Monte Carlo calculation of the current-voltage characteristics of a two dimensional lattice

We have studied the nonlinear current-voltage characteristic of a two dimensional lattice Coulomb gas by Monte Carlo simulation. We present three different determinations of the power-law exponent $a(T)$ of the nonlinear current-voltage characteristic, $V

MonteCarlocalculationofthecurrent-voltagecharacteristicsofa

twodimensionallatticeCoulombgas

HansWeber1,MatsWallin2,andHenrikJeldtoftJensen3

1DepartmentofPhysics,Lule aUniversityofTechnology,S-97187Lule a,Sweden

arXiv:cond-mat/9601040v1 13 Jan 19962DepartmentofTheoreticalPhysics,RoyalInstituteofTechnology,S-10044Stockholm,SwedenofMathematics,ImperialCollage,LondonSW72BZ,UnitedKingdom3DepartmentAbstractWehavestudiedthenonlinearcurrent-voltagecharacteristicofatwodimen-sionallatticeCoulombgasbyMonteCarlosimulation.Wepresentthreedi erentdeterminationsofthepower-lawexponenta(T)ofthenonlinearcurrent-voltagecharacteristic,V～Ia(T)+1.Thedeterminationsrelyonbothequilibriumandnon-equilibriumsimulations.We ndgoodagreementbe-tweenthedi erentdeterminations,andourresultsalsoagreecloselywithexperimentalresultsforHg-Xethin lmsuperconductorsandforcertainsin-glecrystalthin- lmhightemperaturesuperconductors.

TypesetusingREVTEX

We have studied the nonlinear current-voltage characteristic of a two dimensional lattice Coulomb gas by Monte Carlo simulation. We present three different determinations of the power-law exponent $a(T)$ of the nonlinear current-voltage characteristic, $V

I.INTRODUCTION

Intwodimensionsthesuperconductingtransitioninzeromagnetic eldisaKosterlitz-

Thoulesstransition.[1–3]Thishasbeenveri edovertheyearsinbothexperiments[4]andin

manymodelsofsuperconductorsliketheXY,Villain,andCoulombgasmodels[1,5,6].The

importantdegreesoffreedominasystemundergoingaKosterlitz-Thoulesstransitionare

thermallyexcitedvortexpairs.TheKosterlitz-Thoulesstransitionissometimesalsoreferred

toasavortexunbindingtransition,asfortemperaturesbelowthetransitiontemperature

Tcallvorticesareboundinneutralpairs.ThesepairsstarttounbindatandaboveTc.

AtypicalwaytolookforaKosterlitz-Thoulesstransitioninexperimentsonthinsu-

perconducting lmsistoprobethecurrent-voltage(IV)characteristic[4,7,8].Boththe

linearandthenonlinearIVcharacteristicshavespeci c ngerprintsidentifyingaKosterlitz-

Thoulesstransition.VorticesdeterminetheIVcharacteristicforthefollowingreasons:If

avortexisdraggedacrossthesystemavoltageisinduced.Henceresistanceiszeroonlyif

therearenovorticesavailabletomoveacrossthesystem,andonlythenthesystemistruly

superconducting.Vorticesthatareboundinneutralpairsareunabletomovefreelyandto

causedissipation.Howeveranexternalappliedin-planesupercurrentyieldsaperpendicular

Lorentzforceactinginoppositedirectiononvorticeswithdi erentvorticity.Thisgivesa

net uxofvorticesacrossthesystem,whichshowsupasnonlinear(i.e.currentdependent)

resistance.

BelowtheKosterlitz-Thoulesstransitiontemperatureallvorticesareboundinneutral

pairsbythelogarithmicvortexinteraction,andthelinearresistanceisthuszero.Therefore

thesystemsuperconductsbelowtheKosterlitz-Thoulesstransition.Thelinearresistance

dropstozeroattheKosterlitz-Thoulesstransitionwithanexponentialfunctionalform,

R～ξ 2withlnξ～|T Tc| 1/2[3].Thisisconsistentwithexperiments,althoughthe

logarithmisacomplicationforquantitativecomparisonbetweentheoryandexperiment.A

niteappliedcurrentgivesapower-lawnonlinearIVcharacteristicoftheformV～Ia(T)+1.

Thecriticalcurrentisthuszero.AttheKosterlitz-ThoulesstransitiontheIVexponenta(T)

We have studied the nonlinear current-voltage characteristic of a two dimensional lattice Coulomb gas by Monte Carlo simulation. We present three different determinations of the power-law exponent $a(T)$ of the nonlinear current-voltage characteristic, $V

assumestheuniversalvalue2,soV～I3atT=Tc.ForT<Tconehasa(T)>2,andfor

T>Tconehasa(T)=0(forsmallenoughcurrents)[9].Experimentson,forexample,thinHg-Xealloy lms[4]andalsoforcertainsinglecrystalhightemperaturesuperconductors

[7,8],amongsome,havecon rmedthis.

