THE UNlVERSITY OF MANITOBA

PROXIMITY EFFECT & THE THERMODYNAMIC PROPERTIES OF SUPERLATTICE SYSTEMS-

A Thesis Subxnitted to the F a d t y of Graduate Studies in Partial Fulnllnent of the Reqwrments

for the Degree of

MASTER OF SCIENCE

DEPARTMENT OF PHYSICS

WINNIPEG, MANITOBA

P-8PZEZ-Z CS-O

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Abstract

A st~crightfonucrnl niethod based on the standard Bor~oliubov eqiatiorrs cris-

ing in the Banleen-Cooper-Schrieffer (BCS) appronch to supercorrdwtivitg is

genemlized to the case whieii the qsteni is not Isomogeneoiis i7h q u e e e . The

gcip epntion is obtnined from the self-consistency condition imnposed on the

pair amplitude for systems composed of layers of diflerent superconducting and

nomal thin films. Some properties of the srjstern are determineci in tenns of

certain parameters ara'sing in the theory, the behavior of which aflo~cis a consis-

tency check on the rnethalî used. S'&al attention is given to the calcuZntiorr

of the transition tempmrture for a single layer, which lead one to conclude that

this method is ualid only for selatively thin superconducting films.

Acknowledgment

I a m cleeply grczteful to mg si~peruisor. Dr. R. L. Ko bes for his eencorrr<ir~emerrt

nnd gdnrrce throt cg fr this research project. He hnd nct+velvJ invo~ved in t fris

research crnd nlso made a~bstcantial contribution to the completion of this thesis.

I have benefited p a t l y in nmny w<qs 6 y workïng with him.

Tllanks is &O due to Mr. .J. Wang for man3 helpfirl discicssions. I (rnr also

qratefiil to the financial stcpport r e c e i d /rom Nntional Science and Engineer-

ing Research Cound of Canada and Department of Phvsics, University of

Manitoba.

Finally, I would like to express rny gratitude to m y fantily for their constant

support thmughout the dumtion of this research.

Contents

1 Introduction 2

2 Minoscopic Theory 9

1 B c ~ i c Properties of Siipercontliictors . . . . . . . . . . . . . . . 10

1.1 A New Conclensecl State . - . . . - . , . . . . . . . . . 10

1.2 M i e E t . . . . . . . . . . . . . . . . . . . - 12

1.3 E-xistence of Energy Gap . . . . . . . . . . . . . . . . 15

2 BCS Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . - 17

2.1 Reditceci Hamiltonian , . . . . . , . . . , . . . . + . . 20

2.2 CooperPairs . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Superconducting Ground State . . . . . . . . . . . . . 26

2-4 Cdculation nt Finite Temperature , . . . . . . . . . . 29

3 Pioximity Effect 36

1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . 37

1.1 'Leakage' of Cooper Pairs . . . . . . . . . . . . . . . . 37

1.2 Cooper Liuiit . . . . . . . . . . . . . . . . . . . . . . .

1.3 Mague t ic Aspcc ts of t ke Prosiuù t - EErc-t . . . . . . . .

1 4 Naguetic Field Behnvior . . . . . . . . . . . . . . . . .

2 De Genries- Werthamer Solution for Dirty Materials . . . . . .

3 McMÏllan Ttinnelùig Mode1 . . . . . . . . . . . . . . . . . . . .

4 Single Superconducting Film

1 The Energy Eigenstates . . . . , . . . . . . . . . . . . . . . .

2 The Pair -hplitiicie and the Self-Consistent Gap Eqrtatiou . .

3 Thinlcness Depenclence of the Critical Temperature . . . . . .

3- 1 -4 Calcidation Close to the Criticd Temperatrire . . .

3 -2 Wunerical Result . . . . . . . . . . . . . . . . . . . - -

4 The Thichess Dependence of the Gap Parameter . . . . . . .

5 Finite Superlattice System

1 The Solution of Bogoliubov Equation and Gap Eqitation . . .

2 Reducetl Thickness Depenclence of the Rechced Cri t ical Tem-

perature . . . . . . . - . . . . . . . . . . . . . . . . . . - . . .

3 The Behavior of the Energy Gaps . . . . . . . . . . . . . . . .

3.1 A Calculation of &')/A(') at Zero Temperature . . . .

3 -2 A Calculation of h ( j ) / ~ ( j - ~ ) and A ( ~ ) / A ( ~ ) at Finite

Temperature . . . . . . . . . . . . . . . . . . . , . . . .

6 A Wew Model" of the Sinde Layer

1 Phpic-a1 .lIoctel . . - . - - . - - - . . . - . - - . . . . . . - - . 4 T1ii~:kness Depeucleucc of the Self-Cousistent Gap Eqiiation .

2-1 A Cdctilation of the Critical Thichess at Zero Tem-

peratiire . . . . - - - - . - - - - - - - . - - - - - - . . - 2.2 Thickness Dependence of the Critical Temperature . .

3 Cornparison ancl Disciission . . . . - . . . . . . . - - . . . . .

3.1 Critical Tliichess . . . - . . . . - . . - - . . . . - - . . 3.2 Infinite Supcrlat tice . . . . . . . . . - - . . - . . - . - .

4 S i t m n i ' ~ . . . . . - . . . - . - - - . - - * . - . . - - . * - . -

List of Figures

2.1 The electronic specific heat C of a sriperconcluctor (in zero mag-

netic field) as a fimction of temperature . . . . . . . . . . . .

2.2 The bfesissner effect . . . . . . . . . . . . . . . . . . . . . . .

2.3 The magnetization curve of two types of superconchctors . . .

2.4 A tunneling junction between normal metal and a supercondiic-

tor When S is sitperconcliicting (T << Ta) ancl also when S is

normal (T > T). CVhen T << To, to extract one electron from

the superconducting condensate require a minimum energy A

2.5 Electron-phone interaction attractive . . . . . . . . . . . . . .

2.6 Curve showing A(T)/A(O) vs.T/Tc . . . . . . . . . . . . . . .

3.1 Tunneling mode1 with a normal metal and a superconductor

separated by a a potential barrier and with the BCS potential

. . . . . . . . . . . . . . . (dash lines constant in each metal)

4.1 Single superconcluctor layer . . . . . . . . . . . . . . . . . . .

4.2 The assu.mec1 form of the pair amplitude . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . Fiuite sitpcrlnt ticc system

The -4ssti1riccl fonn of the pair amplit ticlc . . . . . . . . . .

The clepenclence of the recliicecl traosition temperat iwe T/Tc on

t lie t kickncss alac . soiici line represents single layer. Dash linc

represents ?ï = 100, Dash-dot line represents N = U), Dash-

dot-clottecl h e represents N = 10 for b/ck = 0.05 Siiperlattice

Tlie clepenclence of the recliicecl t rami tion temperat lire T'/Tc

ou the tliickness n/u,. Solicl Line represents single layer. Dash

line represents bln, = 5, Dash-clot line represents b la , = 0.5.

Dasli-dot-dot ted line represents b/n, = 0.05 For N= IO finite

siiperlat tice - . . . . . . . . . .. . . . . . . . . . . . . . . . .

The gap parameter ratio A ( ~ ) / A ( O ) forN = 40, bla, = 0.01, at

zero temperature . . . . . . . . . . . . . . . . . . . . . . . . .

The gap parameter ratio A(j)/&O) forN = 10, bla, = 0.01, at

zero temperat lue . . . . . . . . . . . . . . . . . . . . . . . . .

The gap parameter ratio A ( ~ ) / A ( O ) ~ forN = 40, bln, = 0.5, at

zero temperature . . . . . . . . . . . . . . . . . . . . . . . . .

The gap parameter ratio forN = 10, bla, = 0.5, at

zero temperature . . . . . . . . . . . . . . . . . . . . . . . . .

.i.9 Tlw gap paraiiic8tt8r r ~ t i o A ( ; ) / ~ ( J - ' ) for-\- = 40. 6 / r t , . = 0.01.

at fiiiitc tc8rupcrcztirw . . . - . . . . . . . - . . . . . . . . . . .

.3.LO The gap parnuirter ratio L(J)/A(') forS = 40. blci, = 0.01. At

Fiiiite Terirpemttwe . . - . . . . - . . . . . . . . . . . . . . .

5.1 1 The Gap Panmeter Ratio A(j)/LdU' for:\- = 40. bln, = 0.5. at

finite temperature . . . . . . . . . . . , . . - - . . . . . . . .

G.12 The gap paraiiieter ratio A(j)/&*) forN = 10. blcc, = 0.5. at

finite temperature . . . . . . . . . . . . . . . . . . . . . . . .

6.1 Gap paranieter tlepenclence on spatial . . . . . . . . . . . . . .

6.2 The critical temperature clepencleucc on the thickness for New

Moclel . , , . . . . . . . . . , . . . , . , - * . . * - . . - - . -

6.3 The dependence of the recliicecl transition temperature T/Tc

on the Thickness a/ac for iiifhite superlat ticeSolid line repre-

sents single layer. Dash iine represents b/a, = 5. Dotilash line

remesents bla, = 0.5. Dash-dot-clotted re~resents bln, = 0.05

Chapter 1

Introduction

The p r o M W effect, which is the mutual idhience of a superconducting

aud normal film being in metallic contact, has been stiicliecl extensively. both

t Leoretically as well as experimentally. The fin t esceriments stiggesting the

existence of a proximity effect on a scale ,$ - l000A were perfomccl very

early by Meissner [l], who measured supercurrents flowing between two su-

perconducting wires separated by a thin gold layer (cieposited on one of the

wires) and found the superconducting transition temperature decreased below

the bulk value with decreasing film thickness. Then Smith et d[2] reported

that the Tc of Ag-Pb superimposeci films decreased with the thickness of the

normal hg film, dN, the thickness of the superconductor, clS, being kept con-

stant. They also showed that the induced gap in the normal metal is much

smdler than that of a pure superconductor's vahe. However Rose Innes ancl

Serin [3] pointed out from their work on Au/% sanclwiches that this effect

niiglit he clue in p ~ r t to the infliieuce of intcmctallic mupoiintk. -4 \-arir& of

ot livr phenouieua t hat are cai~sed by t Le prosiuiity effect Liaw dso beeu Co tiud

(e.g. the transition temperature of site11 a contact d l be n fitnctiou of the two

Eihs' t hickness involvecl and thnt it c m vasy continuoiisly fiorn the transition

temperature of one film to the one of the other film).

Mter t hese e.upcrimentaI observationst several t heoreticd t reat nients of

vanous aspects of the pro.uimity effect were proposecl. The k t a r i c l siuplest

case is clisciissecl by Cooper of a thin Normal (N) and Siiperconcliicting (S)

films [4]. Since Cooper-pairs as weU as impairecl electrons WU clrift fiom one

film into the other with such a contact. a theorist wishing to describe sirch

systerns is confronted wit h the inherent inhomogeneity of the sitiiat ion, which

leacls for instance to a spatially varying pair potentid. However, due to its

non-hem natitre, the BCS theory is cLifFcult to solve for geometrîes other

t han the infinite homogeneous case. As this is a problem in which the spatial

inhomogeneity of the system is essential, the Gov'kov propagator formalism

[5], which based on Landau-Ginzburg theory , is particdarly appropriate, for

instance, in calculating the cntical temperature as a function of film thickness

[6, 7, 8, 9, 10, Il], but it is only valid near the transition temperature.

In order to treat the inhomogeneous problern in general, two general a p

proaches c m be used. One is that the pair amplitude is assumed to have

some convenient hm, and a solution to the Bogoli~ibov eyuations is obtained.

Properties of the system c m be then foiml in terms of the free parameters

assoriatccl nit li the a ~ ~ ~ i t u e c I auptit ucle. Ln uitich of t lie Literat lire t lir- as-

siiuircl Fomi of the auiplitucle is an appropriate cous t w t over a certain regiou

[12. 13. 14. It, 161. siucc t k s pemits a relative- stnightfomC~cI solittiori.

Iriiplicit in this approack is the couclition that the assimiecl <mcl cdcitiat-

cd mplitiicles are close in some sense, so that the solution is "neasly self-

cousistent" [lï. 181.

. b o t her somewhat more generd technicpe which explkit ?y incorpora t es

a self-consistency conclition is a varïationd type of approach. Here. one begins

as More by assiunhg a particuliu form of the pair amplitude and so l~ing

the residtwit ecpiations. The free parameters associatecl with the assumeci

amplit iide are t hen cleterminecl by a conclition siich as t lie minimizat ion of t Le

free energy [l9] or that the assitmect ancl calc~datecl pak amplititde agree as

closely as possible. In both of those approaches it is possible to consider more

general forms for the pair amplitucle by employing a W C 8 type of expansion

[20], which has the effect of reducing the Bogoliubov equations to a system of

first-order equations, t hese being generally easier to hanclle.

