termodinamik superkonduktor
TRANSCRIPT
-
8/7/2019 termodinamik superkonduktor
1/46
Chapter1Introduction
1.4Thermodynamics
ofsuperconductors1.4.1ThermodynamicalcriticalfieldHcmFromtheMeissnereffectit
followsthatthesuperconductivityisanewphasestateofthemetalbecause
thepathwhichbringsthesampletotheSCstateisnot
important.Superconductingtransitionisaphasetransition.Wecanusethermodynamicstostudy
it.Weassume
-
8/7/2019 termodinamik superkonduktor
2/46
thatSCswitchestoSCphasebecause
thisphaseisenergeticallylessexpensive.
Consideracylinderin
aparallelmagneticfieldH(nofieldfocusing,demagnetizingfactor0).IfH
-
8/7/2019 termodinamik superkonduktor
3/46
is-MdH=0HdH.Whenthefield
ischangesfrom0toH0thesiurceoffieldmakesa
workequalto
.H0
H2A=0
HdH=00.
2
0
This
workisequaltothefreeenergyaccumulatedinSC.
H2
Fs(H)=
-
8/7/2019 termodinamik superkonduktor
4/46
Fs(0)+0.(1.6)
2
WhenthefieldisequaltoHcmthenFs(Hcm)=Fnandfor
SCmakesnosensetostayinSCphase.Thus
H2
B2
cmcm
Fn-Fs(0)=0=
,(1.7)
220
Hcmsayshowmuchthesuperconductingstate
isbetterthan
-
8/7/2019 termodinamik superkonduktor
5/46
thenormalstateintermsofenergy.
1.4.2EntropyofthesuperconductorGeneralrelations.Firstlawof
thermodynamicssaysdQ=dA+dU,wheredQheatenergy,dAwork
donebythesample,dUincreaseoftheinternalenergy.By
definitionthefreeenergyF=U-TS,(1.8)thendF=
dU-TdS
-
8/7/2019 termodinamik superkonduktor
6/46
-SdT.ForreversibleprocessesdQ=
TdS,wehave
dU=TdS-dA;(1.9)dF=-dA
-SdT.(1.10)From(1.10)itfollowsthat...FS=
-.TA,(1.11)
-
8/7/2019 termodinamik superkonduktor
7/46
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
-
8/7/2019 termodinamik superkonduktor
27/46
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)
-
8/7/2019 termodinamik superkonduktor
28/46
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
-
8/7/2019 termodinamik superkonduktor
29/46
-
Tc
Estimation
of.(0):AtT=0ns=n~1022cm-3.
Using(2.18)weget.(0)~50nm.
-
8/7/2019 termodinamik superkonduktor
30/46
-
8/7/2019 termodinamik superkonduktor
31/46
UsingEq.(2.19)wegetjc=
Bc0.L160MAcm2.c.f.,J=640A/cm
calculatedbefore.
-
8/7/2019 termodinamik superkonduktor
32/46
2.1Londonequations29ProblemSolutionsuperconducting
films:thickvs.thinAhomogenousexternal
magneticfieldBa
isappliedparalleltoathinsuperconductingfilmofthicknessd..L.
Findthefieldandthecurrentinsidethefilm(x
-
8/7/2019 termodinamik superkonduktor
33/46
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
-
8/7/2019 termodinamik superkonduktor
34/46
-=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
-
8/7/2019 termodinamik superkonduktor
35/46
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
-
8/7/2019 termodinamik superkonduktor
36/46
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
-
8/7/2019 termodinamik superkonduktor
37/46
2.L
-
8/7/2019 termodinamik superkonduktor
38/46
Chapter2Linearelectrodynamicsofsuperconductors
ProblemSolutionSuperconductingfilmwithcurrentConsiderafilmx
-
8/7/2019 termodinamik superkonduktor
39/46
.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
-
8/7/2019 termodinamik superkonduktor
40/46
cosh(x/.)sinh(d/2.).Weknowthat.+d/2
-d/2js,ydx=J,sowecanfindBI=J/20.
Notethatwhenthefilmisthickd..thecurrentflows
inthesurfacelayer.Ifthethefilmisthind.
.,thecurrentflowsalloverthefilmandthefieldisa
linearfunctionof
-
8/7/2019 termodinamik superkonduktor
41/46
x:B(x)-BI2x/d,seeFig.
ProblemSolutionWhathappensifwecombinetheconditionsoftwoprevious
problems,i.e.,considerafilmwithcurrentinamagneticfield?TheLondonequationsare
linear,i.e.,alinearcommbinationofthesolutionsisalsoasolution.
SeeFig.below.ThefieldinbetweenthefilmsisBin=2BI=
J0.
-
8/7/2019 termodinamik superkonduktor
42/46
ProblemSolutionSuperconductingstripof
thethicknessd>.andofwidthwcarryingthecurrentIis
placedatadistanceb.wabovethesuperconductingplane.Findthe
magneticfieldbetweenthestripandtheplane.Weusetheimage
technique.Sincethefieldisparalleltothesuperconductorweremovegroundplave
andintroduceanother
-
8/7/2019 termodinamik superkonduktor
43/46
stripatx=-bwiththe
currentflowingintheoppositedirection.Thentheproblemisreducedto
thepreviousone.B=J0=I0w
-
8/7/2019 termodinamik superkonduktor
44/46
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.
-
8/7/2019 termodinamik superkonduktor
45/46
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).
-
8/7/2019 termodinamik superkonduktor
46/46