Collective modes of spin, density, phase and amplitude in exotic superconductors

The equations of motion of pair-like excitations in the superconducting state are studied for various types of pairing using the random phase approximation. The collective modes are computed of a layered electron gas described by a $t-t'$ tight-binding ban

Collectivemodesofspin,density,phaseandamplitudeinexoticsuperconductors

D.vanderMarel

LaboratoryofSolidStatePhysics,DepartmentofScience

UniversityofGroningen,Nijenborgh4,9747AGGroningen

Theequationsofmotionofpair-likeexcitationsinthesuperconductingstatearestudiedforvarious

typesofpairingusingtherandomphaseapproximation.Thecollectivemodesarecomputedofa

layeredelectrongasdescribedbyat t′tight-bindingband,wheretheelectronsexperiencebesides

thelong-rangeCoulombrepulsionanon-siteHubbardUrepulsionandanearest-neighbourattractive

interaction.Fromnumericalcalculationswesee,thatthecollectivemodespectrumnowbecomes

particularlyrich.Severalbranchescanoccurbelowthecontinuumofquasi-particleexcitations,

correspondingtoorder-parameter uctuationsofvarioussymmetriesofpairing,andcollectivespin-

density uctuations.Fromthecollectivemodesofteningnearthenestingvectorsitisconcluded,

thatinthed-wavepairedstateaninstabilityoccurstowardtheformationofaspin-densitywave.

arXiv:cond-mat/9406097v1 24 Jun 1994MaterialsScienceCenterInternalReportNumberVSGD.94.6.7PACSnumbers:74.25.Gz,78.30.Er,71.45.Gm,74.25.NfI.INTRODUCTIONAwell-knownresultofBCStheoryisthevariationalwavefunction,describingthegroundstateofasuperconductor. →0thisfunctioncanbeeasilyextendedtodescribeasuperconductormovingatasmallandInthelimitQ uniformvelocity v=(2me) 1¯hQ|Ψ>= dRe ·R 3 iQ d3 rφ( r)ψ↑(R + r/2)ψ↓(R N/2 r/2)|0>(1)ThisfunctionhasthemathematicalshapeofaBosecondensateofpairs,wherethewavefunctionφ( r)describingthe ·R )relativemotionofelectronsformingapairistheFouriertransformof[1+( k/ k)2] 1/2 ( k/ k),andexp(iQisthemacroscopicwavefunctiondescribingthecenterofmassmotionofeachpair.ThesimilaritytoaBosecondensatewavefunctionissomewhatmisleading,asalsothewavefunctionofagasofuncorrelatedfermionscanbewrittenin

thisform,inwhichcaseφ( r)isanon-trivialfunctionwithanr 2tail.Inthelimitofaweake ectiveinteractionφ( r)hasanalgebraictailjustasforthefreeelectrongas.Iftheinteractionisstrong,φ( r)canbeinterpretedasawave-functiondescribingtherelativemotionoftwoelectronsformingaBose-condensedpair.[3]Ifthee ectiveinteractionisanon-siteattraction,theelectronspair-upinasinglet-wavefunctionwithanenhancedprobabilitytooccupythesamesite.Clearlyiftheelectronsexperienceastrongon-siterepulsion,thetendencytowardspairingdisappears.Withanetattractionbetweenelectronsoccupyingneighbouringsitesinthelattice,itisstillpossibletoformapairedstate,butφ(r)hastobeconstructedsuch,thattheparticlesavoidthesamesite.Thisconditionisforexampleful lledwhenwhenφ( r)hasa niteangularmomentum.

OnemaywonderwhethertheanalogytoBose-condensationcanbedrawnfurther,andconsidertheenergyspectrumofpair-likeexcitationsasafunctionofpair-momentum.Thisproblemwas rsttreatedbyBogoliubov[4],andAnderson[5].Iftheelectronsexperienceanon-siterepulsion,withanearest-neighbourattraction,thecollectivemodespectrumbecomesparticularlyrich.Itturnsoutthatseveralbranchesoccurbelowthecontinuumofquasi-particleexcitations,correspondingtoorder-parameter uctuationsofvarioussymmetriesofpairing[6].Theexistanceoflow-lyingcollectivemodesmaybeimportantwhenattemptingtoidentifyasuperconductinggapintheinfrared,Raman,orinelasticneutronscatteringspectraofthesematerials.

Collectivemodesinsuperconductorshaveinthepastattractedtheattentionforavarietyofreasons:(1)Bogoliubovpredictedtheexistanceofalongitudinalcollectivemodewithasound-likedispersion[4].LongrangeCoulomb

The equations of motion of pair-like excitations in the superconducting state are studied for various types of pairing using the random phase approximation. The collective modes are computed of a layered electron gas described by a $t-t'$ tight-binding ban

interactionsmakethespectrumidenticaltotheplasmonsofanormalFermigas,aswasshownbyAnderson[5].

(2)ThecollectivemodespectrumnaturallyfollowsfromagaugeinvariantformulationofBCStheory[5],andaconsistentexplanationoftheMeissnere ectrequiresthatthewholeinteractionHamiltonian(asopposedtothereducedBCSHamilitonian)istakenintoaccount[5,7].(3)Ascollectivemodesmediateelectron-electroninteractions,plasmons[8–10]andspin- uctuations[11–14]havebeenconsideredaspossiblecandidatesforapairing-mechanism.

(4)Certainmodes,inparticularcondensatephase- uctuationsnearorbelowthepair-breakinggap,areimportantforthethermalbehaviour,notablyTc,ofthesuperconductor[15].(5)Aninstabilityofthegroundstateandanincipientphasetransitiontoastatewithalowerenergyfollowfromthesofteningofcollectivemodes[16–18].(6)Collectivemodesmayshowupinexperimentalspectra,suchasinoptical[19–21]orRamanspectroscopy[22,23].(7)Asthereisnointer-planehoppinginalayeredelectrongas,thek-dependentplasmonspectrumbecomesgapless[24],whichmaygiverisetoaninterestingbehaviourintheregionformomentumandfrequencyvalueswherethecollectivemodecrosses2 [25,10].(8)Ifthereexistsanelectron-electroninteractioninchannelswitha niteangularmomentumL,excitonswiththecorrespondingsymmetriescanexist[6,16,19].

Usuallymodesofphaseanddensity[25,10]aretreatedseparatelyfromamplitudemodes[22,23],andspin- uctuations

[12,11,26].Aswewillseebelow,especiallyforanon-vanishingmomentumacouplingexistsbetweenthefourcollective-modechannelsofspin-density,charge,phaseandamplitudeoftheorderparameter.Theaimofthisstudyis,toderivegeneralexpressionsforthecollectivemodesinthesuperconductingstate,usingauni edapproachincludinge ectsof nitemomentumpairing.Inthelastsectionexamplesaregivenforthecollectivemodesandthegeneralizedsusceptibilityinthesuperconductingstate.Itisshownthatad-wavesuperconductormaybecomeunstablewithrespecttotheformationofaspin-densitywave,orpossiblyamixedSDW-waveplussuperconductingstate,ifanon-siterepulsionistakenintoaccountinadditiontohavinganattractiveinteractioninthed-wavechannel.Moredetailedcalculationsofvariousresponsefunctionsandthecomparisonthereoftomeasurementsonspeci cmaterialswillfollowinafuturepublication.

