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THEORY OF SUPERCONDUCTIVITY

A Primer

Helmut Es hrig

Dire tor offorthe Institute for Solid State and Materersitals Researhnologyand Professor Solid State Physi s, Dresden Univ y of Te

These le ture notes are dedi ated, on the o assion of his eighties birthday on 4 July 2001,

to Musik Kaganov

who has done me the honor and delighpartof his friendship ver de ades and whom I we gr at of an attitude to ards Theoreti al Physi s.

Contents
1	INTRODUCTION
2	PHENOMENA, LONDON THEORY
		1		Phen	. .	. . . . . . . . . .
	2	2		Londomenatheory		. . . . . . . . . .
3	2	3		Gauge symmetry, London gauge
3	THE THERMODYNAMICS OF THE PHASE TRANSITION
		1			ergy .
		2		Fr e Enthalpy
	3	3		The the		i riti al eld
4	3	4		Heat aparmodynait jump
4	THE GINSBURG-LANDAU THEORY; TYPES OF SUPERCONDUCTORS
		1		La dau theory .
		3		Gin burg-Landau par meter
		2		intermed			equ tions
		1		intermed				ofboundaryMIXEDype s pe ondu tor
	4	4		phas boundary . . . . . . . .							. . . . . . . . .
5	4	5		energy of the phase							. . . . . . . . .
5	INTERMEDIATE STATE,										STATE
	5	3		ux line iate stateype			IIsuper ondu tor . . . .
	6	2		Mixed state of type II
6	6	2		The . . Josephson e e t
6	JOSEPHSON EFFECTS								, quantum interferen e
		1		d		ts			, quantum interferen e
7		1		Slater		ts
7	MICROSCOPIC THEORY: THE FOCK SPACE
		3		O upationdeterminanumber representation
	7	2		The F	k spa e
8	7	4		Field operators . . . . . . . . . . . . . . . . . .
8	MICROSCOPIC THEORY: THE BCS MODEL
	8	1			Fermi liquid as					quasi-parti le gas
	8	4		normalgoliubov-Valatin transformation
		2		Cooper problem .				. . . . . . . . . .
9		3			Hamiltoni			. . . . . . . . . .
9	MICROSCOPIC THEORY: PAIR STATES
		1		BCS ground state
		2		The pair fun tion . . .				. .
	9.3			Non-zero temperature				. .
10 MICROSCOPIC THEORY: COHERENCE FACTORS
			1		modynam state
			2	harge and spin moment densities
			3	Ultraso	i attenuation			. .		. . . . . . . . 3. . . . . . . . . . . . . . . . . . . . . . . . . .
	10.4			The spin sus eptibility				. .

1372

176

253

3278

375

420

512

553

INTRODUCTION

the phenom nologi al and

theo y of

r ondu tiv-

These le tu

intr du e in

ity.

Nowhere anynotespe i

le trto

the mi ros opi

ha ism of super ondu tivity is

although

few o asions

ioned as an example.

The

presup

ositions

are ex lusiv ly guided by

andmek ptqualitativeto minimum in order to a rivmadeat

presen

there are indassumptionphonona of -phenomenaphonontera tion

ivity,

yet

is no

rd proof up to now.

The whole of this treatise w hanismsply to any me

indi ated m

di ati

ns, for

anner

of the order parameter di er

t from isotroppossiblywhtheoreti th

resultsha been h

for thesimplesak f simpli it .

readily

phen

reasonablyogi al approa h the te hni al presentation is standard throughout, so that

prime . For

hanistreated. Referen es

This is

onsidered phenomenon,ouldly the simplest as

are giv

i al

to the most

importansymmetryseminal

riginal

paperssuper. Despitondu the abo

mention

to the neededmenovel before they are

ed. Basi notions of Q

antum Theory

introdu ed

van ed

tools

eld quantization and the quasi-parti le on ept

properChaptery of the quantum ground state. Thermoelthe tri s, ele trodynami s andsupergaugeonduopertiestivitThermodynamiars

strimpat

the existing li eratu .

aftertheoretishortal

essential

h nomena of

enum r

leading tothe

of all

introduto yp ondensI II. The

phasethermodynamiagram of

London

he ryStatistiderivedal from

sole assumption that the super urren

ele tri al urr

(as well as

Physi s in a few o asions) are presu

osed as known.

With

help of simple

thermo

ynami

lations, the

ion

nergy, the

riti al

eld and the spe i heat are onsidere

in Chapter 3.

dis ussed.

for spatially inhomogeneous s tuations,

In Chapt r 4, the Ginsburg-Landau theory is

The

lassifourationpters aresuperdevotedondu torsthe simpl st p

enomen logsimplestal weak

Josephson e e ts

re qualitatively onsi

ered

the basis

G nsburg-Landau theory in

ype II

is obtained in Chapter 5.