SinceIVcharacteristicsarehardtocalculateanalyticallycomputersimulationisauseful

tool.IVcharacteristicsofvortexsystemshaverecentlybeencalculatedsuccessfullywith

MonteCarlosimulations[10].LinearandnonlinearIVcharacteristicsofvortexglasssuper-

conductorshavebeenreportedinRefs.[10,11].InarecentMonteCarlosimulationofthe

CoulombgasthelinearresistancewasusedtolocatetheKosterlitz-Thoulesstransition[12].

ThenonlinearIVcharacteristicsattheKosterlitz-Thoulesstransitionhasbeencalculated

inRef.[13],anda nite-sizescalinganalysisaccuratelyveri edtherelationV～I3atthe

Kosterlitz-Thoulesstransition.

InthispaperwestudytheIVcharacteristicsofalatticeCoulombgasmodelbyMonte

Carlosimulationsofvortexdynamics.WecalculatetheIVexponenta(T)oftheCoulomb

gasinthreedi erentways:(1)BydirectMonteCarlocalculationofthenonlinearresistance,

(2)byaselfconsistentlinearscreeningconstructionfortheenergybarrierforcurrentin-

ducedvortex-pairbreakinggivingthermallyactivatedresistance,and(3)bya nitescaling

constructionfromdataforthelinearresistance.AllmethodsarebasedonMonteCarlo

simulations,andweapplybothequilibriumandnon-equilibriumsimulations.Thesethree

methodsgivethesameresults,givingusaconsistentandsimplepictureofnonequilibrium

responseinthissystem.Furthermore,wecompareourresultsfora(T)withexperiments.

Scalingargumentsgivethata(T)isauniversalscalingfunctionofareducedCoulombgas

temperatureX=T/Tc,andthisisveri edinexperiments[3].We ndcloseagreement

betweenourMonteCarloresultsandtheexperimentaluniversalscalingcurve,andthis

comparisonappearstobepresentedhereforthe rsttime.Theagreementbetweendif-

ferentmethods,andbetweenoursimulationsandexperiments,arethemainresultsofour

paper.SomeofourMonteCarloresultsforthenonlinearIVcharacteristicshavebeen

obtainedpreviously[13],asexplainedabove.

We have studied the nonlinear current-voltage characteristic of a two dimensional lattice Coulomb gas by Monte Carlo simulation. We present three different determinations of the power-law exponent $a(T)$ of the nonlinear current-voltage characteristic, $V

Thepaperisorganizedasfollows:InSectionIIwede nethelatticeCoulombgas

model.InSectionIIIwestudyvariousapproachestotheIVcharacteristics.InSectionIV

wedescribeourMonteCarlomethodsforcalculatingIVcharacteristics.InSectionVwe

presenttheMonteCarloresults.SectionVIcontainsdiscussionandconclusions.

TTICECOULOMBGAS

Ausefulstartingpointforcalculationswithsuperconductorsinthepresenseofcurrents

and eldsistheGinsburg-Landaumodel,withtheorderparameterΨ(r)=|Ψ(r)|eiφ(r)

describingthesuperconductingorderofthesystem.However,thismodeldoesnotfocus

particularlyonvortexdegreesoffreedom.Thevorticesconstitutetheessentialdegreesof

freedomneartheKosterlitz-Thoulesstransition.AnapproximationtotheGinsburg-Landau

modelwhichfocusesonlyonthevorticesisgivenbytheCoulombgasmodel.Herethermal

uctuationsinthemagnitudeofΨareneglected,sincetheyarerelevantonlyclosetothe

mean- eldtransitiontemperature,whichisassumedtobewellabovethevortextransition

temperatureTc.Inoursimulationsthemodelisdiscretizedandputonalattice.The

approximationmadeinthelatticediscretizationwillonlya ecttheshortrangebehaviorof

thevortices,asthelatticede nesthesmallestpossibleseparation.Thecriticalproperties

willhowevernotbee ected.Ingeneral,largelengthscalepropertiesshouldbereasonable

modeledbythelatticeCoulombgasclosetoTc.

ThelatticeCoulombgas[14,15]isde nedbythepartitionfunctionZonasquarelattice

ofsidelengthLusingperiodicboundaryconditions:

Z=Trnexp[ β(H µN)]

H=1(1)

We have studied the nonlinear current-voltage characteristic of a two dimensional lattice Coulomb gas by Monte Carlo simulation. We present three different determinations of the power-law exponent $a(T)$ of the nonlinear current-voltage characteristic, $V

gastemperature[3].Thetraceisoverni=0,±1onallsitesi,subjecttooverallneutrality,

ini=0.GijisthelatticeGreen’sfunctionforthelogarithmic2Dvortexinteraction,

Gij=1

2 cos(kx) cos(ky),(4)

wherekarethereciprocallatticevectors,kx,ky=2πn/L,n=0,...,L 1.

WewillcalculatetheresponsevoltagetoanappliedcurrentimposedontheCoulomb

gas.Theabovede nitiondoesnotincludeanynetcurrents.Howtoincludethemandto

calculateIVcharacteristicsbyMonteCarlosimulationisdescribedinthenextsection.

III.CURRENT-VOLTAGECHARACTERISTICS

Inthissectionwediscussvariousaspectsandapproachestothecurrent-voltagecharac-

teristicsof2DsuperconductorsclosetotheKosterlitz-Thoulesstransition.

