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Study of Properties in Superconducting Nanowires

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Study of Properties in Superconducting Nanowiresof,in,In,Study,study

    Study of Properties in Superconducting

    Nanowires

    Commun.Theor.Phys.(Bering,China)53(2010)pp.777_781

    @ChinesePhysicalSocietyandIOPPublishingLtdVo1.53,No.4,April15,2010 StudyofPropertiesinSuperconductingNanowires

    LIANGFang-Ying(梁芳营),tWANGJian-Hua(王建华),andZHANGZhi-Gang(张志刚)

    InstituteofScienceandTechnologyforOPTO

    ElectronicInformation,YantaiUniversity,Yantai264005,China

    (ReceivedMay15,2009;revisedmanuscriptreceivedJuly15,2009)

    AbstractWeconsiderasimpleapproachofstandardGinzburg-Landaufree-energyfunctionalforawiretostudy

    thepropertiesofsuperconductingnanowires,andanalyzetheproblemofquantumandthermallyactivatedphaseslips.

    Insucilsystemsonecanconsiderapossibilityforphaseslipstobecreatednotonlydueto

    ermalbutalsodueto

    quantumfluctuationsofasuperconductingorderparameter.Weobtainsome,expressionsofthefreeenergy,theentropy,

    thespecificheatandthebiascurrent.respectively.ThebiascurrentIisafunctionofthetemperatureandthelength

    ofsuperconductingnanowires.andhasaquantumphaseslip.Weobtain

    estochasticdynamicsofsuperconductive-

    resistiveswitchinginhystereticcurrent-biasedsuperconductingnanowiresundergoingphase-slipfluctuations,andobtain

    thedistributionofswitchingcurrents.0urresultsCallbeverifiedinmodernexperimentswithsuperconductingnanowires.

    PACSnumbers:74.25.Fy,74.78.Na,74.25.Bt,74.40.+k

    Keywords:superconductingnanowires,transportproperties,thermodynamics

1Introduction

    Recentadvancesinthefabricationofultranarrowsu. perconductingwireshavespurredrenewedinterestin quasionedimensionalsuperconductivity.Inthecaseof superconductingnanowires,ourpredecessorshavedonea lotofworkonsuperconductingnanowires.lillNayana

    ShahandDavidPekkereta1.studiedthestochastic dynamicsofsuperconductive-resistiveswitchinginhys

    theFermienergyandA(T)istheBCSorderparameter) forapartOfthewireofavolume8wheresuperconduc

    tivityisdestroyedbythermalfluctuations(8isthewire crosssectionand()isthesuperconductingcoherence length).

    Here,weconsiderthestandardGinzburg-Landaufree- energyfunctionalforawiretostudythepropertiesofsu- perconductingnanowires.

    tereticcurrentbiasedsuperconductingnanowiresunder- goingphaseslipfluctuations.[DmitriS.Golubevand2Model

    AndreiD.Zaikinanalyzedtheproblemofthermallyac

    tivatedphaseslips(TAPS)whichcandominatethebe- haviorofsufficientlythinsuperconductingwiresattem- peraturesclosetocriticaltemperature.-4_DmitriS. GolubevandAndreiD.Zaikinobtainedwhatifthequan- turnphaseslip(QPS)ratewashigh,superconductivity wascompletelydestroyedbyquantumfluctuationsand thewireresistanceneverdecreasedbelowitsnormalstate value.Fluctuationshaveknowntoplayanimportant roleinsuperconductingstructureswithreduceddimen- sions.Inthecaseofsuperc0nductingnanowires,l36jfluc-

    tuationsmayessentiallydeterminethesystembehavior

    Inawidetemperatureintervalcausing,e.g.,nonvanishing wireresistancedowntotemperatureT=0.

    Atheoryofthisthermallyactivatedphaseslips (TAPs)phenomenonwasdevelopedinRefs.[5,8].This theoryyieldsanaturalresult.thattheTAPSprobability, hence,resistanceofasuperconductingwireRbelowTc aredeterminedbytheactivationexponent

    R(T).cexp[_NoAr2(T)s]7(1)

    whereg0A.()sf()/2istheeffectivepotentialbarrier forTAPSdeterminedsimplyasthesuperconductingcon- densationenergy(N0isthemetallicdensityofstatesat Wesupposethattheleadsconnectedtothesample determinedistinctuniversalityclassesoftheconductance scalingfunction:wireswithsuperconducting(ss),nor

    mal(NN),andmixed(SN)leadshavedistinctscaling functions.6,10]butotherdetailsoftheleadsareunimpor

    tant(seeFig.1).

    nanowire

    NorSNorS

    lead1ead

    -一……

    L…一--j

    Fig.1Asuperconductingnanowiresuspendedbetween twothermalbaths,whichistunedfromasuperconduc- tortoametalby(say)reducingitsdiameter.Theleads onthewireareeithernormal(N)orsuperconducting(S) andtheNN,SN,andSScasesbelongtodistinctuniver- salityclasses.