Chaptisotropi6. Both,

. . and a. . e e ts are treate .

f the phenomupling theoryn. F

understanding of

super ondutivity,tor

46 years after the exp rimental dis overy

of supe

ondu tivity

mi ros opi lev l, the BCS th ory whi h gavthehe rst

tum the reti al

this purpose,remainingChapter 7

Fo k spa e and the

quantizati

is introdu d. Then,

The

ture of thesuperhargedondubosoni ondeinsate, phenomenon eptlogi ally introdu d in

Chapter 2, is de ived

Coop

and the

BCS model are treat ldwith o

upationquanumber operators

of quasi-parti le states whitheoremlatter are

trodu ed

w rking appro

in Solid State

Physi s.

here the essential

sult. The treatise is losed withas onsideration of b

xamples oftemperaturethe impo tant

Chapter

9 as

ondensate of

pairs. The ex itation gap ximationfun tion of

uperpeausingulationsondu tivitpossible

auses of

sup r ondu tivity

shouladditionalprovidetheoristsnew ommer to thetroduele

notion of oherenthe

fa tors.

symmetry, real stru ture feastures of

olid (for

pe ifying more details as lowCooperpoin

pinning

f vort x lines) and many more, a lot

onsideratio instanw ld

be possible

without

sp ifying

mi ro opi me hanism of the at ra tiv

inthera tion

However, thes

are

just

the

one-termtheoretiw -hoursal

le ture

toleading

to the spirit of this kind of theortheti al approa h,notes only

ressing

. In our days of liv

(again not just the theorist) with a safe ground to starit out.

1964,

and R. D. Parks (ed.), Super ondu tivity, vol. I and IIhrieDekker,er,

New York, 1969.

Two lassi s are re ommended f r more details: J. R. S

Theory of Super ondu tivity, Benjamin, New York,

Figure 1: Resistan e in ohms of a spe imen of mer ury

ersus absolute temperature. This plot by

Kamerlingh Onnes marked the dis overy of super ondu tivity. (Taken from: Ch. Kittel, Introdu tion

to Solid State Physi s

Wiley, New York, 1986, Chap. 12.)

PHENOMENA, LONDON THEORY

. By exhausting the

vapor

was st lique ed by Kammerling Onnes at L

den in

Heliumabov the liquid the temperature ould soon

ere down 1908to.5K.

that in purehelium r ury

Shortly aft rwards,

1911, it was f und

laborato y

the ele tri al res stan e disappeared abruptly

elow

riti al temperature,

= . .

the phe-

Deliberately

ron s att ringbey mak

the

impure did not

. Shortly th

eaftthe

he same e e t was

found in indium (3.4K), t

(3.72K)a e t

in lead

in the A15 ompoundsn reasingw found nd igher T -values appeared up

T = 23.2K in Nb3Ge, in 1973.

(7.19K). In 1930,

uper ondutivity was fou

in niobium (T

= 9.2K) and

1940

in the metalli

GeTe, S Tmaterials(T 0.1K,Cohen 1021 m 3) and

SrTiO3

.38K at n

1021

m 3

, ertaT

ompouenond NbN (T = 17.3K)r, and this remai

ed the highest

T until the 50's, when super ondu tivity

These

were all orm l metals

more or

good ondu tors.

.1K for

In 1964, Marvin L.

made theoreti al

predi tiless

of T -values as high as

fermion systems" lik superUBe13ondu(T tivit1K in both ases). The

ear 2000superN belondupri etivitin Chemistry was

doped

and in the same year

the foll

wingmerears,ury

y was found in

0.1K

semi ondu10 tors,m ).

y was f und

everal

ondu ting lyme s as w

l as in \heavy

he early 80's,

dedi ated

tho the

predi tion and realization of ondu ting polymers (syntheti metals) in the late

70's.

Commun.

Phys. Lab. Univ. Leiden, No124 (1911); H. K. Onnes, Akad. van Wetens happen

(Amsterdam)Onnes,14 818 (1911).