II.THEMODELHAMILTONIAN

Inthediscussionofthecollectivemodeswewillmakeextensiveuseoftwo-particlecreationoperators.Wewillseebelow,thatthechannelswithSz= 1,Sz=1andSz=0aredecoupled.IntheSz=±1channelstherearetripletpair-excitations,andspin- uctuations.IntheSz=0channeltherearespin- uctuations,density uctuations,and uctuationsofphaseandamplitudeoftheorder-parameter(singletandtripletpair-excitations).Thecorrespondingoperatorsareinthesameorder

σk(Q)

ρk(Q)

φk(Q)

ψk(Q)

Theremaining6combinationsarec

k+Qσckσ≡≡≡≡ c k+Q↑ck↑ c k+Q↓c k↓ c k+Q↑ck↑+c k+Q↓c k↓ c k Q↓ck↑ c k+Q↑c k↓ c k Q↓ck↑+c k+Q↑c k↓(2)

2V(Q)ρ(Q)ρ( Q)+

1

The equations of motion of pair-like excitations in the superconducting state are studied for various types of pairing using the random phase approximation. The collective modes are computed of a layered electron gas described by a $t-t'$ tight-binding ban

systemaftercarryingouttheSchrie er-Wol transformation.OtherexampleswheresuchtermsoccuraretheRKKYinteractioninmagneticalloys,andthesuperexchangeinrareearth-dopedsemiconductors.Alsotheon-siteHubbardUtermisusuallywritteninthisform,althoughinthiscasethePauli-principlealreadyautomaticallyexcludesoccu-pationofthesamesitewithparallelspins.

Aswewilldiscusstheequationsofmotionofthecollectivemodesforageneralformofthee ectiveelectron-electroninteraction,itisworthwhiletosummarizetheexpressionsforthegapequationandthefreeenergy.Thethermody-namicpotentialatT=0ofaBCSsuperconductoristheexpectationvalueofthegrandcanonicalHamiltonian,andiseasilyobtainedbytakingtheexpectationvalueofEq.3usingthevariationalwavefunctionofEq.1

22 (µ,V,vk1,...,vkN)=2|vk|2(ξk µ)+(4)ukvkλkqu v+|u|V(k q)|v|kqqq

kkq

whereλkqisthepairingpotential.Forthetypeofinteractionintroducedaboveoneobtainsλkq=V(k q)+1

σk(Q)· σq( Q).Thelastterm2U(k+q)wherethelasttermisthespin- ipscatteringcontributioncontainedin

inEq.4correspondstotheexchangeenergy.From oneobtainsthegap-equationbycalculatingtheminimumasafunctionofthesetofvariationalparametersvk1,...,vkN.Thenumberofparticlesinthegroundstateisobtainedbytakingthe rstderivativewithrespecttoµ.Theresultingsetofequationsis

2ukvk quqvqλkq=

2Ek

Wenotice,thatfor →0adecouplingofpairing-channelsoccurs,dependingonthepresenceofo -diagonalelementsinthedecompositionof1/Ek→1/| k|.Asλkqisreal,theset{ψα(k)}canbechosenasrealnumbers.Asaresultalso αisreal.Solutionslike”s+id”[31]becomepossibleifthereisadegeneracybetweensolutionswithadi erentsymmetry.

III.EQUATIONSOFMOTION

]=νO ,whereO isalinearcombinationofpair-operatorsrepresentingTheequationsofmotionareoftheform[H,O

anexcitationofthesystemwithenergyν.Althoughtheseequationshavebeentreatedextensivelybefore,inthepreviouspapersthecouplingtothecollectivespinoscillationchannelhasnotbeenconsidered.Inparticularaspin-dependenttermwasnotincludedinearlierpublications.Asoneoftheaimsofthispaperistodiscusscollectivemodesofspin-densityinthesuperconductingstate,Ire-derivetheequationsofmotionwiththisextendedhamiltonian.Inthesuperconductingstatetheequationsofmotionofspindensity(σk(Q)),chargedensity(ρk(Q)),orderparameterphase(φk(Q)),andorderparameteramplitude(ψk(Q))arecoupledinanon-trivialway.Thecommutatorofeachofthesetwo-particleoperatorswiththeinteractionpartoftheHamiltoniangeneratesproductsoffoursingle-particleoperators,whichareapproximatedbytakingtheexpectationvalueofallcombinationsoftwooftheoperatorsappearinginthisproduct.Theresultingtermsfallintwocategories:thosewhichhavethesamek-value,andthosewhichareaweightedsummationoverk-space.Thelattergiverisetothecollectivemodes.Inthe rstcategoryoneobtains(1)selfenergytermswhichcanbeabsorbedinashiftofthechemicalpotential,(2)exchangeselfenergy

The equations of motion of pair-like excitations in the superconducting state are studied for various types of pairing using the random phase approximation. The collective modes are computed of a layered electron gas described by a $t-t'$ tight-binding ban

whereIintroduced terms,duetowhich kisrenormalizedto k≡ k q|vq|2V(k q),and(3)cross-termsproportionalto k,linkingσktoψk,andρktoφk-operators.Finallythecategoryofweightedaveragesoftwo-particleoperatorsoverk-spaceinvolvesbothdirectandexchangeterms,andisgivenbytheexpressions Hi(k,q,Q)σq(Q)Sk(Q)≡ qσiHρ(k,q,Q)ρq(Q)Rk(Q)≡q (6)iH(k,q,Q)φ(Q)Ak(Q)≡q qφiBk(Q)≡qHψ(k,q,Q)ψq(Q)

iHσ(k,q,Q)iHρ(k,q,Q)

iHφ(k,q,Q)

iHψ(k,q,Q)≡≡≡≡ 12V(Q)+1V(k q)+V(k q)+2U(k11 q)2U(k q)2U(k+q)

2U(k+q)(7)

Withthesede nitions,andusingtherandomphaseapproximationdescribedabove,thecommutatorsofthepairoperatorscannowbederived.Theactualcalculationisastraightforward,thoughratherlaborious,exerciseincommutatoralgebra.AdetaileddescriptionofthevarioustermshasbeengivenbyAnderson,andlaterdiscussedmoreextensivelybyBardasisandSchrie er,whoretainedanumberofverticesintheir nalanalysiswhichwereneglectedbyAnderson.InthepresentpaperallverticesdiscussedbyBardasisandSchrie eraretakenintoaccount.Theexpressionsarehowevermodi edduetothespin-dependentinteractionterminEq.3.Thesetofcommutators,includingtheexchangeinteractions,is