In t his t hesis we shall at temp t two different models of incorporating part ic-

ular geometries other then the W t e homogeneous situation into the kame-

work of the BCS theory to study the prolcimity effect. Chapter Two of the

thesis begins with a brief discussion of the general properties of superconduc-

tors and reviews some aspects of the BCS theory which are generalIy acceptecl

as the standard theory of clescnbing stiperconditcting system.

Chapter Three rontains some geuerai rcuiarh al~oiit t l i ~ prosiriiity

t'ffc~t. atirl cIcsc~-ihcs two mz~jor theorcticd tseatnictits of it. Ouc of tlimc is th<.

approwh ofcle Geunes for c l i r t y niaterids, d o entcusivcly clcveloped a theon

of t lie prosimi- effect n-lùch shows lion- the niii tiial influence of two films whicL

are in coutact can be clescribec1 by appropriate boiinclazy conclitious for the

pair potential A(r) at the interface and at the metal-vaciiuni hoitnclary. It is

especially simple in this approach if the si~perconcltictiiig film is in contact wit li

a magnetic h. siich as Fe, Ni, or Co. Anotlier well-known approacli to the

problem of two clissidar metals jouiecl at a Bat siirface, somewhat clifferent

from t h approaches mentioned in the prececling clisciission, is the tctnneling

mocLeL of Mch1il.lC~ [21]. In this approach a potential barrier is assiimecl to

exist at the interface hetween the two met& ancl the ttinneling thrûiigh the

bamer is treated by means of the transfer Hamihonian methocl [22, 231. The

mode1 h a . the aclvantage that it is not restrictecl to the vicinity of the critical

point.

Starting from Chapter Four, we introduce a somewhat different iine of

approach with regard to the self-consistency of the pair amplitude, which was

successfidly presentecl by Kobes and Whitehead in 1987 [24] The approach

shares some of the characteristics of the methods descnbed in the prececling

discussion but uses a self-consistency condition Iess stringent t han clemancLing

that the assumed and calculatecl pair amplitudes exactly coincicle. To be spe-

cifie, we first work on the simplest geometry: a single siiperconclucting film

iusitkt a uoniial uirtal. Li. prrsrnt t k griierd soltitiou to t l i ~ Bogo1iiil)ol-

qiiiitions. wliirh wi1l dso hc uscd for the Eiuitr siiperlattire systciu to be s t t d

ifcl later. The first appro.shatious made in tLis Chapter is t h wr. assiiui~ the

pair arnplitiicle is coustant nritlua the supercondiicting regiou. and for the self-

consistency condition we recpÏre that the spatial average of the assiuiied ancl

t lie calcdated orcler parameter be eclual. In orcler to sirnplify tlie dgebra we

assime t h the Fermi velocity , the Debye ternperatiires, etc. are the swiie for

hoth metals, ancl we assume the transmission ancl reflection coefficients at the

interface to be imity. This approach. while simple, provides a stnightforwarcl

methocl for the single film system for both critical mcl noncritical properties

with no paranieters outside of those foimd in the BCS theory In the last

section of this Chapter, we solve the gap fimction nnd get the clepentlence of

the critical temperature on the film thichess. As a consequence many of tlie

resiilts containecl in t his analysis can be expressecl in terms of certain riniversal

resiilts which are independent of the particidar material in question.

Although the particular geornetry described above is relatively simple and

has been extensively studied in other approaches, other geometries, such as

siiperlattices composed of alternating layers of normal and supercondiicting

metal, are also of interest, especidy experimentally. In Chapter Five, we

apply the same approach discussed in Chapter Four to a finite superlattice

system. We consider a clean superlattice with 1 > <(T), where 1 is the mean

fiee path and <(T) is the temperature dependent coherence length. The system

is (-ouiposecl ol i&ut ical supcrco~icliictors of widt li 20 srpara t d I - I~L>.CLS of

rioimal uietd of wicltl 26 at the s = O plane. ?\ë clctemint~ soue propcrtics

of the system in tenns of the ratios ci/n, and T/Tc, prticiilarly the rriticd

t eniperat lire. and uw t his to in part icidar examine t lie cpiest ion of when siicli a

firiite system c m reasonabiy apptoxhnate a tnie infinite superlattice. FVe also

examine the behavior of the energy gap parameters. We first consicler the gap

paramter ratio A(~)/A(') at zero temperatine, where j labels the particiilar

film, aud then we consicler this ratio at finite teniperatiu-e. The behavior of

t his ratio nfforcis a consistency check on the met hocls usecl.

In Chapter Six, we clisciiss an approach to the case of a single supercon-

cluc ting Iayer ~Iiscussecl earlier baseci on the met ho& of the previoiis chap ter.

In particitlar. we arbitrarily ctivicle the siiperconcliicting layer into " 21V + 1"

zones, <mcl then self-consistently cdcdate the gap parameter in each zone. CVe

then let 'W' go to infinity and the width "aT7 of each zone go to zero, but keep

the total widt h of the superconductor constant:

An essential advantage in this approach is the hope that the gap parameter

of the single layer can have some "spatial" depcnclence, being peaked near the

center and dying off somewhat towards the edges. We s h d mauily concentrate

on the determination of Tc a s a fimction of the superconducting layerTs width,

particularly aroiuicl Tc = O. We shall finci that the results for the critical

t h - l a t s at T, = O gct cnhariced 1 - a, factor -5-S0 owr that of carlier

ruode1 cliscitssccl prwioiis- It tiims out that the rt.;idts ol~tzlined 1- t h uc8u-

ruocle1 czre uot riuiwrsd. in t i n t they ciepencl ou the particitlcar niaterid in

cpest ion,

The bi ik of the materid of the thesis is embocliecl in Chapters 4, 5, ancl

6. whch is the p r o d t y effect of the siiperconcl~ictor for ciifferent geonietries

iuvcst igatecl. One of t Le purposes of the t hesis is to inclicate how the results ive

fiud compare with other approaches. We d clo siich a cornparison at the end

of t lie thesis. Ultimately we give a brief s i t m m - of the resdts O btaineci

ancl present some conclirsions.

Chapter 2

Microscopic Theory

It is weU known that at stifficiently low temperatures many metals and doys

iinclergo a phase transition to a Superconducting state. In orcler to itncler-

stnncl the mechanism of siiperconcluctivity several macroscopic tlieories, siich

as London's theory, Ginzberg's theory, ancl so on, were presentecl. It took a

long time to k c l the origin of the superconductivity becanse of the extreme-

ly s m d energy Merence between the normal and the superconcliic tive s tate.

In 1950 H.Frohlich [25] suggested that the electron-phonon interaction, which

implies an attractive electron-electron interaction, was responsible for the su-

perconductivity. This was confirmed by the discovery of the isotope effect . In

1956 Cooper showed that the Fermi sea is unstable against pair formation when

there exists any attractive interaction between electrons, no matter how weak.

About a half century later after the initial discovery of superconductivity, a

microscopic theory was completecl by Bardeen, Cooper and Schridffer in 1957,

1 Basic Properties of Superconductors

Si~percoucltictivity was chscoverecl by Heike Kamerlingh Onnes in 19 I l who

foiincl that niercitry loses its resistance at -MI<. Others then foiincl the es-

is tence of O t her siiperconclt~ctors, particidarly mcmy metds ancl alloys. As

the temperature npproaches a critical temperatilie Tc, the met& ancl dloys

tinclergo a phase transition fiom a state of normal electncal resistivity to a

siiperconclucting state. In this state, the properties of the metais are raclicai-

ly clifferent from their properties in the normal state. We sunun&ze t h e e

features which the basic properties of supercondiictivity rely on.

1 . A New Condensed State

FVe take a piece of tin and cool it h m ; at a temperature Tc = 3.71(, we

fincl a specific heat anomaly. The electronic specific heat C,, in the normal

state varies linearly with the temperature. However, the specific heat Cs in

the siiperconducting state initially exceeds C, for T 5 Tc, but below Tc the

Figure 2.1 : The elertronic specinc heat C of il siiperconcliict or (in zero iiiag~-~e t ic

field) as n fi~nction of temperature

specific heat drops below Cn ancl vanisiles exponentially as T approaching zero.

Below T c the tin is in a new thennodynamical state. FVhat lias happenecl?

It is not a change in the ciystallographic stnictiire. as far as ,Y rays c m

tell. It is not a ferromagnetic, or antiferrornagnetic, transition. The striliing

new property is that the tin has zero electrical resistance. (For instance, a

current inducecl in a t h ring has been observecl to persist over times > 1 year)

W'e say the t h , in this particular phase, is a superconductor, and we cal1

the permanent current a supercurrent.

Another important puzzle was the smahess of the condensation energy.

If, as seemed Likely, the condensation was purely electronic in nature, why

was the condensation energy not comparable in magnitude with the Fermi

energy? The free energy Fs in the superconducting phase c m be derivecl h-om

tlic specifir k a t chta. F,. for the normal met&. Thc cliffcreur~ (F, - F,JT=u

is callecl the condensation energy. It is uot of ortlcr kBTu per elrctrou: it

Ls. in f x t . mtch suialier. of orcler (ksTo)2/EF. wlicre EF is the Fenui cnergy

of the çonclttction electrou in the nomal wctd. T_vpirally EF - Tel-' aucl

ksTo - 10-"eV, and o d y a fraction LeTo/EF ( N IO-^) of the metallic electrons

have their energy significantly moclified by the conclensation process. But. iu

spite of the smallness of the energy clifference, many of the properties of the

siiperconcli~cting phase are very clïfferent fkoom those of the nomai phase.

1.2 Meissner Effect

In itself, the vaniçhing of the resistance has not lecl to a goocl unclerstancling

of the phenomenon of siiperconcluctivity- The important tliscovery, in this re-

spect, is the so-callecl Meissner effect, cliscoverecl by Meissner and Ochsen-

felcl in 1933. In the Meissner effect an appliecl magnetic flax was expellecl

from the superconductor, independent of whether the fieIc1 was applied in the

superconduc t ive state ( " zero-field-cooled" ) or already in the normal state(

" field-cooled" ) . This also shows an important ditference between a perfect

conductor and a superconductor in the presence of a magnetic field. The per-

fect conduct ivity, ( R d ) , - alone is not sufncient to describe superconductors.

We also need perfect diamagnetism (B=O).

The behavior observed by Meissner and Ochsenfeldis sigmficantly mers

fiom that which woidci occur at the transition to a state of perfect concluctor,

T'Tc

Figure 2.2: The Mesissner effect

iri which the magnetic state clepencls on history: In the field-cooled case, the

niagnetic BILY present in the normal state woilld be fiozen in, and this would

represent the eqrulibrim s tate. In the zero-field-coolecl c~ase, on the o tlier

hancl, a magnetic field switchecl on in the perfectly conducing state would give

rise to induction currents keeping the interior flitu-free, but this wodd repre-

sent a metastable state: consequently, at the transition back to the nonnal

state, the induction currents woukl die out with Joule dissipation, Le. this

transition would in gened be an irreversible process.

In contrast, according to the Meissner effect, the magnetic state of a

superconduc tor is uniquely determinecl by the applied magnetic field. The flm-

fiee state is the equilibrium state, and the screening currents vanish without

any dissipation when the transition back to the normal state takes place, which

thus becomes completely reversible.

Furthemore, a clear distinction should be made between the Meissner

I &

L A

Meissnet ! i ! -Noma&

B c l Bc B c 2

Figure 2.3: The magnetization cime of two types of siipercondrictors

screening cinent keeping the bulk of the superconcliictor Bu-free, which is

an eqdibrium phenouenon, and the persistent current associatecl with a non-

vanishing BILY through a superconducting loop, which represents a metastable

state, no matter for how long it persists.

Extension of experiments to superconducting alloys led to the discovery of

type II superconductors. Type II superconductor are chaxacterized by two

different critical fields: flux penetration sets in at a critical field Hci(T), which

is mrich smaller than the critical field Ha(T) for fidl restoration of resistivity.

Type 1 superconductor flux penetration and resistive behavior occurs a t the

same cntical field Hc(T) .

.lriot her importiiucr fmt tire of sirpcrroucliir tirity is the esisteuw of a gap

in the Iow energy escitatious. The niagnïtiicle of the cnergy gap is ahotit

10--'eV . Thc existence of this gap Lias been relate<[ by BCS to the fomatiou

of condeosecl pairs of electrous. In most stiperconcliictors it is riecessary iu

order to cteate an electron-hole pair close to the Fermi siuface to fi~rnish au

e u e r s ~~~r > >A, wliere A is the so-cdecl energy gap.

The existence of a gap in 10%- energy e x i t ations was firdy est ablishecl b -

a large nimber of e-cperinients. The five main source of experimental eviclence

For an gap in superconcliictors are:

A (a). The low-temperature specific heat is proportional to exp( - -).

