Melting and transverse depinning of driven vortex

arXiv:cond-mat/0008125v1[cond-mat.supr-con]8Aug2000MeltingandtransversedepinningofdrivenvortexlatticesintheperiodicpinningofJosephsonjunctionarraysVer´onicaI.MarconiandDanielDom´ınguezCentroAt´omicoBariloche,8400SanCarlosdeBariloche,R´ıoNegro,ArgentinaWestudythenon-equilibriumdynamicalregimesofamovingvortexlatticeintheperiodicpinningofaJosephsonjunctionarray(JJA)forfinitetemperaturesinthecaseofafractionalorsubmatchingfield.WeobtainaphasediagramforthecurrentdrivenJJAasafunctionofthedrivingcurrentIandtemperatureT.Wefindthatwhenthevortexlatticeisdrivenbyacurrent,thedepinningtransitionatTp(I)andthemeltingtransitionatTM(I)becomeseparatedevenforafieldforwhichtheycoincideinequilibrium.Wealsodistinguishbetweenthedepinningofthevortexlatticeinthedirectionofthecurrentdrive,andthetransversedepinninginthedirectionperpendiculartothedrive.ThetransversedepinningcorrespondstotheonsetoftransverseresistanceinamovingvortexlatticeatagiventemperatureTtr.FordrivingcurrentsabovethecriticalcurrentwefindthatthemovingvortexlatticehasfirstatransversedepinningtransitionatlowT,andlaterameltingtransitionatahighertemperature,TMTtr.PACSnumbers:74.50+r,74.60.Ge,74.60.EcI.INTRODUCTIONThebehaviorofsuperconductingvorticesinthepres-enceofperiodicpinningshowsveryrichstaticanddy-namicphenomena.1–24Thecompetitionbetweenthere-pulsivevortex-vortexinteractionandtheattractivepe-riodicpinningpotentialresultsinnovelvortexstruc-turesatlowtemperatures.3,8–10,16,17Theequilibriumphasetransitionsofthesevortexstructuresandtheirvariousdynamicalregimeswhendrivenoutofequilib-riumareofgreatinterestbothexperimentally1–9,12andtheoretically.3,10–24Severaltechniqueshavebeendevel-opedtofabricateinsuperconductingsamplesanartifi-cialperiodicpinningstructure:thicknessmodulatedsu-perconductingfilms,1superconductingwirenetworks,2Josephsonjunctionarrays,3–5magneticdotarrays,6sub-micronholelattices,7,8andpinninginducedbyBitterdecoration.9Thegroundstatesofthesesystemscanbeei-thercommensurateorincommensuratevortexstructuresdependingonthevortexdensity(i.ethemagneticfield).Inthecommensuratecase,a“matching”fieldisdefinedwhenthenumberofvorticesNvisanintegermultipleofthenumberofpinningsitesNp:Nv=nNp.A“sub-matching”or“fractional”fieldisdefinedwhenNvisarationalmultipleofNp:Nv=fNpwithf=p/q.Oneofthemainpropertiesofperiodicpinningisthatthereareenhancedcriticalcurrentsandresistanceminimabothforfractionalandformatchingmagneticfields,forwhichthevortexlatticeisstronglypinned.InthecaseofJosephsonjunctionarrays3(JJA)thediscretelatticestructureofJosephsonjunctionsinducesaneffectiveperiodicpinningpotential(theso-called“egg-carton”potential)whichatlowtemperaturescon-finesthevorticesatthecentersoftheunitcellsofthenetwork.4Therearestrongcommensurabilityeffectsforsubmatchingfieldsf=p/q,forwhichthevorticesar-rangeinanorderedq×qsuperlatticethatiscommen-suratewiththeunderlyingarrayofjunctions.Thetran-sitiontemperatureTc(f)andthecriticalcurrentIc(f)havemaximaforrationalf=p/q,whichhavebeenobservedexperimentally.3,5Moreover,aswewilldiscusshere,themodelthatdescribesthephysicsoftheJJAcanbethoughtasadiscretelatticeLondonmodelforthinfilmsuperconductorswithperiodicarraysofholes.However,thiscomparisoncanbevalidonlyforlowsub-matchingfieldssinceitcannotdescribetheeffectsofinterstitialvortices.Theequilibriumphasetransitionsatfinitetempera-turesoftwodimensionalsystemswithperiodicpinninghavebeenstudiedinthepast.11–15ItispossibletohaveadepinningphasetransitionofthecommensurategroundstatesatatemperatureTpandameltingtransitionofthevortexlatticeatatemperatureTM.11–15FranzandTeitel14havestudiedthisproblemforthecaseofsub-matchingfields.For0TTpthereisapinnedphaseinwhichthevortexlattice(VL)ispinnedcommensurablytotheperiodicpotentialandhaslong-rangeorder.ForTpTTMthereisafloatingVLwhichisdepinnedandhasquasi-long-rangeorder.Forhighsubmatchingfields(f&1/30)bothtransitionscoincide,Tp=TM,whileforlowsubmatchingfields(f.1/30)bothtransi-tionsaredifferentwithTpTM.14,15Thenon-equilibriumdynamicsofdrivenvortexlatticesinteractingeitherwithrandomorperiodicpinningshowsaninterestingvarietyofbehavior.18–31Manyrecentstud-ieshaveconcentratedintheproblemofthedrivenVLinthepresenceofrandompinning.25–31Whenthereisalargedrivingcurrenttheeffectofthepinningpotentialisreduced,andthenatureofthefastlymovingvortexstructurehasbeenunderactivediscussion.Themovingvortexphasehasbeenproposedtobeeitheracrystallinestructure,amovingglass,amovingsmecticoramov-ingtransverseglass.25–27Thesemovingphaseshavebeenstudiedbothexperimentally28andnumerically.29–31Mo-tivatedbytheseresults,thedynamicalregimesofthemovingVLinthepresenceofperiodicpinninghasalsobecomeasubjectofinterest.18–24Atzerotemperature,thedynamicalphasesofvorticesdrivenbyanexternal1currentwithaperiodicarrayofpinningsiteshasbeenstudiedinverydetailbyF.Noriandcoworkers.18,19Acomplexvarietyofregimeshasbeenfound,particularlyforNvNpwherethemotionofinterstitialvorticesleadstoseveralinterestingdynamicalphases.Inthispaperwestudythedynamicalregimesofamov-ingVLintheperiodicpinningofaJosephsonjunctionarray(JJA)forfinitetemperaturesintheca