Relate d Publication s fro m Pergamo n
D O N G L U SHI :
H i g h Temperatur e Superconductin g Material s Scienc e 8c Engineerin g (forthcoming )
EVETTS :
Concis e Encyclopedi a o f M a g n e t i c &, Superconductin g Material s
MAHAJA N & KIMERLING :
Concis e Encyclopedi a o f S e m i c o n d u c t i n g Material s 6c Relate d T e c h n o l o g i e s
BIANCON I &MARCELLI :
H i g h T c Superconductor s
K R O T O , C O X Sc F I S C H E R :
T h e Fullerene s
Relate d Journal s
Applie d Superconductivit y
Solid Stat e C o m m u n i c a t i o n s
P A T T E R N O F INDIVIDUA L F L U X O N S I N A T Y P E - I I
SUPERCONDUCTO R
T h i s photograp h show s th e triangula r patter n of fluxons in a type-I I superconducto r
(see Chapte r 12). T h e patter n is reveale d b y allowin g ver y smal l (500 A) ferromagneti c
particle s t o settl e on th e surfac e of a magnetize d specime n (lead—indiu m alloy). T h e
particle s locat e themselve s wher e th e magneti c flux intersect s th e surface . Th e photo grap h wa s obtaine d b y electro n microscop y of th e deposite d particles . (Photograp h b y
courtes y of V. Essman n an d H . Trauble , Ma x Plan k Institu t fu r Metallforschung. )
INTRODUCTIO N TO
SUPERCONDUCTIVIT Y
BY
A. C.
ROSE-INNES
Professor of Physics and Electrical
Engineering
AND
Å. H .
RHODERICK
Professor of Solid-State
Electronics
University of Manchester Institute of Science and
U.K.
SECON D
Technology
EDITIO N
PERGAMON
U.K.
U.S.A.
JAPAN
Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington,
Oxford OX5 1GB, England
Elsevier Science Inc., 660 White Plains Road,
Tarrytown, NY 10591-5153, U.S.A.
Elsevier Science Japan, Tsunashima Building Annex,
3-20-12 Yushima, Bunkyo-ku,
Tokyo 113, Japan
Copyright © 1978 Pergamon Press pic
The cover illustration is reproduced with permission from J. I. Castro
and A. Lopez, Symmetries of Superconductor Micronetworks, Solid
State Commun.
82, 787 (1992).
All Rights Reserved. No part of this publication may be reproduced, stored
in a retrieval system, or transmitted, in any form or by any means: electronic,
electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise,
without permission in writingfrom the publishers.
First edition 1969
Reprinted with corrections 1976
Second edition 1978
Reprinted 1980,1986,1988 (twice)
Reprinted with corrections 1994
Librar y of Congres s Catalogin g in Publicatio n Dat a
Rose-Innes, Alistair Christopher.
Introduction to superconductivity.
(International series in solid state physics; v.6)
Includes index
I. Superconductivity. I. Rhoderick, Å. H., joint author.
II. Title.
QC612.S8R6
1977
573.6'23
77-4811
ISBN 0-08-021651-X Hardcover
ISBN 0-08-021652-8 Flexicover
Printed in Great Britain by BPC Wheatons, Exeter
PREFACE
TO
THE
SECOND
EDITION
edition differs from the first edition chiefly in C h a p t e r 1 1 , w h i c h
has b e e n almost completely rewritten to give a m o r e physically-based
p i c t u r e of t h e effects a r i s i n g from t h e l o n g - r a n g e c o h e r e n c e o f t h e
electron-waves in superconductors and the operation of q u a n t u m interference devices. W e are very grateful to D r . J. Lowell for reading a n d
c o m m e n t i n g on t h e draft of this rewritten chapter. T h e r e are a n u m b e r
of other relatively m i n o r changes t h r o u g h o u t the b o o k w h i c h we h o p e
will improve the presentation.
THI S
W e m u s t also t h a n k D r Lucjan Sniadower for pointing out a n u m b e r
of errors, M i s s D o r o t h y D e n t o n for typing t h e manuscript of this second
edition, and Nicholas R o s e - I n n e s for help in checking t h e manuscript
and proofs.
Uç iversity ofManchester
Institute of Science and Technology
December 1976
NOTE
ON T H E
Å. Ç . R.
A . C . R. - I .
REVISED
SECOND
EDITION
IN this revised second edition we have m a d e some further modifications
to the text and added a short extra chapter dealing w i t h ' h i g h - t e m p e r a ture' superconductors. A vast a m o u n t of research has been carried out on
these since their discovery in 1986 but the results, b o t h theoretical and
experimental, have often been contradictory, and seven years later there
remains little u n d e r s t a n d i n g of their behaviour. W e have therefore been
very selective and have included only material w h i c h we are confident
will have p e r m a n e n t validity.
W e are grateful to D r Geoffrey Ross for c o m m e n t s o n t h e draft of
C h a p t e r 14.
E.H.R.
A.C.R.-I.
May 1993
ix
PREFACE
TO
THE
SECOND
EDITION
edition differs from the first edition chiefly in C h a p t e r 1 1 , w h i c h
has b e e n almost completely rewritten to give a m o r e physically-based
p i c t u r e of t h e effects a r i s i n g from t h e l o n g - r a n g e c o h e r e n c e o f t h e
electron-waves in superconductors and the operation of q u a n t u m interference devices. W e are very grateful to D r . J. Lowell for reading a n d
c o m m e n t i n g on t h e draft of this rewritten chapter. T h e r e are a n u m b e r
of other relatively m i n o r changes t h r o u g h o u t the b o o k w h i c h we h o p e
will improve the presentation.
THI S
W e m u s t also t h a n k D r Lucjan Sniadower for pointing out a n u m b e r
of errors, M i s s D o r o t h y D e n t o n for typing t h e manuscript of this second
edition, and Nicholas R o s e - I n n e s for help in checking t h e manuscript
and proofs.
Uç iversity ofManchester
Institute of Science and Technology
December 1976
NOTE
ON T H E
Å. Ç . R.
A . C . R. - I .
REVISED
SECOND
EDITION
IN this revised second edition we have m a d e some further modifications
to the text and added a short extra chapter dealing w i t h ' h i g h - t e m p e r a ture' superconductors. A vast a m o u n t of research has been carried out on
these since their discovery in 1986 but the results, b o t h theoretical and
experimental, have often been contradictory, and seven years later there
remains little u n d e r s t a n d i n g of their behaviour. W e have therefore been
very selective and have included only material w h i c h we are confident
will have p e r m a n e n t validity.
W e are grateful to D r Geoffrey Ross for c o m m e n t s o n t h e draft of
C h a p t e r 14.
E.H.R.
A.C.R.-I.
May 1993
ix
PREFACE
TO
THE
FIRST
EDITION
THIS book is based to a large extent on lectures w e h a v e given to
u n d e r g r a d u a t e and first-year p o s t g r a d u a t e students. W e intend t h a t it
should indeed b e an introduction
to superconductivity and have selected
and presented our material w i t h this in mind. W e d o not intend the cont e n t s t o b e read as a definitive and exhaustive t r e a t m e n t of superconductivity; several such texts are already published and it h a s not been our
intention to c o m p e t e w i t h these. R a t h e r our object h a s been to explain as
clearly as possible the basic p h e n o m e n a and concepts of superconductivity in a m a n n e r which will be understood by those w i t h n o previous
knowledge of superconductivity and only a modest acquaintance w i t h
solid-state physics.
I n this book w e have concentrated on t h e physics of superconductivity
and, t h o u g h w e occasionally m e n t i o n applications, w e h a v e not treated
these in any detail. W e hope, nevertheless, t h a t this book will be useful
b o t h t o " p u r e " p h y s i c i s t s a n d t o t h o s e i n t e r e s t e d in p r a c t i c a l
applications. W i t h this in mind, w e have used the rationalized M K S
system t h r o u g h o u t . M a n y of those interested in practical applications
will be engineers b r o u g h t u p on the M K S system and, furthermore,
M K S u n i t s are increasingly used in physics teaching. It seemed to u s
that someone should take the plunge and w r i t e a book o n superconductivity using t h e M K S system. T h e M K S approach h a s involved us in
some t h o u g h t about the meaning of  and Ç in superconductors, a point
which is discussed in Appendix A.
W e have, of course, d r a w n on other texts on the subject and, in particular, w e have m a d e frequent use of Shoenberg's classic m o n o g r a p h
Superconductivity.
Because our book is an introduction, w e have not
a t t e m p t e d t o include a complete list of references, b u t have referred t o
some of the key p a p e r s where this seemed appropriate.
W e are grateful to Professor G. Rickayzen and to M r . Ê . E. O s b o r n e
for discussing the c o n t e n t s of C h a p t e r 9 and 11, and to those a u t h o r s
w h o allowed u s to use the original copies of figures from their
xi
xii
PREFAC E
publications. W e m u s t also t h a n k M r s . Shirley Breen, w h o t y p e d t h e
manuscript and w h o , like M a x w e l l ' s d a e m o n , m a d e o r d e r w h e r e n o n e
w a s before. O u r w i v e s and children deserve g r a t i t u d e for their
forebearance d u r i n g t h e t i m e t h i s b o o k w a s being w r i t t e n .
University of Manchester
Institute of Science and Technology
July 1968
E.H.R.
A.C.R.-I.
SYMBOLS
THROUGHOUT this book there is employed, as far as possible, a
consistent set of symbols. T h e following list includes definitions of m o s t
of t h e symbols used.
a
"i
A
si
Â
B
0
B
€
c
c
n
C
s
Qat t
d
e
Å
E
gn
gs
G
G
g
n
s
ç
hi
Ç
H
H
a
t
H
half-thickness of a slab
coefficient of wavefunction 0(p /t, —p/ 0 in pair wavefunction
magnetic vector potential, defined b y  = curl A
area
magnetic flux density
flux density of applied magnetic field (§ 2.1)
critical flux density ( = ìïÇ )
specific heat of normal phase
specific heat of superconducting phase
electronic contribution to specific heat
lattice contribution t o specific heat
thickness of a plate
electronic charge
energy or electric field strength
energy gap of superconductor ( = 2Ä)
G i b b s free energy p e r unit volume of normal phase
G i b b s free energy per unit volume of superconducting phase
G i b b s free energy of specimen in normal state
G i b b s free energy of specimen in superconducting state
Planck's constant -5- 2ð
probability t h a t pair state (p /t, —p/i) is occupied in B C S
g r o u n d state
magnetic field strength
applied magnetic field strength (see p. 16)
field strength inside specimen or field strength d u e to transport
current
t h e r m o d y n a m i c critical magnetic field strength
c
xiii
XIV
H'
H
H
H
i
i
i
J
J
j
J
SYMBOLS
enhanced critical m a g n e t i c field of a t h i n specimen
critical m a g n e t i c field s t r e n g t h at 0°K
lower critical field of t y p e - I I s u p e r c o n d u c t o r
u p p e r critical field of t y p e - I I s u p e r c o n d u c t o r
current
critical c u r r e n t
supercurrent, c u r r e n t of electron-pairs
magnetization ( m a g n e t i c m o m e n t per unit volume) or c u r r e n t
current or c u r r e n t per unit w i d t h
surface current density per u n i t length
volume current d e n s i t y per unit area ( = m a g n i t u d e of c u r r e n t
density vector J )
J
critical c u r r e n t density
J
current density d u e t o n o r m a l electrons
c u r r e n t density d u e to superconducting electrons
J
k
B o l t z m a n n ' s c o n s t a n t or effective susceptibility of plate [eqn.
(8.4)]
l
electron m e a n free-path in n o r m a l state
L
self-inductance or latent h e a t
m
electronic m a s s or n u m b e r of t u r n s / u n i t length of solenoid
Ì
m u t u a l i n d u c t a n c e or total m a g n e t i c m o m e n t of specimen
ç
demagnetizing factor [defined b y eqn. (6.2)] or any integer
n
density of superelectrons
J (å)
density of states for electrons w i t h kinetic energy å
p
m o m e n t u m of an electron or p r e s s u r e
p
F e r m i m o m e n t u m [ = ^(2ôçåñ)]
Ñ
total m o m e n t u m of a C o o p e r pair
q
electric charge or m o m e n t u m of a p h o n o n
R
resistance of specimen in n o r m a l state
R'
flow
resistance of a t y p e - I I s u p e r c o n d u c t o r in t h e mixed state
s
e n t r o p y density or velocity of sound
transition t e m p e r a t u r e in zero field (or critical t e m p e r a t u r e )
T
u
internal energy per unit v o l u m e
v
velocity of superelectrons
V
v o l u m e or m a t r i x element of scattering interaction or voltage
difference
W
kinetic energy
x
thickness of n o r m a l lamina in i n t e r m e d i a t e state
x
thickness of superconducting lamina in i n t e r m e d i a t e state
( C h a p . 3)
a
surface energy per unit area or a p a r a m e t e r = ôç/ì ç â
c
0
cl
c2
c
s
c
n
s
e
s
r
F
n
c
s
n
s
2
¼
5
XV
SYMBOL S
ã
Ä
Sommerfield specific heat constant [eqns. (5.6) and (9.11)]
surface energy p a r a m e t e r defined b y á = %ì ÇÀÄ or a
p a r a m e t e r defined b y Ä = 2hv exp [— [Ë {e )V)~\
equal to
half t h e energy g a p
Ä(0)
half energy gap at 0°K
å
kinetic energy of an electron ( =
p /2m)
e
F e r m i energy
C
attenuation length for tunnelling w a v e function
ç
fraction of b o d y in normal state ( = xj{x
+ x )) or viscosity
coefficient for flux flow (Chap. 13)
î
coherence length (§ 6.9)
coherence length in pure superconductor
î
è
angle or D e b y e t e m p e r a t u r e
ê
Ginzburg—Landau p a r a m e t e r (§ 12.3)
ë
penetration d e p t h [defined b y eqn. (2.2)] or wavelength
ë
penetration d e p t h at 0°K
L o n d o n penetration d e p t h [defined by eqn. (3.13)]
X
ì
permeability of free space (4ð ÷ 10~ H m " )
relative permeability
ì
í
frequency
v
threshold frequency for onset of absorption of electromagnetic
radiation
p h o n o n frequency
v
v
an " a v e r a g e " p h o n o n frequency
ñ
electrical resistivity
p'
flow
resistivity in mixed state
Po
resistivity at 0°K
ó
electrical conductivity
÷
magnetic susceptibility
jy, z, p ) ) one-electron wavefunction for non-interacting electron of
V(p)
' momentum ñ
ö
phase
Áö
phase difference
0
Ã
l
L
F
2
F
n
s
0
0
L
7
1
0
Ã
Q
q
L
o r
\
*i> Pi, tf ,jy , * 2 > p 2 ) two-electron wavefunction for non-inter0(pi» P 2 )
J acting electrons w i t h m o m e n t a pj and p
Ö
magnetic flux
Ö'
fluxoid
[eqn. (11.5)]
fluxon[eqn.
(11.8)]
Ö
0(^1^1 »
2
2
o r
0
Ö ß ^ õ ^ é , ^ é » ^ » ^ » ^) \ two-electron wavefunction for a pair of interor <S>(r r )
J acting electrons, defined b y eqn. (9.4)
lf
2
2
xvi
SYMBOLS
Ö
wavefunction of C o o p e r pair w i t h total m o m e n t u m Ñ
Ø
G i n z b u r g - L a n d a u effective wavefunction (§ 8.5)
^c?( i> 2> · · ·> n)
many-electron
wavefunction
describing
BCS
g r o u n d state [eqn. (9.6)] and formally identical
with the Ginzburg-Landau Ø
electron-pair w a v e function
Ø
Ñ
r
ñ
r
r
INTRODUCTION
SUPERCONDUCTIVIT Y is the n a m e given to a remarkable combination
of electric and magnetic properties which appears in certain metals w h e n
they are cooled to extremely low temperatures. Such very low
t e m p e r a t u r e s first became available in 1908 w h e n K a m e r l i n g h O n n e s at
the University of Leiden succeeded in liquefying helium, and b y its use
w a s able to obtain t e m p e r a t u r e s d o w n to about 1°K.
O n e of the first investigations which O n n e s carried out in the newly
available low-temperature range w a s a study of the variation of the electrical resistance of metals with temperature. It had been k n o w n for m a n y
years that the resistance of metals falls w h e n they are cooled below room
temperature, b u t it w a s not k n o w n w h a t limiting value the resistance
would approach if the t e m p e r a t u r e were reduced t o w a r d s 0°K. O n n e s ,
experimenting with platinum, found that, when cooled, its resistance fell
to a low value which depended on the purity of the specimen. At that
time the purest available metal w a s mercury and, in an a t t e m p t to discover the behaviour of a very pure metal, O n n e s measured the resistance
of pure mercury. H e found t h a t at very low t e m p e r a t u r e s the resistance
became immeasurably small, which w a s not surprising, b u t he soon discovered (1911) t h a t the m a n n e r in which the resistance disappeared w a s
completely unexpected. Instead of the resistance falling smoothly as the
t e m p e r a t u r e w a s reduced t o w a r d s 0°K, the resistance fell sharply at
about 4°K, and below this t e m p e r a t u r e the mercury exhibited n o
resistance whatsoever. F u r t h e r m o r e , this sudden transition to a state of
no resistance w a s not confined to the pure metal b u t occurred even if the
mercury w a s q u i t e impure. O n n e s recognized t h a t below 4°K mercury
passes into a new state with electrical properties quite unlike those
previously known, and this new state w a s called the "superconducting
state".
It w a s later discovered that superconductivity could be destroyed (i.e.
electrical resistance restored) if a sufficiently strong magnetic field were
applied, and subsequently it w a s found t h a t a metal in the superconducxvii
xviii
INTRODUCTION
ting state has very extraordinary magnetic properties, quite unlike those
k n o w n at ordinary temperatures.
U p to t h e present t i m e about half of t h e metallic elements a n d also
a n u m b e r o f alloys h a v e b e e n f o u n d t o b e c o m e s u p e r c o n d u c t i n g a t
low temperatures, t h a t is to say below about 25°K. Recently (1987), h o w ever, it has been discovered t h a t s o m e ceramic metallic oxides b e c o m e
superconducting at m u c h h i g h e r temperatures, i.e. at a b o u t 100°K (see
C h a p t e r 14).
T h o s e materials w h i c h exhibit s u p e r c o n d u c t i v i t y w h e n sufficiently
cooled are called superconductors. For m a n y years it was t h o u g h t t h a t
all s u p e r c o n d u c t o r s b e h a v e d a c c o r d i n g t o a basically similar p a t t e r n .
However, it is n o w realized t h a t there are t w o kinds of superconductor,
w h i c h are k n o w n as t y p e - I a n d t y p e - I I . M o s t of those elements w h i c h
are s u p e r c o n d u c t o r s e x h i b i t t y p e - I s u p e r c o n d u c t i v i t y , w h e r e a s alloys
generally exhibit t y p e - I I superconductivity. T h e t w o types have m a n y
properties in c o m m o n b u t show considerable differences in their magnetic
behaviour. T h e s e differences are sufficient for us to treat t h e t w o types
separately. T h e first part of this b o o k deals w i t h t y p e - I superconductors
and t h e second part w i t h t y p e - I I superconductors.
