Proximity Effect of the Superc onductivity in Met allic Superlattices and a Self-Similar Mullilamellar Syste m

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Proximity Effect of the Superc onductivity in Met allic Superlattices and a Self-Similar Mullilamellar Syste m

B.J.YUAN M.SC.(Plf YSICS) ,TONGJI UNIVF.RSITY

T hesis for Master of Science

Department ofPhysics

MemorialUniversity of Newfoundland

St.John's, Newfoundland

Canada, AlB 3X7

@15 August1989

Abst ract

IIIthe thesis, thevnrio ustheo ries ofillhtlllllp;"IU'III'""\lI)o·t\·"I1.II1l"l ivil~·I"'SI~I onGor'k ov's cquatiolls(CbnptlT 2) nrc Tl,\-jl"",,1il1t'h\llill~.1.·G'111J,"'·\\".'rl!la llwr theory andEilcnh ' Tg"f ("(Inatiolisas ",...II m.th,; r lll'l'lirllliuusI""l u.I~·tllO'I'TllJ\

imitrclfcdof1\biln}'('f ".l""lem.!l.k~Iiij .m·"hUlIu·l1ill':;111' '' \'-\illal",.inl r..<\w, , 1.

The c..18.r;lclmstic:softhe thid,;nM<sdr!.lrud.1 H."neIIII' lrllllsitinn"'u il "".alu n''C inthe thin filmlimit Iromtill'abocetl..·nri.... nrr-.~i"rll...l1l1101wit.III'""1111'"11,1 with theresultsfrommtr cnlntl:1t inll.

TheSll!)(T(,olllh lr t i,-ityorn1i1l1'N'l"h.t1in ·""!lsi",ill':;nfnll"fn alill,!!;:<'11"'''''' '111"...

ingandnorrunll a yc rsllIl" 1)('('11ill\'rs li~nt{,cl(Clmph '!'311»'1111'1111.. ,,(Ill<"1J,,~..1i1l1o,,\·

cquorion.Arclnrlon,which1!l"t!'nn ilit'SlIlt'h'H Il~il,ioll l<'IIII'l'ra l tln'

·r;.,

i~,,1,tnilL''']

fromthesclf-cousl steney 1'llu nliHIIoftill'onh-r I'lIr lllllt'l"r.TIll'llua ly1i"!,1"XI'Io~

siono[thedependenceuftill'tl"lllls iti ull tf'lllJIt'..uturr-1;.rrulIwI.hid\ll'~1<

"r

II.~II poreondue ting lnyr-rilltho thin filmlimitlmalH"'1Iu!t1.nilU'<\t1m'l1~h1111nllll ' ~"~ i~"r theequa t ionand theront pnrison withtln-rl',~IIIl ~fl"olJllIII' Ill_h,']'Illl'nri,'s ,!is"lIs",'d in Clmptcr 21lftsbeendono.ApI'ri'l(!ir"ILl'l"~"1I1011 lt'1L1,'1Il 1n.lufioun'lh'd,ill~til.' pc rlodlc it y o(thestnu-ttu-eIms111'('11IIhl niw,tlwhichj,jil",,,ris,'InIll!"ba ~i"['''I'lilly furtherculculut io nIorthe'pr<',;('llt't'o(alle-xh-run]11In~Il<'Iit'lid,'01"1I HlIit,'tlnkl pnramctcr.

IIIChap ter4,weUPlll i l1.Itilt,\".'rllmU lI'rlIl1'ol")' 1un 'j'lllsi-]H'ri''1li,'~t'olllt'lry withafrectio nnltlill1l'IISiullV.\Ve bnil,1til"l",,'nnl'U,1'wlnlio lluf1,1.:',""'Ili<-i"1,1,,, inthelinearfr;tctiolllll l'~ ''"-nnatitmin h'rlll"t lfUII'Sl'I(-silll ih,ril""ftI",~l'f l1lll"lr)'.

whichprovides n 1»·sl('I0•.•••• 11It:tIUKItndcn! wilhtI ...",ml,lin,I,..11H>llllllill)·''''lilli·

tion.~.The dependeueeofthetruu",iliflll tPllllHTlIh lff' "111...111lI,,' tloio-kll''''''''

"r

II ...

1i\ll'crCOndlld illgIll~'l'1"1111<1frnctinllllldilll('II"j"llV1110'"I.."""lolain...llo)'tI,..",,"nlill!!;

argument inSOIlj(~limitingrus<':lI.TIl<'Rrllliup;llrp;llln"111i",al"",'" ","wl,..ll" 1I..."/l!oO' ofthepr('S('IlCC ofnpcrpr-ndienlarIllltP;lu-ti('lkld.1111'JLIlllwrinll,"alt-lll;,l j" u fur traus itiou tcmpcrnturc III>a flllldiu n,,[till"lhit"lll.~i","'''III,],'tl..!;",,1"']"0\\"11illIII"

correspondi ngfi~llfl'l'.\VeIIIsornlrnlllh'"('ri llin vlIhll"""fT..n,rn",p"lJOliul!;I"II...

givenperruue tcrs , sw:hOl'i,lim" lIs iulIV,tllkk",_,I.lOUt!,"" I...n·II(""1" 'Jtlll{1I(lIl.

usedin theCXpcr illk'1ltIUl.1C1ll1llll1ll' till'tll,,,,rt'l il'llln."u lb,freuu1111"l'llkul:'li"lI with the expcrimcmnl ,'nl twN,Tht:resulttiC11r"" ,mll' "ri';'ll li":;;nt i "f;wl"l")"llull llw discrqllllll"ics rlOIIgillg freuu2'X...10%,rOIl",i",ft11'-willilil"\V" IIIII,u!"r111' '' '1")',

Ackno wledgments

Thi~ tlll~iswas snppor tc.:!bytlll~Fellowship fro mth eSchool oftheGrndu- tlt,l~Stlillil's,M"llIoda lUniversity ofNewfoundla nd, andthe T.A.lindn.A.from

Physil~SDr-purtrucnt, Memorial University,Itismyplcnsu rototha nk Dr.R.L.

K,,' ....~ut.D' !jJ l ll' lll lell l.of Physics,Universi tyof Win n ipeg for hisngre emont which nl1m'lSIll"toOl'p;lIlJi:;.,~t.lLl:published work,which\\'1l6com p lete dcor p orntivclyhy Dr.H.L.Koh,'s,":"..1. P.Whitcheudnud me,intothethcsis(Chn ptcr 2). Es- p,·(-jall)',IwouldUk,-1'1!,uk!·"hi", opport.nni tyto th nuk my supervisor0,'..1.P.

Whil,,·]lt'a dku- hisr-xtrt-un-ly vlllllnh]('aSliistmwl'in theresenrr-hlin dwritin g the t!t''l'lis.

iii

Abstrac t.. . . . Ackn owled gmen t List of Tables . Listof F'igurcs CHAPTER

I.INTRODUCTION

Contents

.ii

.iii

2.THE INHOMOG ENEOUS MICROSCOPI(;

THEORYOFSUPERCONDUCTIVITY . ."

§2·l. GO n'I\OVEQUATION .5

2-1·l.TJIl:Elcct rcn-PhououlutcTiidinllJIlUuill.nniuu •!"

2-1·2.TheEfTI'Cth'l:

DeS

Hruniltouinu . . .

s

z-r-a.Gor'kovEquation . II

§2-2.DE GENNES·WEllTH AMERTHEORY 19

2·2- l.LillclHit:I~1Gill'Equntiou I!l

2-2-2.dcGf~II11C:STheory ill II Dilu)'f,rStl'1\1~'lll'" . 2U

iv

!i2-3.EIL Ei'\ I3EI1GER'S EqUATIOi\ vxnITS

DIflT YL1;\lJTVE:I1SIO;,\ 2&

2·3·1.Eilf·lllwfW'["\;Equation 25

!j2-4.TUNN ELLTNG~IODELINSUPERCm ml"C'TIVI T Y 32

:l. TIlECRITICALTEMPE RATU RE IN

"I ET A LLlCSUPERLAT T ICES 40

!j:l-I .BCI GCI LlUnO\'EqUATIOX .,10

!j:\.2.TII EPERIO DICSO LUT IONAND THE

]~ i'\ EI1G'\'·S P E(,TnU~1 .,13

~j:I·3.Tn.o\NSIT I ONTE r-. I PERAT UR E .,15

.1.THE THANSlTlON TEM PEHATU RE OF ASELF-

SIWLAHMULTILAMELLARSUPER CONDUCTOR 0'

!j.l -1.FCJn:-'IlIL AT IO;.;' 53

·I·I-I.fl<'G"IIIIl'S-\ \" 'I'th auu'rTlwllrr 53 .1-1.2.Eir;I'IIt'(lll ali 'l llHIll]D' I' IlI,lal'rCoudhious 54

!j·l·2.,\PPLI('AT IONTO AFRACTALGEO}.JETnY 57

.1.2.t. Ei~f'II(lIllO'ti" IlS 57

·1-2·2.RI'I·U1"1'I'IWI'RI'la lillll m

4·2·3.C"llll llil l'iSI1l1withIllf'!lkllMll"t't1 \'uhlf's GG

§4-3.SCALINGLAWS

4-3·l.Thin Film Limit 4-3-2.D«tmplinF;of the Stmctnrr-

4-3-3.~nlinF;Lnw

§4-4.THEINFLUEXCEOFTHE PERPENDICULAR MAGNETICFIELD

4.4.t. Eigl·nf\\\l fl i<l1l1lillIll.,Pn 'Sf'lln · .. fp,,"\II'lulk nlar l.fllgu('l ic'Fi"l,1

4.4. 2.LilJ lit.ill~C/I~"~

§4-5.DISCUSSION

5.CONC LUSIO NS APPENDI XA:

APPENDIXD:

APPENDIXC:

APPENDIXD:

DIDLIOGRAPHY:

GO

G$

,0

,5

OJ

ts

[«,

st

0.

111I

Ill:!

List of Tables

'D. bleI. TIll' tlJ('(>rdkn[ lIud experhnentul vnluNiof the:sllJ)ercoIHhtctiu g 1./"llllsitj"l1tt'IlJlwHIt.ll n'Sflfthe J.;:l'Clllwtrieswithdim~l'elltfrnctnldimensions. Gj

Fig.1. Self-Energy

List of Figur es

• ;1.:0

Fig.2.¹⁸ Sdf·Ellrr~·orlh.-I<UIX'TI'IlIUluflufj"11\1'MlIlI"f1111' ""'("1I.1·unl'T phouon :u III11Iliudliugn>1l1rillul i"I1~_Th.,G..."'tl·l'!(nne-linu s IlN"I"h- rm illt,,1",·If

COIl SilitClll l) 'fWllIlh .·Sl"Ir-I.. ll'r gi... :lli

Fig.3.²⁶ ThedC'jM.'lllI{"Il("cofII\("n'(lllf'N\"'I1II>l'f nllln ' Irlll\ s iti,·,

" 1 ;., ",/,1/'1:.,

on the thic kness(//(1, (n) II/fI r

=

0.01,(h)I>f" r ::::ILl.(I')

j,I",.