A.Linearresistance

Abasicexperimentonasuperconductoristomeasurethelinearresistance.Suchmea-

surementsonthin lmsofbothconventionallow-Tcsuperconductors[4]andsinglecrystal

high-Tcmaterials[7,8],havebeensuccessfullyinterpretedintermsofthermallyexcited

vortex uctuationsanalyzedbyuseoftheCoulombgas[3].

Thelinearresistivityisde nedbyρ=E/jforj→0,wherejistheappliedsupercurrent

densityandEistheresultinginducedelectric eld.Somewordsaboutnotation:Since

resistanceandresistivityhavethesamedimensionintwodimensionsandoursystemis

homogeneous,theyarethesame,andtheywewillbothbedenotedbyR.Rwillbe

reservedforlinearresistance,andwillnotbeusedtodenotenonlinearresistance.An

appliedsupercurrentisdenotedbyI=jL,andvoltageisV=EL.

TodeterminethelinearresistanceinsimulationsoftheCoulombgasfromE/jforsmall

jhasitslimitations,aswehavetorepeatthecalculationatasequenceofcurrentdensities

j,tomakesurethatjissmallenoughtobeinthelinearregime.Ifthepurposeisto

We have studied the nonlinear current-voltage characteristic of a two dimensional lattice Coulomb gas by Monte Carlo simulation. We present three different determinations of the power-law exponent $a(T)$ of the nonlinear current-voltage characteristic, $V

measureonlythelinearresistance,andnotEasfunctionofj,adi erentapproachisto

usetheNyquistformula[16],

whichrelates

the

linear

resistancetotheequilibriumvoltage

uctuations:

R=1

2T

GiventheJosephsonrelationweseeimmediatelythattheKuboformulaequalstheNyquist

relation.

Thelinearresistancehasbeensuccessfullyusedinasimulation[12]tolocatethe

Kosterlitz-ThoulesstransitiontemperatureTcofthe2DlatticeCoulombgas.They nd

the nitesizescalingrelationatTc:

L2R1+ ∞ ∞dt Iv(t)Iv(0) canbeused.1

dt～τ 1,where φisthegradientofthephaseoftheGinsburg-Landau

orderparameter[18].Therefore,weexpectthelinearresistance,Eq.(5),toscalelike

R～ξ 2atTc.AtTcthecorrelationlengthdivergesandiscutofbythe nitesizeLofthe

latticeandhenceRL2=constatTc,tolowestorder.Thescalingrelationhasalogarithmic

correctionwhichhasbeenincludedinEq.(6).Thiscorrectionisreadilyobtainedfromthe

correspondingcorrectiontermsfor1/ andλ[19].

We have studied the nonlinear current-voltage characteristic of a two dimensional lattice Coulomb gas by Monte Carlo simulation. We present three different determinations of the power-law exponent $a(T)$ of the nonlinear current-voltage characteristic, $V

B.Thermallyactivatedresistance

Theabovescalingargumentledtoa nitesizescalingformulawhichisusefulforlocating

thetransitiontemperaturefromMonteCarlodatafortheresistanceof nitesamples.Here

wewilldoamoredetailedanalysisthatwillalsoleadtothesameformula.Theanalysishere

doesnotdirectlyinvolvescalingarguments,butconsiderstheinteractionsbetweenvortices

intheCoulombgas.Theanalysiswillgiveexpressionsfortheresistancefromthermally

activatedfreevorticesintheCoulombgasinthepresenceofanappliedsupercurrent.This

moredetailedanalysiswillbeusefulinlatersectionswhenweanalyzeMonteCarlodatafor

theCoulombgas.

AccordingtotheJosephsonrelationthevoltageVcausedbyvortexmotionis

V～d φ

I～nF(7)

Tomakeanestimateofthedensityoffreevorticesweproceedbythefollowingsimplemodel.

TheenergyE(r)ofavortexpairofseparationr>r0inthepresenceofacurrentIis[17]:

E(r)=E0+E1ln(r

2π

(k)

We have studied the nonlinear current-voltage characteristic of a two dimensional lattice Coulomb gas by Monte Carlo simulation. We present three different determinations of the power-law exponent $a(T)$ of the nonlinear current-voltage characteristic, $V

Hereλisthevortexscreeninglength,and (k)isthepartofthedielectricfunction, (k),

describingthepolarizationoftheboundpairs.Thetwo arerelatedby

1

(k)k2

2

2πVl(k)eik·r.

WecanobtainanapproximateexpressionforVl(r)bymakinguseofthefactthat (k)only

dependsweakly(inmostofkspace)onk.Foragivendistancer,theFourierintegralpicks

upitsmaincontributionfromthekvaluesaround2π/r.Hence

Vl(r) Vl(r=1)≈ 1

(2π/r)ln(r/λ)(9)

whereweuse insteadof ,asthetemperatureisbelowTc.