    Considerauniformsuperconductingwirewithcross sectionsandlengthL.Ourstartingpointisthestandard

    Ginzburg-Landaufree-energyfunctional:[4-5] SupportedbytheNationalNaturalScienceFoundationofChinaunderGrantNo.10974167

    tE-mail:Liangyangrning882@126.com

    LIANGFang-Ying,WANGJian-Hua,andZHANGZhi-GangVl0l_53 即??oL(I2TTc

    Here,()isthephaseoforderparameter()andI

     istheexternalcurre~appliedtothenanowire.Tde

    notesthetemperature.Tciscriticaltemperature;e= 1.6x10-10co~omb;(3)?lI2,Q=(A/3)isthe

    diffusionconstant.isapracticallyimportantlimitof smallelasticmeanfreepaths,Vristheeffectivevelocity ofmovingquasiparticleattheFermienergy.[8,11

    Thesaddle-pointpathsforthisfunctionalaredeter

    minedbythestandardGinzburg-Landauequation-1_5J

    +

    T-:re

    8T0x2+

    81r2T2=0.'..'

    Foranygivenvalueofthebiascurrent.[4

    =s.

    3CharacteristicsofSuperconducting

    Nanowires

    l

    E

    

    ,

    lead0Ilead;I

    nanowires

Fig.2@)sketchmap,supposethattheaverage

    potentialenergyofleadsis?fpositionofzeropotential

    energy)andtheaveragepotentialenergyofsupercon- ductingwiresis.

    (3)

    (4)

    +I)0)]

    Wesupposethequasiparticlemovesfromlefttoright withenergyE>0,theaveragepotentialenergyofleads is(here,supposethe?ispositionofzeropotentialen- ergy)andtheaveragepotentialenergyofsuperconducting nanowiresis(<0)(seeFig.2),

    WedefinethereflectioncoefficientRcastheratio ofthedensityinthereflectedwavetothatinthein- cidentwave,anddefinethetransmissioncoefficientD ofthequasi?-particleastheratiooftheprobabilitycur-? rentdensityinthetransmittedwavetothatinthein- cidentwave.FromQua~umtunnelling...twecan

    supposethemovingquasiparticleincidencewaveofthe leftleadis~IliL.ft=exp[ikx+exp[z,themov-

    ingquasiparticletransmissionwaveoftherightleadis 0ght=,,/De~p[ikx],thewavefunctionofsupercon- ductingnanowiresis=Aexp[iKx]+Bexp[-iKx].The D,A,andBhave[1213]

    .=[+r,

    A:[+]g)l,

    B:[一妻]eXp【?酬,

    Here,Rc+D

    ,

    (5)

(6)

    1,k=?历,K=

    istheeffectivemassofamov.

    ingquasiparticle.isplanckconstant.Taking= Aexp[iKx]+Bexp[-iKxintoEq.(2),weobtain

    F:s?0rc~

    TDK2[(+)1-k2/~sin(22KKL)]+s?0[(+)+(Kk_~22)sin(2KL)] +s

    Take=Aexp[iKx

    [(1+)+(Kk__~44)sin(2KL)+()4KL+_87(4KL)

    +Bexp[-iKxintoEq.(4),anygivenvalueofthebiascurrentis

    :

    7reNof~

    s

    {(+)+(一芸)cos(]

    )(0).(8)

    (9)

    Here0zL.

    Weknowmainquantityofthermodynamicsfromgeneralprincipleofthermodynamicsandst

    atistics.Forexample

    freeenergyF=UTS,innerenergyU=FT(OF/OT),specificheatCv=(OU/OT)v=T(OS/OT)v,andentropy

    S=-OF/OT.

    If=?=0(K=),soD=1.WhenKL=w7r(here?andw=l,2,3.),wehaveD=1, E=E=w7rh/2mL1I,L=w)~/2,()(0)=WTl-.

    WhenD=1(here?andw:l,2,3.),fromEqs.(8)and(9),wehave F:s?0w2vr2h

    

    2-

    7r2

m

    

    L2[~]

    L

    IT-Tc.+W27r27r~+

    w

    

    2r

    

    2h2-mL

    

    21V21

    i7~(3)]

    

    [W2T-221~1sin2(叫不兰)(10) N0.4StudyofPropertiesinSuperconductingNanowires

    Thebiascurrentis Theentropyis

    =W2T2

    2W27r21~2T

    2

    JL?