CHAPTER 1
ZERO
RESISTANCE
T H E ELECTRICAL resistivity of all metals and alloys decreases w h e n
they are cooled. T o u n d e r s t a n d w h y this should be, w e m u s t consider w h a t
causes a c o n d u c t o r to have resistance. T h e current in a conductor is
carried b y " c o n d u c t i o n electrons" w h i c h are free t o m o v e t h r o u g h the
material. Electrons have, of course, a wave-like n a t u r e , and an electron
travelling t h r o u g h a metal c a n b e represented b y a plane w a v e
progressing in t h e same direction. A metal h a s a crystalline structure
w i t h t h e a t o m s lying on a regular repetitive lattice, and it is a property of
a plane w a v e t h a t it can p a s s t h r o u g h a perfectly periodic structure
w i t h o u t being scattered into other directions. H e n c e an electron is able
t o p a s s t h r o u g h a perfect crystal w i t h o u t any loss of m o m e n t u m in its
original direction. I n other w o r d s , if in a perfect crystal w e start a
current flowing (which is equivalent t o giving t h e conduction electrons a
net m o m e n t u m in t h e direction of the current) t h e current will
experience n o resistance. H o w e v e r , any fault in t h e periodicity of t h e
crystal will scatter t h e electron w a v e and introduce s o m e resistance.
T h e r e are t w o effects which can spoil t h e perfect periodicity of a crystal
lattice and so introduce resistance. At t e m p e r a t u r e s above absolute zero
the a t o m s are vibrating and will b e displaced b y various a m o u n t s from
their equilibrium positions; furthermore, foreign a t o m s or other defects
randomly distributed can interrupt t h e perfect periodicity. B o t h t h e thermal vibrations and any impurities or imperfections scatter the moving
conduction electrons and give rise t o electrical resistance.
W e can n o w see w h y t h e electrical resistivity decreases w h e n a metal
or alloy is cooled. W h e n t h e t e m p e r a t u r e is lowered, t h e thermal
vibrations of t h e a t o m s decrease and t h e conduction electrons are less
frequently scattered. T h e decrease of resistance is linear d o w n to a
t e m p e r a t u r e equal to about one-third of the characteristic D e b y e
t e m p e r a t u r e of t h e material, b u t below t h i s the resistance decreases
progressively less rapidly as t h e t e m p e r a t u r e falls (Fig. 1.1). F o r a
perfectly pure metal, w h e r e t h e electron m o t i o n is impeded only b y t h e
3
4
INTRODUCTION
TO
SUPERCONDUCTIVITY
t h e r m a l v i b r a t i o n s of t h e lattice, t h e resistivity should a p p r o a c h zero a s
the t e m p e r a t u r e is reduced t o w a r d s 0°K. T h i s zero resistance w h i c h a
hypothetical " p e r f e c t " specimen would acquire if it could b e cooled t o
absolute zero, is not, however, t h e p h e n o m e n o n of superconductivity.
Any real specimen of metal c a n n o t b e perfectly p u r e and will c o n t a i n
some impurities. T h e r e f o r e t h e electrons, in addition t o being scattered
by t h e r m a l v i b r a t i o n s of t h e lattice a t o m s , are scattered b y t h e i m purities, and t h i s i m p u r i t y scattering is m o r e or less i n d e p e n d e n t of
t e m p e r a t u r e . A s a result, there is a certain "residual resistivity" (p F i g .
1.1) w h i c h r e m a i n s even at t h e lowest t e m p e r a t u r e s . T h e m o r e i m p u r e
t h e metal, t h e larger will b e its residual resistivity.
09
Temperatur e
FIG . 1.1. Variatio n of resistanc e of metal s wit h temperature .
C e r t a i n metals, however, show a very r e m a r k a b l e b e h a v i o u r ; w h e n
they are coolea their electrical resistance decreases in t h e usual way, b u t
on reaching a t e m p e r a t u r e a few degrees above absolute zero they
suddenly lose all trace of electrical resistance (Fig. 1.2). T h e y are t h e n
said t o have passed into t h e superconducting
state.t T h e transformation
t I n thi s book we us e th e ter m superconductor
for a materia l whic h show s superconductivit y if
cooled. W e us e th e adjectiv e superconducting
t o describ e it whe n it is exhibitin g superconductivi ty , an d normal whe n it is not exhibitin g superconductivit y (e.g. whe n abov e it s transitio n
temperature) .
ZER O RESISTANC E
5
to t h e superconducting state m a y occur even if the metal is so i m p u r e
t h a t it would otherwise have h a d a large residual resistivity.
1.1.
Superconductin g Transitio n Temperatur e
T h e t e m p e r a t u r e at w h i c h a superconductor loses resistance is called
its superconducting transition
temperature
or critical temperature;
this
t e m p e r a t u r e , w r i t t e n T , is different for each metal. T a b l e 1.1 s h o w s t h e
transition t e m p e r a t u r e s for metallic elements. I n general t h e transition
t e m p e r a t u r e is n o t very sensitive to small a m o u n t s of impurity, t h o u g h
c
0
T
c
Temperatur e
FIG . 1.2
Los s of resistanc e of a superconducto r at low temperatures .
magnetic impurities tend to lower the transition t e m p e r a t u r e . ( W e shall
see in C h a p t e r 9 t h a t ferromagnetism, in w h i c h the spins of electrons are
aligned parallel t o each other, is incompatible w i t h superconductivity.)
T h e s u p e r c o n d u c t i v i t y of a few m e t a l s , s u c h a s i r i d i u m a n d
molybdenum, w h i c h in the p u r e state have very low transition
temperatures, m a y b e destroyed b y t h e presence of m i n u t e q u a n t i t i e s of
magnetic impurities. Such elements, therefore, only exhibit superconductivity if they are extremely pure, and specimens of these m e t a l s of normal commercial purity are not superconductors. N o t all p u r e metals h a v e
been found t o b e superconductors; for example, copper, iron and sodium
have not shown superconductivity d o w n to the lowest t e m p e r a t u r e t o
which they h a v e so far been cooled. Of course, experiments at even lower
t e m p e r a t u r e s m a y reveal n e w superconductors, b u t there is n o fundamental reason w h y all metals should show superconductivity, even at
6
INTRODUCTION TO SUPERCONDUCTIVITY
absolute zero. Nevertheless, it should b e realized t h a t superconductivity
is not a rare p h e n o m e n o n ; about half t h e metallic e l e m e n t s are k n o w n t o
be s u p e r c o n d u c t o r s and, in addition, a large n u m b e r of alloys are superconductors. It is possible for an alloy t o b e a superconductor, even if it is
composed of t w o m e t a l s w h i c h are n o t themselves s u p e r c o n d u c t o r s (e.g.
B i - P d ) . Superconductivity can b e s h o w n b y c o n d u c t o r s w h i c h are n o t
m e t a l s in t h e ordinary sense; for example, t h e semiconducting mixed
oxide of b a r i u m , lead and b i s m u t h is a superconductor, and t h e c o n d u c ting polymer, polysulphur nitride ( S N ) , h a s b e e n found t o b e c o m e
superconducting at about 0-3°K.
X
T A B L E 1.1.
T H E SUPERCONDUCTIN G E L E M E N T S
T is th e superconductin g transitio n temperatur e
is th e critica l magneti c field at 0°K (see Chapte r 4)
c
H
0
# 0
7V(°K )
Aluminiu m
Ë
·
·
/
f
C
C
Amenciumj^ p
c
Cadmiu
Galliu
mm
Indiu m
Iridiu m
Lanthanu m j
Lea d
Luteciu m
Mercur y j ^
Molybdenu m
Niobiu m
Osmiu m
Protactinium
Rheniu m
Rhodium
Rutheniu m
Tantalu m
Technetiu m
Thaliu m
Thoriu m
Ti n
Titaniu m
Tungste n
Uraniu m {*?
\r
Vanadiu m
Zin c
Zirconiu m
a
1-2
1·1
0-79
0-52
11
3-4
Oil
4-8
4 9
7-2
01
4-2
40
0 9
9-3
0 7
0-4
1-7
3-3 x 1 0 ^
0-5
4-5
7 9
2-4
14
3-7
0-4
0 016
0 6
(Am p m )
(gauss )
- 1
0-79 x 10*
0-22
0-41
2-2
013
÷
÷
÷
x
10
10
10
10
6-4
2-8
3 3
2 7
÷
÷
÷
÷
10
10
10
10
99
4
4
4
4
4
4
4
4
30
51
276
16
803
350
413
340
Type-I I (see Chap . 12)
-63
- 0 5 x 10
Type-I I (see C h a p . 12)
201
1 6 ÷ 10
4
0-05
66
0-53 ÷ 10
830
6 6 ÷ 10
Type-I I (see Chap . 12)
171
1 4 x 10
162
1 3 x 10
306
2-4 ÷ 10
4
4
4
4
4
4
4
0 0096 x 10
4
1-2
1-8
5-4
0 9
0 8
Type-I I (s©eChap . 12)
53
0 42 ÷ 10
47
0 37 ÷ 10
4
4
7
ZER O RESISTANC E
N i o b i u m is the metallic element w i t h the highest transition temperature (9*3°K), b u t some alloys and metallic c o m p o u n d s remain superconducting u p to even higher temperatures (Table 1.2). For example, N b G e
has a transition temperature of about 2 3 °K. T h e s e alloys w i t h relatively
high transition temperatures are of great importance in the engineering
applications of superconductivity. Recently some ceramic metal oxides
have been developed w h i c h are superconducting u p to temperatures as
high as about 100°K. T h e s e are of great interest because such t e m p e r a tures, unlike those w i t h i n a few tens of degrees of Absolute Zero, can be
obtained easily and cheaply (by, for example, t h e use of liquid nitrogen).
3
T A B L E 1.2.
SUPERCONDUCTIN G T R A N S I T I O N TEMPERATURE S O F SOM E
ALLOY S AND M E T A L L I C COMPOUND S COMPARE D
WIT H THEI R C O N S T I T U E N T E L E M E N T S
T (°K)
C
T (°K )
e
Ta-N b
Pb-B i
3Nb-Z r
Nb Sn
3
Nb G e
6-3
8
11
IS
23
3
Nb
Pb
Ta
Sn
Zr
Bi
Ge
9 3
7-2
4-5
3-7
0-8
not s/c
not s/c
O n cooling, the transition to the superconducting state may be extremely
sharp if the specimen is pure and physically perfect. For example, in a good
gallium specimen, the transition has been observed to occur within a t e m perature range of 10 degrees. If, however, the specimen is impure or has a
disturbed crystal structure, the transition may be considerably broadened.
Figure 1.3 shows the transition in pure and impure tin specimens.
Pur e
r
/ Impur e
I
3 70
1
3 72
I
3 74
1
1
3 76
Temperatur e ('Ê )
FlG . 1.3. Superconductin g transitio n in tin .
3 78
8
INTRODUCTION TO SUPERCONDUCTIVITY
1.2
Zer o Resistanc e
E v e n w h e n t h e transition is spread over a considerable t e m p e r a t u r e
r a n g e t h e resistance still seems t o d i s a p p e a r completely below a certain
t e m p e r a t u r e . W e naturally ask w h e t h e r in t h e superconducting state t h e
resistance h a s indeed b e c o m e zero or w h e t h e r it h a s merely fallen t o a
very small value. Of course, it can never b e proved b y experiment t h a t
t h e resistance is in fact z e r o ; t h e resistance of any specimen m a y always
be j u s t less t h a n t h e sensitivity of o u r a p p a r a t u s allows u s t o detect.
However, n o experiment h a s b e e n able t o detect a n y resistance in t h e
superconducting state. W e m a y look for resistance q u i t e simply b y
passing a current t h r o u g h a w i r e of s u p e r c o n d u c t o r a n d seeing if any
voltage is recorded b y a sensitive voltmeter connected across t h e e n d s of
the wire. A m o r e sensitive test, however, is t o start a c u r r e n t flowing
r o u n d a closed superconducting ring a n d t h e n see w h e t h e r t h e r e is any
decay in t h e c u r r e n t after a long period of t i m e . S u p p o s e t h e selfinductance of t h e ring is L\ t h e n , if at t i m e t = 0 w e start a c u r r e n t i(0)
flowing r o u n d t h e ring (for w a y s of doing t h i s see § 1.3), at a later t i m e t
the current will h a v e decayed t o
i(t) = i(0)e-(
(1.1)
R/L)t
y
w h e r e R is t h e resistance of t h e ring. W e can m e a s u r e t h e m a g n e t i c field
t h a t t h e circulating c u r r e n t p r o d u c e s a n d see if t h i s decays w i t h t i m e .
T h e m e a s u r e m e n t of t h e m a g n e t i c field d o e s n o t d r a w energy from t h e
circuit, and w e should b e able t o observe w h e t h e r t h e c u r r e n t circulates
indefinitely. Gallop h a s b e e n able t o show from t h e lack of decay of a
current circulating r o u n d a closed loop of superconducting wire t h a t t h e
resistivity of t h e s u p e r c o n d u c t i n g metal w a s less t h a n 1 0 ~ o h m - m e t r e s
(i.e. less t h a n 1 0 ~ t h e resistivity of c o p p e r at r o o m t e m p e r a t u r e ) . I t
seems, therefore, t h a t w e are justified in treating t h e resistance of a
superconducting metal a s zero.
26
18
1.3.
T h e R e s i s t a n c e l e s s Circui t
A closed circuit, such a s a ring, formed of superconducting metal h a s
an i m p o r t a n t and useful p r o p e r t y resulting from i t s zero resistance. The
total magnetic flux threading
a closed resistanceless
circuit
cannot
S u p p o s e as in F i g .
change so long as the circuit remains resistanceless.
1.4a, a ring of metal is cooled below its t r a n s i t i o n t e m p e r a t u r e in an
applied field of uniform flux density B . If t h e area enclosed b y t h e ring
a
9
ZER O RESISTANC E
is s4 an a m o u n t of flux Ö = s/B
will t h r e a d t h e ring. S u p p o s e t h e
applied field is n o w changed t o a new value. By L e n z ' s law, w h e n the
field is changing c u r r e n t s are induced and circulate r o u n d t h e ring in
such a direction as t o create a flux inside t h e ring w h i c h t e n d s to cancel
the flux change d u e to the alteration in t h e applied field. W h i l e t h e field is
changing there is an e.m.f., —sidBjdty
and an induced current i given
by
y
a
I *
w h e r e R and L are the total resistance and inductance of the circuit. In
a normal resistive circuit the induced c u r r e n t s quickly die away and
the flux threading the ring acquires t h e new value. In a superconducting
circuit, however, R = 0 and
*
dt
Li + siB
so t h a t
a
dt'
L
— constant.
(1.2)
Ö = AB.
B.
(a)
<«»
FlG . 1.4. Resistanceles s circuit .
B u t Li + siB
is t h e total m a g n e t i c flux threading t h e circuit; w e h a v e
therefore d e m o n s t r a t e d t h a t t h e total flux threading a resistanceless circuit cannot change. If t h e applied magnetic field strength is changed an
induced current is set u p of such a m a g n i t u d e t h a t it creates a flux w h i c h
exactly c o m p e n s a t e s t h e change in the flux from the applied magnetic
field. Since t h e circuit is resistanceless t h e induced current flows for ever
and the original a m o u n t of flux is maintained indefinitely. Even if t h e
external field is reduced to zero, as in Fig. 1.4b, the internal flux will b e
maintained b y t h e induced circulating current.
a
10
INTRODUCTION TO SUPERCONDUCTIVITY
T h i s p r o p e r t y c a n b e m a d e u s e of w h e n solenoids w o u n d from superconducting wire are employed t o g e n e r a t e m a g n e t i c fields. I n Fig, 1.5
current to t h e refrigerated superconducting solenoid 5 is derived from
t h e d.c. p o w e r supply P. O n c e t h e c u r r e n t h a s been adjusted b y t h e
rheostat R to a value w h i c h gives t h e desired m a g n e t i c field strength, t h e
superconducting switch XY can b e closed. XY and S n o w form a closed
resistanceless circuit in w h i c h t h e flux m u s t r e m a i n c o n s t a n t . H e n c e t h e
field strength generated b y S d o e s n o t vary in t i m e and w e can, if w e
wish, disconnect t h e p o w e r supply a n d t h e field will b e m a i n t a i n e d b y
t h e current i flowing r o u n d t h e resistanceless circuit XYS. A superconducting solenoid operating in t h i s fashion is said t o b e in t h e persistent
mode.
I
é
FIG . 1.5. Superconductin g solenoid .
N o t e t h a t , a l t h o u g h t h e total
a m o u n t of flux e n c l o s e d in a
resistanceless circuit r e m a i n s c o n s t a n t , t h e r e can b e a c h a n g e in t h e
flux density  at any point d u e t o a r e d i s t r i b u t i o n of t h e flux w i t h i n
the circuit. T h u s in Fig. 1.4b t h e flux density h a s b e c o m e stronger
near t h e wire and w e a k e r in t h e c e n t r e of t h e enclosed space c o m p a r e d
to t h e uniform d i s t r i b u t i o n in F i g . 1.4a. In b o t h cases, however, t h e total
flux ( = J j B · ds/) is t h e same.
W e have seen t h a t if a closed circuit of s u p e r c o n d u c t o r is cooled below
its transition t e m p e r a t u r e while in an applied m a g n e t i c field, t h e flux it
ZER O RESISTANC E
11
encloses r e m a i n s constant in spite of any changes in t h e applied field. O n
t h e other hand, if the circuit is cooled in t h e absence of an applied
magnetic field, so t h a t there is initially n o flux inside, and an external
magnetic field is subsequently applied, t h e net internal flux r e m a i n s zero
in spite of the presence of t h e external field. T h i s property enables u s to
use hollow superconducting cylinders t o shield enclosures from external
magnetic fields. T h e shielding is only perfect for the case of a long hollow
cylinder, in w h i c h case t h e induced c u r r e n t s generate a uniform c o m p e n sating flux density t h r o u g h o u t its interior. F o r other configurations, such
as a short ring, it is only the total flux w h i c h is maintained at zero and
the local magnetic flux density generated b y t h e induced current will not
be uniform w i t h i n the ring. H e n c e the flux density d u e to t h e persistent
c u r r e n t s will in some places b e stronger and in other places weaker t h a n
that of an applied field, and there will not everywhere b e exact cancellation. I n other w o r d s , t h o u g h jJB · dstf = 0, Â itself is n o t necessarily
FIG . 1.6. Division of curren t betwee n tw o paralle l paths .
everywhere equal to zero. I n practice, however, superconducting
shielding can b e used to give very good screening against magnetic fields.
W e n o w consider w h a t d e t e r m i n e s the distribution of c u r r e n t s in a
network of resistanceless conductors. Consider, for example, t h e simple
circuit of Fig. 1.6. If the ring ABCD is resistanceless, h o w will t h e
current i divide b e t w e e n b r a n c h e s  and D ? Clearly K i r c h h o f T s laws are
of n o help, because b o t h p a t h s have zero resistance and t h e second law
will b e obeyed for all possible divisions of t h e current i. H o w e v e r , t h o u g h
the t w o b r a n c h e s have n o resistance, they d o contribute inductance t o
the circuit. W e shall n o w show t h a t the division of current is determined
by these inductances. T h e voltage difference b e t w e e n A and C equals
12
INTRODUCTION TO SUPERCONDUCTIVITY
Ô » + Ì
»~dt
di
- L
"
~dt
d i u
+ M
d i D
~dt
BD
D
B D
dV
w h e r e L and L are the i n d u c t a n c e s of t h e b r a n c h e s  and Z>, and
is t h e m u t u a l inductance b e t w e e n t h e m . R e a r r a n g e m e n t gives
B
D
( ^ - M
M
) ^ = ( L
- M
D
B
O
M
BD
) ^ ,
and integrating t h i s we o b t a i n
(L
B
If i = 0 = i
B
D
- M )
BD
i = (L
B
- M )
D
i
BD
D
+ constant.
at t = 0, t h e n t h e c o n s t a n t = 0, and w e h a v e
*D
L —
B
M
BD
and it can b e seen t h a t t h e division of t h e c u r r e n t is controlled b y t h e ind u c t a n c e s of t h e p a t h s . It often h a p p e n s t h a t t h e m a g n e t i c coupling
b e t w e e n t w o parallel p a t h s is small so t h a t their m u t u a l i n d u c t a n c e can
b e neglected. U n d e r t h i s c i r c u m s t a n c e w e can d e d u c e t h e rule t h a t in
resistanceless n e t w o r k s the currents carried in parallel
paths are inversely proportional
to the self-inductances
of those paths.