Accordingtothisdiscussionthecoe cientE1isgivenby[20]

E1=1

We have studied the nonlinear current-voltage characteristic of a two dimensional lattice Coulomb gas by Monte Carlo simulation. We present three different determinations of the power-law exponent $a(T)$ of the nonlinear current-voltage characteristic, $V

TheenergyE(r)inEq.(8)hasa

maximumatseparationr =E1/Iandtheenergy

neededtoseparateavortexpairtothisdistanceis[4]:

E=E(r ) E(r=1)=E1ln(r ) I(r 1)

E=E1ln(E1(11)

nF.

AssumingthatΓisdeterminedbyactivationoverthebarrier Ewegetthefollowing

productionrate[17]

Γ∝e E

2T∝

∝

2Te 1I) E1+I) E12TeE1

keepingtheimportanttermforsmallbut niteIwearriveat:

R∝ E12T(15)

AgivencurrentIgivesrisetoa“currentlengthscale”r fromthemaximumconditionin

Eq.(8).Asthelatticeofthesystemhasa nitesize,thissetsanupperlimittothe“current

length”andhencealowerlimittothecurrentproducingnonlinearresistance.Thesmallest

currentgivingnonlinearresistanceisI =E1/r withr =Landhenceforcurrentssmaller

We have studied the nonlinear current-voltage characteristic of a two dimensional lattice Coulomb gas by Monte Carlo simulation. We present three different determinations of the power-law exponent $a(T)$ of the nonlinear current-voltage characteristic, $V

thanI theresistancewillbecuto bythe nitesizeLofthelatticeandtheresistance

becomesohmic.TheNyquistresistanceiscalculatedwithI=0andhence

R∝ 12T.(16)

ThismeansthatwecanscalethelinearresistanceRfromtheNyquistrelationEq.(5)withtheexponentE1/2T.Thisexponentispreciselya(T),theexponentofthenonlinear

IVcharacteristics(seeEq.(18)below),hence:

f(T)=RLa(T)(17)

shouldcollapseontoasinglecurvefordi erentlatticesizesL.I.e.f(T)shouldnotdepend

onlatticesizeL.Theresistanceweuseforthisscalingwillbetheonedeterminedfromthe

voltage uctuationsEq.(5).Theexponentdeterminedfromresistancedataatzerocurrent

willbedenotedaR(T).

C.NonlinearIVexponent

Wearegoingtomakeuseofacoupleofdi erentexpressionsforthepowerlawexpo-

nenta(T)ofthenonlinearIVcharacteristics.

characteristic:

V∝ E12TFromEq.(15)wegetthenonlinearIVI∝Ia(T)+1(18)

TheexponentcalculatedbymonitoringthevoltageresponseVasafunctionofanapplied

supercurrentIwillbedenotedaIV(T).Ona nitesystemwewillobtainanonlinearvoltage

responseonlyabovea niteappliedcurrent,givenbyI ～E1/L,suchthatthecurrentlength

r isshorterthanthesizeLofthesystem,asdiscussedabove.

D.SelfconsistentIVcharacteristic

AnotherexpressionfortheIVcharacteristicisobtainedifweincludetherdependence

inE1inEq.(10).Thelengthdependencecanina rstapproximation(inanexpansionin

We have studied the nonlinear current-voltage characteristic of a two dimensional lattice Coulomb gas by Monte Carlo simulation. We present three different determinations of the power-law exponent $a(T)$ of the nonlinear current-voltage characteristic, $V

derivativesofE1(r))beincludedsimplybyreplacingE1inEq.(15)by1/ (2

π/r )intheextremumequationI=E1/r .Ourrationaleforthischoiceisthatattheseparationr the

vortexpairisbrokenapartandwethereforeusethesti ness1/ (r)ofthesystematthis

separation.We ndtheappropriate (r)bysolvingselfconsistentlytheequation

I=1

(k )2π(19)

Theselfconsistent obtainedbysolvingEq.(19)willbedenoted .Therelationbetween

theexponenta(T)andthedielectricfunction isaccordingtoEqs.(15)and(10)givenby

theexpression(seeAmbegoakaretal.[17])

a(T)AHNS=1

T 2(21)

AsoneimmediatelyrealisesEq.(21)isnotconsistentwiththeactivationargumentused

toderiveEq.(20).InordertoreconcileEq.(21)witharateequationlikeEq.(13)

Minnhagenetal.havemadethefollowingsuggestion.Theyassumethattheactivationis

correctlyrepresentedbyΓinEq.(14).Therecombination,whichinEq.(13)isrepresented

1+bbytheinnocentlylookingtermn2F,isontheotherhandsupposedtobereplacedbynF

withb=2/(E1/T 2).Thesoleargumentforthisreplacementisunfortunatelysofar

simplytheobservationthatonethencanderiveEq.(21)fromanequationlikeEq.(13).

Nonetheless,weshallseebelowthatfortemperaturesbelowTcEq.(21) tsthesimulation

datamuchbetterthanEq.(20)does.Howeveramotivationforarecombinationterm

di erentfromtheoneinEq.(13)hasnotbeenpresented.AtTcbothrelationsreproduce

thesameexponenta(T=Tc)=2.

We have studied the nonlinear current-voltage characteristic of a two dimensional lattice Coulomb gas by Monte Carlo simulation. We present three different determinations of the power-law exponent $a(T)$ of the nonlinear current-voltage characteristic, $V

IV.MONTECARLOSIMULATION

Inthissectionwedescribehowwecalculatecurrent-voltagecharacteristicsbyMonte

CarlosimulationofthelatticeCoulombgas.