=loll(,I)iuli llitr·

h. ·17

ami(Ii )(1lflr=5O.11•• IS

Fig.5.1'; Thed('II(' III I''1I("1"'

ne

th.' 1l'f ]II .",..l lrllllsjl j' l1l"',II" 'n lll ln'"/:·(", " 1/,1:.

nnthl'thirkll'...,1/11..furtI =

'I ,

^(;,IT'l/Td=11.1.(IIIT..~rr.-1

;;;

^11.111.I' ll "

{cIT,...J!Tcl

=

O. . ·19

Fig.G. Gl"OlIlctric lllt'flllfi,r;nratiuIIlIf11(f m"lal"lfllrl lm',

Fig.

r.

^TIl('clf'JM'lIflrllct:of

,!I"

"'(llIn ..1In" lliitj" ""'lIIJ "·I'lIl.Il~·'1:/,1:.<;"11II...

thicknessJ

=

^ds/~DforT~N

=()

111111V=111(2)/ 11 ,(:1)whh"I'll,·!",,(II.,·(n...I;,1~I.

(a)M=1, (h)M= 2,(r).\I= 3, (II)M=4,ulHl(l'jM",.rl• • Gr,

Chapter 1 Introduction

illSl1P"IT'Hllh ll'lill~hysl" IIIS.)'Iostoftln-"M])'litndi,'softI...proxim it.ycffect were n,rw"rlll',1witt,'Ill"IrHllsiti"1 1I"l ll])('I'IlII11"t'(orSlIP"IT ()lJ<!llctingbilnyorstruct ure awl\\"'1-"Ims"d011tl...Gill~,ll<'l'~.Lmldml(GL )till,,!')'ofinhomogenoonsSlI Jl"I"COn - dll"1idl. .I'or<>11IlwIiIH'ariz"dGor' l;m'"lJlla ti o ll s{'I,111 pnrt.ir-ular <it'G,'IlIll'S['1I

I:, Id"ri\',·<!;,u,'xl'l"pssi"n fortIll'k"l"u('1illt\1I'lim-uriz r-dGlll"!i:OYl"t,1H11i o lihase,!

"IIrlu-solutiuunf1111',]iI1'usi' lllequution tllK,·t lw rwithtill' Illlp HI!lrilitcbouudru-y

1II11l"h lllilH·dIIlTsul1"'lllivlIl" mto thatofOIl('-Fn '(I'U'Il"Y llJlp rOl\r l!fouudin de

Au "xl"lIs j" U"f,Ill'1l1'1'lkal,iollofIIII'II.,Gl'IlIlt's-\\",rthaull'1'tlwu rr to till' mon' ""mpli"lIlt'll,I!;,'olLlPtryi~~traip;htfo r \\'al'(lpl'll\'irh, dtIl"hOll lHla ry coudltious

laY"r~of~1l1"'IT"IHhwlill,C;nnduonnnlnu-ml(5:\ 5., . SXS)ormor eSV'lwn llly~)'5"

h'm"\'ollll'ri"ill,l!;Ill"'rll1lt ill,l!;layl'rsof sIIP"l'l'o/H!lwlillp; ruun-riul s with diff"l"l"'ul11111"

ll'rialdHlriu'lc'risti."s(55'S, .. 55'5)nnrltuenyh-ngthseulcs.:\!Ild loftIll' interest

onlytwolcngt.hscull'S, Experimentalst udies onsuch"lljH'd aUi l'l's lla\'"\'Olll'l'1'Il.·d lnrgclythedcpcudc ucc"f till'!'~l P('lTUll" l1d i ll ~truusitiuufc·lll ]l" l'll tllrt·<Illth, 'n·ll1- tive thicknessofthe IIl)"'!'s nnd tlu-modulatiouk\l~thof tlu- !'Ill'.'rl a ttin' ,lr,II';I (1Jf8lun dth e cffct.t sufappliednmgneric til'1ll,lqI [10I[III [I~I (I:lISnchstud- ies rcvonlnrichvariety ofphcuorucun nss.)('illh ·,1withtil<'routph-xiute'l'tWtiOll..I' compe tingmechanisms1\1111,ill t.hcr-nsr-ofHIl'r-rit k n\ lil'ltiull':I!'lln· Ul<'lIls .till'ofh' l1 sub t le interplaybetween thelIl11gnl't,kh'u glhsl'al.,:lwll lll'illholl1"P;"IU'itrillllu- superconducting coherencelengthussoot'in\.l'dwiththl'supl'rla Ui,'"sl.nw tlll'C,[ 11I, Since the deGeuues ·\\'ertilBIllCl' tlieur>' isrnno-rned with!.IU'('nh'n lati oll of1.111'k.-r- nclin thelint'lll'ii\cdG01"km' cquntious ,it,isnpplit'abl,'onlydos<'to HIl' tnl1lsil ioll tcmporat nr ..,whr-rr-the!'npt~fcnJl( lu..tillg onh'r!IlII'a lllC'l.<'1'"nuisl",s,

Ifone wisllI'sto exploreth eSIlP"1'l'o llcind inp; prop t' I,ti,'Sofslwhs~'SI \ 'I IISIIII'IlY fromthetransit.ion tcrupcrnturc nne 1\'0nl,Ib.'iuvolvedilltil<'"Ollll'liC'll l.illllof llu- Gor'kovr-quut.io nsduo totht·non-lmonrfr-nlurr-.A uuu-hsill1l'lifk .lv,'rsioll ort.1,..

Cor'kovr-quntinnatheEill'u!lerj!;"I'("jlllltiullS[I~'II1(;Ip rt>\' i'l''l<tln-bilsisofst,udy ingsuchpropcrtlr-s ofth:s~S t,f'IlL, \Vcwill olltlitll'till' hl,s isof1,11t'Eill'ul.('I'/!;'·r formal lsm. Closetothetrans i t lont.<'lUjlt·l'i1tlll"',itr-un1)(',~IIOWllthut1.1l<'Eil"II' ber gerequationsroducoto the de GI'IlIWS-\V"rl.lmull'l'(''1llal.iol lsillt.11C~"l'propl'i:II.(' limitsandrecent qunntluu ivet:aklliati oh 'lof the111'1'('1'<Titi,'ulli"ldhav,·1"" '111'1'0' s,'uledhilsl'll011t.111~Eilcnlx-rgert1lt'l)t'~,1Ir,II1<;II17 I,IJ"w"¥<'I" illP;"lwl' nl,

u...

cnlculnt lon s 1I!'Cf'xtI'CllldJ'difficnlt1.0IHlIl, IlI',Heno-, lIh"I'lw tiw 'lIPI'IHJ(illlUI,i"u

AnotherapprunchisIlHs"l! ontill'Do~()li lllIOV.,tjl,;otioll s,wlunh with I'»uif.uldy simplenSSlllllptiol\ofthe fonn oftllt~pa ir ]lotl'lIt.ialO'UllI,,~1'"1\,,,,1111 1'\vur io"s1'1''' ]1' ert iosoftlwsy st emthen-byohtlljllC'tillitlt!«l1'111II'll I,Tlu~rdll!.ioll l",lwl"'ll thetransitiontomporntnremnltill'r,',h1t'I~(1thil:bwss is1'lI!'ily"I,l.lIiw ..l, Ill'W"\-"1

uny

Iurther'~alr:lllnliolliuvolviIih1\fini teslll){,I'Coochlctingorder parameter

n

or the JlI'f:S'~1I<" ~ofthoeriticulrnllli;lJ'~ticfield will:)'~involvedillthecomplicationof the ]l,'ri,,(li('sll"lldlm!ofthe ,'Il{'rg)'-lIlolileutlllllspeclrn m,Anot her pheno men ological IlPl'nmdli:;Uwumnellingmodol of McMillnlll18I,wher ea potcminlbarrierisf1S- sumr-d10r'xis!.Illll l('illl('l"fll,~e;tunnellingth ro ughtheburri er then istreated by the tn'll"f, '1'1!;'lIlilt ouihlllildbo d l:nlI nI,,Vhi]cslIr:htheorie-sluwobeenex t remel y

"l1",·(,,~ ,.:r1l1ill ]ll'ovirliugIIqunutitative descriptionof mnuy ofthe ohsl'[\-c(}phc uom-

"W'thr-rr-Ill"<'illtlu'li t('ru t,llJ1'II1l111l1]wrofdntu whicharc not r-onsjstr-nt. withthe pr...did ious of1,1ll' W"rtlHllllertheoryoftlll~prox itn ity effectlr.I {toII 81,Thill.such tJ"." .-i"sfld]illI~H','111"('ofthin lap:'I'>Iand for elcnn systemsisnotsur p risin g giveu lilt,Ilp p roxi" llll ilJllfiiu\"oh-('rlilltlu-tlr-rfvutionofthr-Wcrthnmcr form ula'2·J1[1 G1 I~,:I.

()l l r" 1"p;'~ ll l r"' l"il':iwhichhaYf'bor-nstudied include on" dimcnsionnlqunsipcri- ,,,lit'I"tnwtnn",,1nI'Illtlsr'lffiill lila r o rfra ct nl geouw1.ric fi[2~I.Theinterest.illsuch rl<1\',.(,g;'~l]lll' tril'fifil<'IIlSill\1l'(,1.from the hopethat despileth e1'1'1111h'dy complex uutnrc-"ftIl<'p;'~l]rrdri ,'sillquosf.iun,itis ucvcrt.lu-lcsspossihlc10 understandninny quulit.utivr- ;'filJCl'l soft.lu-ir lu-hnviour intermsof fairl y gr-ncml nrgumcnts. In the

"i,H"

ofti lt's,,\fsimilarp;'~l\ndl'rUle[[\ct.thatthest ruct u re1'('])('aI5il selfOW'\'ru nny I('up;thfil'akfis'I~W'fi lsthnt OIl('sltouh l lwublotodescrihecert a in aspe cts ofth eir I~'hil\'iolll'iuI"I"IIISofcortninsl'alinp;nrguux-ut s . Th ishasilldCl'dproved tohe lIlt',' 111""illt\nmulx-rDfITI','ulfitlulil'l<011III"dos"lrrdll te ,!problemsinvolvin g fiU]ll'ITOlllh ll'l ill.c;illJ<IIIOrJlml fradalll"tworks[1911'10I,

Intltifi,l ll'si,,;,WI'rnh-ulnted" IIl~ll(1<'ll n'of lhetransitiontemperatureT~011 rlu-tlli"kll"Hfioftln-slIp,'n'()w!udinghlyl'l"fur both theperiodicnnd qunsiperiodic

,L::'~'llll'lri,'sof supr-rlnt.tiro The outline-ofth othesisisasfollows: InCuaptcr2,

\\','rr-vic-whridlyIltl'Hillinutjcrnscopic thcoriesstarti ngIroni th oeffectiveDeS

Hamiltonia n and diOKlIS! th eell' Gl'nnes''''''TlllllllwrIlwury. lilt' Eilt'uhl'l',l:;"r'''Illa- tionsandtheMcllillNllllod ..1.\V..dHlIi~'tilt' I\nlnt i.'lulillthi"dml'h'T frullI Iiuu- to timeillor der 10follcwtill'lIot/llion.. illtI"IS('"l'ip lla lpall"·Pl r it,..1ill ,Ill'dl11 l', ..-r sothat one enn ellsily Ct>IllP "f'('till' l'l'S1\It.. illtilt'thl",i,.wirhIh,,,,,,·intlu- l'n l' l-nt.111 cha pt e r3,we Ilp pl)· th ..DnJ;uliuhcl\'1'l(lliltilluto1I11t'Ti...lir p;l" '"wlry wilhII"illll,I,·

assumpt ion of thefonn of thepnirpot r-ntinl,llIulllwlIrll!rulnh·tlwpnir-1",It'u t i:,1 self-co nsiste ntly whiehnllnw..II"toobtainthr-n'!ati llwlhil'IM·h\....