1.4.
A.C . Resistivit y
T h e fact t h a t a superconducting metal h a s n o resistance m e a n s , of
course, t h a t there is no voltage d r o p along t h e metal w h e n a c u r r e n t is
passed t h r o u g h it, and n o p o w e r is generated b y t h e passage of t h e
current. T h i s , however, is only strictly t r u e for a direct c u r r e n t of c o n s t a n t value. If t h e c u r r e n t is changing an electric field is developed and
some p o w e r is dissipated. T o u n d e r s t a n d t h e reason for t h i s w e m u s t
first discuss briefly some aspects of t h e b e h a v i o u r of c o n d u c t i o n elect r o n s in superconductors.
M a n y of t h e properties of s u p e r c o n d u c t o r s can b e explained if it is
supposed t h a t below t h e transition t e m p e r a t u r e the c o n d u c t i o n electrons
divide into t w o classes, some b e h a v i n g as " s u p e r e l e c t r o n s " w h i c h can
pass t h r o u g h the metal w i t h o u t resistance (i.e. suffering n o collisions),
t h e r e m a i n d e r behaving as " n o r m a l " electrons w h i c h can b e scattered
and so experience resistance j u s t like conduction electrons in a normal
metal. T h e fraction of superelectrons a p p e a r s t o decrease as t h e
t e m p e r a t u r e is raised t o w a r d s t h e transition t e m p e r a t u r e . At 0°K all con-
ZER O RESISTANC E
13
auction electrons b e h a v e like superelectrons, b u t , if the t e m p e r a t u r e is
raised, a few begin t o behave as normal electrons, and on further heating
the proportion of normal electrons increases. Eventually, at the transition t e m p e r a t u r e , all the electrons have b e c o m e normal electrons and the
metal loses its superconductive properties. H e n c e a superconductor
below its transition t e m p e r a t u r e appears to b e permeated b y t w o electron fluids, one of normal electrons and one of superelectrons. T h e
relative electron density in the t w o fluids d e p e n d s on t h e t e m p e r a t u r e .
T h i s "two-fluid m o d e l " is suggested b y t h e r m o d y n a m i c a r g u m e n t s
based on the results of specific heat and similar m e a s u r e m e n t s on superconductors, w h i c h will b e discussed in C h a p t e r 5.
In a superconducting metal t h e current can in general b e carried by
b o t h the normal and superelectrons. H o w e v e r , in t h e special case of a
constant direct current all t h e current is carried b y t h e superelectrons.
W e can see t h a t t h i s will b e so b y noting that, if the current is to r e m a i n
constant, there m u s t b e n o electric field in t h e metal, otherwise t h e
superelectrons would b e accelerated continuously in this field and the
current would increase indefinitely. If there is n o field there is nothing to
drive the normal electrons and so there is n o normal current. W e see,
therefore, t h a t for a constant value of total current all t h e current is
carried b y t h e superelectrons. A superconducting metal is like t w o conductors in parallel, one having a normal resistance and the other zero
resistance. W e can say t h a t t h e superelectrons "short circuit" t h e normal
electrons. T o p u t this another way, if we suddenly apply a voltage
source, such as a battery, across a superconductor, t h e current t e n d s to
rise t o infinity b u t is in fact limited b y t h e internal resistance of t h e
source. While t h e current is changing, an electric field m u s t b e present to
accelerate the electrons. Electrons do, however, have a small inertial
m a s s and so t h e supercurrent does not rise instantaneously b u t only at
the rate at which t h e electrons accelerate in t h e electric field. If w e apply
an alternating field, the supercurrent will therefore lag b e h i n d the field
because of the inertia of the superelectrons. H e n c e the superelectrons
present an inductive i m p e d a n c e t and, because there n o w is an electric
field present, some of the current will b e carried by the normal electrons.
T h e current is not, therefore, carried entirely b y the superelectrons as in
the d.c. case. Of course, t h e normal electrons also have an inertial m a s s
but their resulting inductive reactance is completely s w a m p e d b y the
t Thi s inheren t inductiv e impedanc e is, of course , quit e distinc t from , an d is additiona l to, th e
ordinar y inductanc e of th e conducto r du e t o it s geometry .
14
INTRODUCTION TO SUPERCONDUCTIVITY
resistance resulting from their being scattered in t h e metal. W e can, in
fact, represent t h e bulk p r o p e r t i e s of a s u p e r c o n d u c t i n g metal b y a
perfect i n d u c t a n c e in parallel w i t h a resistance.
T h e fraction of t h e c u r r e n t diverted t h r o u g h t h e n o r m a l electrons d i s sipates p o w e r in t h e usual w a y . T h e m a s s of electrons is, however,
extremely small, so the i n d u c t a n c e d u e t o their inertia is also extremely
small. T h e i n d u c t a n c e in h e n r y s of a typical s u p e r c o n d u c t o r d u e t o t h e
inertia of i t s superelectrons is only a b o u t 1 0 " of i t s n o r m a l resistance in
o h m s , so at 1000 H z , for example, only a b o u t 10~ of t h e total c u r r e n t
is carried b y t h e n o r m a l electrons a n d there is only a m i n u t e dissipation
of power. Nevertheless, t h i s c o n t r a s t s w i t h t h e absolutely z e r o resistance
in t h e d.c. case.
1 2
8
If t h e frequency of a n applied field is sufficiently high, however, a
superconducting metal r e s p o n d s in t h e s a m e w a y as a n o r m a l metal.
T h i s is because, as w e shall see in C h a p t e r 9, superelectrons are in a
lower energy state t h a n n o r m a l electrons, b u t , if t h e frequency of t h e
applied field is high enough, t h e p h o t o n s of t h e electromagnetic field
have enough energy to excite superelectrons into t h e higher state w h e r e
they behave as n o r m a l electrons. T h i s h a p p e n s for frequencies greater
t h a n about 1 0 H z (i.e. greater t h a n the frequency of very long w a v e
infrared). T h e b e h a v i o u r of a superconductor at optical frequencies is
therefore n o different from t h a t of a n o r m a l m e t a l and t h e r e is, for e x a m ple, n o c h a n g e in the visual appearance of a s u p e r c o n d u c t o r as it is
cooled below its t r a n s i t i o n t e m p e r a t u r e .
11
I t is t e m p t i n g t o suppose t h a t t h e superelectrons in a s u p e r c o n d u c t o r
behave like electrons in a v a c u u m . T h e electrons in t h e b e a m of a
c a t h o d e ray t u b e , for example, are resistanceless in t h e sense t h a t they
flow w i t h o u t u n d e r g o i n g any collisions. T h e r e is, however, a significant
difference b e t w e e n the t w o cases. I t is possible t o m a i n t a i n a potential
d r o p along an electron b e a m while t h e c u r r e n t r e m a i n s at a c o n s t a n t
value. T h i s is because, t h o u g h t h e c u r r e n t m u s t b e t h e s a m e all along t h e
p a t h of t h e b e a m , t h e electron density need not r e m a i n c o n s t a n t . H e n c e
t h e electrons accelerate from t h e c a t h o d e t o w a r d s t h e anode and t h e electron density is relatively h i g h near the c a t h o d e and decreases as t h e
anode is approached. H o w e v e r , t h e p r o d u c t of t h e electron density and
electron velocity, i.e. t h e current, r e m a i n s c o n s t a n t along t h e b e a m . T h e
fact t h a t t h e electrons are able t o accelerate allows u s t o m a i n t a i n t h e
electric field. I n a superconductor, however, c o n d i t i o n s are different. T h e
metal m u s t r e m a i n everywhere electrically neutral and, since t h e positive
metal ions are fixed in t h e crystal, t h e electron density c a n n o t vary
ZER O RESISTANC E
15
t h r o u g h the material. H e n c e for a constant current t o b e maintained
t h r o u g h t h e metal, t h e velocity of all t h e electrons along t h e current p a t h
m u s t b e the same. T h e electrons, therefore, d o not accelerate and an electric field cannot exist in t h e metal.
I.T.S.—Â
CHAPTER 2
PERFECT
2.1.
DIAMAGNETISM
Magneti c Propertie s of a Perfec t Conducto r
WE HAVE seen in t h e previous c h a p t e r t h a t a s u p e r c o n d u c t o r below its
transition t e m p e r a t u r e a p p e a r s t o have n o resistance. L e t u s n o w try t o
deduce t h e magnetic properties of such a resistanceless conductor.
S u p p o s e t h a t w e cool a specimen w h i c h , below its t r a n s i t i o n
t e m p e r a t u r e , b e c o m e s perfectly conducting. T h e resistance a r o u n d an
imaginary closed p a t h w i t h i n t h e metal is z e r o ; and therefore, as s h o w n
in t h e previous chapter, t h e a m o u n t of m a g n e t i c flux enclosed w i t h i n t h i s
p a t h c a n n o t change. T h i s is t r u e for any such imaginary circuit and t h i s
can only b e so if t h e flux density at every point w i t h i n t h e metal does not
vary w i t h time, i.e.
6
= 0.
Consequently the flux distribution in the metal m u s t r e m a i n as it w a s
w h e n the metal b e c a m e resistanceless.
Consider n o w t h e behaviour of a perfect c o n d u c t o r u n d e r v a r i o u s circumstances. S u p p o s e t h a t a specimen loses its resistance in t h e absence
of any magnetic field and t h a t a m a g n e t i c field is t h e n subsequently
applied. Because t h e flux density in t h e metal c a n n o t change, it m u s t r e m a i n zero even after t h e application of t h e m a g n e t i c field. In fact t h e
application of t h e m a g n e t i c field induces resistanceless c u r r e n t s w h i c h
circulate on t h e surface of t h e specimen in such a m a n n e r as to create a
magnetic flux density w h i c h everywhere inside t h e metal is exactly equal
and opposite to t h e flux density of t h e applied m a g n e t i c field.t Because
t W e mus t defin e her e exactl y wha t we mea n b y an "applie d magneti c field". An "applied* '
field is a field generate d by som e agenc y (solenoid , permanen t magnet , etc.) externa l to a
specimen . It s strengt h H an d flux densit y B ar e thos e whic h woul d b e measure d if th e specime n
wer e not there . In th e cas e of a unifor m applie d field th e strengt h an d flux densit y ar e th e sam e as
thos e measure d far awa y fro m th e specimen , i.e. wher e an y perturbin g effects du e to th e
magneti c propertie s of th e specime n ar e negligible .
a
a
16
17
PERFECT DIAMAGNETISM
these c u r r e n t s d o not die away, t h e net flux density inside t h e material
r e m a i n s at zero. T h i s is illustrated in Fig. 2.1a: the surface c u r r e n t s i
generate a flux density B t h a t exactly cancels t h e flux density B of the
applied magnetic field everywhere inside t h e metal. T h e s e surface
c u r r e n t s are often referred to as screening
currents.
T h e flux density created b y t h e persistent surface c u r r e n t s does not, of
course, disappear at the b o u n d a r y of t h e specimen, b u t t h e flux lines
form c o n t i n u o u s closed curves w h i c h r e t u r n t h r o u g h the space outside
(Fig. 2.1a). T h o u g h t h e density of this flux everywhere inside the
specimen is equal and opposite to the flux from the applied field, this is
not so outside the specimen. T h e net distribution of flux resulting from
the superposition of the flux from t h e specimen and t h a t from the
applied field is s h o w n in Fig. 2.1b. T h e p a t t e r n is as t h o u g h the sample
(
a
•
Flu x of applie d fiel d
•
Flu x fro m magnetizatio n
~~
\
/
Ã
Net flu x distributio n
\
2 5
y
^
(a )
t
r-
Ã
(b)
FlG . 2.1. Distributio n of magneti c flux abou t a perfectl y diamagneti c body .
had prevented entry into it of t h e flux of t h e applied field. A sample in
which there is n o net flux density w h e n a magnetic field is applied is said
to exhibit perfect diamagnetism.
If w e n o w reduce t h e applied magnetic
field t o zero t h e specimen is left in its original unmagnetized condition.
T h e above sequence of events is illustrated in Fig. 2.2 a - d .
L e t u s n o w consider a different sequence of events. S u p p o s e t h a t the
magnetic field B is applied t o t h e specimen while it is above its transia
18
INTRODUCTION TO SUPERCONDUCTIVITY
FIG . 2.2. Magneti c behaviou r of a "perfect " conductor . (a)-(b ) Specime n become s
resistanceles s in absenc e of field, (c) Magneti c field applie d t o resistanceles s specimen ,
(d) Magneti c field removed .
(e)-(f) Specime n become s resistanceles s in applie d magneti c field, (g) Applie d
magneti c field removed .
tion t e m p e r a t u r e (Fig. 2.2e). M o s t m e t a l s (other t h a n t h e special
ferromagnetics, iron, cobalt and nickel) h a v e values of relative m a g n e t i c
permeability very close t o unity, a n d so t h e flux density inside is virtually
the same as t h a t of t h e applied field. T h e specimen is n o w cooled t o a
low t e m p e r a t u r e so t h a t it loses its electrical resistance. T h i s d i s appearance of resistance h a s n o effect on t h e m a g n e t i z a t i o n , and t h e flux
distribution r e m a i n s unaltered (Fig. 2.2f). W e next r e d u c e t h e applied
field to zero. T h e flux density inside t h e perfectly c o n d u c t i n g metal c a n not change, and persistent c u r r e n t s are induced o n t h e specimen, m a i n taining t h e flux inside, w i t h t h e result t h a t t h e specimen is left p e r manently magnetized (Fig. 2.2g).
PERFECT DIAMAGNETISM
19
It is i m p o r t a n t t o notice t h a t in (c) and (f) of Fig. 2.2 t h e sample is u n der t h e same conditions of t e m p e r a t u r e and applied magnetic field, and
yet its state of magnetization is q u i t e different in the t w o cases. Similarly
(d) and (g) show different states of magnetization u n d e r identical external conditions. W e see t h a t the state of magnetization of a perfect conductor is not uniquely determined b y t h e external conditions b u t
d e p e n d s o n t h e sequence by w h i c h these conditions were arrived at.
2.2.
2 · 2 · 1.
Specia l Magneti c Behaviou r of a Superconducto r
M e i s s n e r effect
In the previous section w e h a v e deduced t h e magnetic behaviour of a
resistanceless conductor b y applying simple and well-known fundamental principles of electromagnetism, and for 22 years after t h e discovery of
superconductivity it w a s assumed t h a t t h e effect of a magnetic field on a
superconductor would be as shown in Fig. 2.2. H o w e v e r , in 1933
M e i s s n e r and Ochsenfeld measured t h e flux distribution outside tin
and lead specimens which h a d been cooled below their transition
t e m p e r a t u r e s while in a magnetic field.t T h e y found t h a t t h e expected
situation of Fig, 2.2f did not in fact occur, b u t t h a t at their transition
t e m p e r a t u r e s the specimens spontaneously became perfectly diamagnetic, cancelling all flux inside, as in Fig. 2.2c, even t h o u g h they
h a d been cooled in a magnetic field. T h i s experiment w a s the first to
d e m o n s t r a t e t h a t superconductors are something m o r e t h a n materials
w h i c h are perfectly conducting; they have an additional property that a
merely resistanceless metal would not p o s s e s s : a metal in the superconducting state never allows a magnetic flux density to exist in its interior.
T h a t is t o say, inside a superconducting metal w e always have
 = 0,
w h e r e a s inside a merely resistanceless metal there m a y or m a y not b e a
flux density, depending on circumstance (Fig. 2.2). W h e n a superconductor is cooled in a weak magnetic field, at t h e transition t e m p e r a t u r e persistent c u r r e n t s arise on t h e surface and circulate so as t o cancel t h e flux
density inside, in j u s t the same w a y as w h e n a magnetic field is applied
t Th e magneti c field used in experiment s of thi s kin d mus t no t b e to o stron g because , as we
shal l see in Chapte r 4, a meta l loses it s superconductin g propertie s if th e applie d magneti c field
exceed s a certai n strength .
20
INTRODUCTION
TO
SUPERCONDUCTIVITY
after the metal h a s been cooled (Fig. 2.3). T h i s effect, w h e r e b y a superconductor never h a s a flux density inside even w h e n in an applied
magnetic field, is called ( w i t h injustice t o Ochsenfeld) the Meissner
effect.
 = Ï
Roo m
temperatur e
Coole d
Coole d
Lo w
temperatur e
Â. —
Ï
B -^0
a
FIG . 2.3. Magneti c behaviou r of a superconductor . (a)-{b) Specime n become s
resistanceles s in absenc e of magneti c field, (c) Magneti c field applie d t o superconduc tin g specimen , (d) Magneti c field removed .
(e)-(0 Specime n become s superconductin g
magneti c field removed .
in applie d magneti c field, (g) Applie d
F o r convenience w e call a hypothetical metal w h i c h simply h a s n o
resistance and would behave as shown in Fig. 2.2 a "perfect c o n d u c t o r "
in c o n t r a s t t o s u p e r c o n d u c t o r s w h i c h in t h e s u p e r c o n d u c t i n g state never
permit a magnetic flux density t o exist inside t h e m (Fig. 2.3). T h e state
of magnetization of a "perfect c o n d u c t o r " w o u l d d e p e n d o n t h e o r d e r in
w h i c h t h e final c o n d i t i o n s of applied m a g n e t i c field and t e m p e r a t u r e
w e r e obtained, b u t the m a g n e t i z a t i o n of a s u p e r c o n d u c t o r d e p e n d s only
on t h e actual values of t h e applied field a n d t e m p e r a t u r e and n o t o n t h e
w a y they w e r e arrived at.
21
PERFECT DIAMAGNETISM
2.2.2.
Permeabilit y an d susceptibilit y of a superconducto r
S u p p o s e a m a g n e t i c field of flux density B is applied t o a supercond u c t o r . t In order t h a t we m a y neglect demagnetizing effects (see
C h a p t e r 6) w e consider a long superconducting rod w i t h t h e field applied
parallel to its length. An applied magnetic field of flux density B
produces in t h e material a flux density equal to ì Â ,
w h e r e ì, is t h e
relative permeability of t h e material. Metals, other t h a n ferromagnetics,
have a relative permeability w h i c h is very close to unity, i.e. ì = 1, so
the flux density within t h e m d u e to the applied magnetic field is equal to
B . However, as w e h a v e seen, t h e total flux density inside a superconducting b o d y is zero. T h i s perfect d i a m a g n e t i s m arises because surface
screening c u r r e n t s circulate so as t o produce a flux density B w h i c h
everywhere inside the metal exactly cancels t h e flux density d u e to the
applied field; B = —B. A rod-shaped superconducting specimen
therefore b e h a v e s like a long solenoid w i t h circulating current t h a t
creates a flux density exactly equal in m a g n i t u d e , b u t opposite in direction, t o the flux density d u e to t h e applied m a g n e t i c field. T o create a
flux density of—B, the m a g n i t u d e of t h e circulating surface current per
unit length m u s t , from t h e ordinary solenoid formula, b e l / l = Â /ì .
In
other w o r d s l / l = H , w h e r e H is t h e applied field strength.
W e can, however, describe t h e perfect d i a m a g n e t i s m in another way.
Because we cannot actually observe t h e surface screening c u r r e n t s w h i c h
arise w h e n a magnetic field is applied, w e could suppose t h a t t h e perfect
diamagnetism arises from some special bulk magnetic property of the
superconducting metal, and w e can describe t h e perfect d i a m a g n e t i s m
simply b y saying t h a t for a superconducting metal ì = 0, so t h a t the
flux density inside, Â = ì Â ,
is zero. H e r e we d o not consider t h e
mechanism b y w h i c h the d i a m a g n e t i s m arises; the effect of t h e screening
currents is included in the s t a t e m e n t ì = 0. T h e strength H of the
applied magnetic field is given by
a
a
Ã
á
Ã
a
t
{
a
a
á
a
0
a
Ã
Ã
á
ô
a
"'-To'
and t h e flux density in a magnetic material is related to t h e strength of
the applied field b y
 = ì
0
(H
a
+
/),
t As explaine d in Appendi x A, we tak e th e view tha t a magneti c field is bes t describe d b y it s
flux density .