As mentionecl previoiisly, at low temperature the contrïbiition of the con-

cliiction electrons to the specific heat in the supercondiicting state is propor-

tionai to e - A / f ' ~ T . It is very hard to understand such a temperatiue clepen-

ctence except as a consequence of an energy gap. If there is an energy gap then

as the temperature is raised, the electrons are thermally excited across the

gap and for each of these electrons an amount of energy equal to the energy

gap A. is absorbed in the process. When the temperature increases, and A.

decreases, the gap vanïshes at T = Tc. The width of the gap is of the order of

~ B Z -

(b). The electromagnetic absorption in the far infrarecl occurs only for

photons of enrrgy 2 U.

(c). The spin-lattice relnsritiou of uiiclear spius procrrcls prerlouùuuitl-

iu an Uiteract ion with the concliirtiori elect rom and Lias fi-eqiieucy pmport ional

to erp( - l / k s T ) .

(d) . Iu tilt rasouic at teniiat ion, the phouon is of low kequency ancl cimiio t

clecay by creation of a pair of excitations. Biit it czm be absorbecl by collisions

with a preexisting excitation. T h process is proportional to the nimbes of

J prcesis ting excitations. ancl t hiis to e z - ( - ) - (e). Tunneiling effect.

The most direct nieasurement of the energy gap has been proviclecl by the

work of Giaever (1960). who essentiaily measiml the width of the gap with

a voltnieter. He accomplished this by observing the tunneliug of electrons

between a siiperconcliicting film ancl a normal one across a thin insiilating

barrier. Quantiun mechanically, an electron on one side of such a barrier has

a finite probability of tunneliig through it if there is an dowed state of equal

or smail energy available for it on the other side. But, the expriment shows

that t here is no current at low temperature unless we apply a voltage V across

the junction such that energy gain eV is larger than A. The current voltage

characteristic has the form shown in Fig(2.4)

IR 1950, the study of different critical temperatures of vaxious isotopes of

the sarne element [27,28] gave the result that the critical temperatiue for those

This is callecl the isotope effect . This relation h l r s the siiperconcliicting

properties of the nietal to the masses of the ions in thr lattice even thoiigh

the 1at tice itself cloes not change cl~lring the transition. and strongly supports

the belief that the electron-phonon interaction is the origin of the supercon-

cliict ivity.

2 BCS Theory

Before the BCS theory it took a long time to h d the interaction responsible

for the superconcliietivity.esconclcti E-xperimental eviclence pointed to the fact that in

the transition to the superconducting state the lattice and its properties were

essentidy unchanged whereas some of the properties of the condiiction elec-

trons were changed radically. In the first instance, it was to assume that the

transition was caused by a change in the state of the electrons alone. However,

attempts to find a theory based on an independent particle mode1 of the con-

duction electrons did not have the possibi l i~ of evplaining the funclament al

property of superconductors, which is their infinite conductivity.

The firs t successful mode1 of superconductivity was developed by Bardeen,

Cooper, and Schrieffer. They presentecl a theory of superconductivity basecl

normal metal

e O

superconductor

oxide layer

D

normal

Figure 2-4: A tunneling junction between normal metal and a superconductor

When S is superconducting (T << To) and dso when S is normal (T > T).

When T << To, to extract one electron fiom the superconducting condensate

requise a minimum energy A

Figure 2.5: Electron-phone interaction attractive

on electronic pair correlations which was able to accoiint for neariy dl of the

csperuiiental phenornena whicli n siiperconcl~ictor exhïbits. It also answers the

cpestion of wiiat speciilc-particle interaction is responsible for siiperconcliic-

t ivi ty.

An electron in a lattice c m interact with another electron by exchanging

acoustic cluirnta, called phonons. This is referred to as an eiectron-electron

interaction mediatecl by phonons. The exchanged phonons are said to be vir-

tua1 because they are only in existence during the exchange from one electron

to the other, but do not have the posibility of passing away fiom the electron

into the lattice as real phonons.

Such processes can be clescribed as follows: suppose that an electron with

wave vector ki emits a phonon q, and the electron is scattered with a new

wave vector ki - q. The emitted phonon is absorbed by another electron with

wave vector 4, and after absorbing the phonon, the second electron has a new

wavr wctor k2 + q. Of coiuse. it is possible that t lie otlier srai trriug prowss

ocxiirs, wliere the k t electiron absorbs a phonou Mth w w e vertor -q aucl M

scat tcrecl Eroru state ki to ki - q. This phonou is emitted froru t h scroud

rlrrtron wlùch is scattcrecl from state 4 to k2 + q.

Uncler certain conclitions, which occiir in siiperconc~iictors, ths attractive

interaction between two electrons by meam of phonon exchange ctooiiliates t lie

iisital repiilsive Coulomb interaction. Cooper was able to show that if these is

a uct attraction, hotvcver we&, betnreen a pair of electrons ahove the Fenui

surface, these electrons can form a stable superconclucting state.

2.1 Reduced

Let iis recall certain

Hamilt onian

relevant aspects of the BCS theory. BCS begin their cou-

siclerations wit h the effective Hamiltonian for elec tron interaction in a me tal.

The Hamiltonian may be written as

where Hph is referred to as the interaction mediated by the phonons, norrndy

written in the form

where

To cornplete the Hamiltoninnn which clescribes the systern WC need to aclcl the

part which ciescribes the inclepenclent Block elec t rom:

The total H d t o n i a n then is

BCS remarkcl t hat Ec1.(2 -1) corresponds to an attractive interaction for

elec trons of sii8icient ly close energies. The important electronic s tates of in-

terest are t h e very close to the Fermi surface, such that

Hesf can then be written as

In order to describe the ground state of a superconductor, we consider the

case of zero total momentum. The only part of the interaction which clescribes

the scattrruig of the pnrtiçln fiom o w pair of statrs (k T.-k i). to iuiotlier

CVe let bl = C & C : ~ , ancl bk = C-kiC-kr, wLi& are the creation aucl

annihiiation operator for electrons in Block statrs.

Thtis, we can m i t e the Hamiltoni;m as

BCS ccde<l this the reclticed Hamiltoniiui. Vkkl is composecl of two ternis:

the Brst clescribing the at trative electron -electron interaction mecliatecl by

phonons d the second restdting from screenecl Coulomb repulsion. BCS.

however, solvecl the mode1 with the strong simplifying assumption that VkJi

is isotropie ancl can be replaced by a constant attractive interaction V within

a characteristic energy tw of Fermi surface such that

so that V,&t is attractive in a shelI of wickh 2fwD centered at the Fermi

2.2 Cooper Pairs

It was Cooper who considerecl what woulcl happen if, to a system of electrons

filling the Fermi Sea, we add two more. He supposed that the electrons within

solw similar to t hc proldeiii of two p.articles hteract ing iu &ce space. the iiiaiii

tliffermre heing that becstise of the exclusion principk they c m uot bc foitiid

in the cl~iasi-particl states alreacly occupiecl. Siil> ject to t hese Limitations. t ke

most geueral state iu which these particles can he foimcl is a Iinear siiper-

position of pairs of qtiasi-particle states. Now. as the Hamiltonian conserves

mornen t~u and spin. ive can choose the wave fimction to be an eigenstate of

moment uni anci spin. The lowest state we expect the niorneutun to be is zero.

Hence, if we suppose k t that the two electrons are in the singlet state, then

we cxpect the groiuicl state of the particles to be

where the state 1 F > represents the Fermi sea ml the coefficient 1 crk 1 are

numbers yet to be cletermined. The expectation value of the energy in the

state 1 O > is

For E to be a minimum when a(k')' varies, subject to the restriction

the frinction

Heuce the energy A is cleterniinecl b r u the equation

The eqiiation has one solution with negative X if V is positive ad hence. the

interaction attractive. To fincl the solution with negative A. we c m replace

the srim by an integral ecpation.

We have assumed the density of state is slowly varying in the interval

O < ~k < hwD and have approximated it by N(O), the clensity of single-electron

states of one spin orientation, evaluated at the Fermi surface. Findy, we can

From Eq.(2.5), one has for weak coupling N(O)V << 1

Thus. au allowetl twerg- state esists with E < 0. Whrn thr iutrractiou

\- is applitul to a gas of fret* electrous. the Fermi Sca of drctmu is iiustai~h~

iiud the clectrons Iicar the Fermi Sca tcud to 11hd iu Cooper pairs. Severai

rcniark c m be macle.

(1). If there is a net attraction, Iiowever WC<&, betweeu a pair of electroris

jiist above the Fermi stuface, these electrons c m fom a boiuid state.

(2). Since the euerg?- E is rueastircd bom the Ferrui sidace. the energ-

of the two electrons at the F e d sidace is zero- E 5 O represeritiug t lie bounct

state has a lower energy th.m a pair of kee electrons? and the state is this

iinstable against siich elect ron pair formation.

(3). 1 E 1 is the bincliug euergy of a Cooper pair, and hence 2 h w ~ exp [- &]

is recloirecl to break up a Cooper pair, a fact which suggests that there shoiilcl

be a gap in the energy spectnun. Becaiw of the particular form of E, it can

not be expancled in s power series in V when V tends towards zero. This

explains why a microscopic theory of superconductivity could not be obtainecl

by convent ional perturbation t heory-

(4). The size of Cooper Pair: The concept of the coherence length was

introdiiced by Pippard in order to accotmt for the feature of the London equa-

tion for superconcluctivity. The correlation distance of the superconducting

electrons Q is related to the range of momentum 6p by c06p - h

In the condensation process the electrons involved are t hose wit hin a dis-

tnrice kBTo of the Fcmui surface. i-e.,

This coherenre Co czui aiso be regardecl as the average size of a Cooper

pair aucl we can say with great sirnplincatiou tliat in a pure stipercoodiictor

a Cooper pair Lias an average size of 100 nm to 1000 nm. This size is Ia rg~

comparecl wit h the mean dis tance between two concltiction electrons whicli

smoirnts to a few tenths of 1 nm. The Cooper pairs thiis overlap very greatly.

In the regions of one pair there are 106 to 10' other electrons which are them-

selves correlatecl into pairs. One tvoiilcl inttutively expect that the ensemble of

a groiip of such particles that penetrate so much into each other wodcl poses

interest ing propert ies.

2.3 Superconducting Ground State

In the BCS formulation of the theory of superconductivity, the ground state

is constmcted by zero-momentum singlet electron pairs (k i,-k 1). We denote

the probability amplitude for unoccupation of a pair as u k and by vk the

occupation of a pair. The wave huiction is then

The recLticecl Hclmiltonian of the groiincl statc of many Cooper pairs in t h

BCS moclel is

'H = C 2 ~ ~ b : b k + C Vkktbcbk- (2.7) k kk'

In orcler to cleterniine the eigen-energies of the Hamiltonian. one c m iise

the either Schrieffer's variational niethocl [29] One mininiiaes the expectatiou

value and fincls

which resitlts in

In t his equation, the " energy gap" parameter 1 A k ( satisfies the self-consistent

eqiiat ion

where p is the chernical potential. This result suggest that Er. is always

nonvanishing and reaches a mininurn Ek = A for electrons placed on the Fermi

surface, when ~k = p. Therefore the meaning of the gap can be understood: It

is the gap for single electron excitations fiom the superconducting (conclensed)

statt- to a fiw elcrt ron state. The tliemally excitecl elect rous .î<so.oss the gap < [ O

tauce. The same Liok tnre ewn for the stiperconclticting - tem for whicli t lie

gap mmïshes in some direction of k space.

Now. we brie@ summarkze the residts that may be obtainecl with this

(1). Gap energy

From the previotis restilts, we h o w the Hruniltonian of the groimcl state

of many Cooper Pairs in the BCS nioclel is Eq.(S.T) If V, ,t is approximated

by a negative constant, then

By assuming a constant clensity of state with an energy hw around the Fermi

energy, the summation in Eq.(2.11) can be replaced by an integral:

where N(0) is the clensity of electron states for one spin direction at the Fer-

mi surface. One notes a striking similai@ between Eq.(2.6) and Eq.(2.13).

In everything that folIows we are only interestecl in the we& coupling Lirnit

lV(0)V «: 1.

For uiost stipcrcouciiictom. X ( 0 ) V is of tLr orclcr 0.2 - 0.G iiucl - (2 ) . Condensed state energy

On(-e 1. is Irnown- one c m csplicitly calcidate t lie b e t i c aucl potential

ruergies. In the weak coupliug b d t , one bnds the energy of the ground state

Finaily, we can get

Obviously the conclensetl state energy is Iower than that of the normal state

system in which the electrons are indepenclent. Thus the energy gain is, how-

ever, very small, of the order A'/E~ per particle. Therefore the transition

from the normal states to superconducting state will certainly occur.

2.4 Calculation at Finite Temperature

(1). The critical temperature Tc

The transition temperature is simply defined as the temperature for which

A vanishes, so E at Tc is just the normal state energy E. Thus, upon integrating

S ~ C P kBTc < hD. which corresponds to a wnib: electron-phouou-clectrou

coupling defiriecl by X ( O ) V < 1. the integrai coverges rapicuy, a n c l we cari

replace the opper Litnit by in£in.ity. The intergal is a number wliich can be

wdiiatecl in ternis of Etiler's constmt 7, as - ln(2ylrr). Thus we have tkr

fiuiiotis BC'S criticd temperat ure relation,

Sirice the most important attractive p u t of the interaction Vkkt is chte to

phonous of high frecpency, we expect hw to be on the order of k 0 ~ . ln actual

pract ice bw is assrunecl to be approximately akûD, anci so one often sees t hat

the crit ical temperature equation as

where OD is the Debye temperature.