'1.

U... h""II"it i"lI temperaturennd the thick ucsaoftilt',.iup;l..h'JI'f.'mllt..l,l,..\ illU1l'""IIt' rll1lti<',', An energy-momentumspec t ru mbasbceu..luniuod withtilt'npl'lit'lltinuofBl"t'l.'"

theorem, \Vecompare nut!coutmstthe r.·"\llb.nht nitu..\t,hrclll,l:;hI,bisI,·t' lllli'lll"

withthoseobtainedbythenthl'l"Hppmxiumtioll l<t"1J1'llIl '" ill\'ariOllHlilllil.ill,l:;«hun- tiona. InChapter4,We'cnlculatcthetrnll"itilllLtl'IJL!Il'I"u llln'

"r

1,111''IIU",il"' ri",li,' geome try withfr nctilll\ul dillu'w,i"1\1,)'t.lU!,I:;'·\I.'fnli~,·.1\\'c'rl!ulL! lI'f1,IL'~ 'I")' lm'l"""~\

b)'TnkalmshiundTuchikil~II.A compurisou ofrlu-1I1,,,,rl'l i,'nll' l"I..li"' i,,"~willI themeasured\"llhlt'l\ln Iis1'l'f'S.'lLt....1uudtlu'r.,I:;T<'t'II11'l.t "hnwlL In I" ,fill ih'S:11i~- fnctof)·.In rwo limitiug CllSl-'lI thnt tilt,l{,lJ~hS('lI\I'IIfIh,'wll..\I'

rr:...

Lu]"t roU'l...' L» ~nndL« !witb! bcingthe('(lhf'l'f'lIr"I':II~t",tJlf'nulll)'t ir:11t"f"'llllJ<..rtlll' sca li ng lawwhiehrctI.ccb thethickll~tlf:lloMcdM.n ·"r till'Ir llll,.jt i" l1lA'IIII"'r llt nro' areobtained.In CIlJlptcr5, we sum maria...tlll'n'Sul t!.;

fir

till'wnrkI'n.,...'lLh·d Ilud offer somecon clusions millt1i""I1~~olJnJllr lTUiup;tlu-ru.

Chapter 2 The Inhomogen eous Microscopic

Th eory of Sup er conductivity

IIIt.his{'h alltf'!',W( ~introduce theeffectiveDeSHamilton ianth atarises as a n·~ illlt.oftill'df~droll-phoIlOliiul.ernet ion[J2Jtogether wit htheGorko v'aequ a tion s foJ'r.1]l~Gr,,,·lJ'sfuud-ious.!1IInorderto apply Gor'kovequ at ionsto theinhomogc- 11<'<lIl"lllljlf'l'C'om!' lcl.OI",some furthertheor-ieswithupproprjn tonpproximnt.ionahave t,uI" ,iut.r<>< h\(·cd . Neill' thetransit ion tempera tureT~,one cnn linear izethe Gor ' ko v '·'Jlllll.io lLSilll'CUIl'onkl'pnnuuoter- isinfinltosimullysm all.ForII bilaye r syste m ('OIup us('dofthen]I.,'rllntiligSIIPf'l'coud lldi ngnndnormnlmotnlIaycr s,de Genncs nss lllll''''\thut Ill/'kl'l'Ill'1illti ll' linenrlar-dCor' kovcquat.ion satisfiestlw diffusio n r-qnntionalldiurroducod the luuurduryconditi on sHI. freesurfaceandinterfacesso rhutt.lll'trall~it.i(llltruupr-ratum runheco mplet ely solvedillvari o uslimiti n g cases.

\\"·I"lllHlLl" I"dl'f i\"l'd thl'smut-resultus thatfoundilldeGennos' theory in more stn,ip;lLtf "I'Wnn lwnyrutln-r t,11H1Jd"lllwith thediffusionequa tion.Awnyfrom the lnllisi ti "ll1'·IIIIK ·l"II 1.lll"' ~ 'a cnk-ulutlou sche-mesuitableforafinite ord er pammot or is1lI~..b-d. Eiknlll'rp;"!' simp lifiedtheGor'kov equatio n through red ucingtheun- ImowlIGre"'II',; fuuct.ions from four to twonnelthetheory hnsbeen widelyused to Sillilytln-df,'d,oftill'Pl"l'SCIU'eof nexte rna l ma gneticfield.Anot he r npproneh is tln-t'\lIllll'llinp; nro.h-lHumlltouln n proposedbyl\Icl\'!illHIlwhich intrc ducosittrans- f,'rHniniltouiaunndtn' ntsttu-iutc rfncc asallenergybarrier.Thevarious theo rie s 1ll<'lltioll,..1 aho Y<'willb,'disl'ns."l..lbrh'Hybelow.

~2· 1Coe'kovEquation

!i2- 1-1'r ileElccu-on-P hon ouInt e rn c ti onHnmiltoulan

III11",s,',.,JIld Illla util.ul iollrepresen ta tion ,themodelHnmiltouinnofthe dec-

tron-phouo nsystemmay be writtenns[aaJ

H=Ho+1I •.

wl...Hois the Hamiltonian for Ieeed("('tnl\l~llIul fm'phClnulL~.

H.

= L l'''''f11~f + ^{Lffc{, rt" .}

f t,..

andHIistheint ernct ionHamiltoni anoftil,·dl'drllllMmnltill' !,!II1Il,.n,.

HI

= 2:(Df«;Ct.,acr,,,. + D~,,1-rL(,,,cr, ..)·

{2·1·:1l

r.f,"

Theindicesijand

k

denotethoW1l\'Cvec torsoftiL('phOIlOllllmillHit,1,1,·,·tl'oIlH respectively,tt".,

f,;-

I'is th e{'lwrgyof nlIillp;lc'eln-tronilltIl"1Ila h'

r

n,lllti\'"

to thechemicalpotentialI',Ill.<Jifdenotes t111't'lLl' C,lO'"fIIplllmnllilltil"Hlnl<'

'i:

rr isthespi n ind ex of fill electron ;D.,(Di'}i>lth('dc,<,tr uu- ph" l1<111rUlip lillR '·"Illltnlll..

whichdepends011theillU'rllCtinn plltcn ti al, unrl

"~ (II;>

^nlLcI

4 . ..

^{(ri... )IIr,·till'}

creation(annihilation)ope re-oraofthe pllUlIUlI>l:lnclc·\.-rlrtlllll.whieh s:.lisf)'li,t·

elgebm

(lI~n

^..J

= lI~i'

^{-1Ii' 1I}}

=

bf f• (2-1-<1 1

{cl..,CI-. .. ^J

⁼

et.c ;.. .... + '-"'... . c t..

^=!l

r

^f.

b.... .

^(:!·I·!',)

respoe rivoly,

StarringfromtheHmniltlluiun (2-1-1 ),F!{lhlidl[ ]lll1l' ri\...~\ tIJl' l'lr,~·ti\·.'llllllli\· tonianbymeansnfthe cnnonica ltn llLsf fl l' lll ll1ill li

f2· \· f))

where th eoperatorSischcensuchthatthetrl1IISfllnl l"'\llnmiltoniunJI.•IIHs111,- slime eigenvaluespeerrmuFIllth atufII.

H._11+ [H,S]

+ ~1I11, S] ,SI + "

~

^II.

+

(II,

+

[H., SJ)

+ _~[(II, +

III" S]) , SI (2-1-7)

+~[lI"SI + ~[II' ⁺

[H"SI,S J

+...

\VI'('1111elhulnntothoiutcructionto JOWl'storderifwechoose S to sntisfythe

"'jll flt i ull

HI

+

[JIu,S]=0, (2-1-8)

~"tlmlthereisnofirstorderelectron-phononinterac tionand weobt ain thefol-

H~ :: H I)+ i IHI,SI + ~ [Hl t lH

i,S},Sj + ·

«u;

+ ~[HI, SJ .

lft,]ll'nlM'r Ht "rSiH lIHHll111f'c1 t.oheoftill:form

5""

L

(A1-(lq..

4 +,r,,,cr.<t + D.r,t4~q.,/r . .,. ),

I.i;.f7

1111'1' I1sillj!;oqnution(2.1.8 ),we can determinethecoefficientsAq and

nif

{

.4,1'"D,n "r

+

11Wif -fk+ij)-I, n,; =D~(fr-!lwq -f' _ii) - I ,

(2-1-D)

(2-1- 10)

(2-1- 12)

S=

L

(D, a,jl J+f,./'k',., tD!. nt4 _<7",cr,(I' ).

r,y.", 'r

fIH

+

fl...·

v

^q

fEr

ff_if hWq~

ToohtulutIn-dfc·!"t,jwHnuriltonianwhirl!de scribesthesca tte ringprocessofthe (·J,·(·tn llIHthnm)l;h{'X{'llallp;,,'(Ifour-phollon,we CUlltok ethecxpcct nt.iouvn luc of

tiLl'tenu1/ ..witht'c";J!''fI,tothophonon vacuumstate10>,whichisdefinedns

(IoriO>::0, ^{(2- 1-13)}

togive 11~1I=<

olH.lo

>

= ~ L L: {<

0IHd"f><

".-ISIO

>-<01511/oj'><"illlllOc-}

- n~ , (2-1-1-1)

= ~ -

2 :{< OIH ,Il(

.

