromone‐channelBTKfit → Fitdoesnotsufferfromunphysicaltemp.dependenceofΓDynes.

• Generalityoftwo‐fluidbehavior(Curroetal.,Yang&Pines)andreducedAR&cond.Asymmetry →OurmodelmaybegenerallyapplicabletootherHFS.

• BTK‐likecalcula@onbasedontwo‐fluidpicture(Araujo&Sacramento,PRB77,134519(2008)):claimbothchannelsshouldbeputimplicitlyintokernel(interference),butnoaccountforasymmetry

High-Temperature Deviation

• DonotfittoaLorentzianbuttoaFanoline‐shape.

Fano Effect in Kondo Lattice? f (e)=(qF+e)2 / (1+e2)

•  Conjecture: Fano interference effect between two conduction channels: heavy-electron band and conduction electron band. •  Fano factor can have negative value (interference), and peak position below Fermi level can mean the Kondo resonance above Fermi level. •  Underlying microscopic picture is being investigated, which should provide valuable insight into the Kondo lattice physics.

Conductance Model based on Fano Formula

F(ε)=(qF+ε)2/(1+ε2),ε≡(E‐E0)/(Γ/2),dI/dV=C∙F(ε)+G0

•  qF=‐2.14,E0=2.23meV,Γ/2=11.13meV,C=0.0061Ω‐1,G0=0.164Ω‐1

• nega@veqFvalue‐interference;posi@veE0‐KondoresonanceaboveEF;largeG0‐largepor@onisnotinvolvedininterference.

• Fanointerferenceeffectbetweentwoconduc@onchannels,intoheavy‐electronbandandconduc@onelectronband.

Fano Resonance

Electron-Helium inelastic scattering Probability ratio for transition to discrete and continuum

(q+e

)2 /

(1+e

2 )

e, relative to eres

Fano / Kondo Resonance in Single Impurities

Co atoms on Au(111)

V. Madhavan et al., Science 280, 567 (1998)

Other groups: Schneider, Eigler, Lieber, Kern, Zhao, Berndt, …

A: coupling to atomic orbital, direct or indirect via virtual transitions involving band electrons

B: coupling to conduction electron continuum

Fano Resonance in Quantum Dots

K. Kobayashi et al., PRL 88, 256806 (2002)

“The Fano effect is essentially a single-impurity problem describing how a localized state embedded in the continuum acquires itinerancy over the system.”

Conclusions

  StrengthofthePCARSmethod ‐Firstspectroscopicdemonstra@onofdx2‐y2symmetryinCeCoIn5‐Densityofstateseffectsmeasured!(energy‐dependentDoS;peak)

  KondoLafceProper2es: ‐Two‐fluidmodel

‐Energy‐dependentDoSgivenbyaFanoresonancepossiblyduetotheinterferenceofthef‐electronswiththeconduc@onelectrons.