Since N ( 0 ) V is typically 0.2, the order of magnitude of the transition tem-

perature for many superconduc tors is correc tly predicted and an explmat ion

is providecl for why Tc is much Iess the Debye temperature.

These results suggest that the critical temperature of a superconductor

depends on three factors: the Debye kequency of the lattice vibration, B D ,

the cleusity of statcs of thr clcctrons at the Fermi sirrfarr X(0). sucl tlir iu-

terartiou coustant Iietween the electrou wid plionou. V. -4 niiiterid ml1 show

siipercouciiic ting heliaçior only if t lie uet interaction I~etweeu the elec-t rons

rcsiiltuig honi the coml~ination of the plionon-incliieetl and Coidonil> interac-

tiou is attractive. TLis is why normal good concliictors like silver and copper.

wliich have a n-enb: elect ron-phone interaction, do uo t exhibi t superconclirc-

tivity at the lowest temperatiire achievecl to date. However some materids

that are poor condiictors nt normal temperatiues, but which have a strong

eiec t ron-phouon interaction, are superconductors at some finite temperat iire.

Intleecl, the poorer the çondiictivity at nomal temperat lues the Ligher t lie

crit icd temperature nt low temperature. This siipports the viewpoint that

the electron-phonon interaction that causes resistivity at normal temperatures

gives rise to siiperconductivity at low temperatitres.

(2). Variation of the gap with temperature

The self-consistency condition is foiuicl

From Eq(2.16) the dependence of the energy gap on temperature can be com-

pu tecl near T 4 Tc; t his equation can be reduced to

For s m d T, h(T) is very nearly constant. Also we see that A(0) is h i te and

non-zero, and that as T + Tc, A + O with iDfinite slope and A* vanishes with

finite slope, so that the ratio A(T)/A(O) will serve as an order parameter <.

The form A(T)/A(O) is given Fig 2.6 A decreases as T increases, and finally

vanishes at a certain temperature Tc.

Bq' coui~wring Eqs.(2.13) and (2.15) WC h d tliat the iDutbrgy gap giws. for

al1 siipcrçouditc-tocs whcu iV(0)F << 1.

n-here ko is Boitzman's coustant. Ec1.(2.17) preclicts t h the critical temper-

atiire is simply rclatecl to the energy gap at T = 0.

(3) .Calcdation of t hermodynamic functions

(A). Specific heat anci Entropy.

The BCS theory leacis to an expression for the free energy from mbch

t lie transition temperature and the thermoclynamic properties can be clerived.

This energy (in zero magnetic field) is

w here

2 112 Ek = (EZ + Lik)

and

From this equation and the relation

we obtain the entropy (per unit volume)

This esxmsion For the entropy is of esactIy the sanie forrn as the entropy of

n #as of iuclepenclent fermions.

Since the gap parameter tends to zero as T -t Tc, the entropy is contin-

iioiis at the critical temperature. There is, therefore, no latent heat involvd

at the trruisitiori. ancl this result agrees with ex~eriment. One also can sec

from Eq-(2.18) t hat as the temperattire approaches absolrite zero. the entropy

A hecomes exponentially soiall, being proportional to exp( - E).

The specific heat per mit voliune of the electrons c m be calculatecl fiom

In the weak-coupling limit N(0)V < 1 1Mmhlschlegel[30] lias solvecl Ec&L19)

numerically ancl has tabulated values of C,/C, as a function of S. The simple

BCS mode1 predicts that in the zero magnetic field the specific heat is not

continuous there, and one has

At lower temperat tues the specific heat decreases rapidly, and because of the

gap in the spectnun of excitation it becornes proportional to exp[-A(O)/kT]

at temperatures below about Tc/lO-

(B) . Critical Magnetic Fielcl

One CHU then show that. necar nl~solu te zero.

where Hc is the tliemodynamic critical field.

For an appliecl field H 5 He, supercondiictivity is clestroyecl becaiise the

spin singlet boiuicl state is clestroyed by thermal Biictiiations. The pair bincling

eoergy is then effectively overcome by the magnetic energy, so that the pairs

break up into single electrons. This type of behavior chnracterizes type I

si~perconcliictors.

Chapter 3

Proximity Effect

Wihen a normal metd N is clepositecl on top of a si~perconclt~ctor S ancl if

t lie electricd contact between the two is good, the two rnetals infltience each

other. The properties of the electrons on both sicles of the boimlary are then

moclifieci; cg., in the superconclucting sicle the energy gap is recliicecl, ancl in

extreme cases the system may even lose its superconducting properties. On

the other side of the boundary, Cooper pairs can exist in the normal metd

side which may have superconclucting behavior incluced in it. This is the so

called proximity effeet for superimposed thin metal films.

Thece are &O a nurnber of works that have been done using clifferent

approaches to investigate theoreticdy the properties of the proximity effect

in superconducting a thin film, such as the Ginzburg-Landau (GL) theory, the

rnicroscopic approach based on Green's functions, and the McMillan Tunneling

Model.

1 Theoretical Background

1.1 'Leakage' of Cooper Pairs

The 'proximity effect' is relatecl CO the " le&geTv of Cooper pairs fioui a sii-

perconchictor S to a normal nietal N. S is the procliicing 'reservoir' aad N is

the 'receiving' one. As the thiclsess Ii-L of the lealizge region on the N sick

is typicdly 10"A - 10%A, the 'proximity effect' is of a relatively long extent.

Wë k t consicler a simple case, where (a), we neglect dl electron-eiectron

interaction in N; m c t (b), there is no magnetic field.

(1). If the metal N is 'clean', i.e tN > eiv , where IN is mean

frce path in N and b is the coherence length, then the probability amplitude

F =< 9, q31 > for finding a Cooper pair at distance 1 x 1 fiom the N-S borindary

has the asymptotic form

where 4(x) is a slowly varying function of x ancl

where UN. is the electron velocity in N and T is temperature. However, when

T + 0, the decrease of F is not exponential any more, but becomes slowly

I F - - ( 1-21

x + x1-

(2). If N is 'dirty' (Lv < & ) , the leaicage of t h pairs is cont rollcd by a

clifhision process [6]. It is then convenient to introcliice the cliffi~ion coefficient

1 D = (for one electron at the Fermi levcl in the normal statc). Then the

asymptotic from Eq.( 1.1) still holcls, but now

1.2 Cooper Limit

The first ancl simplest case is clisciissecl by Cooper of t hin N ancl S films in

1961. He pointecl out a physical argument which can be presentecl as follows.

In the BCS theory the range between the attractive electrons is very short,

biit the size of the wave packet or the correlation clistance of the attractively

boimcl Cooper pairs is of the order of the coherence length. Due to this longer

coherence length, the Cooper pairs can extend a considerable distance into

a region in which the interaction between electrons is not attractive. As a

result , the g r o d state energy of this thin bimetallic layer is chamcterized by

some average of the interaction parameter N(0)V over both metals, which in

tuni cletermines the energy gap of the layer and its transition temperature.

4 n electron spends part of its time in N and part in S. Due to the different

clensities of states in N ancl S (NN, SS ) the normalizecl times spent can be

whrrc the l i d t <IL\*, il3 << < is callecl tlic Cooper E t . The critical tcnipt8ra-

tiirc is giveu by the ustial BCS expression [4].

Experimentidiy it is cLifficidt to obtain a very thin film of weI1 clefincd thick-

ness. This linLit is. however. very important for funclamenta1 stiiclies of the

iuteraçtion constants. .Go, as pointecl out by Bassewitz and Minnigerode.

some films (CIL) deposited at low temperature have properties very ciiffereut

from the biilk,

1.3 Magnetic Aspects of the Proximity Effect

Experimental results of the contact between two layers show that the critical

properties of the superconducting thin f ih are different from the bulk super-

conductor: the critical thickness, which is aiways s m d e r than the coherence

length of the bulk superconductor, decreases with a decrease of the effective

mean fkee path. The deposited films have a very short electron mean free path

in the normal state and their coherence length is modifiecl.

A magnetic metal in contact with a superconductor metal breaks Cooper

pairs and hence suppresses superconduc t ivi ty, which is the magnetic aspect of

the prosimity effwt. If the n o m d nietal is uingurtic-. it shoiilrl have niow pro-

uoiiuc-td effects che to the ackiitioual hterac-tion of spius nith tlic coucli~rtiou

i*lwtroas. Filieu N is a ferronitigriet ( i.e. Fe) or au .uitZerrouiaguet ( hlu. Cr).

es+e"icnts [31. 34 show that the orcler paranieter A essentidy muishes at

the N-S iuterface-

(a). For the ferromagnetic case. this is what is expectecl [33], since the

two nienibers of a Cooper pair (with opposite spins) see very differerit exchange

potential (wlierc ï - O&), the thickness of penetration is of orcler heuFr.

ancl is very srnail-

(b). For the antiferromagnetic case, the penetration is probably coo-

trokcl by c~efects in the magnetic structure (non compensatecl spin) which act

very much iike isola ted magnet ic impiui t ies.

1.4 Magnetic Field Behavior

We now consider the situation where a magnetic field is applied pardel to the

S-N interface, where many interest ing effects appear.

(1). Under certain conditions, N displays a Meissner effect in low mag-

netic field. In incrensing fields, the superconducting properties incluced in N

by the proXimi@ of S (in particular, the Meissner effect) are progressively

clestroyecl. In pract ice, t his destruction occurs in fields significantly smaller

than the critical fields. It turns out that experimental investigation of this

<lestniction leads to valuable information on the 'superconducting properties'

of 3,

(2). Upper mitiral field HL ùi the Cooper Luit.

The critical field of S (ancl, in partictdar. the ttpper critical fielcl). rila)-

I>e strongly nioclifiecl by the prosimity of PI. ;Uso. the beliavior of S near its

cl-it ical fiekcl is niocMecl by t Le proximity effect.

2 D e Gennes-Werthamer Solution for Dirty

Mat erials

It is weil kiown t hat the behavior of sitpercoac~ttctors in a magnet ic field in

t lie neighborhoocl of the criticd temperature is quditat ively well describecl

by Ginzbitrg and Lünclau's phenomenological theory, which Gov'kov [5] hm

shown can be clerivecl by exploithg the fact that the gap or orcler parmeter

approaches zero. There has been much theoretical work on proximity-effect

sanclwiches in the G-L regime, in particiilar the calculation of the critical

temperature by de Germes. In this section, we wiU summarize the result of de

Gennes-Werthamer Mode1 of " proximity effect" .

For sirnplicity the present account will be restncted to the case of a two-

Layered system in zero field. We assume that the transition temperature Scws

of our N-S layer corresponds to a second-order phase change, as is incleecl

obsenred in most cases. The layers are assumecl to be thin in cornparison with

the coherence length ci, cz. Then, for T = TcwsT the Cooper pair amplit iicles

( 1 ) = ( ( r ) >. nliirli is the probabilitj- aniplitiiclt~ of fincliug twc~

~~lcctroris iu the rouclenstcl state at point r. are snid eve~whcrc. h ei~~Li

iuetal we m.&e au approshatiou of the %CS type. TLic cic.c-trous iuterzirtious

arc clescril~ed by a couphg constant k*( I - ) . wliere

The interaction is citt off at a fieqiiency d D ( . c ) = dos. is positive aud

attractive. Zn the N regions, qv may be of either sign, clepencling on a clelicate

balance between the Coiilomb repdsion anc l phonon-inclricecl attraction. In

the siihcritical regirne (T 5 Tc) A(r) is smaU and can be obtairiecl from au

expansion of the self consistent eqiiat ion

where

and the range of the kernel in S and N is given by

From the self-consistency equation, one can derive the discontinuities in vdue

and slope of A(o) at the bounclary and obtain the following conditions by the

pair potentid.

The conctitiou E c l . ( U ) is rather general. but the conclition Eq.(2.'i) is rliarac-

\.Ve now present some remarks macle in the introcitiction concerning the

SC& of these proxiruity effects.

(1). FVe consicler k t the case Kv = O, where &(S) vanishes in N but

the concIensation F ( r ) = h/V has a finite tail in N due to the contributions

of the pair amplitucle in S. Iieeping only the lwgest range solution

one obtains

Flv (x) = F1v (O) ezp( -Kx) , 12.9)

with the Ieakage length = EN. The approximation does not apply well to

thin films where al1 contributions to the kernel have to be considered-

(2). If we consider another situation where there is a nonegligible at-

tractive interaction VN between electron in N: i.e, N is itself a superconductor,

with a low transition temperature TcN , then IC1 diverges for T -, Tclvs- This

divergence is a 'critical opalescence' property typical of second-order transi-

tions. When T > %, it c m be shown that the asymptotic form of F in N is

again exponential. Of course, for T < TcN, F tends to a nonzero limit in N.