> <

1~510

^{>-<01511(> <}

1,~lIdn

^»}.

wh erewehave usedtheecmpletcuessofII... phllllourj~"I~tlltl'S

L t

n..><Iliil

=

1,

·fo'

an dthe prope rtyof theopcnuoeS

c1/<115\0>=0 (or II,;"

'#

107'

Substituting(2-1012)und(2-1-3)iu t o(2-1-14)IlllflrfllllJllr,tiuJ!:lli.,1I1J!:dm. ~·n,kll' latio n, weobtainth eclfret iveinternctionH:uuiltClllia ll

whichdescribesth ein~nction!lctwct'Utwol'k...lrnll~Ihrcl1lJ!:htill"Hur·phllllllll- exc hangeproecss Thc othcrtermsdNrri h in,:; 1II11Ilip lllln'.1I'run...>;f'l"":1111",IImit- teddill'to theMigdnl'.thcorem.l^{3 3}I

§2-1-2 TheElfective

ne s

Hnm il tonin n

TIleeffectiveclcc tmn-p henouink rllr-tjouIIllly h.:wtilt "11I'N

wh e re

~2-1-18)

WIWll l)uthoftheek-ctrons senuc rodb)'exchanginga phononare veryclose toth e n~rllli S\lr ra,~ (:dden n iw!<!lIy ChClllicll-lpoten ti al/I, thediffer en ceof theenergjcs sl, tisfipstil!'rOIHlit iou

h -

fk+t"1<hwij~hW/J(w,,,istheDcbycfrequency )so l.llllt tIl('....lIJllillJ!;constunt1.-"1:,9"<0,whlehyield s

'ill

attructive interac tion. Ifthe Ililli'n'J1"I~oft. he,'llPrp;il'Sis11ll'W'r thanI,....'f),the coup li ngconsta ntVt .ofi~positive sutlllllIl,,,!isIII"f!Jlulsi\'(!lutcrnc t.io nwh ichdecreasesrapidlywit h inercoeing

111tIl" Mljlf'I'l'OIlf lucl.illP:; stlltl',onlythe electronsoCl'up~-ilJgthe stu ll'Swith

"IWI',!!;.\'ill1.1..•l"HllK"ofI'±"w/Jrrmh/~seal,t/'rell to thenewstntestlll1lugh the phOllllIH'x d wllJ!;/' procl'S,', Thoseo('cupyingstates farbelowthl' Fo rmi su r face euu Ill'h'1'ilh'dnafl"'"d''fll' u lIS whichprohibi totlm-electronsfromocctlJl)'ill~the sam" stnh'sby HwPnuliexc-lusion prin ciple,Thus,in theapplication of(2.1-HI) InSllpl']"('''lldll d il'i t,y. Cuupr 'r{:11Itookthenppmpri nte assumption th ntforthe ,'J,'",mllsillt.IL,·statt·sfr<".../Jt,lm cO\l]llillp;imluccdh)' ph onon» is a con s tant F>()nud[,,1'UlUSf'illthost.atcsf(>huJj)thecoup " n gvnms h cs.Soth ceffect ive Iia in ilioniull1)'~"Jl ll"S,illthe DCSnppnedmnt.ion,

(2-1-20)

TIll'approximut.iounbt n ill\'rl hereisvalid oul)'fora"wonkc01\pling"supcr-

"llluhll'lllr,",illt",thofonu o[theHe!!impliestwo nssu mpdo ns:

(illllasslI ll\ill!!;11,' olll'lillg roustau twhlchis illd.,pPJH I"UIofthceucrgy vnna b les

fr

HI,,1h'~·ij.\\','lilli'"IWIl:Ir~'h'<1tIl('effectof rctnrdntion,

(iilAll1,111'1'I'OI" 'SS''Sofahsllll,t.iolLnudem ission ofphon ons1Jrcr('nling mHI illlll ihi ll\ li ll ,~apair ofC\lliiSi-1l1Ulidl' s11I1\'('b(,(,,11llcgb,tedsothatthequusi-particlcs lln ' ll"l'll!f'11as til''''' ' withinfinitrIifct.imo.

'1'.."lllls id ..rbotli of III(':ll Hll('IH'I'SllLl'ntiD)J{'(I,above,nstro ngccu p llngthe-orr slll.u l,llll' 1111i1111UdWI'willdiSf'l\sSit,illthelast sedio n of th is chnptcr hrid ly.

In applyingthe theory to,h(' illholllo gnll 'U1\1lcn>ll"itill rtl1\\'I'lIil'lll lnIrnll~r,'rlll theHellin tothe roor d illnlcrcpresentntionh~' 11I"fm.~urIII.,In\ll~r''''Il\llti''Il,

(:!-I-'2I)

whic h maybeinvert edtoyield

(2-1-22 )

wherewehavechosentlwvolumeasunit.y,TIll' llpl'rllior tlJ'~(,n^/lull

J .!(.rj

1'1Il1I..·

show n to satisfythe nnti.cmlllll1lthlingnlgo'hrn

The effect iveHemil to uinn in theDeSIll' prnximl1tillll llLll)'I", writt e-uill1''1"111'' ortheoperators~and

Jot

111'1

H'II_

V- -~V ... L J ^{"l(x) " l(X)}

>i, (i) >i. (i) .t'J

=

-V

J ^.ptC%) ~t(;) ~l(r) ~I':;)

f".r,

(2-'·:N )

exclusio n principle,whichprohihi h. twocl''CIIllIl ~Inuu"IlIyillp;at111" "HilII'1<1:,11., Lc.

FortheHnmiltoni un(2.1.2)oftherl'{'l!pllOlIUU!'lJ'll'l"h:d.fIlll!'l,WI' llmittill' the free phonontennsinceinsllIWrCo lllhu:t ivit.y,till:physir-al IJrup"rtil'SnrIISIl·

10

t1Jl~IihOIH'IIH"Illyinurol(~whichIn ducesa ncw attractiveIntcmctionH, j j ,The fr"l 'pnrt ofllH~Humiltouiun,

H

uwaytheniJf~written lIS

(2-1-25)

wilh

_1i2 \J 1 f{iV ) =~-II.

CUlIIhillinp; VI,(2 -1 -2'1) 1111d(2-1-25),wefiurdlyobtaintheDeSHamiltonian

i:

k=ko+f' ,

(2-1-20)

wliif,llis11I'll'll definedtheo ryincl u dingth .. phonon-induc ed nttrnct!..c internc tien

pllnllllf''''!"SillSlll11'ITOIHlllCt.ivilyandexplaintile cxpcrlmcntnlph-noruenn.

!j2·1-3Gce'ko v Eq uati o n

TIH'tlll~'Q'ul.l nilwd ill);,.~t.subscc rio u ennbo'-'lIsily gf'lIeflllize<llnin clude till'illrlll" !W''S ofbothf'xt,('rnnIIlHlp;ul'iicHeld nnd theexistenceof non-magnetic impllriti,'s-,Followingtill'methodofA.L,FetterandJ,D.'Vlllf~eka[ J~I,weobtain

k""l~',.+':',

s ; ""

JfP'l'J'1U;'){

~1-

_~/11

i'I,, + =

..:r(;)12-II+UWl}Jon (i).

C (2- 1-27)

f'

= -~ F L J ^{1;·.t(J) J.t (i )}

^\4/1U)

~i,,,( "r)(tl.r,

",II

wlwn'"I(,r)isthe-ver-hn-)In!t'nlial lllJ(!U(F )isIh~impttri t.y potcutiulwhichlSof Itl<' fill"lll

[i(.r)=

L

1I(,r- r.,) , with.f'"Il<'ill~tlu-Pfl~iliol1of,,11, illlJll1l'itr,

11

(2-1-28)

Gencr nll r, onecnnoht ni u0non -]illl ',ll' Ilitf"I"l'Ul illl"lllllliiolldl'S<Tihill~thr-"p a tialnndtempora l evolutionoftheop er atorlit'hls~:''''(.ilhy1l\t':Ill~oftln-1l11111ilt u·

nian(2-1-27 ). Ho we ver, itis teocompllcnn-dtohr-,,01\"('(1"n l1ll'l,'h'I~'"0IIuu-nn fieldapprox iumtiou hasto!>('tukeu, III1hi" Illll'roxil1lnlinu.ow'l'n llc1,'rHl1Il" 's,' theproductofIour fit,ld opt'ra1on;illtill'lntorartlon tenu \.lntotllC'follmrill~

\i"":l('elf

=

-1/

J

^IP;r{<

^J.krIJ,t

^(1)>

~i,d;)Jtl{,r)

^-I-

~;,r

^{U:'jt"rU')..::}

~"IU~h'; tlJI ~ l ·

(2·1 -:.!r11 Thereason forIIlll ki n gsuchadl'<.-o Ill Jlnsitin ll iNtluuau"Ns" lIt,illldlal'Jw "'riN Ii,-

"r

Sll p cl'colldncti vityistheIorrrmtiouurIICuup' -!'pnirshytwo"\,,.-1ruliSwit.hul' l' 0sik spin s, which 1'1-'»l1lt Nill II... ap jlt'J1 I"I1lH'" Olf nil »nh-r 1'111"il11i.-t,-r, sim-,'

<~tJ,>#O.TIll'othert('n us "Olllill j( {mill till'Hal'tro~'-F'H'k,k""l1lllO ,sil i" 1l slwll us<J'!(,f)Ij..,(1)>ami<J.,~(,r}J'!1(.r)^{>ha n11l"'1I}nl1littl~1Silll"" (lilly1110'ditrl'!". cnccbetweensllp('I'('olld \ld j ll~stun-nnduormnlslat l'snn'Iff jllt!'l"I'sl;ru-vr-rflu-h-ss, thosl,Hnrtrce-Fbck tCi'lUSnr-c-ns.'mllll'flto 1)(·tjn-sallll' illl."tl1urtIl<'stlilt'.S111111 the yhnvcnoinfluenceonthe-«uupnrisouIJI't\\'I~'1lt!WSI' t.woSt.HlPS.

\Vit hthemenufidol l\ppn ~~i lllnti"lI ,thl' df,,{·t.i\"I'IIUlIIilhlllinlllllllV1...I·n IIWS

(2·1·aO)

andthepairumpli t ndeiatlu-,lr"-OIllp nsi tiolJi~d,'filll',1il ~

(2·1-:llj

with

Eq.(2-1-31)proVif\cSliSwithusdf·l;fllJsistl'lI1.ddiuitj 'm'ff tllf'IHlil';LIIII /IiI,IlII,·wlli.-ll dependsOiltheeffoctivoH!1IUih ollilllliU"!lI<liull;11]f~JlllirUlu lllit u,],'its..]f,

12

II'JII'I'l'\\'1'umy r"wlt'l!Tn~nnthe imagillaryrlme.

\VitllI.lJiI',j,'fillili" ll,OW'runf'~t ahli~hthecquntlon ofmotion forth...field

"1"'l"a l"I'.~fnuurln-l"ff'.'d ive Hruuilroniun

k.,

11.~ingtheHelscnborgequation (2-1-33)

1I'1l"I"('(),,. il<alIl'hilrury0]>1'1"111.01"ddil1e~!hy

Oi , =

cli ·llfOc-k·llf .Byme'nUl<

Ilflilt,111lIi' ("<lIl111111tlltilll!;all!;,'IH'n(2-1·23 )WI'ohtuiutil<'"f!l\i\t,io llof motiouforthe 1i,·l,101'l'l"lItorl<Ill<f"llO\\'l<

{

"-OD_r

J. " I

=-1-.!....(-_2m ibV'

+ ~..r(.r})2

_c

- /1 +

U(J l]J." l-F<

,7" l!~[

_. ^>

l!,j."

_,

'lf~;'1- 1 "" _{[~(ih \'} + ~ .

.f(;))2 -/1

+ U(;)j~L

^{- l '<}

~I, t I;\t

>

~i'I\I '

r ' . ! t l l r: (2.1 .3,1)

;':otl'tlmttlU'pairmuplitudcslIl11yhf~expressedill tenusofth eH(i~(,llh(>rgfield

"I"' mt ,, rsln llll<fol'lWlti'l ll, i.l'.

{

<

~~1 ,l ,t

>=<

t~L(T)J't ,(r ) » .

