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    Chapter1Introduction

    1.4Thermodynamics

    ofsuperconductors1.4.1ThermodynamicalcriticalfieldHcmFromtheMeissnereffectit

    followsthatthesuperconductivityisanewphasestateofthemetalbecause

    thepathwhichbringsthesampletotheSCstateisnot

    important.Superconductingtransitionisaphasetransition.Wecanusethermodynamicstostudy

    it.Weassume

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    thatSCswitchestoSCphasebecause

    thisphaseisenergeticallylessexpensive.

    Consideracylinderin

    aparallelmagneticfieldH(nofieldfocusing,demagnetizingfactor0).IfH

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    is-MdH=0HdH.Whenthefield

    ischangesfrom0toH0thesiurceoffieldmakesa

    workequalto

    .H0

    H2A=0

    HdH=00.

    2

    0

    This

    workisequaltothefreeenergyaccumulatedinSC.

    H2

    Fs(H)=

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    Fs(0)+0.(1.6)

    2

    WhenthefieldisequaltoHcmthenFs(Hcm)=Fnandfor

    SCmakesnosensetostayinSCphase.Thus

    H2

    B2

    cmcm

    Fn-Fs(0)=0=

    ,(1.7)

    220

    Hcmsayshowmuchthesuperconductingstate

    isbetterthan

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    thenormalstateintermsofenergy.

    1.4.2EntropyofthesuperconductorGeneralrelations.Firstlawof

    thermodynamicssaysdQ=dA+dU,wheredQheatenergy,dAwork

    donebythesample,dUincreaseoftheinternalenergy.By

    definitionthefreeenergyF=U-TS,(1.8)thendF=

    dU-TdS

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    -SdT.ForreversibleprocessesdQ=

    TdS,wehave

    dU=TdS-dA;(1.9)dF=-dA

    -SdT.(1.10)From(1.10)itfollowsthat...FS=

    -.TA,(1.11)

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    1.4Thermodynamicsofsuperconductors23Forourparticularcase.Weuseformula(1.7)tocalculatethediffrenceofen-

    tropiesofnormalandsuperconductingstates.Substituting(1.7)into(1.11)wegetSs-Sn=0Hcm.Hcm.TA,(1.12)Consequences:

    AccordingtotheNernsttheoremS

    .0ifT.0.From(1.12)followsthat(.Hcm/.T)T=0=0,i.e.,

    Hcm(T)haszeroderivativeatT=0.fromexperimentweknow

    that(.Hcm/.T)0.Bz=B0

    =0H0isapplied.Inthis

    problemthefieldhasonly

    z-component,andchangesonlyinx-direction.ThesecondLondonEq.(2.12a).2B(x)

    .x2-1.2B(x)=0.(2.16a)

    Theboundaryconditionsare:B(0)

    =B0and

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    B(8)=0(Meissnereffect).Thesolution

    satisfyingthisb.c.is

    -x/.

    B(x)=B0e.(2.17)

    Figure2.1:Penetrationofthefieldintobulksuperconductor.Thefield

    onthesurfaceisequaltoH0

    m

    .=Londonpenetrationdepthofmagneticfield(2.18)

    nse20

    Thescreening(Meissner)

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    currentsalsodecayonthelengthscale

    ~..Indeed,since0js=rotB,forour1Dgeometrythis

    gives0jy,s(x)=-dB/dx,i.e.,from(2.17)wehaveB0

    jy,s(x)=e-x/..

    (2.19)

    .0Since.(ns)andns(T),then.(T).Goodempirical

    formulais.(0)

    .(T).4.(2.20)T

    1

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    -

    Tc

    Estimation

    of.(0):AtT=0ns=n~1022cm-3.

    Using(2.18)weget.(0)~50nm.

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    UsingEq.(2.19)wegetjc=

    Bc0.L160MAcm2.c.f.,J=640A/cm

    calculatedbefore.

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    2.1Londonequations29ProblemSolutionsuperconducting

    films:thickvs.thinAhomogenousexternal

    magneticfieldBa

    isappliedparalleltoathinsuperconductingfilmofthicknessd..L.

    Findthefieldandthecurrentinsidethefilm(x

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    thickfilm(d.L)bythefactor2.L/d.