[H,σk(Q)]

[H,ρk(Q)]

[H,φk(Q)]

[H,ψk(Q)]==== kQρk(Q) kQψk(Q)+ kQσk(Q) kQφk(Q)+ +kQψk(Q) kQρk(Q)

+kQφk(Q) kQσk(Q) +zkQRk(Q) b kQBk(Q) ++zkQSk(Q) bkQAk(Q)+ b+kQRk(Q) zkQBk(Q)

+ b kQSk(Q) zkQAk(Q)(8)

k, k,bkandzkwerealreadyde nedintheprevioussection.ForthesakeofcompactnessofnotationIintroduced±±bkQ≡bk+Q±bk,zkQ≡zk+Q±zk, ± ± k+Q± k.kQ≡ k+Q± kand kQ≡ The rsttwotermsofallfourcommutatorscorrespondto(1)thekineticenergywithexchangeself-energycorrections(Fig.1a),and(2)Boguliobov-Valatinparticle-holemixing(Fig.1a’).Theremainingtwotermsineachoftheseexpressionscanbebetterdescribedwithreferencetothede nitionofthecollectivecoordinatesinEqs.6and7.Letus rstconsiderRk(Q)andSk(Q).TheV(Q),U(Q)andU(k q)-termscorrespondtothepolarizationvertexinthecommutatorsofσkandρk(Fig.1b).InthecommutatorsofφkandψktheV(Q),U(Q)andU(k q)-termisapolarizationvertexcombinedwithaparticle-holetransformationononeofthelegs(Fig.1b’).TheV(k q)-termscorrespondtotheexchangescatteringvertexwithout(commutatorsofσkandρk,Fig.1c)andwithparticle-holetransformation(commutatorsofφkandψk,Fig.1c’).

FinallyAk(Q)andBk(Q)correspondtothedirectparticle-particlescatteringvertexwithout(commutatorsofφk(Q)andψk(Q),Fig.1d)andwithparticle-holeconversion(commutatorsofσk(Q)andρk(Q),Fig.1d’).

Ifweapplytheequationsofmotiontoageneraloperatoroftheform

O=[v1,k(Q)σk(Q)+v2,k(Q)ρk(Q)+v3,k(Q)φk(Q)+v4,k(Q)ψk(Q)]

k

we ndthattheycanbewritteninmatrixformas

H0(k,Q) vk(k,Q)+Hi(k,q,Q)Γ(q,Q) vq(Q)=ν vk(Q)

q(9)

TheinteractionHamiltonianHicontainsthematrixelementsofEq.7onthediagonal,andiszeroelsewhere.Wefurthermoreusethezero’thorderHamiltoniandescribingnon-interactingquasi-particles

0 0 kQkQ 0 +0 0kQkQ (10)H(k,Q)≡ ++ 0 kQ0 kQ 0 +0kQkQ

The equations of motion of pair-like excitations in the superconducting state are studied for various types of pairing using the random phase approximation. The collective modes are computed of a layered electron gas described by a $t-t'$ tight-binding ban

andthedimensionlessmatrixcontainingcoherencefactors

0zkQ0 b kQ + zkQ0 bkQ0Γ(k,Q)≡ ++ 0 bkQ0 zkQ+ b 0 zkQ0kQ (11)

Thecollectivemodescanbefoundbylookingforpolesinthecorrelationfunctions,inparticularthedensity-density ′,0)>>νand<<Tσ( ′,0)>>ν,whereρ( andthespin-spincorrelationfunctions<<Tρ( r,τ)ρ(rr,τ)σ(rr,τ)etc.are rρ( theHeisenbergrepresentationoftheoperatorsQr)whichwerede nedintheprevioussection.Inthe expiQ·

′,0)>>νand<<Tψ( ′,0)>>νbecomerelevant.Togetherwithsuperconductingstatealso<<Tφ( r,τ)φ(rr,τ)ψ(r

thesixo -diagonalcorrelationfunctions,thefourdiagonalfunctionsde nea4×4two-particleGreen’sfunction0matrix.ThematrixKkq(Q,ν)=(ν H0(k,Q) i0+) 1δk,qcorrespondstotheLehmannrepresentationofthisGreen’sfunctionintheabsenceofresidualinteractions(i.e.withHi=0).Asthisdescribestheresponseofagasofnon-interactingquasi-particlestherearenopolescorrespondingtocollectivemodes.Thegeneralizedsusceptibility 00(Q,ν).IfwenowincludeHi,wecancalculatetheGreen’sfunctionsintheRPAbyχ(Q,ν)=k,qΓ(k,Q)Kk,qapplyingtheDysonequation

00Kkq(Q,ν)=Kkq(Q,ν)δk,q+Kkk(Q,ν)Hi(k,k′,Q)Γ(k′,Q)Kk′q(Q,ν)(12)

k′

Wecanusethesamepartialwavedecompositionasintroducedinthepreviousparagraphwherewediscussedthegapequation.Itisstraightforwardtoshow,thattheaboveDysonequationhasthesolution

i0 1χα,β(Q,ν)=(13)χ0(Q,ν)1 Hχα,γγ,βγ

whereIusedthepartialwavedecomposition

χα,β(Q,ν)= kqψα(k)Γ(k,Q)Kkq(Q,ν)ψβ(q)

withsimilarexpressionsforχ0,andHi.Thecollectivemodescorrespondtothezero’softhedeterminantof

i0δα,βδi,j Hα,µ;i,lχµ,β;l,j

µ,l(14)

whichcanbedeterminednumerically,andinsomelimitingcasesalsoanalytically.TheexpressionoftheresponsefunctionEq.13correspondstocalculatingtheseriesofdiagramsdepictedinFig.1.Itispossibletoimprovefurtherbytakingintoaccountthescreeningofthevertexinallofthesediagrams,exceptinthepolarizationverticesofFigs.1band1b’,asthiswouldleadtodoublecountingofthevertexcorrections.(N.B.:Althoughinthispaperthepairinginteractionisintroducedasanindependentmodelparameter,oneshouldkeepinmind,thatforanelectronicmechanismofsuperconductivitysuchasaspin- uctuationorplasmon-intermediatedinteraction,thepairingarisespreciselyfromthesevertexcorrections.)ThisprocedurewasproposedbyAnderson,RickayzenandalsobyBardasisandSchrie er.Moreover,inthenextsectionwewillsee,thatinthenormalstatetheσandρchannelsarecompletelydecoupledforallvaluesofQ.Thisimplies,thatthesumoverdiagramsforthecharge uctuationsdoesnotcontainanyvertexcorrectionduetothespin uctuationsandvice-versa.Hence,itisnecessaryinthiscasetoscreenallverticesinthecharge- uctuationchannelwiththespin- uctuations,andviceversa.AshasbeenshownbyRickayzen,inthesuperconductingstatethescreeningpropertiesarebasicallythesameasinthenormalstate[7].