22
INTRODUCTION TO SUPERCONDUCTIVITY
w h e r e / is t h e m a g n e t i z a t i o n (often called t h e " i n t e n s i t y of
magnetization") of t h e material. T h e m a g n e t i z a t i o n of a superconductor,
in which 5 = 0, m u s t therefore b e
and t h e magnetic susceptibility, i.e. t h e ratio of t h e m a g n e t i z a t i o n t o t h e
field strength, m u s t b e
T h e t w o descriptions are entirely equivalent because, as s h o w n in
Appendix A, t h e m a g n i t u d e of / is equal t o t h e equivalent surface c u r r e n t
density j .
W e n o w s u m m a r i z e t h e t w o alternative w a y s of regarding t h e perfect
diamagnetism.
(i) Screening-current
diamagnetism
T h e material of t h e superconductor, like o t h e r m e t a l s , is n o n magnetic, and an applied m a g n e t i c field p r o d u c e s a flux density B in t h e
metal. H o w e v e r , screening c u r r e n t s g e n e r a t e an internal flux density
w h i c h everywhere is exactly equal and opposite t o t h i s flux density and
consequently t h e net flux density is zero.
a
(it) Bulk
diamagnetism
T h e material can b e considered to h a v e a relative permeability ì = 0,
so t h e flux density produced in it b y an applied m a g n e t i c field is always
zero. T h e material b e h a v e s as t h o u g h in a m a g n e t i c field it acquires a
negative bulk m a g n e t i z a t i o n / = — H .
A p p e n d i x A s h o w s t h a t these t w o w a y s of regarding t h e perfect
d i a m a g n e t i s m are entirely equivalent. W e shall m a k e use of b o t h d e s c r i p tions, t h e choice in each case d e p e n d i n g on w h i c h is m o r e convenient for
t h e particular situation u n d e r discussion.
Ã
a
2.3.
Surfac e Current s
T h e fact t h a t a superconducting metal d o e s n o t allow m a g n e t i c flux t o
exist in its interior h a s an i m p o r t a n t effect o n any electric c u r r e n t s t h a t
flow along i t ; currents cannot pass through the body of a
superconducting
metal, but flow only on the surface. T o see w h y t h i s should be, let u s take
t h e first viewpoint of a superconductor expressed at (i) above and treat
the material as having t h e same relative m a g n e t i c permeability as a n y or-
23
PERFECT DIAMAGNETISM
dinary metal, i.e. ì = 1. At a n y point in a material of unit relative
permeability t h e relation b e t w e e n t h e m a g n e t i c flux and current density
is, from Maxwell's equation,
Ã
curl  = y u j .
(2.1)
0
If t h e metal is superconducting  is zero inside, and so curl  m u s t also
be zero inside.t It follows, therefore, from M a x w e l l ' s equation that, as a
consequence of  being zero, t h e current density J m u s t also b e zero
within t h e superconductor. T h e r e is, of course, n o reason w h y  m u s t be
(a ) Charge d surfac e
(b ) Perfec t diamagneti c
carryin g curren t
FIG . 2.4. Analog y betwee n distributio n of electrostati c charg e an d surfac e current .
+ + + electri c charge s
. . . curren t perpendicula r to plan e of page .
zero outside t h e superconductor so, if there is a current, it flows not
t h r o u g h the metal b u t on the surface. T h i s is true b o t h of c u r r e n t s
passed along the superconductor from some external source such as a
b a t t e r y (we call these " t r a n s p o r t " currents), and of diamagnetic
screening currents. Any t r a n s p o r t current will flow all over t h e surface,
creating a magnetic flux outside b u t not inside t h e conductor. If a
magnetic field is applied, the diamagnetic screening c u r r e n t s w h i c h flow
so as to cancel the flux density inside also circulate on t h e surface.
T h e r e is an interesting and useful analogy b e t w e e n t h e distribution of
current on the surface of a superconducting metal and the distribution of
electrostatic charge on a conducting body. Consider part of t h e surface of
a charged c o n d u c t o r as shown in Fig. 2.4a. I n t h e equilibrium state Å =
0 inside the conductor but, if t h e b o d y carries a surface charge, this
charge will produce an electric field outside t h e conductor. T h e c o m p o t As explaine d in Appendi x A, we tak e th e view tha t J affect s  bu t not Ç . So th e vanishin g of
 doe s not necessaril y impl y tha t Ç is zero .
24
INTRODUCTION TO SUPERCONDUCTIVITY
nent of t h e electric field parallel to t h e surface, 2?,,, is c o n t i n u o u s across
the surface and therefore, since Å = 0 in t h e conductor, 2?„ m u s t b e zero
j u s t outside the surface. T h e electric field lines m u s t consequently m e e t
the c o n d u c t o r at right angles. T h e surface itself is an equipotential and
the electric field lines are orthogonal t o t h e equipotentials. It can b e seen
t h a t the field lines are c r o w d e d t o g e t h e r w h e r e t h e b o u n d a r y h a s a h i g h
convex curvature, so t h e electric charge, being proportional t o t h e normal c o m p o n e n t of the field, will b e c o n c e n t r a t e d into these regions.
F i g u r e 2.4b s h o w s a section of p a r t of a superconducting metal carrying
a current in a direction n o r m a l t o t h e plane of t h e paper. I n s i d e t h e
perfectly d i a m a g n e t i c material w e h a v e  = 0, b u t if c u r r e n t is flowing
on t h e surface t h e r e will b e a m a g n e t i c flux density outside. T h e c o m p o nent of  n o r m a l to t h e surface is c o n t i n u o u s across t h e b o u n d a r y , so
j u s t outside t h e surface t h e flux lines m u s t b e parallel t o t h e surface. T h i s
flux density is proportional t o t h e surface c u r r e n t density. I n fact t h e
external magnetic field d u e to t h e surface c u r r e n t h a s t h e same form as
the equipotentials d u e to surface c h a r g e s in F i g . 2.4a. T h e m a g n e t i c field
lines are c r o w d e d together close t o t h e region w h e r e t h e surface h a s high
convex c u r v a t u r e , so t h e surface c u r r e n t density m u s t b e greatest at
these regions. W e m i g h t expect, therefore, t h a t t h e distribution of surface current o n a perfectly d i a m a g n e t i c b o d y h a s t h e s a m e form as t h e
distribution of electric charge o n a charged c o n d u c t o r of t h e same shape,
and a proper m a t h e m a t i c a l analysis s h o w s t h a t t h i s is indeed so.
2.3.1.
Hol e throug h a superconducto r
I n later c h a p t e r s w e shall o n several occasions need t o consider t h e
properties of a superconducting b o d y w i t h a hole right t h r o u g h it.
T h o u g h t h e b o d y of a s u p e r c o n d u c t o r is perfectly diamagnetic, flux can
exist inside a hole.
C o n s i d e r for example a long hollow b o d y a s shown in Fig. 2.5. T h i s
forms a closed circuit. W e h a v e already discussed t h e p r o p e r t i e s of
closed resistanceless circuits in § 1.3, b u t , w h e n t h e b o d y is a superconductor, w e m u s t also take into account t h e perfect d i a m a g n e t i s m of t h e
material itself.
F i r s t suppose t h a t t h e b o d y s h o w n in F i g . 2.5 is cooled below i t s
superconducting t r a n s i t i o n t e m p e r a t u r e in t h e absence of any applied
magnetic field a n d t h a t , after it h a s b e c o m e superconducting, a m a g n e t i c
field of flux density B is applied (Fig. 2.5 a). T h e superconducting
material is perfectly d i a m a g n e t i c so t h e r e will b e n o flux density in t h e
a
25
PERFECT DIAMAGNETISM
b o d y of t h e material. T h e perfect d i a m a g n e t i s m of the superconducting
material is m a i n t a i n e d by c u r r e n t s i which circulate on the outer surface
so as to cancel t h e flux in t h e metal. H o w e v e r , t h e flux density generated
by these diamagnetic screening c u r r e n t s also cancels the flux density d u e
to t h e applied field in the hole, so in this case there is n o flux density in
t h e hole. I n t h i s situation t h e superconducting b o d y is behaving n o
differently from a merely resistanceless body. In b o t h cases the current
induced on the outer surface b y t h e application of the m a g n e t i c field
cancels the flux inside.
d
B
id
i
e
'd
d
id
(a)
(b )
FlG . 2.5. Hollo w superconductor , (a) Magneti c field applie d whe n materia l is in th e
superconductin g state , (b ) Materia l become s superconductin g in an applie d magneti c
field.
N o w consider a situation in w h i c h a superconductor behaves
differently from a merely resistanceless body, or "perfect" conductor.
Suppose t h e magnetic field is applied before t h e b o d y is cooled below its
transition t e m p e r a t u r e . Above t h e transition t e m p e r a t u r e the magnetic
flux passes b o t h t h r o u g h the b o d y and the hole. In the case of the perfect
conductor this flux distribution will not change w h e n t h e b o d y loses its
resistance, and n o c u r r e n t s will appear on the surface. A superconductor,
however, behaves differently; below t h e transition t e m p e r a t u r e t h e
material becomes perfectly diamagnetic b u t t h o u g h , there is n o flux
26
INTRODUCTION TO SUPERCONDUCTIVITY
t h r o u g h the material, flux r e m a i n s in t h e hole (Fig. 2.5b). C u r r e n t s m u s t
circulate to m a i n t a i n these differences in flux density. A s w e h a v e j u s t
seen, t h e d i a m a g n e t i c surface c u r r e n t s i w h i c h cancel t h e flux in t h e
superconducting material would also cancel the flux in t h e hole, so if
there is magnetic flux t h r e a d i n g the hole, t h i s m u s t b e generated b y
c u r r e n t s i circulating in t h e opposite ( " p a r a m a g n e t i c " ) direction a r o u n d
its periphery. W e have, therefore, the result t h a t flux threading a hole or
normal region t h r o u g h a s u p e r c o n d u c t o r is always associated w i t h a
current circulating a r o u n d the b o u n d a r y b e t w e e n this region and t h e
superconductor.
N o t e t h a t the net circulating current i — i is j u s t t h a t w h i c h
generates a flux density equal to t h e difference b e t w e e n t h e flux density
in t h e hole and t h e flux density o u t s i d e the superconducting b o d y .
As we saw in C h a p t e r 1, t h e flux t h r e a d i n g any resistanceless circuit
cannot change. C o n s e q u e n t l y once flux h a s been established t h r o u g h a
hole in a superconducting body, this flux, and the circulating c u r r e n t i
associated w i t h it, will persist even if t h e applied m a g n e t i c field s t r e n g t h
is changed or reduced to zero.
d
p
p
d
p
2.4.
Penetratio n Dept h
In § 2.3 w e h a v e seen t h a t t h e perfect d i a m a g n e t i s m of a s u p e r c o n d u c tor p r e v e n t s electric c u r r e n t s flowing t h r o u g h the b o d y of t h e material.
O n the o t h e r h a n d , c u r r e n t s c a n n o t b e confined entirely to t h e surface
because, if this w e r e so, t h e current sheet would h a v e n o thickness and
the current density would be infinite, w h i c h is a physical impossibility.
In fact t h e c u r r e n t s flow within a very thin surface layer w h o s e t h i c k n e s s
is of t h e order of 1 0 " cm, although t h e exact value varies for different
metals. W e shall see that, t h o u g h t h i s thickness is so small, it plays a
very i m p o r t a n t p a r t in d e t e r m i n i n g t h e p r o p e r t i e s of s u p e r c o n d u c t o r s .
5
W h e n a superconducting sample is in an applied m a g n e t i c field, t h e
screening c u r r e n t s w h i c h circulate to cancel the flux inside m u s t flow
within this surface layer. Consequently, t h e flux density does not fall
abruptly to zero at the b o u n d a r y of t h e metal b u t dies away w i t h i n t h e
region w h e r e t h e screening c u r r e n t s are flowing. F o r t h i s reason t h e
d e p t h within which the c u r r e n t s flow is called t h e penetration
depth,
because it is t h e d e p t h to w h i c h the flux of the applied m a g n e t i c field
appears to penetrate. T h u s , although w e speak of a s u p e r c o n d u c t o r as
being perfectly diamagnetic, t h e r e is in fact a very slight p e n e t r a t i o n of
magnetic flux, the flux density dying away at the surface as s h o w n in
PERFECT DIAMAGNETISM
27
Fig. 2.6. ( T h i s is s o m e w h a t like the "skin d e p t h " to w h i c h high frequency alternating fields penetrate in a normal conductor.) Consider t h e
b o u n d a r y of a semi-infinite slab, as shown in Fig. 2.6. If at a distance ÷
into the metal the flux density falls to a value B(x), w e can define the
penetration d e p t h ë by
CO
j B(x) dx = ëÂ(0),
ï
(2.2)
where B(0) is t h e flux density at the surface of the metal. I n other w o r d s ,
there would b e t h e same a m o u n t of flux inside the superconductor if t h e
flux density of t h e external field remained constant to a distance ë into
the metal.
FlG . 2.6. Penetratio n of magneti c flux int o surfac e of superconductor .
T h e L o n d o n theory of superconductivity, which w e shall discuss in
the next chapter, predicts t h a t in a specimen which is m u c h thicker t h a n
the penetration d e p t h the magnetic flux density decays exponentially as
it penetrates into the metal, i.e.
B(x) =
B(0)e~ .
x/A
However, in simple calculations it is often
proximation t h a t the flux density B(0) of t h e
stant to a distance ë into the metal and then
Because the penetration d e p t h is so small,
sufficient to use the a p applied field r e m a i n s conabruptly falls to zero.
w e do not notice the flux
28
INTRODUCTION TO SUPERCONDUCTIVITY
penetration in magnetic m e a s u r e m e n t s on o r d i n a r y sized s p e c i m e n s , !
and these appear t o be perfectly d i a m a g n e t i c w i t h  = 0. W e shall, for
convenience, c o n t i n u e t o refer t o reasonably large superconducting
bodies as being "perfectly" d i a m a g n e t i c , t h e very small m a g n e t i c flux at
the surface being included in t h i s t e r m . T h e p e n e t r a t i o n of t h e m a g n e t i c
flux b e c o m e s noticeable, however, if w e m a k e m e a s u r e m e n t s on small
samples, such as p o w d e r s or thin films, w h o s e d i m e n s i o n s are not m u c h
greater t h a n t h e p e n e t r a t i o n d e p t h . I n t h e s e cases t h e r e is an appreciable
magnetic flux density right t h r o u g h t h e m e t a l ; t h e r e is no longer perfect
diamagnetism, and consequently t h e p r o p e r t i e s are r a t h e r different from
those of the bulk superconducting metal. W e shall deal fully w i t h the
special features of thin specimens in C h a p t e r 8.
2.4.1.
Variatio n wit h temperatur e
T h e penetration d e p t h does n o t h a v e a fixed value b u t varies w i t h
t e m p e r a t u r e , as s h o w n in Fig. 2.7. At low t e m p e r a t u r e s it is nearly in3x1 0
í
2
1 ÷
Temperatur e (°K )
FIG . 2.7. Variation with temperature of penetration depth in tin (after Schawlow and
Devlin).
dependent of t e m p e r a t u r e and h a s a value ë characteristic of t h e p a r ticular metal ( T a b l e 2.1). Above about 0-8 of t h e transition t e m p e r a t u r e ,
however, the p e n e t r a t i o n d e p t h increases rapidly and a p p r o a c h e s infinity
as the t e m p e r a t u r e a p p r o a c h e s t h e transition t e m p e r a t u r e .
0
t T h e methods by which such measurements can be m a d e are discussed in § 4.4.
29
PERFECT DIAMAGNETISM
T h e variation of penetration d e p t h w i t h t e m p e r a t u r e is found to fit
very closely t h e relation
2—
(2.3)
(i-< )'
4
where t is the t e m p e r a t u r e relative to the transition t e m p e r a t u r e , t =
T/T .
e
Perfect d i a m a g n e t i s m does not, therefore, occur in specimens which
are very close to their transition t e m p e r a t u r e . T h e decrease in penetration d e p t h is, however, so rapid as the t e m p e r a t u r e falls below T t h a t
c
T A B L E 2.1.
SOM E V A L U E S O F T H E PENETRATIO N D E P T H AT
In
A (cm)
0
6-4 x 1 0 -
Al
6
5 0 ÷ ÉÏ"
0°K
Pb
6
3-9 x 1 0 -
6
any large d e p a r t u r e from perfect d i a m a g n e t i s m would b e extremely
difficult to detect in bulk specimens because of the difficulty of holding
the t e m p e r a t u r e sufficiently constant during a m e a s u r e m e n t . F o r example, t o observe a penetration d e p t h of about 1 m m a specimen would, according t o (2.3), have to b e held at a t e m p e r a t u r e only 10~ per cent
below the transition t e m p e r a t u r e 1 F u r t h e r m o r e , we would require a
specimen so p u r e and uniform t h a t the transition to the superconducting
state would take place within a t e m p e r a t u r e range which w a s less t h a n
1 0 " per cent of t h e transition t e m p e r a t u r e . T h e experimental relation
(2.3) h a s been obtained b y observing relatively small increases in t h e
penetration d e p t h as a superconducting metal is w a r m e d t o w a r d s its
transition t e m p e r a t u r e . In one experimental a r r a n g e m e n t ! a rod of p u r e
superconductor is surrounded b y a closely fitting solenoid. W h e n the
t e m p e r a t u r e of t h e metal is lowered so t h a t it becomes superconducting
there can be no magnetic flux in the metal, except j u s t below the surface
within the penetration depth. Because the solenoid fits t h e rod very
closely, its self-inductance is largely dependent on the m a g n i t u d e of this
penetration d e p t h . A capacitor is connected across the solenoid, the c o m bination forming the tank circuit of an oscillator of about 100 k H z
frequency. W h e n the inductance alters, because of a change in penetration depth, the oscillator frequency shifts. T h e frequency can b e
measured with great precision on an accurate frequency meter. In the
7
7
t A. L. Schawlow and G. E. Devlin, Phys. Rev. 113, 120 (1959).
30
INTRODUCTION TO SUPERCONDUCTIVITY
experiment of S c h a w l o w and Devlin a frequency shift of 0 1 H z w a s
equivalent t o a c h a n g e of 4 ÷ 10~ c m in t h e p e n e t r a t i o n d e p t h in t h e
metal core. S u c h an experiment needs t o be d o n e w i t h great care b e c a u s e
spurious effects, not d u e t o t h e change in p e n e t r a t i o n d e p t h , m a y alter
the r e s o n a n t frequency of t h e solenoid and capacitor. By t h i s m e t h o d t h e
t e m p e r a t u r e variation of t h e p e n e t r a t i o n d e p t h in tin w a s observed, and
t h e results are t h o s e s h o w n in Fig. 2.7. T h e very rapid increase in
penetration d e p t h as the transition t e m p e r a t u r e is a p p r o a c h e d is in good
accord w i t h (2.3).
U n l e s s w e specify otherwise, w h e n in t h i s book we speak of t h e
penetration depth, w e m e a n t h e value ë to w h i c h ë t e n d s w h e n t h e
metal is at t e m p e r a t u r e s w h i c h are appreciably below its transition
temperature.
T h e penetration d e p t h in a s u p e r c o n d u c t i n g metal also d e p e n d s on the
purity, the p e n e t r a t i o n d e p t h increasing as t h e metal b e c o m e s m o r e impure. F o r example, in tin containing 3 % i n d i u m t h e penetration d e p t h is
twice t h a t of p u r e tin.