(3) Let 11s rousiclw t tLr simple Y-S sanclwivh la>-t-rd stnirtiirc mith ap-

pn)p~iate boiuiclary c-ouclitions iu the Cooper b i t . Da. Ds < <. Our mu fiiitl

for tks sitiiation

rvhere T, = T,(r) is the biillr transition temperature <lefinecl by

Note the cornparison with Cooper's formula

(4). Anotlicr intcresting geometry for the proxïmity effect in the S-N-

S sanclwich. cle Gennes showecl that in this system a stipercrirrent c m travel

t hrotighout the normal Iayer. The theory is quite similar to that of the .Joseph-

son ciment, but it is foiincl t hat the maximrun supercurrent is proportional to

(Tc - T)'. This is in contrast to the (Tc - T) factor in the Josephson ci~rrent.

Due to the fact that the theory agrees with many experimental results,

t his met hod is evtensively usecl to investigate properties of the proxïrnity effect .

3 McMillan Tunneling Mode1

Ln the foLlowing we shall concentrate on another simple theoretical mode1 of

the proximity effec t between superimposed normal and superconducting met al

films. This is the McMillan Tunneling bfode1[21], which has an attractive

Figure 3.1: T~uuieling niocle1 with a normal metal ancl a siiperconcluctor sepa-

ratecl by a a potentid barrier ancl with the BCS potential (clash lines constant

in each metal)

simplicity that ailows a complete solution and has the advantage that it is uot

restricted to the vicinity of the cntical point. In this model, a potential barrier

is assumecl to exist between the nIms (Fig 3.1) and the tunneling through the

barrier is treated by means of the t r a d e r Hamiltooian method.

The tunneling Hamil tonian can wri t ten as

where T,,r is tunneiing mat& element, which we take equal in magnitude

between every state 4s in S and every state 4; in N.

The Hamiltonian for the sanclwich is the sum of the Hmniltonians for the

Treatirig the elect ron-phonou interaction aud the tiuuiclhg H i d t 0 n . h to

second-orcler. the self-consistent ecluation for the BCS potential is

15h = R R ~ [ A , ( E ) / ( E ~ - I + ( E ) ) ~ ~ ] ~ E . (3.16)

(1). For a giveu poteotial A$ and il;', by computing the excitations, one

cm gct some generd features as follows.

(a). \men the pairing interaction vwiishes in N, the energy gap is jiist

which means there exists an energy gap in the excitation spectnim eqiial to

1 . Since the relaxation time r L ~ for tunneling from N to S is the s m e

for each state & in in, where rLv is the average tirne an electron spencis in

N before penetrating the barrier and escaping into superconcluctor layer, the

same energy gap is observed in tunnehg into the S or N side of the sandwich.

(b). The electronic density of states is

for both superconducting and normal metals. We know this electric density of

states is measured clirectly by placing a tiinneling junction on one or the O t her

side of t h SN sa~clmcli nucl uieasiuiug the u o ~ ~ a ~ c d first <lt*ii~i~ti\*c~ of thr

t ituneiiug c-iment \*ersits voltage by t iuinehg t h o -

For the tiuinelirig mode1 WC have assiunecl that the path Lcugth or rc-

laxatiou time is the same for each state anci there e'iists a sharp gap in t h .

excitation spectnuii. It ki clear that the electronic density of states qualita-

tively is in accord wit h the experimental clensity of states, which is somewherc

between t hese limits-

(2). We consicler the self-consistency of the BCS potential with a finite

pairing interaction in N. One fincls ûn enhancecl contribution to the measttrecl

energy gap, which may provicIe an e-xperimental probe of the pairing interac-

tion-

If one considers A" for a thick normal metai ancl talies Tzr = O, Froni

Ec1.(3.16). one can get the self-consistency equation for the BCS potentid in

Sas

where the BCS potential for the bulk superconcluctor is

A h i t e pairing interaction Xx is assumed in the normal metal. The presence

of the superconductor induces pairing in the normal metal and ïncreases A$'

from its buk value. The energy gap is approximately

For the SN sariclwîch, T b shows au eali;mcecl conti-ibittiou to the rucasi~rtd

cnergy gap. which provicles au estimate of the piairing interaction.

(3). Asstmiug that each metal is stdîiciently thin, ancl the ratio of the

me~m Free pat h to the film t hickness remains constant =mcl c 1, one can gct the

transition temperattwe of the SN sanclwich as a fimction of the parameters.

Since the theocy is linear at the transition temperature, if we consider

A N = 0. one gets the self-consistency ecpation for ~f~ at Tc as

The transition temperatrire can be determinecl by

where C> is the digamma function and T, denotes the transition temperat lire

of the bulk superconductor, which satisfies

There are two Limits, the Anderson limit and Cooper

est.

lirnit , which are of inter-

(a). Making use of the asymptotic for +(t) - h(z ) for large z in the

Anderson limit of large r, one h d s

svliicti differs somewhat the solution given in (a).

in the correct treatment of tlie probleni the two sicles are tightly couplecl

o d y for energies < r rztther tban for dl energies iip to the ciit off Frecliienm

as Cooper's treatment ~Lssiimes.

Chapter 4

Single Superconduct ing Film

There h.as been mitch theoretical work on the proximity-effect iising the de

Gennes-Wert hamer Moclel anc l the McMillan Moclel. In this Chapter a met hoc1

basecl on the Bogolitibov equations, with a somewhat cwerent line of approach

regarchg the self-consistertcy of the pair amplitucle, will be disciissed. Wë have

moclifiecl this approach by using a self-consistency condition less s t ringent t han

the exact one which demands that the assumed and calculated pair amplitude

exactly coincide. This approach gives the solution of the temperature and

tliickness dependence of the gap, which is obtained by solWig the gap equation

fkom the self-consistency condit ion imposed on the pair amplitude. CVe start

with the simplest system, which is a superconducting thin film of uniform

thickness surrounded by a normal metal and also, but later this approach will

be used to analyze a superlattice composed of alternathg Iayers of normal and

siipercondiicting met&.

To IIP sprifir. N-e iuaktl the follorviog asstiuiptious nlmrit the sj-strui wc

iutcucl to clesc-rihc:

(1). We stiicly cleiu sitpeirolicliicting cmcl clean iiomal nietds. By rleau

we h p l y I » HT). whc-re 1 is the ue.m free patk ancl c(T) is tempcratiire

dependent coherence iengt h.

(2) A mode1 with interactions is treatecl in the mean-field nppro2Einia t ion,

mcl we assime the phonon-inclucecl BCS interactions me given by

Ir Siiperconc~t~ct ing p- =

(3). W e start by assiiming the pair amplitiicle is constant within the

superconclucting region, thris enabling iis to constnict e'cplicit solutions to t lie

Bogoliiibov ecpations. For the self-consistency condition we recpire that the

spatial average of the assumecl ancl the calculatecl orcler parameter be ecltia.1.

This conclition follows by rninimizing the free energy with respect to the gap

parameter-

(4). In orcler to simpilify the algebra for the gap equation, we assume that

the Fermi velocity, the Debye temperature, etc. are the same for both metals

and we assume the transmission a d reflection coefficients at the interface t O

be imity.

1 The Energy Eigenstates

aiid the normal metais f i h g the remnintlcr of space 1 1 I> CL (geonietii~s

which v<a.ry iu ouc clirection the x dimension). This is illustratecl in Fig. 4.1.

Figure 4.1 : Single sitperconcluctor layer

LVe clenote the operator of supercondiictor by 1 ancl that of the normal

metal by 2. The Hamiltonian for the system will, in general, be of the f o m

with Hi, i = 1,2, given by

where #=(r) is the second quantized electron field operator and

Tlir c-ocfficients 1.; arp the phone-iucliicecl BCS intcractiou. Trwtiug th(- iii-

terwt iou tcms sppearhg ( 1.2) iu the uiezlu-field npprosiuiatiou rccliiccs t lit*

Haiiiiltoninn to t lie following espression:

where F(x) is the real pair amplitricle, whicli is

F ( r ) c m be obtained by using the cpasiparticle creatîon operators c&' and b t

whick operate on the groiuicl state 1 O). h tems of the Namhti doublet

we therefore ob tain fi-om ( 1.3) the Bogoliubov ecpations in the superconcluct-

ing region and normal metal region

[i; - q(-iV)r3 + VF(x)r11q5(i, t ) = 0, 1 x I< a

[à$ - ~ ( - i ~ ) r & ( r , t ) = O, I 4 > a

.ri are the Pauli matrices and q5(r, t ) = ediHiq5(r)eiHt.

The system can now be quantized by, in addition to imposing the equal-

time anticommut ator

spwif-iug appropria tr boiuidq couclitious for the soltition of ( 16). Thsr

coudit ious arisr dite to continiiit- of t lie crvrent acros t lie Uiterfarw. aiid tutD

Wé iucitrcle the effect of transnùssion ancl refiection coefficients at the

interface with rf = rn2/rnl, where 5 -r O+. Then. one reaclïly solves theni

siibject to the boim<lq conditions by writing the field operator m(r. t ) as a

siini of positive and negative fiequency terms.

where E > 0 , I = (ky, k,) ancl p = (y, r) . The index i = 1, ..., 4 represents the

four linearly independent solutions which will be obtained from Eq (1.6). The

wave hinc t ions di) and v ( '1 sat isfy

for 1 .r I> a. The nomnlization conctitiou iuc

J ' ~ x $ ~ ) ( E , [ , .r)dj)(E, ~ , s ) = 2dq6(E - Er)

JFm &z$~'(E~ . r ) ~ ( ' ) ( E. lT X) = 2xaij6(E - Er)

Jzm d d ( E, 1, x)dj) (& l . 2:) = O

wliile t lie .mt icommiitation relation imphes

with al1 ot her anticommutators vanisbg. Now. the bocmclary concIitious

for the wave fiuctions are

To find the solution, we introduce the approximation that the pair-amplitiide

in the domain of the thin film is homogeneous:

Figure 4.2: The assumed form of the pair amplitude

For convenience, let 7 = t = 1 and assume that both the superconducting

and normal metals have the same parameters m and kF. We can then write

the Bogoliubov Equation (1.10) as

Lrt i r s c ~ c h c

u(E.1.x) = \ ( ~ . l . . r ) e ' ~ ~ .

If \( E, 1, r) is smooth on the atomic scale, WC can chop the second <lerivative

te~m in the above ecption, which resiilts in

with A(x) given in (1.15). .b(E, L, :c) has to fiilfill the nonnalization conclition

( 1.11 ) and the following boimclaxy conditions:

Here we have introcliiced the following notations:

4

' k* = q*k

p * = ( 1 1 + p

q = (kt - 1 y

k = mE/q

P = mEylG?

6 = -AIE

. r = 8 ( E - A)(1 - 69'/* + @(6 - E ) i ( b - 1)'l2

T h soltttiour; arr of the

for

Note thst . given the solution t r i ( E, 1, s). we can inuuecliately mite down the

solution ri(& 1, ) hy iising

2 The Pair Amplitude and the Self-Consistent

Gap Equation

In the last section, we obtained the eigenhctions which can be used to com-

pute the pair amplitude F ( x ) as a function of the order parameter. The order

parameter is t hen de t ermined by the self-consis t ent gap equat ion:

Tlic siil>script *W" refers to the 1-2 component of the matLu formecl from the

oiiter pro<luc-t of m(r) with &r) which is tlefined in (1.5). Since m-e niiI work

at finite temperatiue. the .mgtda.r bracket in F(x) denote a themai average

and the operator a' ancl bi satisfies

represents the thermal ~Listnibiition fimction (0 = 5). With this, oor pair

aniplitiide wïll be

,C3E ( i l ) d i ( i l x ) x t h - ) (2.22) 2

Ln principle, the pair amplitude shotdd be computed from the ecpation

above by using the exact eigenfunctions, but this calcuiation is quite ciifficult

and will not be attempted here. One limiting case in which the expressions

sirnpliS. is obtainecl if we set

For this approiuuiatiou. WC helievr tktit the r ~ s i i l t s sliotiicl giw thr uiost

gïvcs for the pair .uiiplititcle F(x). for 1 .r I< a.

Froni the self-consistency gap requiremcut (2.21). we finally get the g,~p

ecluat ion of

n-here M = mkF/2$ is the clensity of states at the Fermi surface of the normal

met al for one spin projection-

3 Thinkness Dependence of the Critical Tem-

perature

Although we have obtained a relatively simple gap equation, except for certain

Iimiting cases it is not possible to obtain analytical results and recourse must

be macle to m e r i c d solutions. In this section, we first consider the case close

to the critical temperature of the film, a d then away fiom the critical point

to examine the temperature and thickness dependence of the gap parameter.

