Phase-locking of driven vortex lattices with trans

arXiv:cond-mat/0105526v1[cond-mat.supr-con]28May2001Phase-lockingofdrivenvortexlatticeswithtransverseac-forceandperiodicpinningCharlesReichhardt1,AlejandroB.Kolton2,DanielDom´ınguez2,andNielsGrønbech-Jensen3,41AppliedPhysicsDivisionandCenterforNonlinearStudies,LosAlamosNationalLaboratory,NewMexico875452CentroAtomicoBariloche,8400S.C.deBariloche,RioNegro,Argentina3DepartmentofAppliedScience,UniversityofCalifornia,Davis,California956164NERSC,LawrenceBerkeleyNationalLaboratory,Berkeley,California94720(February1,2008)Foravortexlatticemovinginaperiodicarrayweshowanalyticallyandnumericallythatanewtypeofphaselockingoccursinthepresenceofalongitudinaldcdrivingforceandatransverseacdrivingforce.ThisphaselockingisdistinctfromtheShapirostepphaselockingfoundwithlongitudinalacdrives.Weshowthatanincreaseincriticalcurrentandafundamentalphaselockedstepwidthscalewiththesquareofthedrivingacamplitude.OurresultsshouldcarryovertoothersystemssuchasvortexmotioninJosephson-junctionarrays.PACSnumbers:74.60.Ge,74.60.JgI.INTRODUCTIONWhenanexternalacdriveisappliedtoadcdrivensys-teminteractingwithaperiodicpotential,resonancebe-tweentheexternalacfrequencyandthefrequencyofmo-tionintheperiodicpotentialcangiverisetophaselock-ing.Thisphenomenonisfoundinawidevarietyofnon-linearsystemsincondensedmatterphysics.Aparticu-larlywellknownexampleistheac/dc-drivensinglesmallJosephson-junction[1,2]andJosephson-junctionarrays[3],whereShapirostepsareobservedinthecurrent-voltagecharacteristics.Shapirotypephaselockinghasrecentlyalsobeenstudiedinexperiments[4]andtheory[5]forvortexmotionintwo-dimensionalsuperconduc-torswithperiodicarraysofpinningsites,aswellasinsuperconductorswherethevorticesaredrivenoverape-riodicpotentialgeneratedbythicknessmodulations[6].Moreover,drivensystemswithmanydegreesoffreedominthepresenceofquencheddisordercanexhibitphase-lockingwhenthereisadynamicallyinducedperiodicity.Examplesofthisarechargedensitywaves[7]andvortexlatticesinsuperconductorswithrandompinning[8,9].Inthecaseofvorticesinsuperconductorswithperiodicpining[4–6],avortexcanbeviewedasanoverdampedparticlemovingalongatiltedwashboardpotentialwherethetiltisproducedbythedcforcefromanappliedcur-rentaswellasbythesuperimposedaccurrentsinthesamedirection.Animportantdifferenceofvortexarraysfromchargedensitywavesystemsandsingledegreeoffreedomsys-tems(likeasmallJosephsonjunction)isthatthedis-placementfieldactingonvorticesisatwo-dimensionalvector.Thismeansthatdisplacementscanbeinducedintwodifferentdirections,andthereforeanewkindofphaselockingispossiblewhenanacforceisappliedtransversetothedirectionofthedcforce.Inthisworkweshowanalyticallyandnumericallyhowphaselockingphenomenamayoccurwhenvorticesaremovinginaperiodicpotentialandtheacforceisap-pliedtransversetothedirectionofthelongitudinallyap-plieddcforce.Thetypeofphase-lockingobservedinthiscaseisqualitativelydifferentfromtheShapirotypephase-lockingobservedwhendcandacforcesareinpar-allel.Wefirstshowanalyticallythepossibilityofthephaselockedstatesforcertaincommensuratevortexcon-figurationsandpredictthatthecriticalcurrentincreasesquadraticallywiththeacamplitudeandthatthewidthofsomeofthephase-lockedstepsscaleasthesquareoftheacamplitude.Wealsofindscalingofthecriticalcurrentaswellasthefirstphaselockedregionasafunctionoftheacfrequencyandthepinninggeometry.Thepredictionsoftheperturbationanalysisareconfirmedwithnumericalsimulations.Weanalyzethevalidityoftheperturbationapproachanddemonstratehowdeviationsoccur.Ourresultssuggestthatmorepronouncedtransversephase-lockingmaybeobservedatlowercommensuratemag-neticfields,suchasB=1.5BφorB=1.25Bφ.Thegeneralmodelandperturbationresultsareeasilygen-eralizabletoothersystemswhichexhibitShapirostep-likephaselockingsuchasJosephson-junctionarraysandFrenkel-Kontorovatypemodels.Ourstudiesaredirectlyrelevantforseveralcontempo-raryeffortssincevortexmatterinteractingwithnanos-tructuredpinningarraysofholes[10–12]anddots[13,14]hasbeenattractingincreasingattentionduetotheeas-ilytunablepinningproperties.Pronouncedcommensu-rationeffectsareobservedinthesesystemswhentheden-sityofvorticesmatchestointegerorfractionalmultiplesofthedensityofpinningsites.Inadditiontosquarepin-ningarrays,recentexperimentshavebeenconductedonrectangularpinningarrays[14].Simulationshaveshownseveralinterestingdynamicalphasesofdcdrivenvorticesinperiodicpinningsystems[15,16].Imaging[12]andtransportexperiments[11,4]alongwithsimulations[15]havefoundthatthevortexmotionabovethefirstmatch-ingfieldcanoccurbytheflowofinterstitialvorticesbe-tweenthepinningsites.Thesevorticesstillexperienceaperiodicpotentialcreatedbythevorticeslocatedatthepinningsites.Recently,phaselockingwasobservedfordcandacdrivenvorticesinteractingwithperiodic1pinningatB=2Bφ,whereBφisthefieldatwhichthenumberofvorticesequalsthenumberofpinningsites[4],wherethedcandacdrivewhereinthesamedirection.ThevaluesofthevoltageresponseinRef.[4]stronglysuggestthatisistheinterstitialvorticesthataremobilewhilethevorticesonpinningsitesremainimmobile.II.THEMODELWewilladoptasimpleoverdampedparticlemodelfordescribingvortices.Theover