A Model to Describe Transport Properties in $Bi_2S

arXiv:cond-mat/0004124v1[cond-mat.supr-con]8Apr2000AModeltoDescribeTransportPropertiesinBi2Sr2(CazPr1−z)Cu2O8+yE.C.Bastone∗,A.S.T.PiresandP.R.SilvaDepartamentodeF´ısica,InstitutodeCiˆenciasExatas,UniversidadeFederaldeMinasGerais,BeloHorizonte,CEP30.123-970,C.P.702,MG,Brazile-mail:erika@fisica.ufmg.brABSTRACTApseudo-spinmodelisproposed,asameanstodescribesometransportprop-erties(resistivityandHallmobility)inBi2Sr2(CazPr1−z)Cu2O8+y.Ourmodelisbasedinadouble-wellpotentialwheretunnelinginagivensiteandinteractionbetweendifferentlatticesitesareallowedonlythroughtheexcitedstates.DopingofthepuresystembytheadditionofPrincreasestheratiobetweentheactivationenergyandthetunnelingconstant.ThemodelHamiltoniandisplayssomefeatureswhicharepresentinthehydrogen-bondedferroelectrics.Itsdynamicsistreatedintherandomphaseapproximationandthecharacteristicfrequency(time)isusedinaDrudeformulainordertoobtainsometransportpropertiesofthesystem,namelytheelectricresistivityandtheHallmobility.ThequantitiescalculatedinthisworkarecomparedwiththeexperimentaldataofB.Beschoten,S.Sadewasser,G.G¨untherodtandC.Quitmann[Phys.Rev.Lett.77,1837(1996)].Pacsnumber:74.20.De,74.25.Dw,74.25.Fy,74.72.Hs1.IntroductionThecontroversybehindthemechanismwhichgovernshightemperaturesuperconductivity(HTS)incupratesisasubjectofcurrentinterest,ascanbeverifiedintherecentliterature[1,2,3].Anderson[2]attributesthenovelphenomenologypresentincupratesmaterialstoasecondkindofmetallicstate,namely,theLuttingerliquid.Zhang[3]hasproposedaSO(5)theoryofcupratesuperconductivitywhichconsidersthatthephenomenologyofthesematerialsmightbefundamen-tallyduetoaconflictbetweendifferentkindsoforder.Chianchi,MorettiandPiazza[4]havepresentedaphenomenologicalmodelfortheformationofCooperpairsincupratesuperconduc-tors:intheirdescription,couplingismediatedbylargeanharmonicoscillationsalongbcdirectionoftheapicalions.Ontheotherhand,Beschotenetal[5]haveinvestigatedthechargetransportininsulatingandsuperconductingsamplesofBi2Sr2(CazPr1−z)Cu2O8+y,intermsoftheHallmobility.ThemainmotivationbehindBeschotenetalwork[5]wasthattheinterplayoflocaliza-tionandsuperconductivityisoffundamentalinterestinthephysicsofelectrons(holes)instronglycorrelatedanddisorderedsystemssuchasthehigh-temperaturesuperconductors(HTSC).∗workpartiallydevelopedatDCNat-DepartamentodeCiˆenciasNaturais,FUNREI-Funda¸c˜aodeEnsinoSuperiordeS˜aoJo˜aodelRei.Pra¸caDomHelvecio,CEP:36.300-000,S˜aoJo˜aodelRei-M.G.Brazil1Theaimofthispaperistoproposeaphenomenologicalmodeltoexplaintheresistiv-ityandtheHallmobilitybehaviorsinferredfromBeschotenetaldata[5],performedinBi2Sr2(CazPr1−z)Cu2O8+y.ThestrategyofourworkisfirsttoproposeamodelHamiltoniantodescribetheBi2Sr2(CazPr1−z)Cu2O8+yandthentostudyitsdynamicsintherandomphaseapproximation(RPA).Afterdoingthat,weusetheobtainedcharacteristicfrequency(time)intheDrudeformula,inordertoobtaintheresistivityofthesystem.TheHallmobilitywillalsobestudiedbyusingtheaboveresults.GaonaandSilva[6],inspiredintheGorter-Casimir(GC)two-fluidmodelofsuperconductivity[7],aswellintheformalismusedtotreatthephasetransitionsinhydrogen-bondedferroelectrics[8],haveproposedaneffectivetransverseIsingmodel(TIM)todescribebothmetallicandHTSsuperconductivity.WethinkthatanextensionofthispreviousmodelcouldbeusefultodescribesomebasicfeaturesofBeschotenetal[5]experiment.Letusfirstmakesomeconsiderationsabouttwowell-establishedphenomenologicaltheoriesofthesuperconductivity.Oneofthemaincharacteristicsofthesuperconductingstateisthemacroscopicphasecoherenceofthewavefunction.In1950,GinzburgandLandau[9]introducedtheirphenomenologicaltheoryofsuperconductivity,wheretheorderparametermustbeidentifiedwiththemacroscopicwavefunctionΨ.TheystartedfromtheideathatΨrepresentssomeeffectivewavefunctionofthesuperconductingelectrons.Ψcanbenormalizedinsuchawaythat|Ψ|2isequaltotheconcentrationnsofsuperconductingelectrons.ItseemsthatthepioneeringtheorytotreatthethermodynamicbehaviorofthesuperconductingstatewasproposedbyGorterandCasimir(GC)[7].Thetwo-fluidmodelofGorterandCasimirwasproposedtoprovideabasisforunderstandingthethermodynamicofthesuperconductingstate.Inthisphenomenologicaltheoryitisassumedthattheelectronsinasuperconductoraredividedintotwointerpenetratinggases:thesuperconductingelectrons,whichhavezeroentropy,andthenormalelectrons,whichhavetheusualpropertiesofelectronsinanormalmetal.Thenormalelectronsgiveupthecondensationenergyβwhentheybecomesuperconducting.TheauthorsproposedthatbelowTcafractiony,whereydependsonthetemperaureToftheconductionelectronsoccupyasetoflower”condensed”energystateswhichareassociatedwiththesuperconductingproperties,whilethefraction(1−y)remainsuncondensed.Thefreeenergyfunctionforthewholeelectronicassemblyismadeupoftwoterms:one,FN,issimilartotheelectronicfreeenergyofanormalmetalandtheother,FS,correspondstothecondensationenergyassociatewiththecondensedstate,FN=−12γT2,FS=−β(1)whereγisthenormalelectronicspecificheatcoefficient,andβisaconstantrepresentingthecondensationenergy.GorterandCasimirassumedthatthephaseswerenotindependent,andproposedthefollowingfreeenergyfuncti