Onehastobecautiouswiththisprocedureofscreeningthevertices,as,bymakingtheRPAbeforecalculatingthesumoverdiagrams,certainclassesofvertexcorrectionsareomitted.Asaresultinconsistenciesmayarise,ascanbeseenfromthefollowingexample:IfweconsidertheHubbardUmodel,theon-siteinteractioncanbeintroducedeitherusinganon-sitespin-independent(V)orasinglet-only(U)termasde nedinEq.3.Theexpressionsfortheequationofmotionshouldbeindependentofthischoice,asthePauli-exclusionprincipleautomaticallyprojectsoutthedoubleoccupancyofthesamesitewithequalspins.Indeed,wecancheckfromEq.7thatthisrequirementissatis edaslongaswedonotintroducescreening.Ifwefollowtherecipe,thatinthe rsttwolinesofEq.7thepolarizationdiagramsU(Q),V(Q),andU(k q),butnottheexchangediagramV(k q),shouldbereplacedwiththebareinteraction,wearriveatadi erentresultdependingonwhetherweintroducetheon-siteinteractionthrough

The equations of motion of pair-like excitations in the superconducting state are studied for various types of pairing using the random phase approximation. The collective modes are computed of a layered electron gas described by a $t-t'$ tight-binding ban

asinglet-onlyoraspin-independentinteraction.iThisinconsistencyisremoved,ifwereplacethedirectandexchangetermsinHσwiththechargescreenedvalue.Inthesamewayscreeningwithspin uctuationsshouldbeintroduced’byhand’inthedirectandexchangetermsiniiiHρ.FinallyallthreetermsinHφandHψshouldbereplacedwiththechargeandspin uctuation-screenedvertices.

0LetusnowcalculateKbyinverting[ν Hqp].Thedeterminantis

22222|ν Hqp|=ν4 2ν2(Ek+Q+Ek)+(Ek+Q Ek)=

(ν2 (Ek+Q+Ek)2)(ν2 (Ek+Q Ek)2)(15)

Thezero’thordertwo-particleGreen’sfunctionisthen

K0=|ν Hqp| 1×

ν(ν 2+2 2+2) (ν2 +) + + ν(ν2 +) ν 222 ν ++ν + ν2 ++( + + + )

ν(ν2 )

ν 22 + ++( + ν2) ν + + ν +( ν2) + +

ν(ν2 +)2222 (ν2 +)++ ν ++ν + + +

+( +2

(16) ν2 ++( + + + ) ν + + ν ν) 2 +( ν2)+ + 2The4×4matrixK0ΓbecomesK0Γ=|ν Hqp| 1 ν2(z +b ) ++(b +z +) ×+ + + ) ( z ν3+νz ( ++ +)+νb+( + +) ν2(z +b+ +)+(b+ +z +)+ + ×( + )νb

νz ( + +)

νb+( 23+22ν2(b+ z+ )+++(z b+ +)+ + ×( + )b+ν3+νz+( + + ) νb+( ++ ) ν2(z+ ++b+ +)+ +(b +z+ )+ + ×( + )

z+ν3

2+222b ν3+νz+( + +) νb ( ++ +)ν2(b z+ +)+(z+ b +)+ + ×( + )νz3+2222z ν3+νz ( + )+νb ( + + )+2 2(17)ν2( b ++z +) +(z +b )+ + ×( + )b ν3 νz ( + + )

νb( 2+ 2) νz+( + ) νb ( + + ) ν2(z+ ++b )+(z+ +b +)+ ×( + + )+ +2) ν2(b+ + z ) (z + b+ )+ + ×( + ) νz+( + )+νb+( + + )

Frominspectionofthematrixelementsitturnsout,thattheyallcontainthefactor(ν2 (Ek+Q Ek)2)inthenumerator.Asthesametermappearsinthedenomenator,thesefactorscancel.K0Γturnsouttobesymmetric,andtheexactresultis

ΓK0=K0Γ=Ek+Q+Ek

Ek+Q+Ek k+Q k k k+Q

νEk+Q k+Ek k+Q νEk+Q k Ek k+Q

Ek+Q+Ek Ek+QEk k k+Q+ k k+Q

Ek+Q+Ek Ek+QEk k+Q k k+Q k

νEk+Q k+Ek k+QνEk+Q k+Ek k+Q

Ek+Q+Ek k+Q k k k+Q

The equations of motion of pair-like excitations in the superconducting state are studied for various types of pairing using the random phase approximation. The collective modes are computed of a layered electron gas described by a $t-t'$ tight-binding ban

k= 2t(cos(kxa)+cos(kya)) 2t′cos(kxa)·cos(kya) µ(19)

whereabandcarethelatticeparameters.Thetandt′-termsareduetonearest-neighbourandnext-nearestneighbourhoppinginasquarelattice.Ift′=0athalf llingoftheband,suchadispersionrelationhastheremarkablepropertythattheFermisurfaceformsaperfectsquare,withadiverginge ectivemassovertheentireFermisurface.Inpracticethissituationwillneveroccur,astherewillalwaysbesome nitecouplingbetweennextnearestneighbours.ThiscausesabulgingoftheFermisurface,whicheventuallytransformsintoarotatedFermisurfaceif|t′| |t|.InallexamplesIwillrestrictthediscussiontosystemswhereelectronshaveanon-siteattractionorrepulsion,anearest-neighbourinteraction,orboth,aswellasthelong-rangee2/rrepulsiveinteraction.MoreoverthediscussionislimitedtothesituationwhereasinglebandcrossestheFermisurface,andtight-bandinglanguagewillbeusedforthedescriptionofthisband.InparticularIwillconsideratightbindingbandonathree-dimensionalsquarelattice,withastronganisotropyleadingtoquasitwo-dimensionalbehaviour.Aconvenientsetoffunctionstobeusedforthepartial-wavedecompositionofHiisthenthesetofharmonicfunctions:

s:

s :

dx2 y2:

px:

dxy:

etc.ψ0(k)ψ1(k)ψ2(k)ψ3(k)====1cos(kxa)+cos(kyb)cos(kxa) cos(kyb)√(20)2sin(kyb)ψ5(k)=2sin(kxa)sin(kyb)

U0ni↑ni↓isthek-independentfunctionU(k q)=

2Thek-spacerepresentationoftheon-siteHubbardUinteractionU0ψ0(k)ψ0(q).Ifweconsiderthenearest-neigbourinteraction1iV(Q)=2V1ψ1(Q),sothatweobtainthepartial-wavedecomposition QV(Q)ρ(Q)ρ( Q)with

(21)V(k q)=V1(ψ1(k)ψ1(q)+ψ2(k)ψ2(q) ψ3(k)ψ3(q) ψ4(k)ψ4(q))

Asinglet-onlynearest-neigbourinteraction1 σ(Q)· σ( Q)]withU(Q)=2U1ψ1(Q),henceithasthesamepartialwaveexpansionasQU(Q)[ρ(Q)ρ( Q) 82V1ψ1(Q).However,fromEq.7wesee,thatthesinglet-onlyinteractionhasotherprefactors,andissummedoverU(k q)andU(k+q)inthepairingchannel.