8
0
CHAPTE R
3
ELECTRODYNAMICS
IN THI S chapter w e shall develop equations which govern t h e behaviour
of magnetic fields and electric c u r r e n t s in superconductors.
3.1.
Consequenc e of Zer o Resistanc e
In a superconducting metal t h e superelectrons encounter n o resistance
to their motion, so, if a constant electric field Å is maintained in the
material, the electrons accelerate steadily u n d e r the action of this field:
m\
s
= eE
(3.1)
w h e r e v is t h e velocity of t h e superelectrons and m and e are their m a s s
and c h a r g e . t If there are n superelectrons per unit volume moving with
velocity v , there is a supercurrent density
s
s
s
J* =
n ev .
s
s
Substituting this into (3.1) w e see t h a t an electric field produces a continuously increasing current w i t h a rate of increase given b y
ß. = ^ôç ·
Å
(·)
3 2
T o obtain an equation describing magnetic fields we r e m e m b e r t h a t a
magnetic field is related to an electric field and current b y Maxwell's
equations
and
 = - curl Å
(3.3)
c u r l H = J + D.
(3.4)
In t h i s chapter w e wish t o develop equations which explicitly relate the
fields in superconductors to the c u r r e n t s flowing in them, and w e shall
t In thi s book e is use d for th e charg e on th e electron , an d thi s symbo l include s th e negativ e
sign, e = —1-602 ÷ 10~ coulomb .
19
31
32
INTRODUCTION TO SUPERCONDUCTIVITY
adopt t h e viewpoint, discussed in § 2.2.2, t h a t t h e metal of a superconducting b o d y is, like other non-ferromagnetic metals, n o n - m a g n e t i c , i.e.
ì, = 1, so t h a t any flux density in t h e metal m u s t b e d u e t o t h e c u r r e n t s .
W e also take t h e view (see A p p e n d i x A) t h a t while c u r r e n t s in t h e metal
affect B , they d o not affect H , and w i t h i n t h e superconductor w e
therefore replace (3.4) b y
curl  = ì (]
0
+ f>),
5
w h e r e J is t h e current density w i t h i n t h e metal. F u r t h e r m o r e , unless t h e
fields are varying very rapidly in time, t h e displacement c u r r e n t D is
negligible in c o m p a r i s o n w i t h J . H e n c e inside a s u p e r c o n d u c t o r w e can
w r i t e M a x w e l l ' s e q u a t i o n s in t h e form
s
5
 = - c u r l E,
(3.3a)
curl  = ì ] .
(3.4a)
0
5
S u b s t i t u t i n g (3.2) into (3.3a) gives
n = -^curlj ,
(3.5)
s
and w e can eliminate J b y m e a n s of (3.4a):
s
 =
ô curl curl 6.
ì çâ
0
2
5
F o r brevity, let u s use a single symbol á for t h e c o n s t a n t ôç/ì ç â ,
last equation b e c o m e s
2
0
 = —á curl curl 6.
5
so t h e
(3.6)
Now
curl curl  = grad div  - V ft,
2
b u t from M a x w e l l ' s e q u a t i o n s div  = 0, so (3.6) b e c o m e s
6 = aV B
2
or
V B=-4.
2
(3.7)
a
T h i s is a differential e q u a t i o n w h i c h  m u s t satisfy. T o see w h a t t h i s
implies, consider t h e plane b o u n d a r y of a s u p e r c o n d u c t o r w i t h a uniform
magnetic field applied parallel t o this b o u n d a r y (Fig. 3.1). S u p p o s e t h a t
33
ELECTRODYNAMICS
the flux density outside t h e metal is B , and let the direction normal to
the b o u n d a r y b e called the ^-direction. Because the applied field is uniform, Â will have t h e same direction everywhere, so we m a y regard (3.7)
as a scalar e q u a t i o n ; also there will b e no gradients of t h e field parallel to
the b o u n d a r y , so in this case (3.7) reduces to
a
dx
2
(3.8)
a
T h e solution of t h i s i s t
**> = A . « P
( ^ )
(3.9)
where B(x) is t h e flux density at a distance ÷ inside the metal and B is
t h e value of  outside t h e metal. T h i s m e a n s t h a t 6 dies a w a y exponenta
FiG . 3.1. Magneti c field applie d paralle l t o boundar y of superconductor .
ially as we p e n e t r a t e into the superconductor. In other w o r d s , changes
in flux density d o not penetrate far below t h e surface, so at a sufficient
distance inside t h e metal t h e flux density h a s a constant value which
does not change w i t h time, irrespective of w h a t is happening to the
applied field.
3.2.
Th e Londo n Theor y
W e have deduced the above behaviour b y applying t h e usual laws of
electrodynamics t o a conductor w i t h zero resistance; however, t h o u g h
(3.9) completely describes the m a g n e t i c properties of a perfect conductt Ther e is an alternativ e solution , B exp (+x/^Ja)
a
f
bu t thi s approache s infinit y as ÷ increases .
34
INTRODUCTION TO SUPERCONDUCTIVITY
or, it does not adequately describe t h e b e h a v i o u r of a superconductor.
T h e M e i s s n e r effect s h o w s t h a t inside a s u p e r c o n d u c t o r t h e flux density
is not only c o n s t a n t b u t t h e value of t h i s c o n s t a n t is always z e r o ; so not
only  b u t  itself m u s t die a w a y rapidly below t h e surface. F . and H .
L o n d o n t suggested t h a t t h e m a g n e t i c b e h a v i o u r of a s u p e r c o n d u c t i n g
metal might b e correctly described if (3.7) applied n o t only to 6 b u t to Â
itself,
V B = -B.
á
(3.10)
2
If this were so, t h e m a g n e t i c flux density  would die a w a y w i t h i n t h e
metal in a m a n n e r similar to t h e b e h a v i o u r of A, as described b y (3.9),
i.e.
B(x) = B
exp
a
(-x/yja).
E x a m i n a t i o n of t h e a r g u m e n t b y w h i c h w e derived (3.7) s h o w s t h a t
w e could h a v e derived (3.10) if e v e r y w h e r e w e h a d replaced  b y B . If
w e retrace the a r g u m e n t , w e arrive at (3.5) w h i c h would n o w h a v e t h e
m o r e restrictive form
 = =5
ne
curl J , .
(3.11)
s
T h i s equation and (3.2), namely
j, = ^ E ,
(3.2)
w h i c h together describe t h e electrodynamics of t h e supercurrent, are
k n o w n a s t h e London
equations.
E q u a t i o n (3.2) d e s c r i b e s t h e
resistanceless property of a superconductor, t h e r e being n o electric field
in the metal unless t h e c u r r e n t is c h a n g i n g ; and (3.11) describes t h e
diamagnetism.
N o t e t h a t these e q u a t i o n s are not deduced from fundamental p r o p e r ties and d o not " e x p l a i n " t h e occurrence of superconductivity. T h e L o n don e q u a t i o n s are restrictions on the o r d i n a r y e q u a t i o n s of electrom a g n e t i s m , and are i n t r o d u c e d so t h a t t h e b e h a v i o u r d e d u c e d from
these laws agrees w i t h t h a t observed experimentally.
T h e L o n d o n e q u a t i o n s lead u s t o replace (3.7) b y (3.10). L e t u s n o w
use this latter e q u a t i o n to d e t e r m i n e t h e d i s t r i b u t i o n of m a g n e t i c flux inside a superconductor w h e n it is in a uniform m a g n e t i c field of flux d e n t F . London and H. London, Proc. Roy. Soc. (London)
A 1 5 5 , 71 (1935).
35
ELECTRODYNAMICS
sity B applied parallel to its surface (Fig. 3.2). In this case w e can use
the one-dimensional form of (3.10),
a
where B(x) is t h e flux density at distance ÷ inside t h e metal. T h i s equation h a s t h e solution
B(x) = B
a
exp
(3.12)
where B is t h e flux density of the applied field at the surface. E q u a t i o n
(3.12) shows that t h e flux density dies away exponentially inside a superconductor, falling to l/e of its value at t h e surface at a distance ÷ = yja.
T h i s distance is called the London penetration
depth, X . N o w
a = ôç/ì ç â , so t h e L o n d o n penetration d e p t h is given b y
a
L
2
0
5
Ê =
(3.13)
^(ôç/ì ç â ).
2
0
5
FlG . 3.2. Variatio n of flux densit y at boundar y of a superconductor .
If w e substitute for m and e t h e usual values of the electron's m a s s and
charge, and take n to b e about 4 ÷ 1 0 m " (i.e. the usual concentration
in metals; about one conduction electron per atom), the L o n d o n penetration depth t u r n s out to be about 1 0 " cm.
W e can n o w write (3.12) in t h e form
28
3
s
6
B(x)
=
B
a
exp
(-x/X ).
L
(3.14)
36
INTRODUCTION TO SUPERCONDUCTIVITY
T h e L o n d o n e q u a t i o n s predict, therefore, a very r a p i d exponential decay
of t h e flux density at t h e surface of a s u p e r c o n d u c t o r . I n C h a p t e r 2 w e
defined, by m e a n s of (2.2), a general p e n e t r a t i o n d e p t h w h i c h w a s ind e p e n d e n t of t h e particular form of t h e decrease of t h e flux density (i.e.
w h e t h e r exponential or otherwise). S u b s t i t u t i o n of (3.14) into (2.2)
s h o w s t h a t t h e L o n d o n p e n e t r a t i o n d e p t h A , defined b y (3.13) and
leading t o an exponential p e n e t r a t i o n law of t h e form (3.14), satisfies t h i s
general definition of p e n e t r a t i o n d e p t h .
L
In t h e previous c h a p t e r it w a s pointed o u t t h a t any current in a superconductor m u s t flow n e a r t h e surface. F o r a uniform m a g n e t i c field
applied parallel t o t h e surface (Fig. 3.1) in t h e s-direction, (3.4a) r e d u c e s
to—
dÂ
Â
e q u a l s — - p e x p (—#/A ), so w e
= ì .5í · F r o m (3.14) dB/dx
0
L
OX
have
A
L
J, = j ^ - e x p
(-x/X)
L
w h i c h w e m a y w r i t e as
W e see, therefore, t h a t any c u r r e n t flows close t o t h e surface w i t h i n t h e
penetration d e p t h .
W e have m e n t i o n e d t h a t according t o t h e two-fluid model of superconductivity (§ 1.4), t h e c o n c e n t r a t i o n of superelectrons n decreases as
the t e m p e r a t u r e is raised, falling t o zero at t h e t r a n s i t i o n t e m p e r a t u r e .
T h e L o n d o n e q u a t i o n s give a p e n e t r a t i o n d e p t h w h i c h is inversely
proportional t o w)(3.13) and so predict t h a t t h e p e n e t r a t i o n d e p t h should
increase w i t h increasing t e m p e r a t u r e , rising t o infinity a s t h e t e m p e r a t u r e a p p r o a c h e s t h e t r a n s i t i o n t e m p e r a t u r e . As w e saw in § 2.4.1, t h i s
is w h a t is observed experimentally.
s
W e can n o w w r i t e the L o n d o n e q u a t i o n s (3.11) a n d (3.2) as
c u r l J i =
-^
B
It is i m p o r t a n t to realize t h a t the L o n d o n e q u a t i o n s d o not replace
M a x w e l l ' s e q u a t i o n s , which, of course, still apply t o all c u r r e n t s a n d t h e
fields they p r o d u c e . T h e L o n d o n e q u a t i o n s are additional c o n d i t i o n s
obeyed b y t h e s u p e r c u r r e n t s .
ELECTRODYNAMICS
37
In the m o s t general case the total current J is the sum of a normal
current and a supercurrent:
J=J* + J*T h e normal current need only obey Maxwell's equations and O h m ' s law,
J. =
ï¸,
where ó' is a conductivity associated w i t h t h e n o r m a l electrons. W e
can n o w bring together t h e special equations which apply to a superconducting m e t a l :
J=J„+J,
(3.15)
(3.16)
curl J, =
-
1
Â
(3.17)
1
(3.18)
F r o m these e q u a t i o n s w e can in principle work out the distribution of
fields and c u r r e n t s in superconducting bodies under various conditions.
In t h e steady state, w h e n fields a n d c u r r e n t s are not changing w i t h time,
the only current is the supercurrent, i.e. J = 0, and w e need only employ
the L o n d o n e q u a t i o n s (3.17) and (3.18). T h e s e lead to
n
1
V ' B = - B .
(3.19)
It should b e pointed out t h a t t h e form of (3.10) introduced to describe
the distribution of magnetic flux density inside a superconductor w a s
only a guess, b a s e d on t h e k n o w n properties of superconductors, and w e
m u s t not expect t h e L o n d o n equations inferred from it to b e exactly correct. In fact, these equations are only approximations, t h o u g h for m a n y
purposes they are sufficiently accurate. F o r example, t h e L o n d o n
equations predict a small penetration d e p t h ; a small penetration d e p t h is
indeed observed experimentally, b u t t h e value is greater t h a n t h e L o n d o n
prediction b y a factor of t w o or more. A discussion of the limitations of
the L o n d o n theory and a description of t h e m o r e refined Ginzburg—Landau theory are given in C h a p t e r 8.
38
3.2.1.
INTRODUCTION TO SUPERCONDUCTIVITY
An applicatio n of th e L o n d o n theor y
W e can in principle use (3.19) t o find t h e distribution of flux density
within any superconducting b o d y b y applying to t h e solution of t h i s
equation b o u n d a r y conditions imposed b y t h e shape of the b o d y and t h e
form of t h e applied m a g n e t i c field. A s an example, w e n o w calculate t h e
distribution of flux density inside a superconducting slab, of finite
thickness, w i t h plane parallel faces, w h e n it is in a uniform m a g n e t i c field
applied parallel to these faces (Fig. 3.3). T h i s is a configuration w e shall
FIG . 3.3. Superconductin g slab in paralle l applie d magneti c field.
need to m a k e use of in a later c h a p t e r ( C h a p t e r 8). L e t t h e thickness of
the slab b e 2a and let t h e jc-coordinate b e n o r m a l t o t h e faces, w i t h the
origin at t h e mid-plane. Because t h e applied field is uniform,
SB
dy
=
0
SB
dz
=
and w e can w r i t e (3.19) as
d B{x)_
1
2
=
dx
1
x\
ilB{x).
T h e solution of t h i s equation is
B(x) = B
x
e ^
+ B
+x/
2
e~ ^
x/x
(3.20)
T h e values of t h e c o n s t a n t s B and B are obtained b y noting t h a t , from
symmetry, t h e variation of  in b o t h t h e +a? and —x directions m u s t b e
x
2
39
ELECTRODYNAMICS
the same. T h e r e f o r e B = B . F u r t h e r , w h e n ÷ = +a> t h e flux density
equals t h a t of t h e applied field, B(±a) = B . So (3.20) b e c o m e s
1
2
a
T h e flux density therefore varies as cosh ( # / ë ) . T h e w i d t h of m o s t
specimens will b e m u c h greater t h a n t h e penetration d e p t h k and then
the flux density will be very small except close t o t h e faces w h e r e
£
L
1*1
-
lei.
CHAPTER 4
THE CRITICAL MAGNETIC
FIELD
W E SHAL L see in C h a p t e r 9 t h a t , if a metal is t o r e m a i n
superconducting, the net m o m e n t u m of t h e superelectrons m u s t n o t
exceed a certain value. F o r t h i s reason t h e r e is a limit to t h e d e n s i t y of
resistanceless c u r r e n t t t h a t can be carried b y a n y region in t h e metal.
L e t u s call t h i s t h e critical current density J of t h e metal. T h i s critical
current density applies b o t h t o a c u r r e n t passed along t h e specimen from
an external source and t o screening c u r r e n t s which shield the specimen
from an applied m a g n e t i c field. W e n o w s h o w t h a t , as a result of t h i s
critical current density, a superconducting metal will be driven n o r m a l if
a sufficiently strong m a g n e t i c field is applied t o it.
c
As w e h a v e seen, t h e perfect d i a m a g n e t i s m of a s u p e r c o n d u c t o r arises
because, in an applied magnetic field, resistanceless surface c u r r e n t s circulate so as to cancel t h e flux density inside. If t h e s t r e n g t h of t h e
applied magnetic field is increased, t h e shielding c u r r e n t s m u s t also increase in order to m a i n t a i n perfect d i a m a g n e t i s m . If t h e applied
magnetic field is increased sufficiently, t h e critical current density will b e
reached b y the shielding c u r r e n t s and t h e metal will lose its superconductivity. T h e shielding c u r r e n t s t h e n cease and the flux d u e to t h e
applied m a g n e t i c field is n o longer cancelled w i t h i n t h e metal, t T h e r e is
therefore a limit t o the strength of m a g n e t i c field w h i c h c a n b e applied t o
a superconductor if it is t o r e m a i n superconducting. T h i s d e s t r u c t i o n of
superconductivity b y a sufficiently strong m a g n e t i c field is one of t h e
m o s t i m p o r t a n t properties of a superconductor.
At any point in a s u p e r c o n d u c t o r t h e r e is a definite relationship
t Strictl y speakin g it is not th e curren t bu t th e meta l whic h is resistanceless . However , th e
expressio n "resistanceles s current " is often used an d we shal l mak e us e of thi s convenien t phras e
t o avoid circumlocution .
t T h e transitio n fro m th e perfectl y diamagneti c superconductin g stat e t o th e non-magneti c
norma l stat e is a reversible transition . Th e screenin g current s d o not die awa y wit h dissipatio n of
energ y an d d o not generat e hea t in th e material . I n fact , a s we shal l see fro m thermodynami c
argument s in § 5.2.2, if a thermall y isolate d specime n is drive n norma l by a magneti c field it s
temperatur e falls.
40
41
THE CRITICAL MAGNETIC FIELD
b e t w e e n t h e supercurrent density J and t h e magnetic flux density,
t h o u g h the values of these are only appreciable within t h e penetration
depth. T h i s relationship can b e obtained from the L o n d o n e q u a t i o n s
(Chapter 3). T h e critical current density will therefore b e reached at the
surface w h e n t h e flux density of t h e applied m a g n e t i c field (i.e. the flux
density at t h e surface) is increased t o a certain value. W e m a y call this
the "critical flux d e n s i t y " B . H o w e v e r , the flux density outside the
metal always equals ì Ç w h e r e Ç is t h e magnetic field strength, so w e
m a y equally well refer to a critical magnetic field strength, H =
 /ì .
N o t w i t h s t a n d i n g our view t h a t  r a t h e r t h a n Ç is t h e basic magnetic
q u a n t i t y (see Appendix A), it is usual in t h e literature t o refer to t h e
critical magnetic field strength H r a t h e r t h a n t h e critical flux density B .
In t h i s book w e shall follow t h i s convention, and refer t o the critical
magnetic field strength H r e m e m b e r i n g t h a t t h i s is always related t o t h e
critical flux density b y H =
 /ì .
s
c
0
9
c
c
€
0
c
c
c
4.1.
€
0
Fre e Energ y of a Superconducto r
W e saw in § 2.2.1 t h a t t h e state of magnetization of a superconductor
depends only o n t h e values of t h e applied magnetic field and t e m p e r a t u r e
and not on t h e w a y these external conditions w e r e arrived at. T h i s implies that, w h e t h e r or not there is an applied magnetic field, t h e t r a n s i tion from the superconducting t o t h e n o r m a l state is reversible, in the
t h e r m o d y n a m i c sense. W e m a y therefore apply t h e r m o d y n a m i c
a r g u m e n t s t o a superconductor, using t h e t e m p e r a t u r e and magnetic
field strength as t h e r m o d y n a m i c variables.