3.1 A Calculation Close to the Critical Temperature

Iu t l i is sert ion. wc wiïiU k t mlcrilatc t ke cIcpencleur.c of the triuisitioii truiper-

:rtiire T , ( r r ) on the tliickncss of the filni. Uncler the limitntiou A -r O . the gap

eqtrntiou bcconies

wkere k = r~rE/q. If we define the fiinctioo

and assiirne that TJa) *: u d , integrating (3.26) by parts residts in

-9 13w 1 = V?v(o) { I (ud ) - - " /a &I(u) cmh -(-)}.

2 O 2

The first term in Ecpiation (3.28) represents the zero temperature contn hiit ion

to the gap fiinction, ancl the second term contains the thermal corrections. The

hinction I (w) may be cdculated as

where

and I ( x ) is given by

1 su iz I ( x ) = Te + lnx + -[- - cosz - ZCi(t) - xSi (2 ) ) .

2 :c

Ci(.!-) amcl Si(.r-) arc t h Cosine and Sim intcgrals[35] nliic-Ii arp clrtiric~tl as

foHows:

and 7, is Eder's constaut- In the limi t 11- - m ancl .c -, 0, mie can 01, tain the

following approiuimat ions:

We next clefine the critical thickness a, as the tbickness below whicli it is

not possible to keep the siiperconchcting state at the any temperature:

This can be determinecl fiom the gap function by setting T = 0, for which

I ( x ) = I(&nd), with ud the Debye frequency. AssAssuming that &Gd » 1, and

using (3.31), in the limit T = O we obtah

where TF = EF/kB clenotes the Fermi temperature and Tc is the bu& super-

conductor transition temperature. This result can be wrîtten in terms of thc

whtre 6 = P ~ / R & ancl A. is the hi& zero temperatrire gap.

In orcler to consider the finite temperatiue case, we rettini to (3.26). The

thichess dependence of the critical temperatiire cm be obtiihecl by using the

cl , given in (3.33):

where Tc@) A Tc@) 2

7, = :i- = --- TF A, Tc T'

From the t hickness dependence of the self-consistent gap iuicler t his b i t at ion,

we can analme the resiilt as T -r O and the asymptoticformof T,(n) as a + W.

(1). In the case T , < 1, we use (3.30) to h c l

T (a) 1 ccc CL l*'- -- w-1- -ae Tc in2 n ccc

Expanding the above equation arotmd a/a, = 1 to first order, we find

(2). In the case Tc » 1, we have

Expanding around T,/T,(a) = 1 to first order results in

3.2 Numerical Result

Let us iiow- irivestigatc t Lc thirhessc .mcl tcniperat tire clepenclcucc of t lir gap

rcpa t ion paraniet er in generd, for which niunericd

Thc p p fiinetion of Eq. (3.26) can he rewritten as

mcthods mitst hc rised.

Assilaiiug Td » CL), we extend the iïmit on the E integrni in the seconcl

tenu of this ecpatioo to W. ancl find the resulting ecpuvalent eqiiations:

In Fig.4.3 We

of the criticd

present the numerical solut ion to (3.42) giving the dependence

temperature over the entire range of thickness. One can also

veriQ in t his figure the asymptotic b i t s of (3.37) aod (3.39). The numerical

ecpivalence of handing the cutoff dependence in this manner wil1 be useful to

evaluate the gap equation with a finite order parameter A.

Figure 4.3: The dependence of the critical temperature on film thichess.

4 The Thickness Dependence of the Gap Pa-

Iri the Iast section. we consiclerecl the gap p;irameter to he zero. Ch+ non-

consicler the case of a h i t e gap. First of d. WC consicler T = O in the genecal

gap equation, ancl hd

where p is clehed iu Eq.(1.19). Some manipidation of (4.43) resiilts in the

foiiowing ecliiation for Ao( a) :

where

anci the fimction d ( b ) is given by

dx s g x J L ( b ) = 4- dr /DD - sin 2x

b~ x2 i2 + sin2 x - -1, 2 2

CVe now corne to the final calcidation in this chapter, that being the thick-

ness and temperature depenclence of the gap, which we define A(n). We start

with tlir fiiU gap tytration (2.25):

I sin 4pcl JE K1- 4pa

) tnnh( +). (4.47) \ --\-(O) I - p c o ~ 2 ( 2 p a ) 2

aud

sin" sin 2s 2 q ( b , ~ ) = [ d z l r z2 Za + rc 11 - -1 (-4.49)

22 l+eq[(z"f)+b2c]'

The nrinierical solution to (4.48) giving the clepenclence of the h i t e tempera-

ttire gap parameter on film thickness can be found in Ref. [2.l].

Chapter 5

Finite Superlattice System

We have presentecl a geneicrl methoci for the self-consistent solution of the

Bogoliubov ecliiations for a class of geometnes which incliicles the case of a

superconclucting thin film. Ln this chapter we stticly the proximity effect in a

finite stiperlat tice system. employing the s m e methods basecl on the Bogoli-

~ibov equations using a step function approximation for the pair amplitude,

the height of the amplitude being determinecl by a minimization of the free

energy of the system. We s h d in particular h d using this method the cnt-

ical temperature of this system as a function of the film thicknesses and the

niimber of layers.

1 The Solution of Bogoliubov Equation and

Gap Equation

The systeui to be iuvestigated consists of iclenticd st~percouclric tors of uviclt li

2a separatecl by I.?yers of normal metal of wicltli 26 at the :r = O plane. while

tlie end of the system is a nomal layer. We choose coordinates for each from

-[(2n+l)c~+2nbj to (2n+l)a+2nb. with n = 0 , l . 2 ,3 . . . , n - 1. If N represents

t Le uiimber of superconclucting layers, then N = (2n + 1). Figure 5.1 shows the

systern. nith S a o c l N representing the supercondiictor layers <mcl tlie nornid

met al layers. respec tively-

Figiue 5.1: Finite superlattice system

For simpiicity it is assumed that a l l physical properties such as the Fermi

velocity, the Debye temperature, etc., are the same for both metals, and the

transmission and reflection coefficients at the interfaces are unity.

For clean materials, the wave function p is

which can be nrritten as foiiows:

CVc note that the gaps for each supercontlucting layer are clifferent. ancl there-

fore we use the label j to intlicate the gap parameter for a particidar layer.

This assumption of the potential term for the system ~mcler consideration is

show in Fig. (5.2).

The origin of the system is at the center so that the system is symmetric

about the origin, a n c l because of parity, we only need to solve one side of

the system (x > O). The solutions can be labeled by their parity &, where

ri yI(x) = f p*(-x). Generally, the solutions can thus be split into 2 paxity

states, and the solutions for the superconducting regions and normal regions

f o r s > O can be written as

Figure 5.2: The Asstunetl fom of the pair amplitude

where d = n + 6 'ami the parameters b i 7 pi, are definecl w ~OUOWS:

l P = h i )

k = mE/q

a(j, = - @ / E

Y(j1 = (1 - b ( j ) 1 ''* Note that , given the soiution x > 0, the solution x < O can be calculated

rp*( -4 = f TLP* (4-

The coefficients in (1.2) are determined by applying the nomalization condi-

t ion

For the huit notrual hyer. ive h d

To simplify the solution, we consider k t order in 6. The wave funct ion

of the system then becomes

With these, and howing for the last normal layer Ncj ) = ~ n / 2 ~ , we c m ,

toget her wit h the boundary conditions, determine d the coefficients.

where "j" labels the layer of superconductor fiom the center of the film to

the right: j = 0,2, J,6 .. .. The total number of layers of the system then is

1V = 2 j + 1, and 1 = j + 2, j +4, .... N are those layen to the right of the jth

wlirre F ( x ) = - ( p + ( r ) y(.,)) p. the stiliscripts intlirating the matLu conipo-

Findly, we get the gap ecpations

sin 2k.a + - sin 2jkcl sin 2k( jd + a) 2kn sinL 2ka + 2

Ar cos 2jkd - sin 26k-rI], 2ka 1 4

where N ( 0 ) = mkF/2a2 is the density of states at the Fermi surface of the

normal metal for one spin projection and V is the coupling constant.

As a check, let j = O and N = 23' + 1 = 1, which is the single supercon-

ducting film. In this case Equation (1.11) reduces to:

1 = p $ taoh(PEl2) /' dp[(l - sin 4 ka

VN(0) O 4ka 11 9

which is the same as that obtained in the last chapter for the single layer.

2 Reduced Thickness Dependence of the Re-

duced Crit i d Temperature

Iu this section we use the self-consis tcnt gap ecliiation jiist clerivecL to cvülii21.t~

the rectiicecl thclmess C L / C L , at zero temperature, ancl then obtain au espression

for the clependence of the transition temperature T,(c) on the thicliness of the

film.

First let u s ciefine a fiinction I ( u ) as

sin 4kcc 4Ea 1

sin 2k-n + - sin 2jkd sin 2 4 jd + a ) . 2kn

so that after integrating (1.11) by parts the gap fimction can be rewritten as

The first term in (2.13) represents the zero temperature contribution to E-

q.(l.ll), while the second term contains the thermal corrections. We consider

T = O first. The function I (ud) is given by

where

where the f u t ion . J (c t ) is definect as foiiows:

sin cr - a cos cr cr .J(n) = [ - -S i (a) - Ci(cr)]

2Ck 3 -

a(l + jy) cosa(l + jg) - sinn(1 + j g ) + 4 1 +J :J ) [ e(l+ j y )

where « = 2AR, y = l+b/a,Ci(n) and Si(cr) are the Cosine and Sine integrals.

If we let xcosx - sinx

K ( X ) = + xSi(x) + 2Ci (x) , 5

Eq.(2.15) wiil be reduced as

We now want to see how the fwiction I ( x ) approaches the Iimit .g

1 1 lim I(n) = 7 + ln& - j y ) - - (1+ jy) h ( l +jy)} .

a-OO 2

which leads to t hc follonring rethiced crit ical t hichess at zero temperat iirc:

wliere < L / C L , is the retliicecl critical thicliuess (Le.. the thckuess below wliick

the si~perconclricting layers become normal at T = O ) . This critical tliicknes

<le. which is clefined for a single siiperconcluctor layei-. will be used as a lengtli

scde. so that the ratio b / a , the width of normal layer and the width of super-

coucluctor layer, can be written as bloc ancl a / c r , , respectivdy. It is also iiscftd

where Bo is the Debye temperature and Te is the bidk siiperconclucting tran-

sition temperature.

To determine the thickness dependence of the critical temperature above

which the superconductors becomes normal, we retuni to Equation (2.13). Of

course we have to consider the second term of Equation (2.13) for the h i t e

temperature case. h u m i n g that hod > 1 With 4 0, (2.13) together

with (2.14) gives

Aft er integration oE the secoud telm. the ecpation becomes

which. iising the e-xpression for T. T/Tc. can be e?cpressecI as follows:

where the fi~nction H(x) is clefined as:

Integrating t his ecption, we obtain TIT, as function of i, bla,, and a/nc as

Numerically evaluating this equation gives results for values of T/T, vs. a/a,

as b/a, and j Vary. Figs. 5.4 Jc 5.3 show typical curves of T/Tc for various

layers. For the h i t e superlattice case, we would expect that as bla, increases,

the results of T/Tc shoulcl be close to the single layer case; i.e., for a large

Figure 5.3: The dependence of the reduced transition temperature T/Tc on

the thickness a/ac . solid line represents single Iayer. Dash line represents

N = 100, Dash-dot line represents N = 40, Dash-dot-dotted line represents

N = 10 for b/a, = 0.05 Superlattice

Figure 5.4: The dependence of the reduced transition temperature T/Tc on

the thickness a/a,. Solid h e represents single layer. Dash line represents

bla, = 5, Dash-dot line represents b/a, = 0.5, Dash-dot-dotted line represents

bfa, = 0.05 For N=10 finite superlattice

3 The Behavior of the Energy Gaps

In t Lis section. we slid inwst igate the t h i c h e s cmcl temperat iirc clepencleucc

of t tic gap parameters, wliich we denote di), for cliffereut 1ayers. In pa-ticiilar.

wt. cousicler the t hicimess depencleuce O t the gap parme ter ratios A(~)/A(')

at zero temperatiue and then consider the case of the finite temperature and

cleterminr the t hiclmess clepenclence of the gap parameter

This is of interest since, in the restdts jiist clenvecl, for T/Tc l e s thnn aroitncL

0.3 the bchwior is im~ts~tal anci for TIT, greater t h u i aroitnci 0.3 the restilts

approach the hitlk limit for al1 bln , and j values. FVe coiilcl then examine how

the ratio

varies with temperature, which can be used to estimate the reliability of the

methods we have employed.

