Superconducting pairing of spin polarons in the t-J model

The pairing of quasiparticles in a CuO$_2$ plane is studied within a spin polaron formulation of the $t$-$t^{'}$-$J$ model. Our numerical solution of the Eliashberg equations unambiguously shows d-wave pairing between spin polarons on different sublattices

Superconductingpairingofspinpolaronsinthet-Jmodel

N.M.PlakidaandV.S.Oudovenko

JointInstituteforNuclearResearch,141980Dubna,Russia

P.HorschandA.I.Liechtenstein

MaxPlanckInstitutf¨urFestk¨orperforschung,Heisenbergstr.1,D-70569Stuttgart,Germany.

(February1,2008)ThepairingofquasiparticlesinaCuO2planeisstudiedwithinaspinpolaronformulationof′

thet-t-Jmodel.OurnumericalsolutionoftheEliashbergequationsunambiguouslyshowsd-wavepairingbetweenspinpolaronsondi erentsublatticesmediatedbytheexchangeofspin- uctuations,andastrongdopingdependenceofthequasiparticlebandwidth.ThetransitiontemperatureTcisanincreasingfunctionofJ/tandcrossesamaximumatanoptimaldopingconcentrationδopt.Forthet-JmodelwithJ/t=0.4weobtainTc 0.013tatδopt 0.2.

arXiv:cond-mat/9510120v1 20 Oct 1995

Recentexperimentalevidenceinfavorofad-wavesu-perconductingpairinginhigh-Tccuprates[1]supportstheoreticalstudiesofmodelswithstrongelectroncorre-lations[2].TheminimalmodeldescribingholemotioninCuO2planeisthet-J[3]ort-t′-J[4]model.Numericalstudies[2,5,6]ofsmallt-Jclusterssuggestad-wavesu-perconductinginstability.Yettoelucidatethenatureofthispairing,ananalyticaltreatmentofthet Jmodelisneeded.Forthispurposeweuseaspinpolaronformula-tionforthet Jmodeldeducedintheregionofsmallholeconcentrations.Anumberofstudiesofthismodel[7,8]predictthatdopedholesdressedbystrongantiferromag-neticspin uctuationspropagatecoherentlyasquasipar-ticles(spinpolarons)withweightZk J/t.Itisquitenaturaltoexpectthatthesamespin uctuationsinducesuperconductingpairingofthespinpolarons.RecentlythisproblemhasbeentreatedintheframeworkofthestandardBCSformalismassumingarigidbandmodelforthequasiparticles[9,10].However,sincethespin- uctuationenergyisofthesameorderasaquasiparticlebandwidth～Jastrongcouplingapproachisnecessary.Inthispaperwepresentthe rstconsistentsolutionofthestrongcouplingspinpolaronmodelat nitetemper-aturesandholeconcentrationsfornormalandsupercon-ductingstates.Anumericalsolutionofaself-consistentsystemofequationsforholeandmagnonGreenfunctionsprovessingletd-wavesuperconductingpairing.Thegapfunctionshowsinterestingadditionalstructureontopofthesimple k= 0(coskx cosky)whichre ectstheFermisurfacegeometry.ThedopingdependenceofTcaroundδopthastheformofaninvertedparabola,similar

max

toexperiment,andaTc～http://doc.guandang.netbiningthesere-sultswithalreadyexistingweakcouplingstudiesfortheHubbardmodel[11,12]wearguethatthespin-exchangepairingisthetruemechanismforhigh-temperaturesu-perconductivityasproposedearlierbyseveralgroupsbasedonmorephenomenologicalapproaches[1,13–15].Wewillstudyaspinpolaronmodelonatwosublattice

antiferromagnetic(AF)backgroundwhichhasbeensuc-cessfullytestedinthesingleholecase[7],[8].Spinless

+

fermionoperatorsh+iandfiareintroducedforholesondi erentsublattices,i.e.onthe↑(↓)-sublatticetheconstrainedelectronoperatorsc iσ=ciσ(1 ni σ)of

+

i↓=h+thet-Jmodelarereplacedbyc i↑=h+iSii,cy ±x

( ci↓=fi+,c i↑=fi+Si),whereSi=Si±Siarespinoperators.ThisrepresentationexcludesdoublyoccupiedstatesandtakesintoaccountstrongAFspincorrelationsintheelectronhopping.

Byemployingthelinearspin-waveapproximation

+

intermsoftheHolstein-Primako operators:Si

++

ai,(i∈↑),Si bi,(i∈↓)andperformingtheBogoliubovcanonicaltransformation:ak=vkαk+

++ukβ k,bk=vkβk+ukα k,weobtainthespinpolaronmodel:Ht J=+

k

kq

+

(h+kfk q[g(k,q)αq+g(q k,q)β q]+h.c.)

+

k(h+khk+fkfk)+

q

+

ωq(α+qαq+βqβq).

(1)

Hereg(k,q)=(zt/

2,γ=1 γkk

1

The pairing of quasiparticles in a CuO$_2$ plane is studied within a spin polaron formulation of the $t$-$t^{'}$-$J$ model. Our numerical solution of the Eliashberg equations unambiguously shows d-wave pairing between spin polarons on different sublattices

spinpolaronmodel(1),weconsiderthematrixGreenfunction(GF)forholesontwosublattices h+= f+

Ghh(k,z)=

k|hk zk|fk zandtheanomalousGhf(k,z)= h++ f++GFk|f k z=anticommutator k|hk z,whereZubarev’snotation[17]fortheGFwasusedwithz=ω+i .Toobtainself–consistentequa-tionsfortheseGF’sweemploytheself–consistentBornapproximation(SCBA)whichprovidedgoodresultsfortheone–holespectruminthenormalstate[7,8,16].InSCBAwegetfortheself-energies

Σhh(k,iωn)= T

Ghh(q,iωm)λ11k,k q(ωn ωm),(2)q,m

Σhf(k,iωn)= T

Ghf(q,iωm)λ12k,k q(ωn ωm).(3)

q,m

wheretheMatsubarafrequenciesωn=πT(2n+1).Theinteractionfunctionsare

λ11k,q(ων)=g2(k,q)D(q, iων)+g2

(q k,q)D( q,iων),

λ12k,q(ων)=g(k,q)g(q k,q){D(q, iων)+D( q,iων)}.

ThediagonalmagnonGFD(q,ω)= αq|α+zeroorderapproximationisgivenbyD(q ωin

the0

q,ω)=(ω ωq) 1withthedopingdependentmagnonenergyωq.ThefullmagnonGFisdeterminedbythematrix

equationD

1(q,ω)=(D 0) 1(q,ω) Π( q,ω)wheretherenormalizationofthemagnonenergyduetoparticle-holeΠ(

excitationsisdescribedbythepolarizationoperatorq,ω).Thisiscalculatedinone-loopapproximationusingthefullyrenormalizedhole-GF.

ThesuperconductingtemperatureTciscalculatedfromthelinearizedformoftheEliashbergequationforthegap-function

φ(k,iωn)=

λ12k,k p(iωn iωm)Ghh(p,iωm)

p

m

×Ghh( p, iωm)φ(p,iωm).

(4)

The rststepisaself-consistentcalculationofthenor-malGFGhh(k,iωn)=(iωn+ k Σhh(k,iωn)) 1with

theself-energyoperator(2)foragivenconcentrationof

holesδ=1

N

k nGhh(k,iωn)

.FIG.1.Thedensityofstates(DOS)fortheholeconcen-trationδ=0.06atT=0.012t.Thes …… 此处隐藏：9297字，全部文档内容请下载后查看。喜欢就下载吧 ……