ThealgorithmtosimulatethelatticeCoulombgasworksasfollows[15]:Firstwepick

anearest-neighborpair(i,j)oflatticesitesatrandom.Thenwetrytoincreasenibyone

andtodecreasenjbyone,thuspreservingoverallvortexneutrality, ini=0.ThisMonte

Carlomoveofinsertinganeutralpairwillbeinterpretedastransferofoneunitvortexfrom

sitejtoi.Iftheenergychangeis Eweacceptthistrialmoveaccordingtothestandard

Metropolisalgorithm[22]withprobabilityexp( E/T).ThesesimpleMonteCarlomoves

canbothcreate,annihilate,andmovevortices.Thermodynamicaveragesarecomputedas

MonteCarlotimeaveragesoverthesequenceofgeneratedcon gurations.

TocalculateIVcharacteristicsworksasfollows[10,11]:Anappliedcurrentdensityj

givesaLorentzforceofjh/(2e)onaunitvortex.TheLorentzforcecanbeincorporatedin

theMonteCarlomoves[10]byaddingto Eanextratermjh/(2e)iftheunitvortexmoves

inthedirectionoppositetotheLorentzforce,subtractingthistermifitmovesinthesame

direction,andmakingnochangein Eifitmovesinaperpendiculardirection.Biasingthe

MonteCarlomovesinthiswaytakesthesystemoutofequilibriumandcausesanet uxof

vorticesinadirectionperpendiculartothecurrent.Thisgeneratesavoltagegivenbythe

Josephsonrelation:

V=h

We have studied the nonlinear current-voltage characteristic of a two dimensional lattice Coulomb gas by Monte Carlo simulation. We present three different determinations of the power-law exponent $a(T)$ of the nonlinear current-voltage characteristic, $V

equilibriumvoltage uctuationsintheabsenceofanynetcurrents.FordiscreteMonte

Carlotimeitisgivenby[16,23],inourunits,

R=1

(k)=1 2π

We have studied the nonlinear current-voltage characteristic of a two dimensional lattice Coulomb gas by Monte Carlo simulation. We present three different determinations of the power-law exponent $a(T)$ of the nonlinear current-voltage characteristic, $V

jumpcriteriontellsusthat1/ (k=0)jumpsfrom4TcatT=Tc to0atT=Tc+[9].Apracticaldi cultyforlocatingTcfromMonteCarlodataonsmalllatticeswiththis

procedureisthatextrapolationtothek=0limitrequireslargelattices,asthesmallest

nonzerokis2π/L.Thecorrespondingquantityto1/ (k=0)inthetwodimensionalXY

modeliscalledthehelicitymodulusγ[5].Bothquantitieshavebeenusedtolocatethe

Kosterlitz-ThoulesstransitiontemperatureinMonteCarlocalculations[5,25,19].

InthedataanalysisinthenextsectionweuseanalternativeproceduretolocateTcfrom

thelinearresistance[12].WeobtainthelinearresistanceRfromtheNyquistformulain

Eq.(5)forasequenceofsystemsizesLandtemperaturesT.AccordingtoEq.(6)data

forL2Rfordi erentsystemsizesshouldbecomesystemsizeindependentatthecritical

temperature,whichisourcriteriontolocateTc.Thiscircumventsthedi cultiesofusing

1/ (k=0).ThetwodeterminationsgivewithinerrorbarsthesamevalueforTc.

V.RESULTS

WepresentMonteCarlosimulationresultsforthreedi erentdeterminationsoftheIV

exponenta(T)forthe2DCoulombgasmodel.Theresultsweshowhereareforthechemical

potentialµ=0.0andlatticesizeL=32ifnotdi erentlystated.

A.NonlinearIVexponent

Our rstmethodconsistsinadirectmeasurementoftheelectric eldEinducedbyan

appliedcurrentdensityj.InFig.1aresultsfortheIVcharacteristicofthetwo-dimensional

Coulombgasareshown.Thedashedlineintheln(E)versusln(j)plothasslopethree

andrepresentstheslopeatT=Tcaccordingtotheuniversaljumpcondition[9].The

solidcurvesrepresentresultsfordi erenttemperatures.Forveryhighcurrentthevoltage

responsesaturates.Thisisbecausewhenallattemptstomovethevorticesinthedirection

oftheLorentzforcearealreadyaccepted,furtherincreasingthecurrentcannotgivemore

voltage.Forlowenoughcurrentthereisacrossovertoohmicresistance,whenthecurrent

We have studied the nonlinear current-voltage characteristic of a two dimensional lattice Coulomb gas by Monte Carlo simulation. We present three different determinations of the power-law exponent $a(T)$ of the nonlinear current-voltage characteristic, $V

lengthequalsthesystemsize,andthenonlineardependenceoftheresistanceonthecurrent

vanishes.Theregimewhereweprobethenon-linearIVcharacteristicisforthis gure

approximatelyfromlnj≈ 1.5upto≈ 0.5.AccordingtotheKosterlitz-Thoulesstheory

theslopeofthelinesshouldbe3atthecriticaltemperature,andthiscriteriacanbeusedto

determineTc.Wewillhoweveruseanindependentdetermination[12]ofTcforthissystem,basedonthe nitesizescalingrelationEq.(6).

