Theory of the Resistive Transition in Overdoped $Tl_2Ba_2CuO_{6+x}$ Implications for the an

TheoretischePhysik,ETHH¨onggerberg,Z¨urich,Switzerland

DepartmentofPhysicsandAstronomyRutgersUniversity,Piscataway,NJ088

3

DepartmentofPhysicsandAstronomy,TheJohnsHopkinsUniversity

3400NorthCharlesSt.,Baltimore,MD212184

LandauInstituteforTheoreticalPhysics,Moscow,Russia

1

V.B.Geshkenbein1,4,L.B.Ioffe2,4andA.J.Millis3

2

arXiv:cond-mat/9801059v1 [cond-mat.supr-con] 7 Jan 1998Weshowthatrecentmeasurementsofthemagneticfielddependenceofthemagnetization,specificheatandresistivityofoverdopedTc∼17KTl2Ba2CuO6+δinthevicinityofthesuperconductingHc2implythatthevortexviscosityisanomalouslysmallandthatthematerialstudiedisinhomo-geneouswithsmall,afewhundred˚A,regionsinwhichthelocalTcismuchhigherthanthebulkTc.TheanomalouslysmallvortexviscositycanbederivedfromamicroscopicmodelinwhichthequasiparticlelifetimevariesdramaticallyaroundtheFermisurface,beingsmalleverywhereexceptalongthezonediagonal(“coldspot”).Weproposeexperimentaltestsofourresults.

Thenatureofthesuperconductingtransitioninhigh-Tcsuperconductorsisthesubjectofpresentcontroversy:someworkersarguethatthetransitionisdrivenbyther-malorquantalfluctuationsofthesuperconductingorderparameter[1,2]implyingthatinawideregimeabovetheresistivelydeterminedTconehaslocalsuperconductingorderwithoutglobalphasecoherence.Insuchasitu-ationonewouldexpectstrongsuperconductingfluctu-ations,i.e.alargeparaconductivity,magnetoresistanceandfluctuationdiamagnetism.Theapparentabsence(orweakness)oftheseeffectsinallbutthemostheavilyun-derdopedcompoundscastsdoubtonthephasefluctua-tionhypothesis[3].

RecentmeasurementsonoverdopedTc0=17KTl2Ba2CuO6+δinamagneticfieldprovideaninterest-ingnewperspectiveonthisissue.ThereisareasonablysharpresistivetransitionwithonsettemperatureTρ(H)[4–6].ApairingonsettemperatureTγ(H)canbedefinedforH<4Tfromspecificheatandmagnetizationmea-surements[7,8].AtH=0,Tρ=Tγ=Tc0,thezerofieldtransitiontemperature,butatH>0itisfoundthatTρ(H)1

inthesampleofsmallregionswithTcmuchhigherthanthebulkTc0.Acloselyrelatedideainvolvingsmallre-gionswithanomalouslylargeHc2wasputforwardinthegeneralcontextofdirtysuperconductorsin[10].Wefirstdiscusstheresistivity,specificheatandmag-netizationdata.AtlowappliedmagneticfieldBthematerialissuperconducting.Theresistivityρ=0andthereisalarge’London’magnetizationgiveninequilib-riumbyMLondon=−Φ0ln(Hc2/B)/32π2λ2.(Realma-terialsarenotinequilibriumbecausethevortexlatticeispinned).AtsomefieldHmelt(T)(visiblee.g.asthefootoftheresistivetransitioninFig2ofref[4])thevor-texlattice(glass)melts.ThemagnetizationisobservedtotaketheLondonvalue[8]andtheresistivitybecomesnon-zero.ForHmelt(T)ThespecificheatexhibitsamaximumwhichbroadensandshiftstolowerTasthefieldisincreased.WedefineHγ(T)fromthetemperatureofthemaximum;itturnsoutthatHγ(T)≈2Hρ(T)inthefieldrange0Apairedstateshouldexhibitalargemagnetizationvanishingatthe’microscopic’Hc2.Theobserved[8]magnetizationisapparentlygivenbythesumofa’Lon-don’termwithaln(1/B)fielddependenceandadiamag-netictermwithalinearBdependence,i.e.M(B,T)=MLondon+χdB.TheLondontermisobservedforarangeoftemperaturesgreaterthanTρ(H),howeverasTap-proachesTγ(H)thesecond,diamagnetictermdominates