It is possible to deduce something about t h e critical magnetic field
by considering w h a t effect the application of a magnetic field h a s on the
free energy of a superconducting specimen. W e are interested in the free
energy because in any system t h e stable state is t h a t w i t h t h e lowest free
energy. In considering the critical magnetic field of a superconductor w e
are interested in t h e Gibbs free energy, because w e w a n t to c o m p a r e t h e
difference in t h e magnetic contribution t o t h e free energy of t w o phases,
superconducting and normal, w h e n they are in t h e same applied
magnetic field (i.e. w i t h the intensive t h e r m o d y n a m i c variable constant).
Consider a specimen of superconductor in t h e form of a long rod. (At
this stage we consider a long rod so as to reduce to a negligible degree
special demagnetizing effects which are due t o the e n d s of t h e specimen.
T h e s e effects are discussed in § 6.2.) W h e n t h e specimen is cooled below
its transition t e m p e r a t u r e it b e c o m e s superconducting. Therefore, below
42
INTRODUCTION TO SUPERCONDUCTIVITY
the transition t e m p e r a t u r e t h e free energy of t h e s u p e r c o n d u c t i n g state
m u s t b e less t h a n t h a t of t h e n o r m a l state, o t h e r w i s e t h e m e t a l would r e m a i n normal. S u p p o s e t h a t at a t e m p e r a t u r e T, and in t h e absence of an
applied magnetic field (H = 0), the G i b b s free energy per unit volume of
the s u p e r c o n d u c t i n g state i s ^ ( T , 0) and t h a t of the n o r m a l state isg (T> 0)
(Fig. 4.1). N o w let a m a g n e t i c field of s t r e n g t h H be applied parallel
a
n
a
,
â,(Ô,Ç)
Applie d magneti c fiel d
·-
FIG . 4.1. Effect of applie d magneti c field on Gibb s fre e energ y of norma l an d
superconductin g states .
to t h e length of the rod. Any substance, w h i c h in an applied field H
acquires a m a g n e t i z a t i o n / , c h a n g e s its free energy per unit v o l u m e b y an
amountt
a
Ä£Ç )
= -ì
á
0
jldH .
(4.1)
a
0
So in the case of t h e field producing a positive m a g n e t i z a t i o n , i.e. t h e
magnetization in t h e s a m e direction as t h e m a g n e t i c field, t h e free energy
is lowered. [ N o t e t h a t (4.1) implies t h a t , w h e n a s u b s t a n c e is m a g n e t i z e d
by an applied m a g n e t i c field, its free energy c h a n g e s b y an a m o u n t
proportional t o t h e area u n d e r i t s m a g n e t i z a t i o n curve ( / v. H ). T h i s is a
useful general result, w h i c h w e shall employ several times.] In t h e case of
a superconducting specimen t h e application of a m a g n e t i c field p r o d u c e s
a negative m a g n e t i z a t i o n which, if p e n e t r a t i o n of t h e field is neglected,
exactly cancels t h e flux d u e to t h e applied field, so t h a t / = — H. T h e free
energy per unit v o l u m e is therefore increased t o
a
g (T,H)
s
= g (T,0)
s
+ ì^
\I\dH .
0
a
ï
t See Appendi x B.
THE CRITICAL MAGNETIC FIELD
43
In fact 11\ = H, so the magnetic field raises t h e free energy density to
ÅëÔ,Ç)
= é{Ô,0)
8
+
ì
0
ø .
(4.2)
So, when we apply a magnetic field to a superconductor, its free energy
increases to this value due to the magnetization (Fig. 4 . 1 ) . T h e normal
state, however, is virtually non-magnetic and acquires negligible
magnetization in an applied magnetic field. Consequently t h e application
of a magnetic field does not change the free energy of t h e normal state
though it raises t h a t of t h e superconducting state. If the field strength is
increased enough, the free energy of t h e superconducting state will b e
raised above t h a t of the normal state, and, in t h i s case, t h e metal will not
remain superconducting b u t will b e c o m e normal. T h i s occurs w h e n
g (T> H) > g (T, 0 ) , which w i t h ( 4 . 2 ) gives
s
n
Ë | > [ ^ 0 ) -
&
( Ã , 0 ) ] ,
T h e r e is therefore a m a x i m u m magnetic field strength t h a t can b e
applied t o a superconductor if it is to r e m a i n in the superconducting
state. T h i s critical magnetic field strength is given b y
H {T) =
C
[g„(T, 0 ) - g (T, 0 ) ] } \
s
(4.3)
T h i s critical magnetic field w h i c h w e have derived b y a t h e r m o d y n a m i c
argument is t h e same as t h e critical field which we discussed on p. 4 0 in
t e r m s of a critical current density.
T h e critical magnetic field strength can b e measured q u i t e simply b y
applying a magnetic field parallel to a wire of superconductor and observing t h e strength at which resistance appears.
4.2.
V a r i a t i o n o f Critica l F i e l d w i t h T e m p e r a t u r e
If the critical magnetic field of a superconductor is measured, its value
is found to depend on t h e t e m p e r a t u r e (Fig. 4 . 2 ) , falling from some value
H , at very low t e m p e r a t u r e s , t o zero at the superconducting transition
t e m p e r a t u r e T . A diagram such as Fig. 4 . 2 can b e called t h e phase
diagram of a superconductor. T h e metal will be superconducting for any
combination of t e m p e r a t u r e and applied field which gives a point, such
as P , lying within the shaded region. As the a r r o w s indicate, the metal
can b e driven into the normal state b y increasing either the t e m p e r a t u r e
or the applied field, or both.
0
c
44
INTRODUCTION TO SUPERCONDUCTIVITY
Temperatur e
+-
FIG . 4.2. Phas e diagra m of a superconductor , showing variatio n with temperatur e of
the critica l magneti c field.
7x10 •
4
800
6 x10 4
700
5x10 -
600
4xt0 -
500
4
4
„
(A
3
3x10 -
400
2x10 -
300
4
4
5
200
1 ÷ 10 ;
4
100
0
1
2
3
4
5
6
7
8
Temperatur e (°K)
FIG . 4.3. Critica lfields of some superconductors .
T h e value of H is different for each superconducting m e t a l ; m e t a l s
w i t h low transition t e m p e r a t u r e s h a v e low critical fields at absolute zero.
So each superconductor h a s a different p h a s e d i a g r a m (Fig. 4.3). F r o m
experiment it h a s been found t h a t t h e critical fields fall off almost a s t h e
square of t h e t e m p e r a t u r e , so t h e critical field c u r v e s a r e closely a p proximated b y parabolas of t h e form
Q
45
THE CRITICAL MAGNETIC FIELD
H =
H [\-(T/T n
C
0
(4.4)
c
where H is t h e critical field at absolute zero and T is t h e transition
t e m p e r a t u r e . E a c h superconductor can be characterized b y its particular
values of H and T , and, knowing these, w e can use (4.4) to find the
critical field at any t e m p e r a t u r e . T a b l e 1.1 lists values of H and T for
the superconducting elements.
W e m a y remark here that there is n o fundamental significance in the
relation b e t w e e n critical field and t e m p e r a t u r e being a parabola. I t h a s
merely been found experimentally t h a t t h e variation can b e conveniently
described to w i t h i n a few per cent b y a parabolic curve. T h e experimental curves are not in fact exactly parabolas, and t o describe t h e m accurately one would need a polynomial expression. F o r m o s t calculations,
however, it is sufficient to use t h e parabola given b y (4.4).
0
c
0
c
0
c
The Cryotron
T h e existence of a critical magnetic field h a s been m a d e use of in a
controlled switch called a cryotron (Fig. 4.4). T h e current J w h i c h is to
be controlled flows along a straight wire of t a n t a l u m called t h e " g a t e " .
Gat e (Ta )
Contro l
(Nb )
•e
FIG. 4.4. Cryotron of tantalum and niobium.
Around this, b u t insulated from it, is a niobium wire w o u n d in a long
single-layer coil called the "control". W h e n cooled to 4-2°K by i m m e r sion in liquid helium b o t h the t a n t a l u m gate and the niobium control are
superconducting, and the gate offers no resistance to the passage of the
current / . If, however, a current i is passed t h r o u g h the control coil, it
generates a magnetic field along the gate, and, if the control current is
large enough, the gate is driven normal b y t h e magnetic field, and t h e
appearance of resistance reduces the current , / . T h e control coil,
however, r e m a i n s resistanceless because the critical field of niobium is
considerably higher t h a n that of tantalum. H e n c e the current J t h r o u g h
the gate can be controlled by a smaller current in the control, and t h e
c
46
INTRODUCTION TO SUPERCONDUCTIVITY
device is analogous to a relay. Small c r y o t r o n s w e r e first developed as
fast acting switches for possible u s e in digital c o m p u t e r s . L a r g e
c r y o t r o n s can b e used to control t h e c u r r e n t s in superconducting m a g n e t
circuits (see § 1.3).
4.3.
Magnetizatio n of Superconductor s
W e n o w discuss h o w t h e m a g n e t i z a t i o n of a s u p e r c o n d u c t i n g
specimen varies as a magnetic field of increasing s t r e n g t h is applied. L e t
u s again consider a rod of s u p e r c o n d u c t o r and imagine a m a g n e t i c field
H t o b e applied parallel to its length. F i g u r e 4.5a s h o w s h o w t h e flux
density  inside t h e specimen varies as w e increase the s t r e n g t h of t h e
a
I = Ï
Superconducto
r
(b )
FIG . 4.5. Magneti c behaviou r of a superconductor .
applied field. N o r m a l m e t a l s (excluding t h e special ferromagnetic metals,
such as iron) are virtually n o n - m a g n e t i c and so t h e flux density  inside
t h e m is proportional to t h e s t r e n g t h of t h e applied field,  = ìïÇ , as
shown b y t h e d o t t e d line in Fig. 4.5 a. A superconductor, however, is
perfectly d i a m a g n e t i c if w e neglect t h e p e n e t r a t i o n d e p t h , and as t h e
applied magnetic field strength is increased, the flux density w i t h i n t h e
specimen r e m a i n s at zero. B u t w h e n t h e applied field s t r e n g t h r e a c h e s
the critical value H , t h e superconductor is driven into t h e n o r m a l state,
and t h e flux d u e t o t h e applied field is n o longer cancelled inside. At all
higher applied field s t r e n g t h s t h e s u p e r c o n d u c t o r b e h a v e s j u s t like a n o r mal metal. F o r a p u r e specimen t h i s b e h a v i o u r is reversible; if t h e
applied m a g n e t i c field is decreased from a h i g h value, t h e specimen goes
back into t h e superconducting state at t h e value H and below t h i s t h e r e
is n o net flux inside.
á
c
c
47
THE CRITICAL MAGNETIC FIELD
W e can describe the magnetic behaviour of a superconductor in
another way. W e have seen that, w h e n a metal is in t h e superconducting
state, t h e r e is n o magnetic flux inside, because surface c u r r e n t s circulate
t o give t h e specimen a magnetization J exactly equal and opposite to t h e
applied field, so t h a t / = — H . F i g u r e 4.5b s h o w s h o w t h e magnetization
of a superconductor varies w i t h t h e strength of t h e applied magnetic
field. W h e n t h e applied field strength reaches H t h e superconductor
b e c o m e s normal and t h e negative magnetization disappears. At higher
applied fields the superconductor h a s , like any normal metal, virtually n o
magnetization. F i g u r e 4.5 is, of course, j u s t t w o equivalent w a y s of
presenting t h e s a m e information. W e need t o b e familiar w i t h t h e form
of b o t h curves, because sometimes it is convenient to consider the internal flux density, w h e r e a s on other occasions it is m o r e convenient to consider magnetization.
a
c
4.3.1.
"Non-ideal " specimen s
T h e magnetic properties w e h a v e been considering so far in this
chapter are those t h a t would b e shown b y " i d e a l " specimens, i.e. those
containing n o impurities or crystalline faults. Any real specimen is,
however, n o t perfect, and i t s behaviour will depart to s o m e extent from
the behaviour w e have j u s t described. Nevertheless, it is possible w i t h
great care to produce specimens so nearly perfect t h a t they have properties very closely approximating t o the ideal. H o w e v e r , t h e greater t h e
degree of imperfection t h e greater will b e the d e p a r t u r e from ideal
behaviour.
An ideal specimen h a s a sharply defined critical field strength and its
magnetization curve is completely reversible. Figure 4.6 illustrates the
Ç
ï
I.T.S.—c
Ç
FIG . 4.6. Magneti c behaviou r of a non-idea l superconductor .
INTRODUCTION TO SUPERCONDUCTIVITY
48
magnetic b e h a v i o u r of an imperfect sample. I t can b e seen t h a t t h e r e is
n o longer a sharply defined critical m a g n e t i c field, t h e t r a n s i t i o n from t h e
superconducting t o n o r m a l state being " s m e a r e d o u t " over a r a n g e of
applied field strengths. F u r t h e r m o r e , t h e m a g n e t i z a t i o n is not reversible;
in decreasing fields the curves follow different p a t h s from those traced in
the original increasing field. W e call t h i s hysteresis. Finally, w h e n the
applied field h a s been reduced t o zero, there m a y r e m a i n some positive
magnetization of t h e sample, giving rise t o a residual flux density B and
magnetization I . W e say t h e sample h a s trapped flux. I n t h i s condition
the superconductor is like a p e r m a n e n t m a g n e t .
W e see, therefore, t h a t a non-ideal specimen m a y s h o w :
T
T
1. Ill-defined critical m a g n e t i c field.
2. M a g n e t i c hysteresis.
3. T r a p p e d flux.
T h e s e three d e p a r t u r e s from ideal b e h a v i o u r d o n o t necessarily all occur
together. F o r example, a specimen m a y not h a v e a sharp critical field and
m a y show hysteresis b u t still not t r a p any flux. Defects involving large
n u m b e r s of a t o m s , such as particles of another s u b s t a n c e or t h e c h a i n s of
displaced a t o m s k n o w n as dislocations, t e n d to give rise to hysteresis
and t r a p p e d flux, w h e r e a s i m p u r i t y a t o m s and u n e v e n n e s s of c o m p o s i tion reduce t h e s h a r p n e s s of the critical field. T h e r e a s o n s w h y different
k i n d s of i m p u r i t y and imperfection p r o d u c e t h e various d e p a r t u r e s from
ideal behaviour are complicated and n o t yet fully u n d e r s t o o d , and so w e
shall not discuss t h e m in detail here. H o w e v e r , these effects are of considerable practical i m p o r t a n c e and w e shall r e t u r n to t h e m again in
C h a p t e r 12.
4.4.
Measuremen t of Magneti c Propertie s
T h e t e c h n i q u e s w h i c h can b e used t o m e a s u r e t h e m a g n e t i c
characteristics of s u p e r c o n d u c t o r s d o not differ in principle from those
used in m e a s u r e m e n t s on ordinary m a g n e t i c materials, such as a ferrom a g n e t i c s ; b u t they m u s t , of course, b e suitable for use at very low
t e m p e r a t u r e s . T h e m e t h o d s fall into t w o classes: those w h i c h m e a s u r e
the flux density  in the sample and t h o s e w h i c h m e a s u r e the s a m p l e ' s
magnetization / (Fig. 4.5). T h o u g h either of these gives t h e full information on the m a g n e t i c p r o p e r t i e s of the sample, it m a y b e convenient t o
choose one or o t h e r m e t h o d of m e a s u r e m e n t , depending o n circ u m s t a n c e . M a n y different t y p e s of a p p a r a t u s h a v e b e e n used, of v a r i o u s
THE CRITICAL MAGNETIC FIELD
49
degrees of complexity depending on the sensitivity, degree of a u t o m a tion, etc., required. However, these are all based on the simple m e t h o d s
we n o w describe.
4.4.1.
M e a s u r e m e n t o f flux d e n s i t y
T h i s is a very simple m e a s u r e m e n t which does not require any moving
p a r t s in the low t e m p e r a t u r e region. T h e principle is to m e a s u r e the
magnetic flux w h i c h a p p e a r s in a specimen w h e n a magnetic field is
applied. O n the specimen X (Fig. 4.7) is w o u n d a " p i c k - u p " coil C consisting of a few h u n d r e d t u r n s of fine wire. T h e ends of this coil are connected to a ballistic galvanometer G outside the low t e m p e r a t u r e a p p a r a t u s . A m a g n e t i c field Ç can b e applied parallel to t h e axis of the
specimen b y m e a n s of the solenoid S, and w h e n the magnetic field is
suddenly applied b y closing the switch, the ballistic galvanometer will b e
deflected by an a m o u n t proportional to the flux threading the pick-up
coil C, i.e. the deflection is proportional to t h e flux density B. H e n c e b y
successively applying stronger magnetic fields w e can follow t h e varia-
D.C.
powe r
suppl y
FIG . 4.7. Measuremen t of magneti c flux densit y in a superconductor .
50
INTRODUCTION
TO
SUPERCONDUCTIVITY
tion of  w i t h field s t r e n g t h H. F i g u r e 4.8 s h o w s a  v e r s u s Ç
characteristic obtained in this way. T h e critical field H at w h i c h t h e
specimen ceases t o b e perfectly d i a m a g n e t i c c a n clearly b e seen.
I n the case of a non-ideal specimen some flux will b e t r a p p e d and so,
w h e n t h e field is switched off, t h e r e will b e a smaller reverse deflection of
c
Oerste d
Å
ï
ù
0
10,000
|
20,000
30,000
Critica l
field
Applie d magneti c fiel d strengt h (A/m )
FIG . 4.8. Experimenta l result s of ballisti c flux measuremen t on a superconducto r
(tantalu m at 3 7°K) . T h e smal l "background " deflectio n è' is du e t o th e flux densit y
produce d by th e applie d field in th e actua l wir e of th e pick-u p coil (afte r Rose-Innes) .
the galvanometer. I t is therefore necessary to n o t e t h e g a l v a n o m e t e r
deflection b o t h w h e n switching on a n d w h e n switching off t h e applied
m a g n e t i c field in order t o observe h o w m u c h flux is t r a p p e d .
4.4.2.
Measuremen t of magnetizatio n
I n this m e t h o d t h e specimen is m o v e d into and o u t of a p i c k - u p coil
w h i c h is connected t o a ballistic galvanometer. T h e t h r o w of t h e
THE CRITICAL MAGNETIC FIELD
51
galvanometer will b e proportional to t h e flux carried b y t h e sample and
this will b e proportional t o its magnetization. T h e m e t h o d is illustrated
in Fig. 4 . 9 . T h e specimen X is m o u n t e d o n t h e end of a sliding rod R so
t h a t it can b e moved from t h e upper pick-up coil A t o t h e lower one B. A
and  are identical coils, except t h a t they are w o u n d in opposite direc-
FIG . 4.9. Apparatu s t o measur e magnetization .
tions. T o m e a s u r e the magnetization of the sample a steady magnetic
field of t h e required m a g n i t u d e is applied b y m e a n s of t h e solenoid S and
the specimen is t h e n moved quickly from coil A into coil B. I n t h e case of
a diamagnetic sample, t h e flux threading coil A will therefore increase
and t h e flux t h r o u g h  will decrease. Since t h e t w o coils are connected in
series opposition t h e e.m.f.s. induced in t h e m will add together and t h e
ballistic galvanometer will swing b y an a m o u n t proportional to t h e
magnetization of t h e sample. T h e m e a s u r e m e n t is repeated at gradually
increasing applied magnetic field strengths, and so w e m a y construct a
g r a p h of sample magnetization as a function of applied field strength.
52
INTRODUCTION TO SUPERCONDUCTIVITY
Because A a n d  are w o u n d in o p p o s i t e directions any e.m.f.s. d u e t o u n intended fluctuations of t h e applied field s t r e n g t h should cancel a n d n o t
deflect t h e galvanometer.
A feature of m e a s u r e m e n t s o n s u p e r c o n d u c t o r s is t h a t they are selfcalibrating. At applied field s t r e n g t h s below t h e critical field strength, t h e
slope of t h e / v e r s u s Ç curve always e q u a l s — 1 .