<

, 1' IJ'1

>=<~~I\·I(T)J,.'I[(T)>.

(2·1·3,))

Silll'<'WI'Ill"<'onlyillll'l""l<k d illfilUliujl;thesolutionofthepairnmplitnrlo.

whk-i,,e;il" '"til('onl"1"I'lltlllllP h'I'ill1Isupcrcrunluctor,mrhcrthantln-detailed r<'[lI"'''l'lll:,tiOll oftill'fll\1lllti;:ptl WII\l'functions

J.

owl

J, t

tht'I)l !i'('h·C's,wointroduce

13

togetherwit hanomalous).Intllubnrnfunctio,~~.whichnrr-dl"O"d~'n'lntnllu tl1<·I>ll;t-

nmplirude.gi"Cll1>"

{

:F{~r.i'T')=-<T,fio,,·,{i'. rh~":I(i',T' )I

».

:Fi (;r, ;'T')

=_

<

T, rt'~L ( i'. r)~~L(t,

^{r'»)> . (:!.\.:Ii)}

Thesclf-consi<;tCllt eXl'rt"l<lliouuftlK'uf,I.-r\Nlmm, 'Il-r;,.cl.,lil"..\l1:<

Thetime-order operatorT,whichnppcnrs intill'd,'fil,: .j"l1Il(1,111"""~lll hllrml11 functionswithrc~p,'('tI,)thl'imnginurr tim.'T is<\d1111..1I1S

<T..(.~{r )b(r'lJ>

=8fT -T')< ..i(r)J1(r' )>

-8(T'-r)<D{ r)•.t(r'l>.

where

.4

azlfl

B

nrcnil)'(t'rmiouoprmt nn<11m )fI(r)j,.IIII'7<1."fll ",.,i"ll {

9(T)= I, 0,

r>0, r<0.

Tof theseddi ncdMab uhnTfl{u ut"tiolls.wr r"'11linn'till'Cnr'ku,''''llmn"uS

1-lIi -

2k(- ;/,V+ ~..r(.i')l 2 ^{+" -}

U{i' )J9(ir•.i"'r',

+

6(.rJ,FI(ir ,..i"'r' !

=Idl(r-i'), I:H:J!J!

(-,, ~

^-

~(-i'IV +~..f(7ll2 ⁺

^II-Ur;,I.1"U'r•.i"r'I-6.(;:Jmj'T,J-"r' ) .,.,0,

(2,1"11))

[ ft -t; - ~(iftV + ;.4(i »2+

/1-U(l /IF f(;r,;'T',-CltCr J{i{.i'r ,j·"r')

=11•

f2·H1J 14

IIIIJ If'fnSl~thnttlwHumilt on innis inll"!ll'oclclltof imagina ryt.im"T,the Ma t.slilmn.functicmsonlydepend ontlJ,..di ~"'rcllcl'oftheirrmginnry time(T -1").

TIl"F' >1II"i"l"tmusfonuutbm oftheseIuuctionswithrcsIlI'c'ttoI'yiel d s

{

9(:I:T, ;'T')

=

({JII)-I

2:::>-i...

^1r-r'lg(i ,i', wn ), .rt(.rT,.r~T' }

=

(f'lh)-lLc-iw,,(r-r' lFt(J7,J~ ,w,,) ,

wll" rl' w"=(2u +1)11'1/'11,;ll1Ilu =O,± 1,±2, "a ndwehuvonsedt hcpcriocliciry

"ft,llI'fI,l..t,su lmnlfllllfti<ln(nlll;iw'ullJ'

9(r<0)=f9 ( r+{1>0),

wlll'n'1IIf'Sill;ll1111111<'rill;ht sicl('c!"l l('llclso ll whethe rthefield operato r-scous h 'tld ing 111"1I1i11sI11,amflllwlinuan'fl'l' lilinllsur lmsolls.l 15 ]

TIll'("llllllio llSofIliotion~:lTtill'Four-ie r«uupourutscan bewrit.tonas

j

[ill """-;f-(.-

_II '

iIIY

+ ~,.i( i»'l

r

+

I' -

U(.rJl9(,r,J~"-'"

1

+

.6.(rjFt (J,;r',w,,)

=M(;-;r'l,

I-i" ....:"-

~(/h" ⁺ ~..r( J))1 +

^11-U(:i")j.rf(;r,i',,,-,u)-.6.f(; )9 (;,;"

, ~,,)

=0,

(2-1-43) Tll ,'wlf-\"' llsis!l'll t rouditiouln'ouue-s

~(,i")=t'<~i'd,rl\i, l{';:")>

=l'F( ~'7Tu+,iT )

=1'(/1(1)-1

L ,-_ . ....

^utF(;'r, ~~, w,,) ,

(2-1-44)

TIll',,:":nl"k, ,\'\-'llHllicllisul.tniucdnbovo Illlwidetheba si s forthe self-cons istent .'a klllll li o!l(IfIlli'onll'rpnruuu-n-r,Our('1111nssum citsim pleform of .6. illEq,(2- I-,t;ll1111,1obtainth,'snilltjull ufF.TIl(' £'1.(2·1-44 )thengivesn lIC'Wformof

.a ,

15

sothatbysubstit ut ingthe ob t ainedDointoEq.{2-1-4 3)ll~llillandrt'llI'ntill~till' sam eproce d ure,onecnnfinally ebtn inthe sclf-eousisteutsolu tionfor~,

The theoryobtainednbovecoversmost, ofIlwl'nr l)'tht~ )ric 'HsudlliSCill~.I,,'r,C;·

Lan dauth eorylindDeStheory .Gor'km·l~6ISUl't'l'l, tl c,,1ill d,'riviu,l!:llit'C-L"111111' lionwhichis validri eurthecriticalt empcrn tnn-T..fr omGor'lw\'1'l[Unti'hls,TIll' der ivation determines thephcnomcnol ogicn l('tll1.~t ll l\!.It1l1'pt'1l1'ill,C;intl lO'(;·LIll<"

oryin termsof themicroscopi ccnlls tnllWl lludt,lll' np l' rnprint,I'nlll,ctf,,1'whidlt.I\O' G-LtheoryisapplicablesothattheG-Lthooryhas:.!~t 'firmhnsisofmk ros l"0 l' i,' theorynnd con be generalized to muchmore('omJllknt.t·r~S)'S!.t' llissuchlISlIlilP;llt'l,i,' superconductorsetc,

Ifoneapplies th e GOI"ka vt'{l'mt.iollsto t.hehulkSIl[l"Il'o lldHl'l tirwhichis SJlIl·

tia llyhomogeneouswithoutthecxtemnl mn g llt'tirfit'll! ,"HI' runI'1lJlid ly[inr]th,' resu ltslIS obtainedfrom theDCSr.1.\ItlU'o r y , Asun ('XlIllIllll''Iflil"1l]']ll it'll1.i c'll'If the Gor'ko vcqllation~,wewillshowt.h,~t!e!"ivlIlioll11I' i "n ~,below.

In a spatially homogenous superconductor,(Ill"1'1111tnke

..rCx)

=

0, UCi)=O ,

~C i) = .u. , so thattheEq,(2-1-43)is oftheform

[i liw" -

~(-i'I'V)~ +

,'lQ(r ,X'w ,,)

+

bo.Ft(r,j,'f,w,,)=',h{.;-/ ) , (2.].4,",)

[-ill wn

-~(il'Vfl +

/IIF t(.T',t,w"J_6

t9(:i"

,j'" ,WIl)" "o, (2 - 1-t\Ci) Thetrnnslntionnlincnrianccof till:ah ove Eqs.illlpli.'litln,.

9( ; ,t ,w,, )=g C7-:t ,w"j , Ft(x, t ,w ,, ) =Ft(i_j:' ,w,, ),

IG

(2-1-471

IIll1lII(~ll<:eWI:may havetheFourier trans form ntlonof the Matsubarafunctions wlthn~sp(~dto spatinl coordinates

0(; -

.f" ,w,,)

=

(211')-3

J

^{dJke ik}.(l - i" IOCk,w,,) , ,rt(;-i',w,,)=(211')-3

J

d1k eif.( l'- I"IFf(k,wn),

(2-1-48 ) (2. 1-4!l) Sllh~tillltilll;thoE'I,(2-1-48)and(2-1-49)into(2-1-45)an d(2-1-46),we have the ,'cJlIllliOlIS

li//w"-cd 9(k,w,,)

+

6.,rt(k,wn)

=

11,

!-ihw"-frIFf Ck,wu)-6.-0(k,wu)

=

^O.

wln-n- till'onll'I'pammctr-rfj,is ofthe form

TIll"s"llIti"u softhoElls,(2-1-50)lind(2-1-lil )are casvto obtain

st>thet<df-l'ollsistel\tc'clual,iollf<>rtheorderparameter is writtenas (2-1-SO) (2-1-51)

(2·1-52)

(2-1-li3 )

(2-1-54)

(2·1-55 )

Elilllillllt,illP; till'rouunoufactor Aill(2-1-5ii) 1"111<\completingthe summation0\"1'1'

..'", \\"1'huvr-

(2-1-56 ) 17

Thisequationca nc1c lC'nnilicthr-tr nusinontempcrutun- nul!urderpimmlt'lt'r~(T)' Wh enT

=

Tc,thesyst em goes tonormalslalt'l'Illhnt~( T~)

=

0, EtI.(2-1-5G)i"

of theIc rm

{2-1-571

Th is equationallowBusto writedowntill'r-xprr-ssien (urIrll11sili.>1\1'''III''Tallll1''/~

illthebulk anpercondnr-torhS

Tc

= ~cxl'{-I/JY(o)n

^(2-I,tJ8)

where"'f is Euler 'sconetnnt.Thisisthel<h':nrtL'l<\\ll'1I111In l n£De SIIH'ur~'"ht llill"l!

bymeansof vnrintionlllC'th ocl.1HJ

An othecinteresting1illlitil\~rusoi.~thr-vnhu- urUlI' ol'll/'c11lLnlm'!" cIll.'1'""fl.

InthilSca se,Eej.( 2·1-60)IX'('UIlLC'S

so that

6(0)=2wnf!xp(-I/N(O)V). (2-1-00)

Themore gene ralrelat ionbetweenthe:uNliT p:mtllll ·!.c·ruud!.c'IlIII11"llblfl·11<.-.1"

numericalealct-Iationand we do nol disnlS'1ith'·re'.