    ThesolutionoftheLondonEq.(2.12a)canbe

    writtenas

    xx

    B(x)=B1exp-+B2

    exp+

    .L

    .L

    Theboundaryconditionsare

    ddd

    B-=B0:B1exp

    +B2exp

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    -=B0

    22.L2.L

    ddd

    B+=B0:B1exp-

    +B2exp+=B0

    22.L2.L

    Fromherewe

    calculate

    B0

    B1=B2=.coshd

    2.L

    Thus,thefinalexpressionforthefieldB(x)

    insidethesuperconducting

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    filmis

    x

    B(x)=B0cosh..L.coshd

    2.L

    Fromj.=-.B=-B/0.2wefindthat

    L

    x

    sinh

    B0.L

    j(x)=-.

    0.Ld

    cosh

    2.L

    Thecriticalfield

    Bcissuch

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    afieldatwhichj(d/2)=jc.Thus

    Bcd

    jc=tanh

    0.L

    2.L

    Fromhere

    jc0.L

    Bc(d)=.tanhd

    2.L

    Andfinally

    d

    limd.8

    tanh

    Bc(d.0)2.L12.L

    Bc(d.8)dd/2.Ld

    limd.0tanh

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    2.L

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    Chapter2Linearelectrodynamicsofsuperconductors

    ProblemSolutionSuperconductingfilmwithcurrentConsiderafilmx

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    .iny-direction.Findthedistributionof

    thecurrentsandmagneticfieldinsidethefilm.Usingthisb.c.together

    withgeneralsolutionB(x)=B1exp.-x.L.+B2

    exp.+x.L.givesB(x)=-BIsinh(x/.)sinh(d/2.).Then

    wecalculatecurrent(userotB=0js)js,y=-B(x)0

    =BI.0

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    cosh(x/.)sinh(d/2.).Weknowthat.+d/2

    -d/2js,ydx=J,sowecanfindBI=J/20.

    Notethatwhenthefilmisthickd..thecurrentflows

    inthesurfacelayer.Ifthethefilmisthind.

    .,thecurrentflowsalloverthefilmandthefieldisa

    linearfunctionof

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    x:B(x)-BI2x/d,seeFig.

    ProblemSolutionWhathappensifwecombinetheconditionsoftwoprevious

    problems,i.e.,considerafilmwithcurrentinamagneticfield?TheLondonequationsare

    linear,i.e.,alinearcommbinationofthesolutionsisalsoasolution.

    SeeFig.below.ThefieldinbetweenthefilmsisBin=2BI=

    J0.

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    ProblemSolutionSuperconductingstripof

    thethicknessd>.andofwidthwcarryingthecurrentIis

    placedatadistanceb.wabovethesuperconductingplane.Findthe

    magneticfieldbetweenthestripandtheplane.Weusetheimage

    technique.Sincethefieldisparalleltothesuperconductorweremovegroundplave

    andintroduceanother

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    stripatx=-bwiththe

    currentflowingintheoppositedirection.Thentheproblemisreducedto

    thepreviousone.B=J0=I0w

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    2.1Londonequations31Figure2.2:

    Ifinparallelsuperconductingplanesthereareequalcurrentsofoppositedirections,

    themagneticfieldofthesecurrentswillbelocatedonlyinbetweenthe

    planes.

    Figure2.3:Magneticfieldcreatedbythesuperconductingstrip

    withcurrentplacedabovethegroundplane

    ProblemSolutionSketchthecurrentdistribu-tioninasuperconductingwire(madeofatype-Isu-perconductor)withcircularcross-sectionandwithradiusr0.L.ThecriticalcurrentIcisreachedwhenthecriticalcurrentdensityisexceededsomewhereinsuperconduc-

    tor.Calculatethecriticalcurrentofawire.Howlargeisthecriticalcurrentofa1mmthickPb-wire?WhatisIcfortwicethickerwire?Comparewiththenormalwire.

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    Weassumethatforr0.Jthepenetrationofthefieldisexponential(thisisnotthecasewhenr~.L).jz(r)=B00.Le(r-r0)/.L=j0e(r-r0)/.LAtr=r0,jz(r)=jc=j0.Ic=2p0r00jz(x)2prdfdr2pjcr0r00e(r-r0)/.L2pjcr0.L=2p0r0Bc400A.(2.21)FortwicethickerwiretheIc.r0istwicelarger.Fornormalwirethemaximumcurrentmaybe4tim

    eslarger.ForthinwiresSCisbetter(realHTSwiresmadeofmanysmallwires).

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