4.4.3.
Integratin g metho d
In this m e t h o d , illustrated in F i g . 4.10, w e again h a v e t w o identical
pick-up coils A a n d  connected in series opposition, b u t t h e specimen is
now p e r m a n e n t l y located in one of t h e m . T h e c u r r e n t t o t h e solenoid S is
÷ -Y
FIG . 4.10. Integratio n metho d of measurin g magnetization .
gradually and smoothly increased so t h a t a continuously increasing
magnetic field is applied t o t h e pick-up coils. A n e.m.f. will b e developed
across each coil, proportional t o t h e r a t e of c h a n g e of t h e m a g n e t i c flux
threading it, b u t , since o n e coil c o n t a i n s t h e sample a n d t h e other d o e s
not, a n d since they are in series opposition, t h e n e t e.m.f. 6 across t h e
t w o coils will b e proportional t o t h e r a t e of c h a n g e of m a g n e t i z a t i o n of
the s a m p l e :
T H E CRITICA L MAGNETI C
FIEL D
53
T h i s e.m.f. is fed into an electronic integrator, a circuit w h o s e o u t p u t
voltage is related to its input voltage b y V oc j V dt. T h e o u t p u t of the
integrator is therefore
oxxt
in
ï
so at any t i m e t h e o u t p u t voltage of t h e integrator is proportional to the
magnetization of the sample.
T h i s m e t h o d of measuring magnetization h a s the advantage t h a t no
m o v e m e n t of t h e specimen is required, b u t t h e electronic e q u i p m e n t is
more complicated t h a n the a p p a r a t u s described in t h e previous t w o
sections.
CHAPTER 5
THERMODYNAMICS
OF T H E
TRANSITION
IN PREVIOUS c h a p t e r s w e h a v e from t i m e t o t i m e m a d e use of
t h e r m o d y n a m i c a r g u m e n t s to derive s o m e of t h e p r o p e r t i e s of superconductors. I n particular, w e have been able t o learn m u c h about critical
magnetic fields b y considering h o w t h e free energy of a s u p e r c o n d u c t o r
is altered b y t h e application of a m a g n e t i c field. I n t h i s c h a p t e r w e d i s cuss a few further t h e r m o d y n a m i c aspects n o t treated elsewhere in t h i s
book.
5.1.
Entrop y of th e Superconductin g Stat e
W e saw in C h a p t e r 4 t h a t t h o u g h t h e free energy density g of a metal
in t h e normal state is independent of t h e s t r e n g t h H of any applied
magnetic field, t h e application of a m a g n e t i c field raises t h e free energy
density & of t h e metal in t h e superconducting state b y an a m o u n t \ì^ÇÀ>
T h e critical field H is t h a t field strength w h i c h would b e required to
raise t h e free energy of the superconducting state above t h a t of t h e normal state. W e have, therefore, in an applied m a g n e t i c field of strength H
a difference in free energy b e t w e e n t h e n o r m a l a n d superconducting
states,
n
a
c
a
gn ~ g*(H ) = \ì (ÇÚ
a
0
- ¹).
(5.1)
As s h o w n in Appendix B , t h e free energy of a m a g n e t i c b o d y c a n b e
written
G=U-TS
+
pV- Ç Ì,
ìï
á
w h e r e U is the internal energy, S t h e entropy, p t h e pressure, V t h e
volume, H t h e applied m a g n e t i c field and Ì t h e m a g n e t i c m o m e n t . If
the pressure and applied field strength are kept c o n s t a n t b u t t h e
t e m p e r a t u r e is varied b y an a m o u n t dT there will b e a c h a n g e of free
energy,
a
dG=dU-
TdS -
SdT
54
+ pdV -
ìïÇ ÜÌ.
á
55
THERMODYNAMICS OF THE TRANSITION
But, b y the first law of t h e r m o d y n a m i c s ,
ì Ç ÜÌ
dU = TdS - pdV +
so
dG = -SdT
0
á
S= -i|§
and
T h e entropy per unit volume is given b y
s= —
Substituting (5.1) into this w e obtain for a s u p e r c o n d u c t o r , t since H
does not depend on T,
a
rj
*+**c
(5.2)
N o w the critical magnetic field always decreases w i t h increase of
t e m p e r a t u r e , so dHjdT
is always negative and the right-hand side of
this equation m u s t b e positive. Hence, b y simple t h e r m o d y n a m i c
r e a s o n i n g a p p l i e d t o t h e k n o w n v a r i a t i o n of critical field w i t h
t e m p e r a t u r e , w e have been able t o deduce t h a t the e n t r o p y of the superconducting state is less t h a n t h a t of the normal state, i.e. t h a t the superconducting state h a s a higher degree of order t h a n the normal state. T h i s
is in agreement w i t h the B C S microscopic description of superconductivity ( C h a p t e r 9) according to w h i c h t h e electrons in a superconductor
" c o n d e n s e " into a highly correlated system of electron pairs.
T h e critical field H falls to zero as the t e m p e r a t u r e is raised t o w a r d s
T , therefore, according to (5.2), the entropy difference b e t w e e n the normal and superconducting states vanishes at this t e m p e r a t u r e . F u r t h e r more, b y the t h i r d law of t h e r m o d y n a m i c s , s' m u s t also equal s at Ô = 0.
An example of t h e t e m p e r a t u r e variation of the entropies is shown in
Fig. 5.1.
F r o m the fact t h a t t h e entropies of the superconducting and normal
states m u s t b e t h e s a m e at Ô = 0 w e can deduce from (5.2) that, since
the critical field H is not zero, dHjdT
m u s t be zero at 0°K. T h i s is in
accordance with t h e experimental observation that, for all superconducc
c
n
s
c
t H does not appear in (5.2) and, as we have assumed the normal state to be non-magnetic
(i.e. properties independent of any applied field), this equation implies that the entropy of the
superconducting state is independent of any applied magnetic field. T h i s is only strictly true if we
ignore the flux within the penetration depth. Equation (5.1), from which (5.2) is derived, did not
take account of any flux penetration into the superconducting state. Equation (5.2) applies to
bulk samples, i.e. those with dimensions greater than the penetration depth.
a
INTRODUCTION TO SUPERCONDUCTIVITY
FIG . 5 . 1 . Entrop y of norma l an d superconductin g tin (base d on Keeso m an d va n
Laer) . 7 , an d T refe r t o th e adiabati c magnetizatio n proces s describe d in § 5 . 2 . 2 .
2
FIG . 5 . 2 . Specific hea t of tin in norma l an d superconductin g states .
57
THERMODYNAMICS OF THE TRANSITION
tors, the slope of t h e H versus Ô curve (Fig. 4.2) appears t o b e c o m e zero
as the t e m p e r a t u r e approaches 0°K.
c
5.2.
Specifi c H e a t an d Laten t Hea t
M u c h of t h e u n d e r s t a n d i n g of superconductors h a s been derived from
m e a s u r e m e n t s of their specific heat. T h e solid curve in Fig. 5.2 s h o w s
h o w t h e specific heat of a typical type-I superconductor varies w i t h
t e m p e r a t u r e in t h e absence of any applied m a g n e t i c field. W e can d r a w
several conclusions from the form of this curve, particularly if w e c o m pare it w i t h the specific heat curve of t h e same metal in the normal state.
T h e curve for t h e normal state can be obtained by making m e a s u r e m e n t s
in an applied magnetic field strong enough to drive t h e superconductor
normal.
5.2.1.
First-orde r an d second-orde r transition s
T h e form of the specific heat curves can be predicted from
thermodynamic arguments.
Since, at t h e transition t e m p e r a t u r e , $ = s and w e h a v e s h o w n t h a t
s = —(dg/dT)
w e have, for t h e superconducting-normal transition
at T ,
n
s
pJia9
c
A phase transition which satisfies this condition (i.e. not only is g cont i n u o u s b u t dgl d Ô is also continuous) is k n o w n as a second-order
phase
transition.
A second-order transition h a s t w o i m p o r t a n t characteristics:
at the transition there is n o latent heat, and there is a j u m p in t h e specific
h e a t . t T h e first characteristic follows immediately from t h e fact t h a t dQ
= Tds and w e h a v e seen t h a t at the transition t e m p e r a t u r e s = s .
H e n c e w h e n t h e transition occurs there is n o change in e n t r o p y and
therefore n o latent heat. T h e second condition follows from t h e fact t h a t
the specific heat of a material is given b y
n
5
(5.3)
t For a discussion of second-order phase transitions, see Pippard, Classical
Cambridge University Press, 1961, Ch. 9.
Thermodynamics,
58
INTRODUCTIO N
TO
SUPERCONDUCTIVIT Y
w h e r e í is t h e volume per unit m a s s , so t h e difference in t h e specific
h e a t s of t h e superconducting and n o r m a l s t a t e s can b e obtained from
(5.2):
C -C„
= íÔì Ç,
s
^
0
+ íÔì
0
( ^ J .
(5.4)
In particular, at the transition t e m p e r a t u r e , H = 0 and so w e h a v e for
the transition in t h e absence of an applied m a g n e t i c field
c
{C -C )
s
n T
= v T ^ { ^ )\
(5.4a)
T h i s is k n o w n as R u t g e r s ' formula, and it p r e d i c t s t h e value of t h e discontinuity in t h e specific heat of a s u p e r c o n d u c t o r at t h e transition
t e m p e r a t u r e . It should b e emphasized t h a t , t h o u g h (5.4a) c o n t a i n s a t e r m
depending on H
it gives t h e specific heat difference in zero applied
magnetic field. dHjdT
is a p r o p e r t y of t h e material w h o s e value does
not depend on w h e t h e r or not a field is actually present. E q u a t i o n (5.4),
on the other h a n d , gives t h e specific heat difference in the presence of an
applied magnetic field in w h i c h case t h e t e m p e r a t u r e of the t r a n s i t i o n is
reduced from T to T. E x p r e s s i o n s (5.4) and (5.4a) provide a useful link
b e t w e e n t h e m e a s u r e d critical m a g n e t i c field curve and the therm o d y n a m i c properties. W e can, for example, derive t h e m a g n i t u d e of t h e
specific heat j u m p at T from m e a s u r e m e n t s of the slope of the critical
field curve. F u r t h e r m o r e w e can provide a check on experimentally
measured q u a n t i t i e s by seeing w h e t h e r they satisfy these e q u a t i o n s .
ci
c
c
T h o u g h there is no latent heat w h e n , in t h e absence of a m a g n e t i c
field, a metal u n d e r g o e s t h e s u p e r c o n d u c t i n g - n o r m a l transition, t h e r e is a
latent heat if a magnetic field is present. T h e latent heat L for a t r a n s i tion b e t w e e n t w o p h a s e s a and b is given b y L = vT\s — $ ), so from
(5.2) w e have
a
1 = -íÔì Ç ^.
(5.5)
£
0
Ë
€
In t h e absence of any m a g n e t i c field t h e t r a n s i t i o n o c c u r s at t h e t r a n s i tion t e m p e r a t u r e and H = 0, b u t if t h e r e is a m a g n e t i c field the t r a n s i tion occurs at some lower t e m p e r a t u r e Ô w h e r e H > 0. T h i s latent heat
arises b e c a u s e at t e m p e r a t u r e s b e t w e e n T and 0°K the e n t r o p y of the
normal state is greater t h a n t h a t of the superconducting state, so heat
m u s t be supplied if t h e transition is to take place at c o n s t a n t
t e m p e r a t u r e . In t h e presence of an applied m a g n e t i c field, therefore, t h e
c
c
c
THERMODYNAMICS OF THE TRANSITION
59
superconducting-normal transition is of the first-order,
continuous, dg/dT
is not.
i.e. although g is
5.2.2.
Adiabati c magnetizatio n
T h e fact t h a t at t e m p e r a t u r e s below T there is a latent heat and t h e
entropy of t h e normal state is greater t h a n t h a t of the superconducting
state h a s an interesting consequence. In an ordinary m a g n e t i c material
application of a magnetic field decreases t h e entropy because the atomic
dipoles become aligned in t h e field. T h i s decrease in entropy w i t h
magnetic field strength is t h e basis of the well-known m e t h o d of
lowering t e m p e r a t u r e by " a d i a b a t i c demagnetization", w h e r e t h e
t e m p e r a t u r e of a thermally isolated specimen falls as the applied
magnetic field is reduced. However, the application of a sufficiently
strong magnetic field to a superconducting metal will drive it into the
normal state and at a given t e m p e r a t u r e this h a s a greater e n t r o p y t h a n
the superconducting state. If the specimen is thermally isolated n o heat
can enter and t h e latent heat of the transition m u s t c o m e from the thermal energy of the crystal lattice. H e n c e t h e t e m p e r a t u r e falls. T h u s , in
contrast to a p a r a m a g n e t i c material, a superconductor is cooled b y
adiabatic magnetization.
T h e t e m p e r a t u r e d r o p to b e expected can b e
deduced from an entropy diagram such as Fig. 5.1. If the superconductor
is initially at t e m p e r a t u r e 7 \ , adiabatic destruction of t h e superconductivity b y the magnetic field will take the material from point 1 to 2 and
the t e m p e r a t u r e will fall to T .
Although M e n d e l s s o h n and his co-workers have d e m o n s t r a t e d t h a t a
lowering of t e m p e r a t u r e can b e obtained by this method, it is not used in
practice t o obtain very low t e m p e r a t u r e s because there are several p r a c tical disadvantages compared to other m e t h o d s of cooling.
c
2
5.2.3.
Lattic e an d electroni c specifi c h e a t s
T h e r e are t w o contributions to the specific heat of a metal. H e a t raises
the t e m p e r a t u r e b o t h of t h e crystal lattice and of the conduction electrons. W e m a y therefore w r i t e the specific heat of a metal as
C — Ci
a t t
+ Ci .
e
However, as will b e pointed out in C h a p t e r 9, the properties of the lattice
(crystal structure, D e b y e t e m p e r a t u r e , etc.) d o not change w h e n a metal
becomes superconducting, and so C i m u s t b e the same in b o t h t h e
att
60
INTRODUCTION TO SUPERCONDUCTIVITY
superconducting and n o r m a l states. H e n c e t h e difference b e t w e e n t h e
specific heat values in the superconducting and n o r m a l s t a t e s arises only
from a change in the electronic specific heat, i.e.
C — C — ( C ) — (C |) .
s
n
d
s
e
n
T h e fact t h a t j u s t below t h e transition t e m p e r a t u r e t h e specific heat is
greater in t h e superconducting state t h a n in t h e n o r m a l state implies
that, w h e n a metal in t h e superconducting state is cooled t h r o u g h this
region, the e n t r o p y of its conduction electrons decreases m o r e rapidly
w i t h t e m p e r a t u r e t h a n if it w e r e in t h e n o r m a l state [see (5.3)]. H e n c e on
cooling a s u p e r c o n d u c t o r some extra form of electron order m u s t begin
to set in at t h e transition t e m p e r a t u r e in addition to t h e usual decrease in
e n t r o p y of t h e conduction electrons w h i c h occurs w h e n a n o r m a l metal is
cooled. T h i s additional order increases as the t e m p e r a t u r e is lowered and
so gives an extra contribution to dS/dT
and therefore increases t h e
specific heat. A s w e h a v e seen, t h e t r a n s i t i o n in zero field is second order
with n o latent h e a t and n o sudden c h a n g e of e n t r o p y at t h e transition
t e m p e r a t u r e . T h e r e is at t h e transition t e m p e r a t u r e only a c h a n g e in the
rate at which t h e e n t r o p y decreases as t h e t e m p e r a t u r e is reduced. T h e
two-fluid model w a s b a s e d on t h e above considerations, it being s u p posed t h a t at t h e transition t e m p e r a t u r e some c o n d u c t i o n electrons
begin to b e c o m e highly ordered superelectrons, the fraction a p p r o a c h i n g
1 0 0 % as t h e t e m p e r a t u r e is lowered t o w a r d s 0°K. T h e n a t u r e of t h i s
more ordered state of t h e electrons in a superconducting metal is d i s cussed in C h a p t e r 9.
F i g u r e 5.2 s h o w s t h a t at t e m p e r a t u r e s well below the transition
t e m p e r a t u r e the specific heat of t h e superconducting metal falls to a very
small value, b e c o m i n g even less t h a n t h a t of t h e n o r m a l metal. W e h a v e
seen t h a t the difference in t h e specific h e a t s of t h e s u p e r c o n d u c t i n g and
n o r m a l states is t h e result of a c h a n g e in t h e electronic specific heat. I n
order to u n d e r s t a n d the difference in specific h e a t s w e need, therefore, to
b e able t o deduce the value of t h e electronic specific h e a t from t h e
experimentally m e a s u r e d values of the total specific heat. T h i s m a y b e
d o n e as follows: for a normal metal at low t e m p e r a t u r e s t h e specific heat
h a s the form
(5.6)
w h e r e A is a c o n s t a n t w i t h t h e s a m e value for all m e t a l s . T h e D e b y e
t e m p e r a t u r e of t h e lattice 0, and t h e Sommerfeld c o n s t a n t y, w h i c h is a
61
THERMODYNAMICS OF THE TRANSITION
m e a s u r e of t h e density of t h e electron states at t h e F e r m i surface, b o t h
vary from metal to metal. W e can d e t e r m i n e the lattice contribution,
C i , as follows. E q u a t i o n (5.6) m a y b e re-written as
att
so a plot of t h e experimentally determined values of C /T against T
should give a straight line w h o s e slope isA/è* and w h o s e intercept i s y .
H e n c e , from m e a s u r e m e n t s on the superconductor in t h e normal state,
i.e. b y applying a magnetic field greater t h a n H , w e can d e t e r m i n e t h e
lattice specific heat, C
= Á(Ô/È). T h e specific heat of t h e lattice is t h e
same in b o t h the superconducting and normal states, so, b y subtracting
t h e value of C
from t h e total specific heat C of t h e superconducting
state, w e can o b t a i n the electronic contribution ( C ) .
It is difficult t o obtain accurate experimental values of t h e specific
h e a t s of superconductors, because at low t e m p e r a t u r e s t h e specific h e a t s
become very small. H o w e v e r , careful m e a s u r e m e n t s have revealed t h a t
at t e m p e r a t u r e s well below t h e transition t e m p e r a t u r e t h e electronic
specific heat of a metal in the superconducting state varies w i t h
t e m p e r a t u r e in an exponential manner,
2
n
c
3
latt
latt
s
el
(C )
d s
s
= ae,-b/kT
w h e r e a and b are constants. S u c h an exponential variation suggests t h a t
as t h e t e m p e r a t u r e is raised electrons are excited across an energy g a p
above their ground state. T h e n u m b e r of electrons excited across such a
g a p would vary exponentially as t h e t e m p e r a t u r e . W e shall see in
C h a p t e r 9 t h a t t h e B C S theory of superconductivity predicts j u s t such a
gap in t h e energy levels of the electrons.
T h e B C S theory also shows that, t h o u g h t h e energy gap is substantially constant at very low t e m p e r a t u r e s , it decreases if the t e m p e r a t u r e is
raised t o w a r d s t h e transition t e m p e r a t u r e , falling t o zero at T . T h i s
rapid decrease in t h e energy g a p j u s t below T accounts for t h e rapid increase in the specific heat of a superconducting metal as t h e t e m p e r a t u r e
approaches T .
c
c
c
5.3.