September, 1993

(under pressure)

160

Liquid

[

August, 1993

CF4

Hg−1223

(under pressure)

140

April, 1993

Tl−2223 [

September, 1992

February, 1988

120

Bi−2223

January, 1988

(K)c 100

Shuttle

Y−123

January, 1987

Liquid

December, 1986

(La−Ba)−214 [

(under pressure)

December, 1986

(under p essure)

Liquid

Nb3Sn

Nb3Ge

pril, 1986

NbN

NbC

Nb−

l−Ge

onin:FigurePhysiB.Batlogg,2:s,TheMaterialsevolutionC.W.andChu,of

Liquid

AppliTWwith.Kations,. ChutimeDWorld(from.U.GubserCS.ientiW. Chu,and,Singapore,KSuper. A. M•ullerondu1996(La,Sr)tivity(eds. . .AbovePro .90KHTSandWorkshopBeyond

V3Si

CuO with T

= 36K,

In 1986, Georg Bednorz and Alex M•uller found super ondu tivity in

an in redible new

1900

1920

1940

1960

1980

2000

Within months,reTord-values in uprates were shooting up, and the re ord is now at T 135K.

YEAR

2.1

Phenomena

Z ro

even for high s attering rates of ondu tion ele trons.

P rsistenresistanurrentse

magn ally indudeteed intableoil of Nb

and wat hed with NMR yielded n

estimate

of the de ay

time stangreater than 10 years!

(From theoreti al estimates the de ay time may

be as large as 101010

ars!)

No S ebe k voltage, no Peltier heat, no Thomson heat is

Absen e

of th rmoele tri e e ts3

dete table

0:75 0:25

= 1. Weak magneti elds are omplet ly s reened w y from the bulk

Id al diamagnetism m

(see next se tion).

the presen e of a weak magneti eld, below

Meissn r e e t

If a super ondu tor is ooled down

ondu tor.

Flux quan

The magneti ux through a super ondu ting

is quantized and onstan

he ld

ompletely expelled from

he bulk of the super ondu tor.

time.

Thistizationphe omenon was theoreti ally predi ted by F. Londonring 1950 and experimentally

veri ed 1961.

Bednorz and K. A. M•uller, Z. Phys. B64 1

86)33 .

5W. Meissn

R. O hsenfeld, Naturwiss. 21,927)(1963)87

3J.

File and R. G. Mills, Phys. Rev. Lett. 10, 93

B. S.

, Z. Ges

K•alteindustrie 34, 197 (

Deaver and

W. M. Fairbank, Phys. Rev. Lett. 7, 43 (1961); R. Doll and M. N•abauer, Phys. Rev. Lett. 7,

(1961).

2.2

London theory

that the super

rren

(at T = 0) is a property of the quantum

ere

(a) and (b) learly

ust be an ele triindiallyateh rged (

quantum

q), hen e omplex bosoni eld

Phenomenaw i o denses in the ground state into hargema ros opi amplitude:

ground state:

is the orresponding eld amplitude.

where

means the bosoni density and

(1)

= j

;

Sin e the eld is ele tri ally harged, it is subje t to

elds (E; B) whi h are usually

ele tromagneti

des ribed by potentials (U; A):

E =

;

(2ba)

The eld amplitude should obey

B =

S h •odinger equation

+ qU

E B

;

(3)

hemi al potential

of the boson eld, sin e what

where the energy is reasonably measured from the

is measured in a voltmeter is rather the ele tro hemi al potential

(4)

than the external potential U, or the

+ qU

e ele tri eld

e e tiv

(5)

As usual in Quan

Me hani s, i~ = r is the anoni al momentum and ( i~ = r qA) = p^

the me hani al

tum.

The

momen

tumdensity is then

iq~

super urren

= q

(6)

term)onst

;

urren( ;t') =

It o

( rst

and

sists as usual of a ` a amagneti

`diamagneti urrent' (se ond term).

In a homogeneous super ondu tor, whe

., we may write

i (r;t)

and have

js = q r A;

(8)

= n q2 :

Sin e in the ground state E = , and E

= i~

t, we alsoBhave

(9)

The

theory

limit, where nB

erives from (8) and (9). It is valid in the London

= onst. in

F.Londonan

Pro . Roy. So .

A149, 71 (1935); F. Lo

don, Pro . Roy. So .

A152 24 (1935); F.

London, Super andassumeui , WileyLondon,

1950.

spa e

These are formal ames: sin e the splitting

into the

t ontributio

s depends on the gauge, it has o

de per phy al meaning. PhysiLondon,ally

positivurrenresponse on an external magneti eld (enhan ing

the eld inside the material) and diamaramagnetimeansmeanegative response.

The time derivative of (8) yields with (9)

;

= Ee

(10)

This is the st

don

freely

t s

an applied v

or, in a bulk

A super urr

with no sup

r urrenquationor awitheleratedsttionary

super urrenoltage,there

is no e e tivsuperele ondutri tore

( on

ele tro hemi al potential).