\Vc have discussedthe theo re ticalclC"tlCriptioll..

r"

slIl,,'cro...lur lll,.Im",~1'11 1 theGor'kov('(l\liltiomwhichis'Iuitrgeneml1111I1iJlI'h ll"'l'ti ...G-t.t11/~'CY1I11'1IICS theo ryasthespecialeases.III mMiliollwera nllpplyIbis1,ll'~ W)'t ..nilio n'""Ill' plientcclensowith theoxu-mnluwgm·ticfil 'hlllllJ!i\ll\lul'itypoh'utialu l",liS1'1"will sh ow,to cunaidcrillhu lllllW ~lwllnsst r uctures. N'1'vl'rtlll'l,OSS,tIll' non-liuvnr(l'lot nl'!'ill theGcr'kcv cquutious mukesii,\'f~CYfliffkn llto""l\'f~Ihf ~"fli lll ti oIlSdin-ct.ly~;"t.llHI

\Vc willdise usethCM~1l1l'lho.ls bridlyhd u w,

18

(2-2-1)

!j2-2de Ocnnes-Werthamer Theory

!i2-2 -1Lineaeleed GnpEquatton

Tu c"klllnh)the trunsit.iemtem pe rat ur eTe,theintegral solutions for hoththe flllwti.,llH9 unel.1"tfrom oquntlons(2-1-43)canbe worked out as 9(J ,i" ,w)

=

g"(,r,i" ,w)

-J

^({I",IrP.T:.!Q"(x,,rl,w)ll.(i'dl'r(,f'2 ,xl '-W)6.

t

(.T2)V" ( X2'i",w), .;dp'·,1",w )=

J

^{(P,rlQl/(.il,x,-w)6}

^tP'·

dQ" (xl,X' ,w)

-.I

^{11'1J' 1t/J.}T'l9"(.il,.i', -w)6 tU:d 9 " (XI , .i 'l,-w).6.(:i"2).1't Ci 2,.i',w), (2-2-2) when-WI'1111\'1' iUI,["IK!nC"l,d theGreen'sfunctionwhich describesthebehaviorof a Hillp;II' 1'[I'I' t nlllilltlu- nonunlstate,Q"(,f', X',w,,),satisfying

WIU'I'I' W

=

w"

=

(21l

+

^{l)rrj jJhandII}

=

O, ± 1,±2. "as before.One can notice

tha t,ne-arthl'f1'iticlll transitiontempe ratur e, theorderpammoterwhichdescr ibes asl\[l,'n'!>I IC!Hl't i ll ~^p1111S11ill supposed to besmallsothat a.linearizntionapproxlma- l.io Bfol'til!'order!Hl1"l\Ul!'tercnn be considered,hence,theself-consistentequation

wln-n-til!' cOllplingconstnu t has boon writtenas a function of,i'since ill the nppli- rutioutnI.h.,inhourogcncous stl\ll'rCOIldllctor,ittnkes differentvaluesfor different rq l;ill1l's nll .lt,lw!';t '1"I1t'I is

(2-2-5) 19

wh ere the barmCl\ USthe n\'c rnge everthernllllUllIlrc1i~lrilllltl'(lillll'nri1rl"lUli~lI­

rationsand theorderpara me terhn!!; been m<sllmrd n',,1with lmlrlu-ump:m1 irtk-ld.

Uone canhave thesolut ion forthekernelh~'nnr1I\1'1lll~,nul' "nil"nln,ln'.'II••, trans itiontemperatureTO'from theEq.(2·2·-1),

§2-2-2 de Gen nesTheoryinaBilayerStruclure

Per hapsthe simplestgeometryoneenuconsider illn1,i1n~"'rlllrl\cl lll'1'wil h .,\".

byerofsupe rco nductorlabeledbyAnod unothcr of Ilnrllllll 1lI,·t nllnh,.],·tl11)'B, Relat edtotheco rr elation(unctiondeGcn ncs[~1,,110\\'1'"Um ttill'l:t'ru d

q...

{.r,til sat isfiesthediffusionequat ion. IIIOI Wdhucusionnl,'nlll',it illo( 1.1ll'form

whereD(x)is thediffusionconstnntandN(r)illtill'tl"lIl"ityufl"lnlt~llI'hrtl...

F(' r misurface.bothofthe t\\'o coWltnllt.'1tnk,!"rrt/lin\~IIIIf'Slll'''ltfllillt;;'11ris ill

the supe-eondueteercgiml or Donllnl metnln-p;iou .Fr"llilltill'£I,.(2-2-6)

w ,·

I..,y"

the generalsolution

{, >0

rur , •

;r:>C1

{ r>(1

fur ,

T'<n {2· :!·1 j

where (.1

=

(~)\ .(IJ=(~)\nndA1111<1

n

nrcthe11l1",lsdl,jill...llll)(,v,·,I"lIot illl1;

thediflcrcnt Inyc ra.Thecndfidc ut.lIIInud>..lIltoll l,!Ill'd"kl"luiIWdbyllwI'I'''P' '1 bound arycond irlo na,

The firstboundaryccndit lou givenbyIi.!GI:llIll"l [~1i..

(2-2 ·8J 20

Thiscondition meansthatthere is no electron flow out of thefree surface, which

-$;Q,..,(x,

i')

=

^{0 atfreesurface. (2-2-9)}

Thesl'mlJf!conditionCO/nI:Sfromthe &1_(2-2 -6)uirectlyif onecomp let es the

illli'~rl\!withrc'liJlf~ttox'illtheEq.,theboundary conditionis

Drib.

VJ;

^continuousatinterface .

TIJ(~ 111,~tboundarycon dition givenbyde Genncs(2I,having considered thedirty limit1<.E.with11Jl'iup;the mcnn-Irccpath,isthat

whichyields

cont.inuouslitinterface .

Fl'OlIIt1lf1Sf'bouudnr yconditions,itis str aightforwnrdto solve the coefficientsIi.

l'llh"l'<'llI'f'kngt.hliE..v(T),E.s(T)sot.1111tthekernelQ",ineach regionCll11betreated

ill<roustnut \\"1'('1111hnvethequite »implcequations forthose orderpnrrunctcrs

p;i " " 1l 11S

{

,;"= I:l'"T~ N""~N,n(N,~",;,,

+N"N,nC>,),

.Il~

⁼

L "~ T-I 'I ~(N:b.ll~ +

N"N.ab.,,),

'" ....·1

,,'+1

~(I

21

(2-2-15)

where1\and b arcthethicknessesor the supc rcoudue tiug 1l!u!not'lllHllny"l'rt'~II1" '"

tivcly linda,b« (0'The thickness dependenceoftlll'lI'i1l1Nili o ll ll'1l1]ll' r nln fl'7;.

rna)"be obtained

(2-2-Hi)

with

beingth e "effect.iveNV"in theDe Sforrunlu forTe, \VIll"lItill'I'UIIl'rl nU h'I'jl' com pos ed of supercondu ctingand pure normalnU't.nl.~(V"=ll),wehnl'\'

muchlargerthan the coh er encelengths . IIIth islOlL>;f',only 1,1ll'lo\\~'sl,fn'ljW'II"yi:<

importan t sin ce~."'⁼(DI2Iwl)~illthe kernelQ...(J,3"')hast.lmuurxinnuu{..."= (DI211'T) ~⁰Thedeta ilsofthe one-frequencyupproxuuut.ion,whichiNvnlielfor1,llis limit,can be fo undindeGcnncs'work!~Inoel we <lilly!ll"l'S('lItIl1'r"til, ' l""Nliltalll.

equation determiningTc

superconductorand normalmctnlrespectively.

§2~2~3Werthamer'sKerneland theTran s it ion Tempc ent ure

Insteadofrelatingth e kernel illthelillcnl"i1.l:'] sd f·e'lIlsi sl ' :lltEq.(2. 2. 4 )1,11 tlll' diffusionequat ion , \Vcrtlwmerl 4Idealtwithtill:k"l1ldill a 1lI0l"')s1.rHi~ht.rl)rwn]'(]

22

WHylindobtninerlthe explicitexpressionasfollows

1

L Ci_. (i'Y 'J ~

^{N(OIlI,,(l.lT..'9D)6(i_ii) -X(x-}

^,)1,

" '

X(x -

y)

=(211" )- 3/ r,k·(i"-li)>.:(e~,:'l)^d·1k, xC,)

~ ,,(~ + ~,) - ~(~),

(2-2-19)

where I/,{;;)iiiadi~Ill11lllnfunctionfind N(O) is the density of srntcs for onespin pmj"d ioll ntFcnui-sur fucc. The kernel obtaine daboveis quitesimilartothat wlJrkl'l! outhy ('" GeJLliI'S[ 2Jthroughthe one-frequency approxim ation. Indeed,

WI'will1'11' 1'ls-lowthat. bothofthekernels yield exactly thesame implicit equntiona

whichdr'iI'I'l11illr~t.bl'tmnsit.ioll temperatureTcof abilayersystem . TIlt'snhstltutlou of r-qnntion (2-2-10)intoth e linearized self-consistentgap l'rjllnli"lL

(2-2-20)

"llsl...til<'illt."gn Jr-qnntioninto the (lilfcfcntinl form

wlwl'"

e ⁼ G~~::;~c ⁼ 211"1~~Tc

^'

[N (O)F (.; WI

=In(I;~;~ ).

(2-2-21)

(2-2-22) (2-2-23) Til,'houndnry ,'umlil.iolls,based011thr- conscrvat.ion ofthe cnrr ..nt when the elcc- tronsnOSHIlll'interface, ure

litfree surfaccs,

(2-2-2') contlnuousarinterfa ce ,

Itworth\lutingthatthr-secondboundaryconditionadoptedhere withoutthe ,lilfuHioll\'odfi l'il'll lDuudtill'densityof sinks ut the Fermisurf aceNis slightly

23

different (romthat ofde Gcnnce.Auimplicit nssumptlon has1H'~' 11IllIlll.,tltnttlu- densitiesofthest at es at Fermi surfacenrct,hl'sutnr-ill Illlthoft.lu-r..,dollsmul thediffu sion cons t nutson ly influence thecohorcuce "'np;ths inenr-hI~'p;i"n_ TIU' detaileddiscussion onthe effectsof illtrodm'illll;the diifllt<innt't>Ilt<lll nlt<nndtht, dondrie s of the sta tes at illt.erfIlCl'S('1111hI:' funn'\intill'workshyS. TulmlllLslli1111<\

11.Tach ikil141,

Ifoneconsi d ersth esolutionsof tiLl' orderparnuu-terillhoth rl'/l;itlils toIll'or the followingform

6.(x)

=

c:W -.r, O< x $D. ,

(2.2.2&) one caneasily find thefoll owingC'llllltiollswith thehtilludn ry t'OlIditiolls

(

,1('1'; )

~

^lUl

¥.'),

X(_~2!:~.) = Ill(~) '

k.tan( !:. n)

=

k"t,nnh (~~ lI b ),

whicharcsufficientto determineT<. Asmentionednhov e,l.llI'lasl" ' IHl1 lillll

or

theabove ,ifoneincluded diffusion cous tant annd(kusilil'l'loft.lll ~ stal<·.~lit tllI'tllI' interfa ce, wouldbe tile snrne as tha t obtained inprev ioussnIJSI'd ioll 1Jy,I..G"III1"S theoryunder theone-frequencyupproximntion,Itil'lillthn.t,;l'llS "tha tWI' ""lI s ir ],' 1"

theWcrt hamer theoryva lid forthickfilm s.