themeasuredmagnetizationsothebehavioroftheLon-dontermisdifficulttodetermine.Thedataareconsis-tentwiththeexpectationthattheLondontermvanishesatHγ(T).Thecoefficientofthediamagneticterm,χdhasmagnitudemuchlargerthantheusualLandaudia-magnetismandpersistsoverawiderangeofT>Tγ(H),indeeduptoT∼2Tco.WeinterpretthediamagnetictermasarisingfromthepresenceinthesampleofsmallregionswithtransitiontemperaturesmuchhigherthanthebulksuperconductingTco.

ThescalesHmelt(T),Hρ(T)andHγ(T)areshowninFig1,alongwithashadedareaindicatingtheregionintheH−Tplanewhereananomalouslylargediamagnetictermχdisobserved.Notethatastraight-lineextrapo-lationofHγ(T)toT=0yieldsaHγ(0)≪Hρ(0).TherelativelysmallvalueofHγ(T=0)impliesthatthelow-TupturnofHρisnotabulkpropertyofthematerial.Wewillshowbelowthatitisduetothesameinhomo-geneitieswhichproducethediamagnetism.

1614121082

0.20.4TTc

-1

intheflux-flowregimeistheBardeen-Stephenformula

σBS=

Hc2

H0.60.81Fig1PhasediagramofTc≈15KTl2Ba2CuO6+δasdeter-minedfromresistivity,specificheatandmagnetizationmeasure-ments.Lowestline:Hmelt;intermediatelineHρ,upperlineHγ.Lineardiamagnetismwasobservedintheshadedregionand(al-thoughnotshowninthefigure)persiststoT≈2Tc.

nV

󰀋γ(θ)󰀊θ

(2)

Fromthephasediagramoneseesawideseparationbe-tweenthelocalpairingscaleHγandtheresistivescalesHρandHmelt.ThisimpliesthatthewholetemperaturerangebetweenTmelt(H)andTγ(H)shouldbedescribedasavortexliquid,albeitonewithunconventionaltrans-portproperties.Theconventionalview[11]oftransportinatypeIIsuperconductorinanon-zeromagneticfieldisasfollows.Onehasvortices,thesemoveinresponsetoanappliedcurrentandthismotioncausesthephaseofthesuperconductingorderparametertobecometimedepen-dentleadingviatheJosephsonrelationtoanelectricfieldwhichcausesdissipationandhenceafiniteconductivity,σV.Thetotalconductivity,σ,isthesumofthevortexpart,σV,andanormalone,σnduetouncondensedcar-riers.Thestandardestimateofthevortexconductivity

Here󰀋󰀊θdenotesanaverageoverthe2DFermisurfaceparametrizedbyangleθand

󰀁2∆max

γ(θ)=2ν(θ)∆(θ)2τ(θ)ln

diagonalswherethesuperconductinggaptheusualBoltzmanapproachσ2

vanishes.In

n≈e

τ0−1+Γθ2

󰀃

(4)

withθtheanglemeasuredfromthezonediagonaland

Γτ0the≫scattering1.Thisformularateisthemaysumbederivedoftwobyterms,assumingaconven-thattionalonewhichisduetotheimpuritiesscatteringwhichweparametrizebyτ0−1

orFermiliquid

andunconven-tionalone,Γθ2

,whichislargeeverywhereexceptforthezonediagonals.EvidenceforalargeΓcomesfrompho-toemissionexperiments[16]whichatT>Tcfinddis-persingquasiparticlepeaksformomentanearthezonediagonals(“coldspots”)butfindonlyverybroadinco-herentstructuresforothermomenta.FromthewidthofthesestructuresweestimateΓ>conductivityimpliedbyEq4is∼0.1eV.Normalcarrier

σ2n=e

τ0/Γ.Itisdominatedbyquasiparticlesnearthezonediagonals;becausethesuperconductinggapvanishesthereσndoesnotchangesignificantlyasTisreducedbelowTc.IntheTl2Ba2CuO6+xmaterialsof)implyingthat