    2mL.Jsin2w不兰)

    ??o[(一去)+w27:21i2_mL2[~

    

    [w271-22_2sin2(叫不兰)] Thespecificheatis =s?0w27c2h2-mL21V2JI"w27r2+wZTczhz_mL21~1

    w'7r'LL'4_'t'7r''

+[w271-22_21~rsin2(叫不

    Here0<<

    4DiscussionandComparisionwithPrevious

    W_0rk

    If=?=0(K=),

    andnanowiresaresame.If

    soD=l:therebytheleads

    0<D1,Itisnotsuper-

    conductingcharacteristicinsuperconductingnanowires. WhenD=1(here?Ylandw=1,2,3.),themov-

    ingquasi--particlespassingoverthenanowiresarenotre-. flected;thereare=(w.h./2mL)Il,A=2L/w

    and()(0)=wTr;thenanowiresareexhibitedthe

    superconductingcharacteristic.

    FromEq.f11),thebiascurrentIisafunctionof thetemperatureandlengthL,andhasaquantumphase slip,avirtualdepletionofthesuperconductingdensity throughwhichthesystemtunnelstoadifferentvalueof thesupercurrent.Wleobtainthestochasticdynamicsof suDercOnductiveresistiveswitchinginhystereticcurrent

    biasedsuperconductingnanowiresundergoingphaseslip

    fluctuations,andobtainthedistributionofswitchingcur

    rents.Phaseslipfluctuations.inwhichthephaseofthe orderparameterbetweentheleftandrightendsofthewire changesbyamultipleof2,havelongbeenknowntobe animportantingredientindeterminingthepropertiesof superconductingwires.1.7JNayanaShaheta1.foundthat itisessentia1toincorporatequantumphaseslips,andnot (11)

    (12)

(13)

    justTAPS.Atevenlowertemperatures,theirmodelsug- gestedthatswitchingiscausedbyasinglequantumphase slip.Theswitchingcurrentdistributionthuspromisesto provideapowerfulprobeforaccessingandtherebyunder

    standingthebehaviorofindividualphaseslips,thermal orquantum.[6,8]

    FromEq.(Ii),thebiascurrentincreasedwithdecreas. ingT.RecentlyBezryadin,Lau,andTinkhamreported[J theresistanceofthesamesampleswasdecreasedwithde- creasingT.

    FromEq.(i0),theFisafunctionoftheTandL,

    hasaquantumphaseslip,avirtualdepletionofthesu- perconductingdensitythroughwhichthesystemtunnels toadifferentvalueofthesupercurrent.S.Khlebnikov hasshownthatquantumphaseslipsgiverisetomulti- piebranchesbandsintheenergycurrentrelationandto

    aninterband"exciton"mode.[14_FromEqs.

    (12)and(13),

    theSandtheCvallarethefunctionoftheTandL,hasa quantumphaseslip.SubirSachdev,PhilippWerner,and MatthiasTroyereta1.obtainedthatthecriticaltheory obeyshyperscaling,whichleadstoauniversal~equency, temperature,andlengthdependenceoftheconductance; quantumandthermalphaseslipsarecontainedwithinthis criticaltheory.[6-8,10]

    FromEqs.(10)(13),wecanobtaintheaveragevalue

    offreeenergyofnanowiresis

    =s

    w27r2h

    2-

    mL2[~J

    L

    IT-Tc.++w27r2h2_mL2[~J

    ..

    s

    4

    r

    

    N

    2

    o~r27rm.

    Theaveragevalueofbiascurrentofnanowiresis =W27i2忍一m.II]叫不.

    Theaveragevalueofentropyofnanowiresis ?o[(朵一去)+]_

    Theaveragevalueofspecificheatofnanowiresis (14)

    (15)

    [w2,ff2(16)

    一?0[+]+(17)

    LIANGFang-Ying,WANGJian-Hua,andZHANGZhiGangVbl_53 Ifw.7r0.=mL.ff(herew=l,2,3,),from

    Eqs.(14)(17),wehavetheaveragevalueofbiascur? rentofnanowiresI=0,F=0,Cv=0,andS=0; thenanowiresmayactuallybecomeinsulating.Khleb

    nikovhasshownthat,whilethestatewithahighden- sityofphaseslipsisnotatrueinsulator(aconse

    quenceofJosephsontunnelingbetweentheleads),fora

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