In principle, the gap parameter A(x) = V F ( x ) is spatidy dependent,

where F(x) = -(v,br(x)+l(x)) is the pair amplitude and @r (x) and $J~(X) are

the bounclecl pair electron wave functions. By the proximity effect, the ieaked

electron pairs may interact with other electron pairs which leak from other

superconduct ing regions, and t hus the electron-electron coupling in the central

regions of the system is expected to be stronger than in those regions away

aud to die off soiutwhvliat towarcls the eclges. as iiliistratccl in Fig(3.2).

lu order to see if the calctilations support tks espectatiou. WC start n-it h

t Le fidl gap ecpation (1.1 1). C V e now define the I ( w ) as follows:

sin 2kc + - sin 2jkd sin 2k( jd + n) 2kn

Taking iuto acconnt the terms involving the gap parameters:

the gap eqitation c m be written as in the Last chapter by intergratiug (1.11)

hy parts:

It is conveaient to define some au,.Ciliary functions as follows, dter using

t rigonometnc identities and element ary integrations:

and

( 1 ) = -2(Iy + jy) h ( l y + j y ) - 2(ly - j y ) h( ly - fi)

84

With thcse, we find (3.26) caii be written as

For Aud > 1 the fitnction I" = O. Ths, in tliis limit, I (n) becomes

Inserting (3.31) into (3.271, the gap ecliiation is

where T = ; z ~ . " " Together with Eq.(2.18), we then fincl the gap eqiiation can

be d t t e n as foilows:

3.1 A Calculation of A ( ~ ) / A ( ~ ) at Zero Temperature

From Eq.(3.33), we can fincl the thickness dependence of the gap parameter

at zero temperature by taking the third term as zero:

whtw d ( x ) is $\-en 1 - (2.E) for -\Qd > 1. For n giwu J . tilt- al>ow qliiatiou

grurra tes j qiiat bus. and uitmerica- solvhg t hese tytiations giws t hr gap

I~ths ior . \P+ o d y d i s r . t ~ ~ ~ A ( J ) / ~ ( " ) which are preseutecl in Figures 5.5 to j.8

for different dries of b/nc and various valites of j.

Figs. 5.5 Si 5.6 represents the cases of bla, = 0.01, LV = 40. iuid b/crc =

0.01. .V = 10. The residts are as espectecl. wliich incikates the method iu this

muge provicles reasonable residts for smail valiies of b/a, as one lowers N at

T = o.

Figs. 5 .ï QL 5.8 represents the cases of blci, = 0.5. .V = -40. and blcr, = 0.5.

N = 10. Both configurations gives unexpectecl behavior and thiis iiiclicate the

methocl in this range does not give reasonable results.

Figure 5.5: The gap parameter ratio A(~)/A(') forN = 40,6/n, = 0.01, at zero

temperat ure

6 8 10 12 Layers

Figure 5.6: The gap parameter ratio A ( ~ ) / A ( ~ ) forN = 10, bla, = 0.01, at zero

temperature

18 20 22 24 26 28 30 32 34 36 38 40 42 Layer

Figure 5.7: The gap parameter ratio &j) /~(*) l forN = 40, b/a, = 0.5, at zero

temperature

Temperat ure

.4rcorcling to t lie crit ical truiperatiire stiicly7 most of the iineiupectecl I)cb.zvior

is ne= S = O, but for T T, the e?cpected (hi&) critical temperature is

rencliecl for al1 cases. To investigate the gap paraaieters at finite temperature.

WC integrate the 1 s t part of Eq43.33):

Mter integrating we can mite down the gap equation as

tqtiation may be solvecl uumerically for different valtics of blci,. T/T, ancl

varioits vdues of j. The temperatiirc clepenclent gap puameter A can be

present the N = 30 ancl blrr, = 0.01 resiilts at finite temperattire. The gap

paranieters clecrease for increasing j, as es~ected, even at zero temperature.

Thtis, we eqect we can 'trust' the residts for N ancl bla, for ail valties of the

Figs. 5.11 Si 5-12 present the N = 40, b l a , = 0.5 .witl LV = 10,b/cr, = 0.5

resiilts. From this %e see that the gap parmeters become better "beLaveci"

thwi they are at zero temperature.

8 12 16 20 24 28 32 36 40 44 Layer

Figure 5.9: The gap parameter ratio ~ \ ( j ) / ~ ( j - l ) forN = 40, bla, = 0.01, at

fini te t cmperature

12 16 20 24 28 32 36 40 44 Layer

Figure 5.10: The gap parameter ratio A(~)/A(') forN = 40, bla, = 0.01, At

Finite Temperatrire

8 12 16 20 24 28 32 36 40 44 Layer

Figure 5.11: The Gap Parameter Ratio @ / A ( ~ ) forN = 40, b/n, = 0.5, at

finite temperature

6 Layer

Figure 5.12: The gap parameter ratio .@/A(*) forN = 10, bla, = 0.5, at

finite temperature

Chapter 6

A "New Modelv of the Single

Layer

We have focincl the critical temperature as a fiinction of the wiclth "a" for

the single layer case where we assumec1 a constant gap parameter over the

entire region in Chapter 4. In this Chapter the cdculation of the transition

temperature for this single layer will be attempted to be improved by a way

which extends the approach presented by the foregoing model by using gap

parameters that have an approximate spatial dependence. It will be shown

that the critical thickness gets enhanced by a factor of 5-50 over that of the

single layer with an assumed constant gap parameter over the entire region.

These two models are in agreement at low temperature, but this new model

cloes not give the expected results at high temperature.

1 Physical Mode1

The central iissiiniption of the prcvîous work is that the order paralueter is

onifonii iu the S layer ancl zero in t hr normal Iayer. Howewr. re.dist ir gap pa-

rauieters are expectecl to have some spatial clependence, being peakcd ueai- t Le

center ascl cl-ving off somewhat towards the &es of a aven super<roucliirting

layer. rUso, in the finite superIattice system, we had shown for valiies of the

pararueters thst codd be tnistecl that the gaps at the central regions me l.zrg-

cr t han in t lie other regions, ancl siibsecpently clecrease away bom the central

layer when the wiclth of the normal Layer "b" is small, which can be foiincl in

Figs. 5.9 Si 5.10. The physical system we are going to clisciiss presently is as

follows.

-4ccorcling to Fig 5.9, if we imagine letting "b" tencl towards zero, then

the situation is For a single Iayer superconductor, where we have a r b i t r d y

divicled it into some "21V + 1" zones. We then self-consistently calculate the

gap parameter in each zone, and Let "N" go to infinity and "a7' tend to zero

but keeping the total width of superconductor constant as

where "a" is width of the superconductor in a particular zone. In this way

the gap parameter over the entire region is not some constant, but can be of

different heights in each region, approxirnating in a sense a spatially dependent

pair amplitude. This potential is sketched in Fig. 6.1.

Figue 6.1: Gap puameter dependence on spatial

2 Thickness Dependence of the Self-Consist ent

Gap Equation

2.1 A Calculation of the Critical Thickness at Zero

Temperature

Ive now use such a theoretical mode1 to find the critical thickness A, below

which it is not possible to maintain the superconducting state. This may be

obtained from the original gap equation that we found for the finite superlattice

in the 1st chapter:

1- letting T = O. Tho. WC c m nrite the gap ecliiatiou for tlus. nt zero

tr cos O - sin cr a( l + 2Xy) c o s [ ~ ( 1 + 2Ng)1 - s i ~ [ ~ ( l + 21&/)] + cl

+ Q

wliere we have definecl

Mth

If one lets

LVe then take iV + oo, a + O such that A is constant, and the gap equation

becomes

then solvc (2.7) with tlus limitation aucl get the critical thichess at zero

This gives the critical thichess in this nioctel for which the siiperconcliict ivity

\-anishes s t T = O . Here Sd is Debye temperature, a, is the critical thickness

at zero temperature assilming a constant gap parameter tlirotighoiit, and 'Tc

is the biilk superconctiic t ing transit ion temperat tire.

The restdt &O may be written in terms of

by recailiug the effective interaction by Fet ter (371 as

2.2 Thickness Dependence of the Critical Tempera-

Havhg founcl the cntical t hichess at zero temperatiire. we cnn inimecLiate1~-

calculate the t hichess clepenclence of the critical temperat i ~ r e by re t iirning to

Eq42.1). This leluls to the gap equation at finite temperature:

+ pci(a) + asi(a)] - (2Ny)[2ci(2~Vg) + 21Vy)si(2ny + l)]

with

& = & A r ac R Tc

1 & = - 2N+L

T = 6~

2A = (2iV + 1)(2a) we obtain the foilowing equation by using (2.7) :

For srud r. ititegratiug the secoticl terni of this qiiation aucl letting Ar 4 CC.

rvt. gct TIT, as a fiuictiou of A/aC . Td ancl Tc:

this b i t becomes

The ecpatiou for the critical thichess foi&

see that the integrai in (2.13) in

Eiom (2.13) may be conveniently

Recdling that the critical thickness at zero temperature is

we can see immediately bom (2.14) and (2.15) that A = Ac when T is zero.

h case worth noting is the limit 6 < 1, for which we obtain the simple

which may be e-yendecl (2.16) mund A/& = 1 to first order to give

Near -41-4, = 1. the r~tic-al temperattire T/r. approacli zero liiiearly. tkis Iw-

Liavior l~eing siniilar to t tic prcvioiis ruoclel clisc-tissecl iu chsp ter 4. h p~-iricipli~.

tliis uen approacli shoiilcl leacL to the s m e wstilt ol~tained bq- the wrlirr one

for d > 1, -4 -. x;. However. tks liniit does not give the c o ~ ~ e c t c-riticril

tenipernture Tc as t h bu& value Tc.

Note that in the above we must solve Equation (2.13) for a partictllar

siiperconcliictor with various values of T, .ancl ri; typicaI such values arc sliown

in the table 1.

Table 1. Tansition Temperture and Debye Temperature

Fig 6.2 presents n~unerical resiilts of the gap equation ancl shows the cliffi-

c d t y in applying the new formula at high temperatine. For low temperatures

the resdts agree with the analytic result of (2.1T), but, as jiist mentionecl, for

high temperattires the results do not approach the correct biilk vahie. This

tinexpected behavior may indicate that the method is valid only for relatively

t hin superconduc t h g films, over which the position-depenclent pair ampli tilde

does not Vary too greatly.

Figure 6.2: The critical temperature dependence on the thickness for New

Model

3 Cornparison and Discussion

3.1 Critical Thickness

Having conipleted the estension of the theory, we now cornpue it nith the

earlier moclel. FVe founcl the ctïtical thichess at zero tempe rat tu^ as

aucl if we recd that g N ( 0 ) % 0.2 - 0.4 for clifferent sitperconductors, Erom

Eq-(3.18) one c m get

In this moclel, the criticd thickness gets enhanced by a factor of 5-50 over t bat

of the single layer approximation usect in Chapter 4 ancl 5.

One coulcl pictiue titis residt as follows. Let us consicler n single layer of

thickness 2A with a constant gap parameter. At T=O lowering il 4 a, results

in the superconductor becoming normal (by definition of a,). One coidd view

this as the Cooper Pairs inside the superconductor (which give rise to the

stiperconductivity) 'leaking out' to the normal metal. The bond between the

Cooper pairs thus gets broken, and the superconductivity disappears.

Cooper Pair

Cooper Pair Broken

Howerer. the gap parmeter is not really constant. in space, wllich WC

~uight approximate as a series of (2N + 1) steps as s h o m by Fig 6.1. Bitt

with stich a fom, the superconcluctivity is 'wealrer' as one goes towwcls the

eclges, ancl hence one might expect the Cooper pairs to 'le& out' faster in

t hese weaker regions. Thus, A is decreased, one might expect in this moclel

the supercondiictivity to disappear qiücker, comparecL to the case where one

assumes a constant gap parameter over the eatire region. In other worcls, in

the present model one could expect A, > a,. and this is what is found.

The critical thickness at zero temperature in the earlier model, (1, =

0.882c0, may be compared with that obtained by Zaitsev [9, 101, who con-

sidered the same problem using the GL theory. The expression for the critical

thickness in that approach is

ivhcr~ c(3) = 1202. The orcler-of-uiagnitttck agreecmcut hctweeu t lie tn-o

approacLcs for t Le vdiic of the criticd t hichess siiggwts the p r o g f i u n-c a w

followhg is a rc.wonabIe one.

We bave &O seen the two limiting cases of the grrp ecpation obtained iu

Chapter Foiw: r *; 1 aucl r » 1. Once agah Rie r n q compare the results

of tliese with that obtainecl by the GL theory worked out by Zaitsev[9. 101.

where it is foiinci

where ci, here refers to ZaitsevTs value for the critical thickness, (3.20). The two

limiting cases corresponcling to o u linùting case (3.37) ancl (3.39) of Chapter

Four are

and

Thiis, there is a cWerence even in the function form of the asymptotic depen-

dence of the cntical temperature as a 4 oo between the two rnethods. In

Fig 4.3 we present the numerical solution of (3.42) of Chapter Four giving the

dependence of the critical temperature over the entire range of thichesses. A

r c )i~q)ai-isoii tc-i t li Zaitsev* solii t ion. as d l as t hat oht ainecl iu t lic t iiirncliug

iiioclrl of McMillau cau Iw foiincl refercucc [-Y. Therc- is not a grmt clif-

ftmmce Iwtween the resdts of the t h e methoch, kclicsthg ouce agi:~h oiu-

approach semis reasouable. However, we me Iookiug at the clccui limit wc-itli

ided boimclaries, and thus we e-xpect the presence of irnpiuities and scat ter-

ing s t the interfaces to rediice the coherence length of the sitperconclucting

electrons. iniplying s siibsec~iient recluction of the cnticai tliichess. This cle-

peucleuce of the critical temperature on the thicliness (3.34)(Cliapter 4) shoiilcl

then be seen as provicling an upper bo~mcl.