FinallywehavetotakeintoaccountthelongrangeCoulombinteraction.HerewewillusethelatticeFouriertrans-formofe2/r.ThescreeningoftheCoulombinteractionforpartoftheverticeshasbeendiscussedabove,andisessential,asabareQ 2interactionisknowntocreateasingularityattheFermilevelwithintherandomphaseapproximation.Iwillusetheconventionintheremainderofthispaper,thatV(Q)isthebareCoulombrepulsionatlargedistances,whereasforshorterdistancesU0andV1aretheprojectionsofV(Q)ontheon-siteinteractionandthespin-independentnearestneighbourinteractionrespectively.Takingallthesetermstogetherweobtainforamodelwitha’singlet-only’nearestneighbourinteraction

iHσ(0,0)iHσ(α,α)iHρ(0,0)

iHρ(α,α)iHφ(0,0)

iHφ(α,α)======ρρ U0 U1ψ1(Q)ρ U1/2(α=1,2)σσσ2V(Q) U0+U1ψ1(Q)σU1/2(α=1,2)ρσU0ρσU1(α=1,2)(22)

iiForallsymmetrieswehaveHψ(α,β)=Hφ(α,β).Theupper-indicesρandσindicatewhetherscreeningwithcharge-

iorspin- uctuationsisimplied.Theminus-signinfrontoftheU0terminHρ(0,0)isnotamisprint.As2V(Q),

’contains’theon-siteHubbardterm,thesumofthesetwocontributionsis+U0.Inprincipleoneshouldalsoincludehigherharmonics,astheexpansionofV(Q)doesnotendatψ4.However,astheexpansiononlyappearsasascreenedinteractioninexpression22,itisreasonabletoworkwithaamodelwheresuchinteraction-termsareneglected.Ifthenearest-neigbourinteractionisspin-independentwemustalsoincludepxandpysymmetriesofpairing,andweobtain

The equations of motion of pair-like excitations in the superconducting state are studied for various types of pairing using the random phase approximation. The collective modes are computed of a layered electron gas described by a $t-t'$ tight-binding ban

iHσ(0,0)iHσ(α,α)iHρ(0,0)iHρ(α,α)

iHφ(0,0)

iHφ(α,α)======ρ U0 V1ρ(α=1..4)σ2Vσ(Q) U0 V1σ(α=1..4)ρσU0V1ρσ(α=1..4)(23)

Intheprevioussectionwehaveseen,thatinadditiontothepartial-waveexpansionofH

i,wealsohavetomakeasimilarexpansionofΓK0.TheexpressionforthisproductisgiveninEq.18.Thepartial-waveexpansionofthisexpressionisingeneralcomplicated,andhastobedonewiththehelpofacomputer.Somelimitingcasesexisthoweverwheretheintegralscanbesolved,especiallywhenanexpansionforsmallQcanbemade.Someoftheselimitingcaseswillbetreatedinthesubsequentsections.InadditionnumericalcalculationswillbegivenatgeneralvaluesofthecollectivemodemomentumQ.

A.Normalstatelimit

Inthenon-superconductinglimitEq.18hasonlynon-vanishingmatrixelementsonthediagonal.Furthermoreonlythechargeandspin-channelsarerelevantintheabsenceofo -diagonalorder.Letusmakethefurtherassumptionthattheelectronsinteractwitheachotherviaanon-siteHubbardUrepulsion,whichisthereforeindependentofk.Afterthesummationoverkweobtainforthetop-leftcornerofEqs.18

Hi(Q)χ0= ( k+Q k)(fk fk+Q)

k

=Q.IfweassumethatwehaveacylindricalFermisurface,withan4πe2/V(Q),whichhasthepropertylimQ→0Q2isotropicFermivelocityvF,andaFermiwavevectorkF,weobtainwith1 =2V(Q)χ0(2,2),andνp≡2e2d 1¯hkFvF

2πdφ =1 2V(Q)2πkFh ¯vF0

2h¯2Q2 vF1 2v2h¯2QF22h¯2Q2 vFcosφ

2 2v2/(2νp1+h¯2Q)F

Q Q 1+3

2νp+... (26)

Thespin-susceptibilityperunitcell( uistheareainthe2Dplane)is

χ(1,1)=22 1/2(1 ¯h2Q2 1 vF/ν)

isthee ectivebandwidth.Wesee,thatinthehighfrequencylimit(ν ¯hQ vF)χHF(1,1)=

22 1EFQ2,andinthelowfrequencylimitχLF(1,1)=(U0 W) 1,HencetheACsusceptibility u[ν+U0EFQ u]

issuppressed,whereasthestaticsusceptibilityisenhanced.AmagneticinstabilityoccursforU0≈W.Theaboveexpressionsarederivedassumingafreeelectrondispersion.IftheFermisurfacehasnestingvectors[12],instabilitiesforspeci cvaluesofQareoftenfound.h¯2π

The equations of motion of pair-like excitations in the superconducting state are studied for various types of pairing using the random phase approximation. The collective modes are computed of a layered electron gas described by a $t-t'$ tight-binding ban

B.s-wavesuperconductivity

Fors-symmetry,andneglectingtheradialk-dependenceofthepairingpotential,thepartialwavedecompositionofχandHiistriviallyachievedbysummingoverallk.Asamodelforthepairing-interactionweadoptU(k q)= g.Asisususallydoneinthegap-equation,onecanlimittheenergy-rangeoftheinteractionsintheseexpressionsbyputting =0forenergieslargerthenascalingvalue(theDebijefrequencyforphonon-mediatedpairing).ForthelongrangeCoulombinteractionwetakeagainV(Q)=4πe2Q 2.Duetothefact,that k+Q= k Qweobtainaftersummationthatχ0(1,2)=χ0(1,3)=χ0(1,4)=0.Hencethespin- uctuationsarefullydecoupledfromtheotherthreeandcanbeconsideredseparately.Theremainingdiagonalando -diagonalsusceptibilituesare nite,andthefollowingexpressionsarerequired0

χ0(1,1)=

χ0(3,3)= (Ek+Ek+Q)(EkEk+Q k k+Q k k+Q)

k

2EkEk+Q(ν2 [Ek+Ek+Q]2)(Ek+Ek+Q)(EkEk+Q+ k k+Q+ k k+Q)

k

2EkEk+Q(ν2 [Ek+Ek+Q]2)(Ek+Ek+Q)S=

M= k

2EkEk+Q(ν2 [Ek+Ek+Q]2)