InFig.1bwedemonstratethee ectsofthe nitelatticesizeforlowdrivingcurrents

attemperaturesbelowTc.ThedatashownareforT=0.15andlatticesizesareL=8

(triangles),12(opensquares),16(stars),24(opencircles),and32( lledcircles).

The nitesizee ectsforthelowertemperaturescanbeunderstoodinthefollowingway.

(SeethediscussionabovefollowingEq.(15).)The nitelatticesizeisimportantbecause

pairexcitationoverthebarriergivenbytheperiodicitylengthLwilladdtothedissipation

duetounbindingofpairsoverthebarriergivenbythepairsizer .Theinducedelectric

eldwillaccordinglybeoftheform

E=R(L)j+constantjE1/2T+1,(26)

wherethe rsttermR(L)followsfromEq.(16)andR(L)→0asL→∞.Thesecond

terminEq.(26)isgivenbyEq.(15)andwillremain niteinthelimitL→∞.Thisis

clearlydemonstratedinFig.1bwhereweseethatthecrossoverinEq.(26)betweenthe

linearandnonlinearregimeappearsatahigherdrivingcurrentforthesmaller8×8lattice

asthecurrentlength(E1/I=ξI～L)associatedwiththecurrentdensityjexceedsthesize

ofthelattice.

InFig.2theexponentaIV(X)isshownasafunctionofthereducedtemperature

X=T/Tc.ThedashedhorizontallinerepresentstheuniversaljumpconditionforaIV(X).

TheplussesrepresentexperimentaldatafromasuperconductingHg-Xe lm[4,26].The

lledcirclesaretheresultsforaIV(T)fromFig.1.Theotherthreedatasetsarefor

latticesizesL=16(stars),24(opencircles),and48(triangles).Asonecanseethereare

noapparent nitesizee ectsinthedata.Inthevicinityofthecriticaltemperaturethe

We have studied the nonlinear current-voltage characteristic of a two dimensional lattice Coulomb gas by Monte Carlo simulation. We present three different determinations of the power-law exponent $a(T)$ of the nonlinear current-voltage characteristic, $V

experimentaldataarereproducedbytheMonteCarlosimulations.

ThereducedtemperaturevariableXusedinFig.2fortheexperimentisalsofromRef.[26]andforthelatticeCoulombgasdataweuseTc=0.218[12]determinedfroma nite

sizescalinganalysisusingEq.(6).

TheinsetinFig.2showsaselectionofexperimentaldataanalyzedalongthelines

describedinRef.[26].Thedataintheinset(plusses)arethesameasinthemain

gure,theotherdataareforBi2Sr2CaCu2Oxsinglecrystal( lledsquares)[7],andfor

Bi1.6Pb0.4Sr2Ca2Cu3Oxsinglecrystal(opensquares)[8].

TheMonteCarlodatapresentedherefora(X)areallforµ=0.0.Wealsodidthesame

analysisforMonteCarlodataforL=32andµ= 0.4, 0.2and0.2.Theclosest ttothe

experimentalresultsisproducedbyµ=0.0.Resultsfordi erentµdi erfromtheµ=0.0

results,bythatµ=0.2hasaslightlylargerderivativeatX=1andthesmallerµare

correspondinglylesssteep.

B.SelfconsistentIVcharacteristic

Inourseconddeterminationoftheexponenta(T)wewillmakeuseoftherelations

between anda(T)inequations(20-21).Theanalysisisbasedontheselfconsistent

solutionofEq.(19).Foragivencurrentdensityj,asetof (k)willbecalculatedfor

di erenttemperatures.TheselfconsistentsolutiontoEq.(19)for isshowninFig.3,

in(a)dataforT=0.18isshownandin(b)T=0.24.Thesolidanddashedstraightlines

represent =k /j2π,givenbyEq.(19),fordi erentcurrentdensitiesj.Theopencircles

represent (k )asafunctionofkfordi erentcurrentdensities.Thechoiceofthedirection

alongwhich (k )isprobedisperpendiculartothecurrentdensityj,ie.paralleltothe

vortexdriftcausedbythecurrentdensityj.Theintersectionbetweena (k )curveandthe

correspondingstraightlineisthesolutiontoEq.(19),thesearemarkedwith lledcircles.

InFig.3aforT=0.18<Tc=0.218weseethattheselfconsistentsolutiondependsonly

weaklyonthechoiceofprobingcurrentaslongasthecurrentisnottoolarge.InFig.3b

We have studied the nonlinear current-voltage characteristic of a two dimensional lattice Coulomb gas by Monte Carlo simulation. We present three different determinations of the power-law exponent $a(T)$ of the nonlinear current-voltage characteristic, $V

forT=0.24>Tcwesee,however,thatthereisnowellde nedlimitingsolutionfor as

j→0.ThisisbecausethesystemisabovetheKosterlitz-Thoulesstemperatureandvortex

pairswillalwaysdissociateirrespectiveofj.InFig.4thefunction isshownasafunctionoftemperature.Thesolidcirclesrepresent fromtheselfconsistentEq.(19)forthe xed

currentdensityj=0.03125.ThedatashownhererepresentstheconstructionshowninFig.