London

uses

ele tro

mi hemia potentialal

ket hy illustrated

Fig. 3. The r

The rst London equation

abs n e of

if the

rmoe

ials of

yields, and of

very

, , are e upledets, . The

e e ts

tare

up r urren

arrying ld to be

onstan

ary situati

. Ifelehetrosup r u

potenar-

rying

ondu tionwith

he ondu ti

ele tronsuperldthermoeleurrenequationwi tri

forming

quantum with

into super urrenrea ts density at

onethermoeleend of

trisamplerelatedand

ba k

temperaturehe other end.

q then the eleele trotrons,h mi al

tentials

ust be

eleas ntrons= . Hen e the ele tro h mi al

pot

tial of the

ele trons

ust also be

no thermopower (Seebeldk v ltage) may

ombineratures,tion of

w metals. If there is no tem erature

transformationthe beginning, but a ur

devhargelop in

super ondu tor. The

t, with

urren

owing due to the

di eren e

n eled by

onduk tionwi g

uous

then

di ere

super urren dev

onstangetherwithtransformadi eren e of

tro

loop of

normal

is f rmed

the jun

keptondua

ele trons-

potenti

ls of the

jun tions, and several forelopsof heat are

everything depdindingeren

maintai

betweenthermoelejun tionstri

develondu tors. This is

how

produPelti metaled,oo

works. In

loop oftemwo

thering (by inserting

pow

supply

to one of th

halfs),

then

tem

hemiural

di eren edu tors

on of

those phenomena

an appear sin die aeren e

n e of el tr

tentials

jun tions be ause of

presen of

supertheonduothertorside. This yieldshemidirethermoeleal t absoluttri

annot be main

ed. Every normawill urren

hortir uit

by super urrents.

superrreno will ow in

normal half without

developing

an el tro hemi al potential di eren e of

If, ho

super ondu tor

B in a loop,

ever, tai ormal metal A is ombined withally

measurement of thethermoele tri theoÆ ients of a single material A.

The url of Eq. (8) yields (with

= 0)

= B:

(11)

This is the se

d London

. It

yields the

ideal

iamagnetism, the Meissner e e t, and the

Take the url of Maxwell'sequation (Ampere's law) and

B =

ux quantization.

B = 0

+ j

;

onsider= 0;

(12)

js + j

;

B =8

B 0

normal ondu tor

super ondu tor

sup r ondu tor

2;el

n 1;el =

thermoele tri urrent

n 2;el

1;el

normal ondu tor

ne ! q

super urrent j

q !

Seebe k voltage

Sin e j( ) = onst.

fT1 ( 1;el)

(

)

js(t) = onst.

2;el

Ee = =q = 0; = 0

The

due to the rst London equation.

e of the Fer distribution

a di eren e of el tro hemi al potentials

fun ti

in onne tion with a

nononstanT

nsity of states results in

due to the

detailed balan e of urrents.

loop of two metals A and B:

no total urrent

the

m's heat +

tri

Thomson heat

no heat

urrmoeleen J

A( T; J)

Peltier heat

heat +

BA(T1; J)

(T2

; J)

ThoOhm'sm on heat

( T; J)

ondu tor

normal

super

Figure 3: Thermoele tri phenomena in9normal ondu tors and super ondu tors.

= ne

js z= L

external eld B0

super ondu

If j = 0 or

Figure 4: Penetration of an external magneti eld into a super ondu tor.

j = 0 for the

urrent inside the supe

then

B =

B ; L = s

= r

normal

onduqtor,

with solutions

B = B

(14)

e n = L ;

n2 = 1;

= 0

several of whi h with

v tors

y be

onditions.

depth.

imposed to ful ll

L is London's

penetration

o zero insidemabulk

state withinboundarys

rfa e layer

Any

eld B is s reened

of thi kness

. It is

ppropriatethaunit(11) does not

tain time derivatives of the eld but the eld

B itself: If

importanapplied eld B

is ooled down below T , the eld is expelled.

metal

Considerexternal

super ondu ting

with magneti ux pass-

ringthrough it (Fig. 5). B ause of

(14) and (12), j

= 0 de

the ring on the ontour C. Hen ,

(10),

insidether .

froF m Faraday's law,super( =ondu) Eting=

BdS = I

E l = 0;

(15)

where A is a surfa e with

bound-

is the magn

Figure 5: Flux through a super ondu ting ring.

th ough A.

andif the

arysu fa e layer of the ring is ti

ing with time (for instan e, if

applied magnetisupereld urrenis hanging wi h time), the ux is not:

The ux through

super ondu ing ring is trapped.

Integrate Eq. (8) along the ontour

dl =

~ I

dl:

A + j

C r

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