(Ek+Ek+Q)( k k+Q)2(28)

k

2EkEk+Q(ν2 [Ek+Ek+Q]2)

Letus rstconsiderthespinsusceptibility.Asnowχ(1,1)=χ0(1,1)

itiseasytoprove,thatχ0(3,3)= 1/g+ν2S/2 M/2andχ0(4,4)=

χ0(3,3) 2 ingtheseproperties,weseethatthematrix1 Hiχ0becomes2Ek)

1 2VQχ0(1,1) 4V 2S

gν S

g T2VQν Sg(νS M)/22 2VQ T gν(T+N)/2

(29) gν(T+N)/2g([ν2 4 2]S M)/2

TofurtheranalyzethisexpressionweneedtomakeaseriesexpansionforsmallQ.InwhatfollowswewillneglectN(∝Q2,butwithavanishingprefactorifthegaphaselectronholesymmetry).Furthermorewenotice,thatwe

220 2canwrite2VQχ0(1,1)≈νp(Q)2(ν2 W2 4 2) 1,M≈S<(v F·Q)>,VQM≈(1+W/4 )VQχ(1,1),and 2T≈ 2µFS,whereµF=EF W/2,andWisthee ectivebandwidth.Retainingonlyleadingordersin<(v F·Q)>

wesee,thatthecollectivemodescannowbesolvedfrom

22222222 22(ν2 W2 4 2 νp(Q)2)(ν2+2 2)<(v F·Q)>=(ν 4 4µF)(ν W 4 )(ν νp(Q))andthecollectivemodescanbecalculatedfromthedeterminant 2 2 02220=(1 2VQχ(1,1))νS M[ν 4 ]S M ν(T N) +4V 2(N2+2TN)Sν2+M(S2[ν2 4 2] T2(30)

The equations of motion of pair-like excitations in the superconducting state are studied for various types of pairing using the random phase approximation. The collective modes are computed of a layered electron gas described by a $t-t'$ tight-binding ban

Hencewesee,thatthenegativeUHubbardmodelpermitsinprinciplefourcollectivemodes:aspin-densityoscillationdiscussedabove,aplasma-mode,andtwoadditionalmodes,whicharehoweversituatedinthetwo-quasiparticlecontinuum,andthereforearestronglydamped.Interestinglytheplasma-likemodecanexistatfrequenciesaboveandbelowthegap,dependingontheinitialvalueofνp(Q)inthenormalstate.AshasbeendiscussedbyFertiganddasSarma[25]alayerdispersionrelationasdiscussedabove,permitstheexistanceoflowlyingplasmonsbelowthegap.Anothermechanismforreducingtheplasmafrequencyinthesuperconductingstateisstrongdampingofthemotionperpendiculartotheplanes,aswerecentlydiscussed[27].

C.s-wavesuperconductivityinalayeredelectrongas.

Ifthesuperconductorisstronglyanisotropic,theplasmaenergyforQ→0dependsonthedirectionofpropagation.Anextremeexampleofthisarises,whenthemassinoneofthethreedirectionsisin nite,resultinginasystemwhichbehavestwo-dimensionalfromthepointofviewofthesignleparticleband-structure,whereastheCoulombforcesarethreedimensional.Asimplemodelexhibitingsuchbehaviourisanin nitestackofatwo-dimensionallayers.Theelectrodynamicsofthissystemwasalreadydiscussedbyseveralauthors[24,9,8]usinghydrodynamiccalculations,aswellaswiththerandomphaseapproximation.Theresultingplasmonspectrumofsuchametalis,inthelimitofa )=νpQ |Q | 1,whichforQ⊥=0saturatesatthevalueνp,whilefor nitevaluesofQ⊥ithaslargewavelength,ν(Q

anaccoustic-likedependenceonQ .

Thisimpliesthatherewehaveasystemwhichontheonehandhasadensityofchargecarrierscharacteristicofametal,and,providedthatthereisapairingmechanism,thereforehasthepotentialofbecomingaBCS-likesuperconductor.Ontheotherhandthedynamicalresponseoftheelectronsinoneofthedirectionsismorecharacteristicofasemiconductororaninsulator.ThiscombinationprovidesuswithanexamplewheretheAnderson-HiggsmechanismdoesnotshifttheGoldstonemodetoahighenergy,inspiteofthefactthattheparticlesinteractthroughalongrangeCoulombforce.HerewewillusethedispersionintroducedinEq.19witht′=0.InthisexampleW=4tisthe1 2bandwidth.ForthelongrangeCoulombforceswetakethelatticeFouriertransformofe2/r,whichhas u4πeQasitslongwavelengthlimitingbehaviour,andposessesthesameperiodicityink-spaceasthetight-bindingband.Ry =e2d 1isthee ectiveRydbergwhich,togetherwiththeFermienergy,setsthescaleoftheplasmafrequency intheplanardirection(νp=2

The equations of motion of pair-like excitations in the superconducting state are studied for various types of pairing using the random phase approximation. The collective modes are computed of a layered electron gas described by a $t-t'$ tight-binding ban

bandwidth.ForT=0nosharpphaseboundariesexist.Somewhatsurprisingly,for|U1|largerthanacriticalvalue(whichdependsonne),thegroundstateisofmixedαs+βdsymmetry,whichisautomaticallyastateofbrokenspatialsymmetry.Howeverthestronginteractionwhichisrequiredprobablydoesnotexistinanyrealisticmodelofsuperconductivity.Itisworthwhiletomentioninthiscontext,thattheregionofsd-mixingalmostcoincideswiththeregionofp-wavesymmetry,ifweuseaspin-independentinteraction(V1)instead.

LetusnowconsiderthecollectivemodesfortheexamplesalongthelineABindicatedinthephasediagram.TheresultisdisplayedinFig.4.Wesee,thatasoft-modedevelopesifweapproachthephaseboundarybetweensandd-wavesuperconductivity.ThetransitiontakesplaceexactlywhenthemodehasdevelopedintoaBogoliubovsound-mode.Ifwekeepimposingthes-symmetryfortheground-state,whileactuallybeinginthed-wavepartofthephase-diagram,wealways ndasoft-modeofphase- uctuatingcharacter,indicatingthatthesolutionisinstable.Ifweallowthegroundstatewave-functiontobecomed-wavepaired,thegapdisappears,andasound-wavephase- uctuationmodeoccursdirectlybelowtheparticle-holecontinuum.BoththeBogoliubovmodeandthelowerboundoftheparticle-holecontinuumaresound-like,sothataccordingtoLandau’sargument[28]asupercurrent- owisstillpossibleinspiteofthefactthatthereisnogap.

E.Phaseversusspin- uctuatingmodesinalayeredelectrongas.