󰀄interesthereσnisoforder

(10µΩcm−1

sin22θτ(θ)

σ=

=

ΛHc2n

󰀂

dθB

Γτ0

Γτ0yield

√

≫5.Ourestimates

48πλ2(6)

G

withλGthegrainpenetrationdepth.Thetotalmagne-tizationisthenxGMG.TheexperimentalresultisthatatT=8KandB=10T(muchgreaterthanthemicro-scopicHc2)thediamagneticmagnetizationxGMGequalstheLondonmagnetizationobservedatthesameTandB=0.1T.Fromthis,Eq.(6),theLondonformula[11]andHc2(T=8K)≈2Tweconcludethat

xGR2

=3×10

4λ2G

10T

∼β20,000˚A2withβ>1.

TheJosephsoncouplingbetweentwosuperconductinggrainsinanormalmetalhostdependsonmanydetailsincludingthetemperature,themagneticfield,thesizeofthegrain,theintergraindistanceandthestrengthoftheelectroniccontactbetweenthegrainandthehostmetal[11].Inthedilutelimitd≫RandinzeromagneticfieldtheJosephsonenergyis

EJ(T,φ)=E0Je

−d/ξn

Fd(φ)(8)

Hereξn=vF/(2πT)isthecleanlimitnormalmetal

phasecoherencelength.Thephasedependenceiscon-tainedinthefunctionFd(φ)whichtendstosinφford/ξnTheJosephson≫1butbecomesenergyscaleasawtoothEJ0is

functionatd/ξn≪1.EJ0=Λ2

vF

lowesttemperatures,thegrainswouldorderatatemper-atureTcgsatisfyingTcg=ZEJ(Tcg)(hereZtheeffectivenumberofneighborsofagivengrain,is∼6atT>T0and∼(T0/T)2forT√Z

Φ0

T

e−T/T0

(10)

OurnumericalestimatesimplythatatT=T0BG∼20Λ2NcT.Inviewofthelargeuncertainties∼1Kwe,regardthisestimateasreasonable.Thetemperaturede-pendence(e−T/T0forT>T0)iscertainlyinqualitativeagreementwiththeexperiment.

Tosummarize,Tl2Ba2CuO6+δmaterialexhibitstwoexperimentalanomalies:pronouncedupwardcurvatureofHc2atlowtemperaturesandavortexliquidregimewithanegligibletemperatureandfielddependenceoftheconductivity.Wearguedthattheformerisnotin-trinsicbutisduetothepresenceinthesampleofsmallgrainswithTchigherthanthebulk.Weestimatedthedensityandthesizeofthegrains.Thelatteranomalyisintrinsicandimportant;itimpliesthatthevortexvis-cosityisunusuallysmall.WeshowedthatsuchasmallviscositycanariseifthequasiparticlerelaxationratehasaverystronganisotropyaroundtheFermisurface,andinparticularbecomesveryweakalongthezonediago-nalwherethesuperconductinggapvanishes.Wecalltheplaceswherethescatteringratevanishes“coldspots”.Finally,wenotethatsmallviscosityimplieslargequan-tumfluctuationsofvorticesand,perhaps,quantumliquidofvorticesatT=0inawidefieldrange.

Acrucialissueinhigh-Tcsuperconductivityistheap-parentcoexistenceinunderdopedmaterialsoflocalpair-

ingoverawiderangeabovetheresistiveTcandameanfieldlikesuperconductingtransition.Ourworkshowsthattherearetwowaysinwhichthismayoccur:thevor-texliquidcontributiontotheconductivitymaybeverysmallorthelocalpairscanbeconfinedto’grains’thatarecoupledonlyviaJosephsonjunctionsthroughanor-malhost.Itwillbeinterestingtoseeifthesepossiblitiesareindeedrealizedinunderdopedmaterials.

Acknowledgements:WethankC.Bergemannet.al.forgenerouslysharingtheirdatapriortopublication.A.J.M.acknowledgesN.S.F.DMR-9707701;V.G.andA.J.M.thankthephysicsdepartmentofRutgersUniver-sityforhospitality.