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arXiv:cond-mat/9512123v210Jun1996VorticesinathinfilmsuperconductorwithasphericalgeometryM.J.W.DodgsonandM.A.MooreTheoreticalPhysicsGroupDepartmentofPhysicsandAstronomyTheUniversityofManchester,M139PL,UK(June7,1996)WereportresultsfromMonteCarlosimulationsofathinfilmsuperconductorinasphericalge-ometrywithinthelowestLandaulevelapproximation.Weobservetheabsenceofaphasetransitiontoalowtemperaturevortexsolidphasewiththeseboundaryconditions;thesystemremainsinthevortexliquidphaseforallaccessibletemperatures.Thecorrelationlengthsaremeasuredforphasecoherenceanddensitymodulation.Bothlengthsdisplayidenticaltemperaturedependences,withanasymptoticscalingformconsistentwithacontinuouszerotemperaturetransition.Thiscontrastswiththefirstorderfreezingtransitionwhichisseeninthealternativequasi-periodicboundarycon-ditions.Thehightemperatureperturbationtheoryandthegroundstatesofthesphericalsystemsuggestthatthethermodynamiclimitofthesphericalgeometryisthesameasthatontheflatplane.Wediscusstheadvantagesanddrawbacksofsimulationswithdifferentgeometries,andcom-parewithcurrentexperimentalconclusions.Theeffectofhavingalargescaleinhomogeneityintheappliedfieldisalsoconsidered.74.20.De,74.76-wI.INTRODUCTIONTheexistenceandnatureofmeltingtransitionsintwo-dimensional(2D)systemsremainsatopicofinterestanduncertaintyinboththeoryandexperiments.Inpartic-ular,themeltingofthevortexlatticeinathinfilmsu-perconductorhasreceivedmuchattention.AcontinuousmeltingtransitionfromanAbrikosovlatticetoavortexliquidviaaphaseofhexaticorder,asintheKosterlitz-Thouless-Halperin-Nelson-Young(KTHNY)theory,wasfirstproposedoverfifteenyearsago.1A“transition”inthecurrent-voltageresponseofthinfilmsuperconduc-torshasbeenseeninseveralexperiments.2–4Thevortexsystembecomesdepinnedabovesometransitiontemper-ature(thedepinnedvorticeswilldissipateenergyastheymove,leadingtoanOhmicresponse)andtheriseoftheresistivityofthefilmispresumedtobecausedbythemeltingofthevortexlattice.However,itseemsthatthistransitiontemperaturevarieswiththestrengthofthepinningcentersinthethinfilms.ForfilmsofamorphousNbOthepinningisweakandnofeaturescanbedetectedinthecurrent-voltagecharacteristicswhichmaybeat-tributedtoa“transition”.ThishasbeeninterpretedasbeingduetotheabsenceoftheAbrikosovvortexlattice.5ThesuperconductingstatearisesfrommacroscopiccorrelationsinthephaseoftheCooperpairwavefunc-tion,orequivalentlytheGinzburg-Landauorderparame-ter.Thisphasecoherenceisusuallyaconsequenceofoff-diagonallongrangeorder(ODLRO),i.e.non-vanishinghψi,whereψisthesuperconductingorderparameter.IthasbeenpredictedthatthermalexcitationsoftheshearmodesofthevortexlatticeshoulddestroyODLROinthecrystallinestateandthelengthscaleoverwhichphasecoherencepersistshasbeenestimated.6,7O’NeillandMoorehaveperformedMonteCarlo(MC)simulationsofthinfilmsuperconductorswithinthelow-estLandaulevel(LLL)approximationusingaspheri-calgeometrytominimizefinitesizeboundaryeffects.7Nofreezingtransitionoutofthevortexliquidphasewasfoundinthetemperatureregioninvestigated.Thecorre-lationfunctionassociatedwiththepositionsofthevor-ticeswascalculatedandtheassociatedlengthscaleoftheselattice-likecorrelationswasfoundtodivergeonlyinthezerotemperaturelimit.ItwasnotedthatthishappenedonasimilarlengthscaleoverwhichtherewasexpectedtobephasecoherencefromtheargumentsofRefs.6and7.IntheworkofO’NeillandMoore,thesuperconductingfilmexistedonthesurfaceofasphere,andthisgeometryistheoneemployedhere.Themagneticfieldperpendic-ulartothesurfaceisgeneratedbyamagneticmonopoleplacedatthecenterofthesphere.8Thestrengthofthemonopolemustbequantizedtohaveawelldefinedsuper-conductingorderparameter,butthestrengthofthefieldatthesurfacemaybecontinuouslyvariedbychangingtheradiusofthesphere.TheLLLapproximation,whichassumesaninfinitemagneticscreeninglength,workswellinthe2DcasewheretheeffectivescreeninglengthisΛeff=2λ2/dbecauseofthemagneticfieldoutsidethesuperconductor.9Λeffbecomeslargeasthefilmthick-nessdisreduced;λisthebulkmagneticpenetrationlength.ItisoftenthecasethatΛeffisgreaterthanthefilmdimensionsforsmalld.Morerecently,strongevidenceforafirstordertran-sitionatafinitetemperatureinanalternativefinitesizeapproximationwithquasi-periodicboundarycon-ditions(QP)onaninfiniteplanehascomefromMCsimulations.10–14ThismodelhasaperiodicamplitudeoftheorderparameterandafixedchangeinthephaseasaunitcellcontainingNvorticesiscrossed(thechange1inphasedependsonthechoiceofgauge).Thequasi-periodicityinthephaseplacessomeconstraintsonthemotionofthevortices,butitisnotobviousthatthiswillchangethethermodynamicpropertiesofthesystem.Theobservedtransitiontemperatureiswithintherangeinvestigatedonthesphere.Thisraisesthequestionofwhetherornottherereallyisnotransitiononthesphere.Ifso,thenwhatarethereasonsthatdifferentboundaryconditionsgivedifferentresults?Ifthereisafreezingtransition,atleastontheplane,doesthiscontradictthetheoreticalpredictionofnoODLROatfinitetemperatures?Apparentlynot,astheworkofSasik,StroudandTesanovic14(SST)hascon-cludedthatinthethermodynamiclimiteventhelowtemperaturestatebelowthefirstordertransitionhasnoODLRO,despitethepresenceof“Braggpeaks”inthec