In this thick films cnsc, onecnn CO!lsider thelituit~-t:I.:;'~::::::d' .11SlIlllll\.till' fun ctionx{z)ca nberep lacedby

{

lll(l+(fZ)}, fOI":::::O , X{ z)... •

"1"-111(1 + : ), for:$() ,

(2-2·27) Choosingtherranslnc nte mpo rn turoTeN=IIfortlll ~lHll"lilllllJl dlll ,tllr't,l'll1lsit.illll temperature for thebilayer syste m isobuducd

(2·2·28)

24

The re·mitsohta il\('c1fromde Geunes-wcrthamer theor y willhe discussed further illnextchapterwiththecomparison to ourcalculation.

!i2-3 Eilenbcrger'sEquat ionand ItsDirty LimitVer sion

~;2 -3.1Elle nberger-tsEq uatio n

SiIWI~(lr~Gf'1lI11'>!-\Vert.lIa me r thcory isbasedonHnearlzed self-consistentequn- I.ioll(2-2-4) ,it is validonlywhenthe tcmpcrnturc is veryclose to thetransition u-mpmuturoT~.To»tudythe behaviorofthe inhomogeneoussystemwiththe ex- l.f'rllallllllll;lldil·til·lll.OlWhnstoconsiderliw finite orderparameter so that a more 1l;1'1Il'l"altln-ory,hut. simpl,'!' tlinnsolvingtheCor'kov equationsisneeded. SuchIl theorywnsl'l'till,lishellbyEileubergcr11ftItIIII.\Vewillonlyoutline tho resu lts

ofEikllllf'l'p;l'r'.~theory below.A somewhatmore detailed discussion cnn be found

inAl'l'f'lldixAnrfro m Eilf'nlK'rgcr'sworksl15 1[¹⁶1.

Eilonlx-rgorintroduc ed the gauge-invaria ntGreen function s

{

G,,(f , .r, £')=-i<T(~..(.i',t )v\t<:t ,O))>.c-UIZ.E') ,

CluJ,.n= i<

T(~l{X,t)~.(.",O))

> ,'''''''', (2·3· 1) uudtheHllOl1ln!n \1;;Gn'C'u's functions

{

F{I"i:',i')=-i<T("~dx,t)~l (;"O))>eil(E,r) , pf(l,.r,.r')

=

i<

T(J,k;,o J.kt ,O))

>e-i/ (i'. i" ), (2-3-2) whore

nud

I(.7"r')

= ~

_{c r}

1,'

^..1(i')d:r.

25

(2-3.4)

To coincide the notations usedillprevious scctlouwith IhuSl'illEilt'llhl'l'~('r'~lI'urk", we have introducedthe transformation from Srhrilclinp;f'rfieldIt>IIf'isl' ll!ll'r,..; H.,!,!

(2-3-3 )inwhichthe timetis real ruther thaninmginury ustv-for e1II1l1dlu'~.'II"=1, Theoperator

R.

1fis definedby(2. 1. 301 11 \1(\WI.'rewrite itlx-low

with

s, = _~(-ih'V

-

;A(i»'l

-I'

+

^{U{;), (2·:}·0)}

The phase IactorIintroducedin thedcfiuitiouof!.lIP Gt'l'f'Il ' 1lf111ll't.iollllUhO\'f' removesthe inHucneeofgauge trnnsformntlonOILtheGI"f~f'lI'lIflllll'l ionsG,,(I,:r,,r) andGl(t,i',i' )buttheFt(t,i',i" )andF(f ,i,:f')an' roruniut-dP;lIU,c;I'dl'lU'lIfJt'lll..

TheFourier transformat ionsofthe CreonfllUel,jllll"G{I,;r,?jart'.!. 'Ii l lf~lm;

(2-:~ -7)

FortheGreen functionsinrllldingth.~JlltlL<;(~ flll~tors,til.!Gor'kov"'llI hti ll ll S 2G

r-an lxleasilywrit ten us

[iw

+

(I-

~(-jV

^-;.4(i)

+

R)2-U(i )]G..,(i, :?')

+

6(X)F.!(x,i')

=6(.i-.t), [-iw

+

II-

~( -iV + _~A(i)

-j{ i -U(.i )]Gt(i , .i')-6

t

( i )F",( x,i' )

=6(x-.t ) ,

[iw

+

JI-

~(-iV

^-

~A(i')

-

ih

²-U(.i))F",(i,i")

+

6(.i)GL(x , X')

=0 ,

(- iw+/I- ~ ( -iV + ~.4(.i) +R /

-V(.i )]F.!(i',X')- 6

t(x)

G",(a"",.f" )

=0, (2-3-8) wlll'n'

(2-3-9)

IIIon1l'I'I,llsolve theabove E(IS,(2-3-8 ),Ellenberger considersIImore general expn- ...lou{Ol't.llt'Grl'<'u'sIrua-tionG",(10; ;/: , ; ')which sntisfics

( iW+/I- r,;;(-iV- iDlo )2- U(i ) 6(.io)

)G

^{(x ;.i')}

-6 f P:'1l) - iw + /I - 1T.;(-iV-i Dlo12-U(.r) ..,

0,·,

(2-3-10)

=

io(x -.i' ), wlu-rr-t1;ois t1u'Eikuh"l'ger'sgauge-Invariantdiffercutintion

{

-iV - 2;..f(.ro) whcu workingOil6(xo) ,

D

^{ro= -}iV + 2; ..f(.r u) whenworking on6f{i'o) ,

-iV whenworking on somefunct ion of1~(xo)12.

(2-3-11) nudG"'( ;Il ' .r, ~;'I)istilt'mntrixformoftheGrCC'Il 'sfunc tions

01wiulIsly\\"'hn \'('

G...(.r,.r')= limG...(To;1,;') ro-F

27

(2-3-12)

(2-3-13)

Eilenbe rger thenassumesthntthen\'('fn!!:C'of til('Grc't'I\'"fl1llt'l ion

'w,,,

th..1':111- doml ydistributed. impuri rlcs restorestilt· :<!",t illltnmsL,liulililiuvnrjnuee!'It.tlnu theaveragedGreen's(unctionG..,(ro;i,r)rnnb.'("l':I' Il'l<."l'tII IS

FUrther,Ellenber ger chougc-stheYllrillhlt'Sfrom

C

^tn(P~,,( l^{wit h(}' " 'in,;lh.'.-IlO't lt\- vnriable(

=

((If-

kpll80

that

G ...

(r;

k)

=

G...(i;k,."O

and definesthe simp lified Greenfunctionsns

1

. ^{f "(- •}

y(w,J.-r,Xl=; -G(w, (,I.-,.,,;) ,

- f

2.

,/(- ' -

!(WJ,J.:r,i1

=-

~F(WJ, (,J.:,."r),

I ' f

^{'/(-1 ' -}

!

(WJ,J.:P, i1

=

;;-F(w.(, (-"',.1').

•w

(2-:1-15)

(2-3-16)

where,on therightside,theargument(d('JlOt ~c'UIiII'If")l:"llf'r l!;Yvllri;II~ ,·whirh sho uldbeilltl'grated O\nalOIl,!!;thecontour(ird iu ll;U... 1',,1,-,("Itill'!Cn "'l1fllurl i"lII', par ameteriisthe replacementfor.io.

StartingfromtheGur'km'('(!lIatinlls(2-3-8)llllf\f" n"willJl;I'ktli"lIsd,·.-iv/,- tion ,Ellenbergerohtnincdeqnntious for thOlll'silllplifi''f lGrc'f'1iflllld iow,.W"'lilly present thefinal resul tshere and leavethedel/,il,,',\.1i~l:m;."i"l1illAl' p"Il<lixA.

(2-:J-17)

28

{2w -!if" (V

+ i~l(x)}ft(w,

kp,x)

=2L:if(xJrJ (w ,

k

r ,i')

+ I

rPif,..W(kF-t1F) }S.-47T

.{y(w, k /..,iJ.rf(w ,tji.·,i)-jt(w,kp,.i')y(w ,qp, x')} , 1LI,,1

TIll'SyS!.I'1lIoff,(Jllll l.iolisis com pletedbytheself-cons istency condition (2-3-18)

(2-3-10)

(2-3-20)

whereV istlu-wnpliup:constantand maybe replacedbyth e standardcutoff

(2-3-2 1)

TIll',·xl'lkit.f'xJll'l'lisiollof .6.(1)in terms oftheshn plifk-d Green'sfunctionsisgiven

(2-3-22)

wlll'n'Sf'Ill'noll'still'Ft-rmi surface.Thesame proceduregivesrise totheequation for tlw'·l1n.·ut.1],'us ity ] ( 1)

wlu-n-IJ,istheIl1nglwti,'fieldinducedhythe motionof the chargedparticlesin a slIllI·rc'ondlld ,ol'.Compared withtheoriginalGor'kov'seqnation ,theEilcnbcrgcr 's

"II11a1ioaslin'lum'lIeasierto handle sincethe numberofthe variables is reduced ttltwoIromfour.Inthe procedure of obtainingtheequations,no assump tionof sma llorderI'lIl'lIml'll'!'has lx'cunindesothnt Eilcubcrgcr's equa t ions arc suitable

ril l"1Ii1'lc\1:"sill~,!I('(,m'ctnfelll'mngncticficld.

20

§2~3- 2Eilenb e r ger ' s Equatio n in Dirt yLimit

Usadell37 Istudied Eilenbcrgce'eequations for1\llirty"'II....ITolltlnl"lufillwhlr-h themean-free pathis very shor tsothntthe1I\otion oftill'd,-rtnlllll illIll'lll"ly isotropicwithrespec ttoFm lli veloc ity

vp = kpl m.

011('t1l1L~r-anrxpllu<lIll., simplified Green funct ion/(w,z,ok)as

/(w,z,iT)=F(w, z>

+; .