Letusnowconsiderthesinglet-onlynearest-neighbourpairinginteractionU1.Inthediscussionoftheresonatingvalencebondstate[29–31]thet-Jmodelhasbeenused,whereJ= U1,andareductionofthedoubleoccupancyofthesamesiteisincluded,eitherbyreplacingthebarehoppingparametertwithane ectiveone,orbyusingmoreelaborateschemes.ItisnottheaimofthepresentdiscussiontoaddressthetJmodel.InsteadweconsideraFermi-liquid,withanon-siterepulsion(U0)whichisnottoostrong,andanattractiveinteractionbetweenelectronsonaneighbouringsite(U1).AstheactualbandstructureinthesesystemsisexperimentallyknowntobebetterdescribedbythethreebandmodelofZaanen,SawatzkyandAllen[32],(whichisagainasimpli edversionoftherealvalencebandstructureinvolving6oxygen2pbandsand5cupper3d-bandsfortheoccupiedstates,aswellasunoccupied3sand3pstates)atransformationtoasinglebandhamiltonianwillinprinciplegeneratebothane ectiveHubbardU0andanintersiteU1[33–35].ExamplesofsuchtransformationscanbefoundintheworkbyEmery[36],andbyJansen[37].However,alsoother,morecomplicatedtypesofinteractionsaregeneratedwhenmakingsuchtransformations,notablythecorrelatedhoppingterm(withsixoperators)which,ashasbeenshownbyHirsch,promotessuperconductivityofhole-carriers[38].TheinteractionconsideredbyJansenaswellasthecorrelatedhoppingtermtreatedbyHirsch,e ectivelyprovideanon-siteattraction,which,whenconsideredonitsown,promotespairinginthe(non-extended)s-wavechannel.AlsoU1termcontainscontributionsfromthevirtualexchangeofspin- uctuations[39,11].AshasbeendiscussedbyScalapino[12],suchprocessesgiverisetoanattractiononnearestneighboursites,andincreasetheon-siterepulsionbetweenelectrons.Astheexchangespin- uctuationsarei-channel,onecouldschematicallyregardU1inEq.22asthevertexcorrectionreallyvertexcorrectionsduetotheHσofU0.Assuchcorrectionsarenecessarilyretarded,andthereforeratherill-representedbythenon-retaredinteractionassumedhere,thepresentanalysiscanatbestprovideaqualitativepicture.

BZA[30]consideredpairingofthes -typenearhalf lling,Emeryconsidereddx2 y2-pairing,andKotliarstudiedboths andd-typepairing.Aswewillsee,thes -typepairingisnotastablesolutionnearhalf lling,andisdominatedbypairingofthed-type.Asthelatteragaintendstobeunstablewithrespecttotheanti-ferromagneticMott-Hubbardinsulatingstateathalf lling,superconductivitycanonlyexistsu cientlyfarawayfromthisregion.AstheoptimalTcwouldhavebeenreachedathalf llingforasymmetricalband,thiswouldleadtotheconclusionthatsuperconductivityisonlyamarginale ectinsuchasystem.However,thehighTccupratesdonothaveanelectron-holesymmetricalband,andtheFermisurfaceisknowntobestronglydistortedfromtheperfectsquarethatarisesfromconsideringonlynearestneighbourhopping.Thisactuallycomestorescue:Asafunctionofband- llingitpullsaparttheregions,whereanti-ferromagnetismandhighTchavetheirhigheststablility.

Thethreecoupledgapequationsare(withx≡kxaandy≡kya)

tanhEk/(2kBT) 1+

k

The equations of motion of pair-like excitations in the superconducting state are studied for various types of pairing using the random phase approximation. The collective modes are computed of a layered electron gas described by a $t-t'$ tight-binding ban

placeforalargevalueofU1.

Istillneedtospecifytheelectrondispersionrelationbeforewecansolvethegapequations.ForthedispersionrelationwenowuseEq.19witht′= 2CuO4andYBa2Cu3O7[40,41].Duetothe nitevalueoft′asigni cantchangeoccursinthedensityofstates(DOS)attheFermienergyasafunctionofthenumberofelectronsperunitcell.TheDOSisnowa-symmetric,andthemaximumisshiftedtothe’hole-doped’sideofthepointwherethebandishalf lled.Ofcoursethedirectioninwhichthisoccursisdictatedbythesignoft′.Witht′<0wemimicthesituationencounteredintheCuO2-planesofthehighTccuprates.

Thephasediagramwitht′/t= 0.7andU1/(4t)= 0.5,andU0/(4t)=0isdisplayedinFig.5.Duetobreakingofelectron-holesymmetry,thediagramisnowa-symmetricaroundhalfoccupationoftheband.Roughlyspeakings -pairingisfavouredfarawayfromhalf llingoftheband,whereasd-wavepairingbecomesthemoststablesolutionnearhalf lling.Wealsonoticefromthisplot,thatthea-symmetryimpliesthatthehighestTc’sandd-pairingsuperconductoraretobeexpectedontheleft-hand(’hole-doped’)sideofhalf- lling.LowerTc’sands-pairingoccurontheright-handside.

Letusnowconsiderthe /Tc-ratiofollowingfromthegapequation.WithinthecontextofBCStheorywehave 0(T)=0atTc,sothatTcfollowsfrom

22kBTc)[cosqxa±cosqya]2(32)

wherethe±signrefersagaintothetwosymmetriesofpairing.Thisequationcanbeeasilysolvednumerically.Theresultis,thatforextendeds-wavepairingtheratio2 0/kBTcis6.5,whereasford-wavepairingitrisesgraduallyfrom4if|U1| W,upto6.5inthelimitwhere|U1| W.Thisisnotsensitivetothevalueoftheparametert′.Weshouldkeepinmindhere,that 0isthemaximumvaluereachedby (k)(respectivelyatthe(π,0)-and(π,π)-pointford-ands -pairing).MFLetusnowlookhowthemean eldestimateofTcdependsonthecouplingstrength|U1|/W.InFig.6Tc/WisMFdisplayedasafunctionof|U1|/Wforthed-wavechannel.Firstofallwenotice,thatfor|U1|>W/4thevalueofTcMFisabout|U1|/4.For|U1|/W<<1thiscrossesovertoaquadraticdependencyTc=4|U1|2/W.Forcomparisonasimilarcurveisdisplayedforconventionals-wavepairing,usingthenegativeUHubbardmodelinabandwithaMFsquareDOS.Wenoticethatthemean eldtransitiontemperaturewiththelattermodelbecomesTc=|U0|/4forlarge|U0|(whichisactuallyoutsidetherangeofvalidityoftheBCSweakcouplingapproach[42,43]),andhasthefamiliarBCS-likeexp( W/|U0|)behaviourforsmallU0.TheTcfortheextendeds-wavepairingliesagainbelowthenegativeU0curve,andisonly niteaboveathresholdvalueof|U1|asdiscussedabove.Letus nallyturntothecollectivemodespectrum.Wecananticipate,thatagaind-wavephase uctuationsexistbelowtheparticle-holecontinuum.Inaddition,becausethereisanon-siterepulsiveU0,abranchofspin- uctuationscanbepulledbelowtheparticle-holecontinuum.InFig.7thecollectivemodespectrumisdisplayed,usingU1/(4t)= 0.5,andne=0.85,andwithU0/(4t)rangingfrom0to1.5.IntheplotforU0=0wealreadynotice,thattheparticle-holecontinuumhas8pointsink-spacewhereittouchesthehorizontalaxis:TheFermisurfacecrossesthenode-lineskx=±kyatthecoordinates(±(π δ)/2,±(π δ)/2),hencetheparticleholespectrumisgaplessfortheQ-vectors(0,±(π δ)),(±(π δ),0)and(±(π δ),±(π δ)).PreciselyfortheseQ-valuesthespin(andcharge)susceptibilityacquiresthelargestvalue,alsointhesuperconductingstate,henceifweswitchona nitevalueoftherepulsiveon-siteU0,aspin-densitywavestartstodevelopearoundthe(±π,±π)pointsontheFermisurface.Clearlytheground-stateisnolongeroftheformofEq.1,andthecorrectionsmaybecomestrongenoughtocompletelydestroysuperconductivity.As,ontheotherhand,thespin-densitywaveexistsaroundaportionoftheFermisurfacewherethegapiszero(andthereforecontributestheleasttotheground-stateenergy),whereasthemaximumgap-valueisatthe[±π,0]and[0,±π]points,theremayactuallybeacoexistanceofsuperconductivityandaspin-densitywaveindi erentportionsoftheFermisurface.