3.

Theresultsfromtheselfconsistentsolutionfor areanalyzedinFig.5.Herethe lled

circlesrepresenttheexponentaIV(T)fromFig.2.Theupsidedowntrianglesrepresent

aAHNS(T)fromEq.(20)withthesolutionfromFig.3andthetrianglesarethecorre-

spondingsolutiontoEq.(21).OnecanclearlyseethattheexpressioninEq.(21),derived

byMinnhagenetal.[21],reproducestheexponentaIV(T)forT<Tc.Notehowever,itis

onlyacoincidencethatEq.(20),derivedbyAmbegoakaretal.[17],worksfortemperatures

aboveTcinthis gureasthelimiting(j→0)solutionfor1/ isnotwellde nedforthese

temperatures,asalreadydiscussedinconnectionwithFig.3b.Asthesimulationdata

aIV(T)( lledcircles)alsomatchedtheexperimentaldatainFig.1wemustconcludethat

belowTctheinterpretationaccordingtoEq.(21)isclearlythemoreappropriate.

C.LinearResistance

Wewillnowturntoourlastdeterminationoftheexponenta(T).Theresultspresented

aboveallreliedonnon-equilibriumMonteCarlosimulations,i.e.witha niteapplied

supercurrentdensityj.Wewillnowpresenttheequilibriumdeterminationforj=0basedon

nitesizescalingofMonteCarlodataforthelinearresistancegivenbytheNyquistformula

(5)togetherwithEq.(17).InFig.6wedemonstrateadatacollapseofthelinearresistance

forseverallatticesizes.FromEq.(17)weseethatthelinearresistancedatacanbe

collapsedontoasinglecurve,thusrepresentingthethermodynamiclimit,byanappropriate

choiceateachtemperatureToftheexponentaR(T).Wedothisinthefollowingway.For

agiventemperaturewe ndtheexponentaRwhichminimizestheerrorofthe tde ned

We have studied the nonlinear current-voltage characteristic of a two dimensional lattice Coulomb gas by Monte Carlo simulation. We present three different determinations of the power-law exponent $a(T)$ of the nonlinear current-voltage characteristic, $V

as

TheobtainedscalingexponentaRasafunctionoftemperatureisshowninFig.7.AsacomparisonwealsoshowdataforaIV(T)fromFig.2,obtainedfromdirectevaluation

oftheIVcharacteristic.The nitesizescalinganalysisinFig.6breaksdownforlow

temperatures.ThiscanbeseenbythedeviationofaR(T)fromthedataforaIV(T)at

T=0.15.InFig.7thisdeviationisalsoevident.Acarefulinspectionofthescalingat

temperaturesT=0.15andT=0.18revealsthattheorderofthelatticessizesisreversed

forT=0.15comparedwiththehighertemperatures.Thismayberelatedtothedi culties

toconvergethesimulationatlowtemperature. aL,L′(R(L)L R(L′)L′a)2.TheconsideredlatticesizesareL,L′=6,8,12,16,24,32.

VI.DISCUSSION

WehavecalculatedthenonlinearIVexponentaIV(T)ofthetwodimensionallattice

Coulombgas.Ourresultsarebasedonthreedi erentdeterminations.Adirectcalculation

parisonwithexperiments

[4,7,8,26]onHg-Xe lmsandsinglecrystalhighTcsuperconductorsshowgoodagreement.

Oursecondmethodisbasedonasimpleselfconsistentcalculationofthedielectric

function attheunbindingseparation,andtheIVexponentcanthenbecalculated.Here

weespeciallyfocusonthecomparisonoftworelationsbetweena(T)and .The rstrelation

Eq.(20)[17]isbasedonordinarydi usionintwodimensionswitharecombinationrate

proportionalton2F.Thesecondexpressionfora(T)giveninEq.(21)hasbeenderivedfrom

ascalinganalysis[21].

We ndthattheexponentdeterminedbyEq.(21)fortemperaturesbelowTcisclose

tothemoredirectdeterminedaIV(T)andwillthereforealso ttheexperimentsforthese

temperatures.

ThethirdmethodisbasedonequilibriumMonteCarlosimulations.Fromthescaling

relationEq.(17)forthelinearresistancewecanderiveaR(T).We ndthatthescaling

exponentaR(T)toahighdegreeofaccuracy tsthedirectdeterminedaIV(T)forabroad

We have studied the nonlinear current-voltage characteristic of a two dimensional lattice Coulomb gas by Monte Carlo simulation. We present three different determinations of the power-law exponent $a(T)$ of the nonlinear current-voltage characteristic, $V

rangeoftemperatures.Thisprovidesalinkbetweentheequilibriumandnonequilibrium

responsepropertiesofthesystem:A nitemesoscopiclinearresistanceina nitesample

belowTcisduetothermallyactivatedvortexmotionacrosssomepotentialbarrier,generated

byinteractionswithallothervorticesinthesystem.Whena nitecurrentisimposedacross

thesystem,newnonequilibriumcon gurationsareaccessedwherevorticesaredrivenaway

fromtheirequilibriumpositionsbythe niteLorentzforce,thusgivingnonlinearresponse.