^F(w,11,

andassume

IFI» IV 'PI

From theEq.(2-3-19),OllChas

9(W,; ,;1)

=

G(w,i')

+ ~

^{·C( w, i'),}

whereG(w ,.ili!ldefinedby

G(w,Zl=

[1- IF(w,ZlI' !'"

andG(w, i1,by

(2-3 -2Cil

(2-3-2G)

(2-3-27)

Substitutin&Eqs.(Z-3-24) and(2-3-25) into theEil'~IlI",r~'r'I<"'llIati"uI<.QY1'"ill lastsubsectio nandcr-uipletiugMJII1Cbnsie rJllculutinll1UI, utI"II;IS Eih'1l1l<'f';""'"

equationsin thedirt}'limit version

2wF(w ,i)-DOIG(w, i )DF(w,i)

+ ~~~:: ~~ VIF(W, ;WI

⁼

2~(;'J(:{w" rJ ,

"(x )In(:f)

+

2,T

Ii ).(' )-

F(w.x il

=

0,

c ...>0 w

Jr,)

⁼2;,N(O),TDL [Ft (w, i )DF(w,i)-F(w,J )(DF(w,i'))tJ

w "

(2, :j-2fiJ

30

where

fJ =

'V

+

2icA(xjawlD=~1)Flis the diffusion cons tan twhich canbe used ill Ntndy iu,l!;tlwdfectoftlil~magneticfieldin inhomoge neousdirtysupe rcond uct or.

mll~i[17I unrl eo-workersappl iedthe equa tionto the SN mulrllnverscasewith I.lll' t"'l lllliliryeoudltions

{

^{F-' ~ FNDF"}

⁼

^{'I DF• N at}SN intcrfnccs .

D, D,

wllO'I"<''}

=

(1,,/(1~with(7Htnndingfor condu c ti vit yofnormalmetal. For the [l1'1"\"'lLdindal' llppr'l' criticalfield,the result may bc summnrizcdl!"Jto

11I(fS)

= _~(4) _ ~(~ +

y(;s» , In(f N)=

J/~( ~)

^-

J'(~ + y(~),

k~=2rrT!I(ls)/hDs ,

~.~

=

-'l rrT y(t ,v )/ h D N , fJstHu(fJsr/s/'l)='If/I\'tnnh(IJNd;..r/'l),

rJ1

=

~'1 - 2~~C2

^,

rJ~' = ~';'" + 'l~cz

^,

wlu-n'fi=

l:;; ,(i

="S,N)isthereducedtcmpc r ntnr oandy(t)is thofuncti on ofthe n'dll""d tt'lIl(lr'l'nlllWdctcnniucdbytheeigenvalueks^lindk;..r,Given the needed I'aill<'>'

"r/...

1.\,.Dx ;Ds.1Ild'/. Oil('ohtniUHthe rchulon between theumguetic field 1I1I,l l" 11I1"' I'a l ul'l'

[2;rTII(ls)/hD....]I/!t.all([ 2 r.T y (t s )/h D s ]1/1,Is/'l )

=1I['l;rTy(IN)/lt D.v]1/ 2tnllh ([21l'T y(t ,v)/IiD N]^1/2tlN/ 'l ) (2-3-31) wlw\'l'

rI,.

amitis111'"till'thicknesses of thesnpcreoml nctingend1l0 1'll1 nJmetal lnycn,ITHI" ·l'l h -d y.Atlilt'eriticnltra nsit ion te mperatureoftho suporlattice,the

31

uppercri ticalfidel\·tmishcs,theEq.(2⁴3.31) rt"tlllfl'!l tn tilt' rt'!lllitof.1.·G,·nu.",.

Wcr tham("l"andco-wo rkersfOTadirt )' SNJlmxilllit ~·IIp, tc1n.

(2.3.32)

§2-4 Tunnelli ngModel inSupe r conductivi ty

In thissection.wewillmninlydiscus!'thetuum-ling1\1...1,,1IInlllillu uiun Hn.1 there s ultantexpr<'5siotJforthetr ansit iontcul llt'r lllu T<'"ht ll;Il...11~' ~It-~Iill;, u,

Ratherthansturfiugfrom theGor'kov('(jllll lillll,Mr-MlllnnsLwli,'<\till'1'1"0 :(' imit ycff<'Ct ofnsandwichb)'menus

or

tIL,'hllLlWllillKllLod, '!Hamiltonian~h"'ll

H=Hs

+

JI,..+11.,., (2+1)

whereUsandH,ydenotethecoutributiou totill: Hlllu ilttllli:lIlIrruulIw,·I,'('llulIil' states inthesUflI'1'condllctingnnt!normalh,)"t,l'!IT<-t<jM'("tiw'I)' :.1It11/.,.d...TiI"...Ill"

precess wherebytheelectro nsrunnel IromOil,:111J ITtittil.·ulll<'r .IIItl'T1IlS ufII.,·

Nambudou ble ts ,

Hs=

L::>~~'1T3~l + L WfO~f +

^L !ll,_tJ(tll , _,.+

"t -t,)'j,1, f~ '~iJ

:; , { ,.ll

of

L Vt<~'t_i~~r, }( 4rt+fT~+ll), i,.r.,'

withthe Nnmbuduu hlctlJ('ill~,\I'filll"ti as

(2·4-:1)

wherecLandcr,oaretill:l'I'("n! jolllIuduuuihilurion"/M'/"IlllI/"SfUl".,In ·I I'HIISill single-particlecigclIst ntc'I'i'oftil.:M1I)t'rr tllldlldnr,till' spi ll1Il1l1tJJt~IM,h,rii':llli" ll

33

ludk-eslIf':suppressed lind (Jt (a)crcatcstannihilatcs]barephononswith energy Wll,V.,istIL"hare Coulomb interacti on,

yf,_',

is the electron-phononcoupling illsllJlf'n:OliCllidor,

f.f

⁼

,N,; -

IIis the energy spectrum ofthe free electronin slll·" n:olld,ld or.

Similarly,HNis of tho form

/fN=

2::>~~'~1f.1 4r1;+

^{L w.;ai n-r+}

E g,~_p,((li;,-/1.

^+nt,_iil

)<lrk,T3>iF

P•

,1 f p,.F.

+ L V~.pt, _/~4<PI )(q.t.Hf~q." , ),

F",7.,f

(2-4-4) IIIf;l<'t.on,'(' 1111ohtnintil{'HNfromHs^{hy replacing}

k

withIjlLllllthe indexS wit hN.

TIll't,'ll llldillP;HuntiltonlnnH'I'iiiof the fo r m

HI' =LTp,I:{ct1c,1,l+ctlc,. I)+ h,c" (2-4-5)

/1,k

wil, 'I'I'till'1.,'1'11141"'7,1indicate sfIprocess of creatingan electron in a state<Ilk iu IHIJl,·n'o llChl,'t.ll"uud IIl1uihilating one in ast ate<PI'innormal meta l.Theprob- IlIJililymllplit.nd,·fill"l'('lIliilat io1\ofsuch nprocessis denoted byT/₇,k' A further

"illlpli fic'fltiollwillIx-IUllllrbytr,'nt.iugthrtunucllugtun tr-ix clcmcotsTji,kof equal lllnp;uit,ll!<'Th" t \\'<"'\1 ev..'tj'state'1>r lindeverystate'1',7'

UsingI,I1l'HnmlltouinnlIsnndH,\,end crunplcring thetransformntlon (2- 1-

a: n.

nile'runt1d i ll<'tiL,·Nnmba 01"('<'11'1;fUlIctioninsuporcondnctivityas

~ls(r.^{r)=- <}Tet~'^r(r)q,1(0)>

_ «

Te(Cr.l(rkt.I(Oll> <T..(cr,l(r)c_r,I(O))>) -- <T..

(C~r}r)cL{o))

> <T..

(c~",I(r)c_,.I (O»

>

(:t-.J.-6)

wIL"!,,,<,..><!f'llOIl 'lo:tilt'm'('rageovert.heGTlI1tI[Carlonic.alBns embie .In a similarWilY,mu-"a n ddi lll'tln-NUl'lhn Green 's func ti onin the normnl metal 3S

33

<Tr(~F.I(r)r_;.I(O))>)

<Tr(r_il.l(r)r-F.I(On>

(:!--1 .j')

TheGreen'sfuncti o nsGS,NdefinedhereMe,infll cl.thC'l'lplltinlFtmric"Tn'IUl'ullI'\jt,.

ofthe Grren'sfunctions appearing in(2. 1·30)and(2.1.37).

If oneonly conRidentherlrdron-p !lOlllluillh'rnd inninJI....NIIUt! 11'l"

/1 .,.

go tozero, theFou riercomponents of theerN'n' "(lI 11r l iull

g

s. '" withrc"l"I)(~'1t"

im agin ary timeTmaybeexpressed,illICOIl!!ofD~'Sll\I"'lllatillll.Ill'

{

GSI(l: .l )

=Gol(k,f ) - t~~(f, f)

9N1h7,d

=

G;I(j;,f) -f;~'hi,d ^{(:H·S )} wheregil·(ji,e)=[e-"pi':J)-1t!CIl('(.(:S till' Gn 'I'IL'SCHud,joll rornfn'('d,'('II'''u in normalmetalfindt~(jr.f)dl'IlOtf'St!lr IIIntrix IIdf"'W-flO'('ulllinv;Inuu lilt' electron- ph onon intcr ncrionshownillFig.!D.o;.Nell'lIo te·lIw Gn "II's£ulld iu lI!I..

r

the pho no ninthercspcct jver<"gioll~

Ifone treatsthe tunneling HamiltoninnH'I't.. Sf'nlllll-nn)'·r .ill,..·If-rn ll,,j,.h·uL perturbationtheory,themlltrixNof.·If"'·llcrE;Yis ulUclilkl'l.Iil l,ltrlll ll lll:,c;r ll ll ylISl<lll...vu in Fig, 2whichgives riseto thema t rix,;e!(..,:ltl·flO'u(til<'(urlll

{

tN(E)

= t';t + 1

²~G...,.<EI.

£5(£)=t~·+T1EGN..,,(E).

',.

(2+9)

where

r

denotesthetrnlU;;t;oll f11l1trix-d"lIlc'Ilt llwlthc'!lJIlI.rix ....·If-' ·llI'f/O'

t';t,N

i:<

assumedtobeEiurlepcudr-ut,IIIg"ucl"a l, til,:2 x 2uuurlx"df-"lI"rI-;Y

t.,>.N

muy he exp nnde rlIlStheliuourcorubi nu tiou oft.IL'~ILlll!.rix!old

Ii ,f'l

withl'IwiUl!; t.llf' Paulimatricesnudjhoi ugthe unit matri x,

(2·4.]{J)

34

Fig.1. Scl£-Energy

35

Fig. 2." Sclf· E ncrgyofthe supercond uc tor isthesumofibesecond-order phon onand tunnt:lJingcont ributions.The Green'sIuneticns eredeterminedself-

eonsis~nt1yfromthesel!-enerp~.

36

(Jill :1:1I1l :-;iHlplify theexpressionbyproperlychoosi ngthegaugeandrcnor- llllLliy.iug the energy sp ect mml38) so thatthe componentsof T2lind T3 canbe eliruinutod.'I'hesd f-eu("rgy mayhI)expressedas

('. 4·11)

Tlw fllur.tiuuZN(EjIlJId<lI(E)should bedctct m i ncdeclf -c onslstent ly, The Green's fliUdio llfor II freeelectronisgivenhy

(2-4-12) Sllhs t i tll tj l l ~(2-4-11) intothe Dyson'sequationgivesrise to

(; _ZN(E)E i

+

fr,TJ

+

<[IN( E )'h N,'r,- Z~ (E)E2_f~,

+

<I'~(E) . P,' r!on llilll!:theSHil lover atntcs, we have

LG = ..E1!.. J

^({lI/ZN(E)Ei

+

fr,T3+<I>N(E)i"1 'r' N"l' (211")·1 Z'j.;(E )E2 f~l'

+

<I>~(E)

=- iW{l N(O)

£1+

^D.N{E)Tl

N [E'-"'ME)] !'

(2-4·13)

(2-4-14 )

(2-4-15 )

II'lll'r l'HN

=

^{Arl o\'}is th eVUhl -forNslab, A linddNdenote theare aand the thk klll' HHofthpnnnunlmetal,N(O)is thedens i tyof thestatesatFermi surface nud

a N

isthe!",'uotllmli1.l'l!order parnmctor defi n edby

(2-4-1G) Suhstil.lltilll: (2 -4-15) into (2-4 -0)and taking:£~=_6~T1t[JI.IIwe obtninthe HI.[r- t 'Ilt'I·~r t'(lllnl iolL~ :

(2-4-li )

37