FromFig.7wecansee,thattheregiontakingpartintheformationofthespin-densitywavequicklyspreadsaroundthe(±(π δ)/2,±(π δ)/2)pointsifU0/(4t)increases,leavingasmallregionaround[±π,0]and[0,±π]fortheformationofasuperconductingcondensateifU0/(4t)=1.ThephasediagramforU0/(4t)=1,andU1/(4t)= 0.5isindicatedinFig.5b.TheshadedarearoughlyindicatestheregionwithaninstabilitytowardsaSDW.InprincipleamixedSDW-superconductingstatemayexistforallconcentrations.Itisnotpossibletodecidefromthenumericalresultspresentedabovewhetherornotthereisasharpphaseboundaryseparatingregionswithamagneticinstabilityfromsuperconductingregions.

The equations of motion of pair-like excitations in the superconducting state are studied for various types of pairing using the random phase approximation. The collective modes are computed of a layered electron gas described by a $t-t'$ tight-binding ban

V.CONCLUSIONS

Auni edapproachispresentedtothecalculationofthecollectivemodesofspin,charge,phaseandamplitudeinsuperconductorswithanon-trivialpairinginteraction.Theexpressionsforthedynamicalspin-andcharge-susceptibilitiesaregeneralizedtotakeintoaccountsuperconductivityatgeneralvaluesofmomentumandfrequency.Severalexamplesaretreated.Notablytheresponsefunctionsofalayeredchargedelectrongas,withapairinginter-actioninthed-wavechannelisconsideredintheabsenceandpresenceofanon-siteHubbardrepulsiveinteraction.Anincipientinstabilitytowardaspin-densitywavefollowsfromthesofteningofthecollectivemodespectrumnear =(π,π)inthed-wavepairedstate.Q

VI.ACKNOWLEDGEMENTS

ThisinvestigationwassupportedbytheNetherlandsFoundationforFundamentalResearchonMatter(FOM)with nancialaidfromtheNederlandseOrganisatievoorWetenschappelijkOnderzoek(NWO).

The equations of motion of pair-like excitations in the superconducting state are studied for various types of pairing using the random phase approximation. The collective modes are computed of a layered electron gas described by a $t-t'$ tight-binding ban

[36]

[37]

[38]

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[42]

[43]V.Emery,Phys.Rev.Lett.58,2794(1987).L.Jansen,PhysicaC156,501(1988).J.E.Hirsch,Phys.Rev.B43,11400(1991).P.Monthoux,A.V.Balatsky,andD.Pines,Phys.Rev.B46,14803(1992).W.E.Pickett,Rev.Mod.Phys.61,433(1989).Z.X.Shenetal.Phys.Rev.Letters70,1553(1993),andpaperscitedtherein.R.Micnas,J.Ranninger,andS.Robaszkiewicz,Rev.Mod.Phys.62,113(1990).D.vanderMarel,andJ.E.Mooij,Phys.Rev.B.45,9940(1992).

FIG.1.DiagramstakenintoaccountintheRPA.Exchangeselfenergy(a),particle-holemixing(a’),polarizationvertex(bandb’),exchangescattering(candc’),anddirectparticle-particlescattering(dandd’).Diagramsb’,c’andd’existonlyinthesuperconductingstate.

FIG.2.Collectivemodespectrumofasuperconductinglayeredelectrongas,assumings-wavepairing.Theparametersare:EF/(4t)=0.35,Ry/(4t)=4.0.Q⊥cisvariedwith0toπwithincrementsof0.2π(toptobottomsolidcurves).ThedashedcurvesaretheboundariesoftheregionofLandaudamping.(a)normalmetal,U=0and(b)superconductingstate,U/(4t)= 0.67.(c)Theamountofρ(solid)andφ(dashed)characterofthecollectivemodesinFig.2asafunctionofcollectivemodeenergy.TheinterruptionoccurwherethemodesbecomeLandaudamped.

FIG.3.PhasediagramintheU1-nplane,wherenisthenumberofelectronsperunitcell,witht′= 0.7andU0/(4t)=0.FIG.4.Phase- uctuatingcollectivemodeversusmomentumforalayeredelectrongaswithlongrangeCoulombinteractions(Ry =20),anon-siterepulsiveinteraction(U0/W=0.5),andanearestneighbourattractiveinteractionU1/W= 0.5.Thenumberofelectronsisne=0.2(a),ne=0.25(b),ne=0.3(c)andne=0.4(d).

FIG.5.(a)PhasediagramintheT-nplane,wherenisthenumberofelectronsperunitcell,witht′= 0.7tandU0/(4t)=0,andU1/(4t)= 0.5.(b)ThesamewithU0/(4t)=1

FIG.6.Solidcurve:Tc/|U1|calculatedforthed-wavechanneloftheexchange-onlymodelwitht′=0and1electronpersite.Thesamecurveisobtainedfort′= 0.7twith0.7electronpersite.Openlozenges:Tcofthes -wavechannelwiththelatterparameters.Dottedcurve:Tc/|U|versus|U|/WforthenegativeUHubbardmodeltakingasquareDOS.

FIG.7.Thecollectivemodesinthed-wavepairedstate,usingU1/(4t)= 0.5,andne=0.85,andwithU0/(4t)=0(a),0.5(b),1(c),and1.5(d).

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