OurdatashowsthatthebarrierovercomebytheLorentzforce,givingnonlinearresponse,

isessentiallythesamebarrierasintheequilibriumcase,i.e.,thepotentialbarrierinthe

caseofa nitecurrentappearstobedeterminedbyequilibriumstatesinthesystem.This

isconsistentwiththescalingansatz,discussedabove,ofacertaincurrentlengthscale(r )

associatedwiththe nitecurrent,suchthatforlengthsshorterthanthecurrentlengthscale

anequilibriumstateisstillattained,whichgivesapotentialbarrieressentiallyequaltothat

intheequilibriumcase.

Aninterestingpossibilityarisesheretomeasure nitesizee ectsonthelinearresistance

inlithographicJosephsonjunctionarrays.Theideawouldheretotakeadvantageof nite

sizeroundings,ratherthanasusualwantthemtobeassmallaspossible,andtryingto

texperimentaldataonverysmallarraystoour nitesizescalingformulas.Thiswould

provideanunusualexperimentaltestof nitesizescaling.Itisalsoimportanttoanalyze

thedataintermsofthereducedtemperaturescale,asTcissampledependent.Accordingto

Eq.(17)itshouldbepossibletoscalethelinearresistanceofsamplesofdi erentsizesonto

asinglescalingfunctionusingtheexponenta(X)fromthenonlinearIVcharacteristics.

Ourconclusionsare:(1)Simulationofnonequilibriumvortexdynamicsallowcalculation

ofalatticesizeindependentIVexponenta(T)asafunctionoftemperatureT.(2)This

curvefora(T)agreesnicelywithexperimentsinanintervalaroundTc,andthisappearsto

bereportedhereforthe rsttime.(3)Thiscurvecanbeobtainedfromasimplephenomeno-

logicaltheoryforthenonlinearIVcharacteristic.(4)Thiscurvecanalsobeobtainedfrom

asimulationoftheequilibriumvortexdynamics.Thisprovidesausefullinkbetweendriven

di usionandequilibriumdynamicsoftwo-dimensionalvortexsystems.

We have studied the nonlinear current-voltage characteristic of a two dimensional lattice Coulomb gas by Monte Carlo simulation. We present three different determinations of the power-law exponent $a(T)$ of the nonlinear current-voltage characteristic, $V

WeacknowledgestimulatingdiscussionswithP.MinnhagenandK.Holmlund.H.W.was

supportedbygrantsfromCarlTrygger,M.W.wassupportedbygrantsfromtheSwedish

NaturalScienceResearchCouncil(NFR)andH.J.J.wassupportedbytheBritishEPSRC.

We have studied the nonlinear current-voltage characteristic of a two dimensional lattice Coulomb gas by Monte Carlo simulation. We present three different determinations of the power-law exponent $a(T)$ of the nonlinear current-voltage characteristic, $V

REFERENCES

[1]J.M.KosterlitzandD.J.Thouless,J.Phys.C5,L124(1972);6,1181(1973).

[2]V.L.Berezinskii,Zh.Eksp.Teor.Fiz.61,1144(1971)[Sov.Phys.JETP34,610(1972)].

[3]P.Minnhagen,Rev.Mod.Phys.59,1001(1987).

[4]A.M.Kadin,K.EpsteinandA.M.Goldman,Phys.Rev.B27,6691(1983).

[5]S.TeitelandC.Jayaprakash,Phys.Rev.B27,598(1983).

[6]J.Villain,J.dePhys.36,581(1975).

[7]tyshev,Pis’maZh.Eksp.Teor.Fiz.51,197-200(1990)

[JETPLett.51,224-227(1990)].

[8]A.K.Pradhan,S.J.Hazell,J.W.Hodby,C.Chen,Y.Hu,andB.M.Wanklyn,Phys.

Rev.B47,11374(1993).

[9]D.R.NelsonandJ.M.Kosterlitz,Phys.Rev.Lett.39,1201(1977).

[10]M.WallinandS.M.Girvin,Phys.Rev.B47,14642(1993).

[11]R.A.Hyman,M.Wallin,M.P.A.Fisher,S.M.Girvin,andA.P.Young,Phys.Rev.

B51,15304(1995).

[12]M.WallinandH.Weber,Phys.Rev.B51,6163(1995).

[13]J.-R.LeeandS.Teitel,Phys.Rev.B50,3149(1994).

[14]J.V.Jos´e,L.P.Kadano ,S.Kirkpatrick,andD.R.Nelson,Phys.Rev.B16,1217

(1977).

[15]J.-R.LeeandS.Teitel,Phys.Rev.Lett.64,1483(1990);Phys.Rev.B46,3247(1992).

[16]F.Reif,Fundamentalsofstatisticalandthermalphysics,McGraw-Hill(1965).

[17]V.Ambegoakar,B.I.Halperin,D.R.Nelson,andE.D.Siggia,Phys.Rev.Lett.40,

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