3.2 Tnfinite Superlattice

In orcler to compare the finite superlattice with an infinite sitpeirlat tice [38] at

the critical temperature, we briefiy present a calcitlation of the critical temper-

at ure for clean infini t e superlat tices which incorporates explici t ly i ts inherent

Bloch nature. The coefficients in Eq (1.2) of Chapter Five are rlrteminecl by

the normalizat ion condition

by continuity at the interface, and by the requirement of Bloch's theorem:

for periodic systems, where d = a + b is the lattice period and q is the Bloch

wavevoc t er.

For nout ririal solutions t lie foUott-ing condition is t L m n~.cssai-y:

iu tkc limit that A approach zero. In this liniit the gap eqii;ztiou also siniplifies.

aucl cleteruiioes the transition temperature Z ( n , 6 ) frorn

For infinite b, the function J(T, r ) vanishes, and an analysis of the oth-

er terms yields the solution T,(a,) = O. We also see from this that J ( r , r )

then represents coherence effects the other layes introduce to the single 6lm

geomet ry-

In Fig. 6.3 we show some results for the transition temperature cletermined

by (3.24). Here we plot the reduced critical temperature TfTc as a function

of a/a, for various values of bla,. We see that for about 6 > a,, the critical

temperature corresponds closely to that of a single film embedcled in an infinite

normal metal, with significant cleviation o c c 6 g only for around a < a,.

0.031 62 0.1 0.31 62 1 3.162 10 31.62 100 aiac

Figure 6.3: The dependence of the reduced transition temperature T/Tc on

the Thickness a/= for infinite superlattice.Solid Iuie represents single layer.

Dash line represents bla, = 5. Dot-dash line represents bla, = 0.5. Dash-dot-

dotted represents bla, = 0.05

AIso. WC firicl for tirùtc h tliat T, (cL-b) = O o d y for a = 0. iudic-nting sri

alxxmcc OF n çriticd thiçliliess .as Lisppens for idbite b.

Now we compare the restllts with o t k r bilaver approximations in tlir

Cooper M t . It is hown that iii tliis limit the cle Genncs- CVertlianier [G. 7-81

approaches are unable to couipletely accoiint for the observecl behavior of the

t rmsition temperat ure[39, 40,41] . Uncler such a conciition FVethamer's restilt ,

clescribecl in Chapter -4, becomes

whereas the de Germes resiilt recluces to that of Cooper [4]:

As we can see from these e-upressioos, for a fixecl ratio blrr, as the thickness

CL tends to zero the crit ical temperature approaches a finite constant. On the

O t her hand, E y .@.%)for small t hickness yields the result

which, again for fixecl bla, approaches zero as a Mnishes. This approach to

zero is of the same form as that fo-md io the MeMillan Mode1 [21] and agrees

qualitat ively with corrections to the de Gennes t heory .

With the preceding discussion in mind we now return to a cornparison of

the finite superlattice results of the critical temperature in Figs.5.3 with that

of the infinite superlat tice in Fig.6.3. Restricting the cornparison to thin films,

for a-hirli the resiilts sceui rcIiziblc., WC coitlcl çoriçliicIe that a fiuite siiperlat tic-c

approsiiiiates w l l au infiuitr siiperlattice for n reasouably suiidI utiiuller of

1 ~ + e i s ciucept For w1-y thin films. for n-hich qiiautitative cki-iatious are sceu For

a smcd nimber of layers. However, for very thin layers we have to takc Lito

account the fnct that the approximation hoa > 1 t a c l in tleriving Ecpatiou

will begin to break clown, where A = 2kFn. We have, thotigh, checlied tliat

the ft~U expressions no t assiiming ARd ) 1 give essentially the same ctwes <as

in Figs.5.3 ancl 5.4 except for the extreme b i t of cil i t , -r O. Thiis, a reliable

conclusion from this analysis seems to be that for thin layers with ci/n, and

bfn, less than about 0.1 one neecls more than around 40 sripercontliicting films

I~efoi-e a finite size superlat tice behaves, as fcw ris the critical temperature is

concernecl, Lilie an iclealizecl infini te stiperlat tice-

4 Summary

It has been shown that a method for the self-consistent solution of the Bogoli-

ubov equation for various systems: the single layer, finite superlat tice syst em

and infinit e superlattice, discussed in t his thesis, give a trac table and reason-

able t heoretical treatment .

For the different models under consideration, we studied solutions of the

gap equation obtained from the self-consistency condition impcwed on the pair

arnplit ude, t hereby ob taining the t emperat ure and t hickness dependence of the

gap pararueter. Sewral Ferrt~ires obtainecl I)y the prortdiire rat1 Iw si~tiiiiiarizcd

f>ri~flj-*

( 1 ). The depeuclence of trmsitiou temperature T(û.b J) for a fininite super-

lat-tice system Lias beeu stticliecl. Under the asstuuption ARd > 1. the results

show- t hat T 4 Tc t Le right Mt is reached for al1 cases, but itnespec t ed restil ts

appear for s m d T. The valiclity of this methocl hw been checkeci in the single

laycr-s case ancl the innnite snperlattice case, ancl has also heen coaipcved with

ot ker technicpes. Particidarly in the case of the b1chiiiIa.n tiinneling moclel

which assiunes the samples are in some sense cLirty, the generd cliialitative

agreement between the various techniques siiggests that this approach and the

cîssi~iptions contain in it are physiccdly reasonable [24]. This provicles tis with

a mat hemat i cdy well-clehed approach for cle terminhg the properties of t hin

film systems ~ising only the parameters containecl in the BCS theory of bulk

siiperconcluct ors.

(2). There are two different approaches to the single layer case that have

been discussed. The first is that the electron pair amplitude in the supercon-

ducting regions is spatially constant over the entire region. The second at-

tempts to give some spatial dependence to the gap parameter in a simple way.

We found that in this latter approach the critical thichess at T = O gets en-

hanced by a factor 5-50 over that of the former approach. The critical thickness

clepenclence of temperattire of this new mode1 was stuclied numerically and an-

dyticdy. The restilts show tliat near A/Ac=l, the critical temperattue T/Tc

approaclirs zero liuearly. Imt it clorsu't givr t h criticai tr~upc*ratiisc T as t h

I~iilk vnlw Tc when the stipcrcondiictor is hiik. This iiiiespectetl I~cli~vior

uiay intlirtztc t h the mctlioct is wlicl only for relat ively thin stipercon<liict-

ing b i s , oves which the posit ion-clepenclent pair tunpliti~cle cloes not van- too

greatly. so that the gap paranieter is approskiatecl reasonably by a coustant.

(3) The thickness ancl temperature clepeudence of the gap paraniet ers

were s tiidiecl niuiiericaiiy in the h i t e stiperlat tice system. The resdts show

the gap parmeters becomc hetter "behavecP' at finite temperature than that

nt zero temperattire. In principle, at a temperature T/Tc of aroiincl 0.8 the

gap paranieter shoiilcl stay constant for al1 layers, since the critical tempera-

tues approaclies the bi& vali1es for this region. which is tnie for the infinite

stiperlattice. One wotdcl thus expect the gap parameter to have the sanie

value for ail j. In fact. we could not extract this conclusion, since in high

temperatures the widt h of the superconducting layer is becoming larger, ancl

the approximation that the gap parameter is constant over the wiclth of the

layer is poorer.

From (2) and (3) above, the procedure outlined in the thesis indicates

that the method is reasonable for low temperatures (thin films) but not too

good for high temperatures ( thick films). This shows that the approximation

of a constant gap parameter is good only for thin layers, where the true gap

fiuiction doesn't have the "space" to vary too much , whereas for thicker layers

the tnie gap function can vasy rapiclly, and so assiuning it to be a constant

Appendix A

Int egrals

From chapter four, Ec1.(3.23), we have

where the ftinction H(x) is c1ehecI as

xcosx - sinx ~ ( x ) = + zSi(x) + 2Ci(x).

x

In this appendix we calculate the integral

We h t consider a general expression with parameter a.

(1). We have

(3). \Ne next have

Lct t ing

for which 2

ch = ~1.r ~inh-~(rn.r/2) u = - lii[tanh(.ni~/4)]. TT

(4). FVe finally consider

Thus, siuiuning the various terms, we have

where t = srzl4, we obtain

Bibliography

[I] H. Meissner, Phys. Rev. 117: 672 ( 1960).

[2] Smith, P. H.. S. Shapiro, J. L. Miles .mcl J . Nichal, Phys Rwktt. 6: 686

(1961)-

[3] Rose Innes, A. C. and B. Serin, P h p . Rev. 1: 278 ( 1961).

[-II L. N. Cooper, Phys. Reu. 6: 689 (1961).

[5] L.P. Gov'kov, Sou. Phys.-JETP 9, 1364 (1959)

[6] P. G. de Gennes and E. Guyon, Phys.Lett. 3: 168 (1963)

[7] N. R. \Verthamer, Phys. Rev. 132: 2440 (1963).

[8] P. G de Gennes, Rev. Md. Phys 36: 225 (1964).

[9] R. O. Zaitsev, Sou. Phys.- JETP 21, 426 (1965)

[IO] R. O. Zaitsev, Sou. Phys.-JETP 21, 1178 (1965)

[il] A. E. Jacobs, Phys. Rev. 162: 375 (1967).

[ E l P. G. clc Geunrs r u c l D. Saiut-.James. Piws. letf 4151 ( 1963).

[13] Et. 41. C l c - Phys. Rev. B5: 67 (1972).

[l4] G. B. . h o ~ c l . Phys. Rev. B. 18: LOT6 (1978).

[15] O. Entio - Woldman ancl T. Bar-Sagi, Phys. Rev. B 18: 3175 (1978).

[16] M. P. Zaitlin. Phgs. Reu. B 25: 5429 (1982).

[E] DS-Fcdk, Phys. Rev. 132: 15ïG (1963).

[18] W. L. McMiilan, Phys. Rev. 175: 559 (1968)

[19] -1. Bardeen, R. k~mmel, A. E. Jacobs, ancl L. Teworclt . Phys. Reu. 187.

556 (1969)

[20] A. F. Andreev, Sou. Phys -JETP 19, 1228 (1964)

[21] W. L. McMillan, Phys. Reu. 175, 537 (1968)

[22] J . Bardeen, Plys. Reu. Lett 6, 57 (1961)

(231 M. H. Cohen, L. M. Falicov, and J. C. Phillips, Phys. Rev. lett. 8 316

(1962)

[24] R. L. Kobes, J. P. Whitehead, Phys. Rev. B36: 121 (1987).

[25] H. Frohlich, Proc. R. Soc. London A 215: 291 (1952).

[26] Bardeen, L, N. Cooper, and J. R. Schrieffer, Phys. Rev. 29: 1175 (1957).

['LS] C. A Reyuobls. B. Seriu. W- H. Wright ,PIgs. Rev. 78: 487 ( 1950).

[-91 d. R. Schrieffer, - Tlieo7-y of s~~perconLictivity*- W. -4. Benjamin. Reading,

Massachiiset ts

[31] J . J . Hamer, C. H Tlietierer. .uicl N. R Pkrthmer, Phys. Rev. 142: 11s

(1966).

[32] R. P Groff ancl R. D Parb, Phgs. lett 22:19 (1966)

[33] P. G. Gnnes, and G. Scanna, J- Appl. Phys. 34, 1380 (1963).

[34] E. Guyon Superconcluctivity eclited by \Vallace 781 (1969)

[35] Hancibook of Mat hematical Functions, Natl. Bur. Stand. Appl. Math. Ser.

No.55, eclited by M-Abramowitz and 1. A. Stegun (U. S. GPo, tvashiag-

ton, D.CJ(l96.L).

[36] J. Wang Prozirnity effect and the themodynamic p~opert2es of Superlattice

System Master thesis 92

[37] A. L. Fetter, J. D. Walecka, Quantum Theory of Many Parlicle System

McGmw-Hill, Inc. (1971)

[38] R. L. Kobes, J. P. Whitehead and B. J. Yuan Phys lett A 132, 182 (1988).

[41] P. R. ;\r»il. .J. R. Ketterson. Proc. 15th Int. Conf. on Ion- tempcratiuiB

physics. .Iap71. J . Appt. Phys 26, Suppl. 26-3(198ï)