M e c h a n i c a l Effect s
B o t h t h e transition t e m p e r a t u r e and t h e critical magnetic field of a
superconductor are found experimentally to b e slighdy altered if t h e
62
INTRODUCTION TO SUPERCONDUCTIVITY
material is mechanically stressed. M a n y of t h e mechanical p r o p e r t i e s of
t h e superconducting and n o r m a l states are t h e r m o d y n a m i c a l l y related t o
the free energies of these states, and w e h a v e seen t h a t t h e critical
magnetic field strength d e p e n d s on t h e difference in t h e free energies of
the t w o states. H e n c e , once it is k n o w n t h a t the critical field c h a n g e s
slightly w h e n t h e material is u n d e r stress, t h e r m o d y n a m i c a r g u m e n t s
show t h a t t h e mechanical properties m u s t b e slightly different in t h e normal and superconducting states. F o r example, there is a n extremely
small change in volume w h e n a normal material b e c o m e s superconducting, and t h e t h e r m a l expansion coefficient a n d bulk m o d u l u s of elasticity
m u s t also b e slightly different in t h e superconducting and n o r m a l states.
It is possible to derive expressions for these effects b y straightforward
t h e r m o d y n a m i c m a n i p u l a t i o n , ! b u t t h e effects are extremely small and
w e shall not consider t h e m further in t h i s book.
5.4.
Therma l Conductivit y
T h e t h e r m a l conductivity of a metal is affected if it goes into
the superconducting state. M o s t of t h e heat flow along a n o r m a l metal
in a t e m p e r a t u r e gradient is carried b y t h e conduction electrons.
In the superconducting state, however, the superelectrons n o longer
interact w i t h t h e lattice in such a w a y t h a t they can exchange energy,
and so they c a n n o t pick u p heat from one part of a specimen and
deliver it t o another. Consequently, if a metal goes into t h e superconducting state, its t h e r m a l conductivity is reduced. T h i s effect can
be very m a r k e d at t e m p e r a t u r e s well below the critical t e m p e r a t u r e ,
w h e r e there are very few n o r m a l electrons left to t r a n s p o r t t h e heat.
F o r example, at 1°K t h e t h e r m a l conductivity of lead in t h e superconducting state is about 100 t i m e s less t h a n t h a t of t h e metal in t h e
n o r m a l state.
If, however, t h e superconductor is driven n o r m a l b y t h e application of
a magnetic field, t h e t h e r m a l conductivity is restored t o t h e higher value
of the normal state. H e n c e t h e t h e r m a l conductivity of a s u p e r c o n d u c t o r
can b e controlled by m e a n s of a m a g n e t i c field, and t h i s effect h a s been
used in " t h e r m a l s w i t c h e s " at low t e m p e r a t u r e s t o m a k e and break heat
contact b e t w e e n specimens connected b y a link of superconducting
metal.
t See, for example, E. A. Lynton, Superconductivity,
Methuen, London, 1964.
THERMODYNAMICS OF THE TRANSITION
5.5.
63
T h e r m o e l e c t r i c Effect s
I t is found, b o t h from theory and experiment, t h a t thermoelectric
effects d o not occur in a superconducting metal. F o r example, n o current
is set u p around a circuit consisting of t w o different superconductors, if
the t w o j u n c t i o n s are held at different t e m p e r a t u r e s below their transition t e m p e r a t u r e s . If a thermal e.m.f. were produced there would b e a
strange situation in which t h e current would increase t o t h e critical
value, no m a t t e r h o w small t h e t e m p e r a t u r e difference. I t follows from
the T h o m s o n relations that, if there is n o thermal e.m.f. in superconducting circuits, t h e Peltier a n d T h o m s o n coefficients m u s t be t h e same for
all superconducting metals, and they are in fact zero.
Because all superconducting metals have identical thermoelectric cons t a n t s they may, in principle, b e used as a standard against which t o
m e a s u r e other metals. T h e absolute values of t h e thermoelectric
coefficients of a normal metal can b e measured in a circuit comprising
the metal and any superconductor.
T h e absence of thermoelectric effects only applies to t h e superconductors considered in Part I of this book. Thermoelectric effects m a y appear
in t h e " T y p e - Ð " superconductors considered in P a r t I I .
CHAPTER 6
THE INTERMEDIATE STATE
S o FAR our discussion of h o w a m a g n e t i c field induces t r a n s i t i o n s from
t h e superconducting to t h e n o r m a l state h a s b e e n confined to cases in
w h i c h end effects are u n i m p o r t a n t . W e e n s u r e d t h a t this w a s so b y considering specimens in t h e form of a long t h i n rod. W e consider in t h i s
c h a p t e r w h a t h a p p e n s if w e relax t h i s restriction and consider specimens
of arbitrary shape.
6.1.
Th e Demagnetizin g Facto r
Consider the case of a superconducting sphere placed in a uniform
magnetic field H . As w e have seen, t h e flux lines are excluded from t h e
interior of the sphere b y d i a m a g n e t i c screening c u r r e n t s and h a v e t h e
form shown in Fig. 2.1 (p. 17). W e shall n o w s h o w t h a t t h e value of the
magnetic field strength inside t h e sphere (H ) exceeds t h e value H w h i c h
would exist if t h e sphere w e r e removed.
fl
{
a
S u p p o s e t h e field H is p r o d u c e d b y a solenoid, a s s h o w n in F i g . 6.1.
O n e of t h e fundamental p r o p e r t i e s of t h e m a g n e t i c field vector Ç is t h a t
its line-integral a r o u n d any closed curve is equal t o t h e n u m b e r of
a m p e r e - t u r n s which link t h e curve (see A p p e n d i x A). Applying t h i s result
to t h e p a t h ABCDEF gives $ H . dl = N w h e r e Í is t h e total n u m b e r of
t u r n s in t h e solenoid and i t h e c u r r e n t t h r o u g h each of t h e m . W e m a y
write
fl
if
BCDEFA
AB
w h e r e H , is t h e field w i t h i n the sphere and H t h e field at any point o u t side. N o w if t h e sphere is removed t h e line integral is still equal t o Ni,
and w e m a y w r i t e
e
H' .dl
e
AB
BCDEFA
64
=
Ni,
THE INTERMEDIATE STATE
65
Solenoid
FIG . 6.1. A superconducting sphere in a solenoid. T h e field strength at a point close to
the sphere, such as X, is less than it would be if the sphere were absent, while the field
strength at a point far away, such as Y, is essentially unchanged. T h e line integral of
Ç around the broken line is independent of whether the sphere is present or not, so the
field strength inside the sphere must exceed the applied field H .
a
w h e r e t h e field b e t w e e n A and  in t h e absence of t h e sphere is b y definition H , and
is t h e field at any point outside AB w h e n t h e sphere is
removed. H e n c e
FL
JH,.</1 +
AB
j
H .dl=
jll .dl +
e
a
BCDEFA
AB
J
H' .dl.
e
(6.1)
BCDEFA
N o w comparing H and H' at a point o n t h e axis such as X (Fig. 6.1), H
is clearly less t h a n H' because t h e effect of t h e screening c u r r e n t s
extends outside t h e sphere and d i s t o r t s t h e flux lines ( c o m p a r e Fig. 2.1).
But at p o i n t s far from t h e sphere, such as Y, t h e presence of t h e sphere
h a s a negligible effect and H = H' . H e n c e H is everywhere either less
t h a n or equal t o H' , and it follows from (6.1) t h a t H m u s t b e greater
t h a n H . I n other w o r d s , although t h e flux density inside is zero, because
of screening c u r r e n t s the magnetic field strength inside the sphere exceeds
the applied field strength
H.
e
e
e
e
e
e
e
e
(
a
a
T h i s is a special case of a well-known p roble m in m a g n e t o - s t a t i c s ;
namely, w h a t is t h e magnetic field inside a m a g n e t i c b o d y of arbitrary
shape exposed t o a uniform m a g n e t i c field? E x c e p t for t h e case of a long
thin body or a toroid, t h e field inside t h e b o d y differs from the applied
field. I n t h e case of a diamagnetic b o d y such as a superconductor, the internal field exceeds t h e applied field, while in t h e case of a ferromagnetic
body t h e internal field is less t h a n t h e applied field. Because historically
the study of ferromagnetism preceded t h a t of superconductivity, t h e
66
INTRODUCTION TO SUPERCONDUCTIVITY
p h e n o m e n o n is referred to as demagnetization.
A m a g n e t i z e d b o d y is
said t o p r o d u c e w i t h i n itself a demagnetizing field H w h i c h is s u p e r i m posed on t h e applied field. So w e m a y w r i t e t h e field H , inside t h e b o d y
as H , = H - H .
F o r b o d i e s of general shape t h e situation is complicated b e c a u s e t h e
demagnetization is n o t uniform, t h e internal field varying in s t r e n g t h a n d
direction t h r o u g h o u t t h e b o d y . F o r t h e special case of any ellipsoid,
however, t h e situation is m u c h simpler w i t h t h e internal field uniform
t h r o u g h o u t the b o d y and parallel t o t h e applied field. T h e internal field is
then given b y
D
fl
D
H/ = H
fl
— nl
(6.2)
w h e r e I is the m a g n e t i z a t i o n and ç i s t h e demagnetizing
factor of t h e
ellipsoid. T h e demagnetizing field is nl b u t , b e c a u s e in a s u p e r c o n d u c t o r
I is negative, t h e internal field is increased.
D e m a g n e t i z a t i o n is significant only in strongly m a g n e t i c m a t e r i a l s
because, as eqn. (6.2) shows, t h e difference b e t w e e n t h e s t r e n g t h s of t h e
internal a n d applied fields is proportional t o t h e m a g n e t i z a t i o n of the
body. A superconductor is, however, a strongly m a g n e t i c m a t e r i a l ; its
susceptibility h a s a m a g n i t u d e of unity, w h e r e a s t h e susceptibility of a
typical n o r m a l metal is 10~ or less. C o n s e q u e n t l y d e m a g n e t i z i n g effects
are very i m p o r t a n t in s u p e r c o n d u c t o r s .
4
A s w e h a v e pointed out, t h e d e m a g n e t i z i n g factor n, and hence t h e
strength of the internal field, d e p e n d s on t h e shape of t h e b o d y . T h e
special case of a sphere is treated in m o s t s t a n d a r d t e x t b o o k s o n elect r o m a g n e t i c theory.f V a l u e s of t h e demagnetizing factor for ellipsoids of
revolution are s h o w n in Fig. 6.2. I t c a n b e seen t h a t for t h e case of a
sphere ç = j . In later c h a p t e r s w e shall b e concerned w i t h t h e effect of
magnetic fields o n long w i r e s of s u p e r c o n d u c t o r ; an infinitely long circular cylinder can b e regarded as t h e limiting case of a long ellipsoid, so
ç = \ if H is perpendicular t o t h e axis of t h e wire, while ç = 0 if H is
parallel to t h e axis. T h e case of a rod w h o s e length is not very great c o m pared w i t h its diameter, in an applied field parallel t o its axis, or of a flat
plate perpendicular t o t h e applied field, can b e a p p r o x i m a t e d q u i t e closely b y replacing t h e rod or plate b y its inscribed ellipsoid. F o r t h e p a r ticular case of a superconductor, / = —Hi and (6.2) takes the form
a
t F o r example, A. F . Kip, Fundamentals
1969, p. 349.
a
of Electricity
and Magnetism,
2nd ed., McGraw-Hill,
67
THE INTERMEDIATE STATE
10 ,
2
0
2
4
6
8
10
LENGTH / DIAMETER
FIG. 6.2. Demagnetizing coefficient ç for ellipsoids of revolution
-L Field perpendicular to axis of revolution
I i Field parallel to axis of revolution
Ç
'=(Ú^) "
Ç
At the surface of the superconductor the tangential c o m p o n e n t of Ç is
continuous, and, since t h e internal field is parallel to the applied field, it
follows t h a t at t h e e q u a t o r the field strength j u s t outside the surface is
equal to t h e strength of the internal field H . Therefore, t h e external field
strength at t h e e q u a t o r exceeds t h e applied field strength, as can b e seen
from Fig. 6.1, and is given b y H /(l — n). In the case of a sphere, the
external field strength at the e q u a t o r is equal to jH , and in the case of a
long cylindrical rod in a transverse field, w h i c h is a case w e shall consider
in C h a p t e r 7, it h a s the value 2H .
{
a
a
a
6.2.
M a g n e t i c T r a n s i t i o n s for ç ö Ï
Consider w h a t h a p p e n s to a superconducting ellipsoid w h e n the
applied field H is gradually raised. At first sight one might expect t h a t
w h e n H reaches a value H' equal to (1 — n)H , so t h a t t h e internal field
H becomes equal to the critical field H , the sphere would b e driven into
the normal state. But if this were to h a p p e n / would b e c o m e zero,
because in t h e normal state t h e susceptibility is zero, and w e should have
a
a
t
c
c
c
68
INTRODUCTION TO SUPERCONDUCTIVITY
H = H = H' , w h i c h is less t h a n H . W e should t h e n h a v e t h e impossible situation of a completely n o r m a l b o d y in a field smaller t h a n H . T h e
p a r a d o x can b e resolved b y noting t h a t w h e n H b e c o m e s equal t o H it
is possible for t h e superconducting and n o r m a l p h a s e s to exist side b y
side in equilibrium, in t h e same w a y t h a t a liquid can coexist w i t h its
v a p o u r if t h e p r e s s u r e is equal t o t h e s a t u r a t i o n v a p o u r pressure. T o take
the simplest possible case, suppose t h a t w h e n H reaches t h e value H
the ellipsoid splits u p into n o r m a l and superconducting laminae parallel
to t h e applied field as s h o w n in F i g . 6.3. S o m e of t h e flux lines avoid t h e
ellipsoid altogether, and o t h e r s p a s s t h r o u g h t h e n o r m a l regions. If it
t
a
c
c
c
t
c
t
c
FIG . 6.3. Ellipsoid split up into normal and superconducting laminae in a magnetic
field.
should t u r n out t h a t the G i b b s free energy for the configuration s h o w n in
Fig. 6.3 is lower t h a n b o t h t h e free energy for t h e purely superconducting
state and t h a t for the purely n o r m a l state, t h e n t h i s a r r a n g e m e n t of
superconducting and n o r m a l d o m a i n s (or s o m e t h i n g like it) will b e the
equilibrium state for H' < H < H . W e shall see t h a t this is indeed the
case and t h a t H r e m a i n s equal to H t h r o u g h o u t t h i s range. S u c h an
a r r a n g e m e n t of normal and superconducting d o m a i n s is k n o w n as t h e intermediate state, and is a n essential feature of m a g n e t i c t r a n s i t i o n s for
any b o d y w h o s e demagnetizing factor is not zero. T h e model w e h a v e
adopted here, in which the normal and superconducting regions are
plane parallel laminae, is an oversimplified model, a n d in general t h e w a y
in which t h e b o d y splits u p is very complicated. Nevertheless, t h i s simple model b r i n g s out surprisingly well m o s t of t h e i m p o r t a n t features of
the i n t e r m e d i a t e state.
c
{
6.3.
a
c
c
Th e Boundar y Betwee n a Superconductin g
an d a Norma l Regio n
Before proceeding to discuss t h e i n t e r m e d i a t e state in s o m e detail, w e
first consider t h e conditions w h i c h m u s t exist if there is t o b e a
69
THE INTERMEDIATE STATE
stationary b o u n d a r y b e t w e e n a superconducting and a normal region.
Suppose t h e field strength in t h e normal region is H, so t h a t t h e flux density  in t h e n o r m a l region is equal to ì^ß. A fundamental property of
the magnetic flux density vector  is t h a t its c o m p o n e n t perpendicular to
any b o u n d a r y b e t w e e n t w o media is continuous, and the t w o phases are
here playing t h e role of t w o different media as far as their magnetic
properties are concerned. In t h e superconducting region  is everywhere
zero, so it follows t h a t there can b e n o c o m p o n e n t of  perpendicular to
the b o u n d a r y in t h e normal region and t h a t on the normal side  and Ç
m u s t b o t h lie parallel t o t h e boundary. I n other w o r d s , b o u n d a r i e s
b e t w e e n normal and superconducting regions m u s t lie parallel to the
local direction of t h e magnetic field.
T h e r e is also an i m p o r t a n t restriction on the value of t h e magnetic
field strength at t h e b o u n d a r y . T h e c o m p o n e n t of Ç parallel to any b o u n dary b e t w e e n t w o different magnetic media m u s t be continuous, and w e
have shown t h a t Ç is in fact parallel t o the b o u n d a r y on t h e normal side.
It follows t h a t the m a g n i t u d e of Ç m u s t b e the same on b o t h sides. At
t h e b o u n d a r y t h e m a g n i t u d e of Ç m u s t equal H . If it w e r e less, the
material on t h e normal side could lower its free energy b y becoming
superconducting. O n the other hand, the field strength cannot exceed H
or t h e material on the superconducting side of the b o u n d a r y would b e
driven normal. W e see, therefore, t h a t a stationary b o u n d a r y will only
exist w h e r e the field strength is exactly H , and w h e r e there is such a
stationary b o u n d a r y the flux density on t h e normal side will b e ì Ç . (It
will appear later o n t h a t this restriction on the value of Ç at the b o u n dary h a s to b e modified slightly if there is a "surface e n e r g y " associated
w i t h the interface between the normal and superconducting regions.)
c
c
c
0
6.4.
€
Magneti c Propertie s of th e Intermediat e Stat e
W e shall n o w discuss the magnetic properties of a b o d y split u p into
normal and superconducting regions as in Fig. 6.3. First of all w e need t o
k n o w the effective value of  inside such a body, which w e will a s s u m e
to b e an ellipsoid so t h a t we can assign to it a demagnetizing factor w.
T h e effective value of  is the flux density averaged over a region whose
dimensions are large compared w i t h t h e cross-sections of t h e normal and
superconducting laminae. I n other w o r d s , t h e effective value of B, w h i c h
w e write as B, is simply the total flux passing t h r o u g h the ellipsoid
divided by its m a x i m u m cross-sectional area. Since the flux density in
the superconducting regions is zero, Â is simply r\B w h e r e B is the
ny
n
70
INTRODUCTION TO SUPERCONDUCTIVITY
local flux density in the n o r m a l regions and ç is t h e fraction of t h e c r o s s + x ).
section w h i c h is in the n o r m a l state. F r o m Fig. 6.3, ç = x /(x
Also,j n t h e n o r m a l regions B = ì Ç
w h e r e H is t h e internal field, so
t h a t  = çì Ç . If w e regard t h e ellipsoid as having an effective relative
t h e n ì = ç. Also, w e can assign
permeability jx such t h a t  — ì ì Ç^
n
n
0
0
ï
n
s
t
{
r
Ã
0
Ã
to it an average m a g n e t i z a t i o n / = (—
\ì
HA.
I
0
N o w Hi = H -
nJ
a
=
or
Ç -ç(ç-1)Ç<
á
# , = Ç /[1 + ç(7?-1)] .
(6.3)
á
W e have already pointed out t h a t t h e n o r m a l a n d s u p e r c o n d u c t i n g
p h a s e s can coexist only if t h e m a g n e t i c field is equal t o H , so w e shall
a s s u m e t h a t t h e b o d y splits u p in such a w a y t h a t H = H . If H = H ,
then (6.3) gives ç for any particular value of H between H' and H . For
all values of H in t h i s range, t h e value of ç so obtained gives a value of
B and hence of / , w h i c h is j u s t right t o m a k e H equal H , so the
a s s u m p t i o n t h a t H = H is self-consistent.
c
t
c
a
{
c
c
c
a
y
t
{
c
c
I t is instructive t o plot t h e variables Ç^ç, Â and / as functions of t h e
applied field strength H . T h e s e are s h o w n in Fig. 6.4a-d. T h e g r a p h (d),
showing / , is particularly i m p o r t a n t in practice because t h e total
magnetic m o m e n t of t h e body, w h i c h is often m e a s u r e d experimentally,
is given by VI.
a
6.5.
T h e Gibb s F r e e Energ y in th e I n t e r m e d i a t e Stat e
T o find t h e G i b b s function G in the i n t e r m e d i a t e state w e start w i t h
and m a k e use of t h e result dG =
H = 0, so t h a t G(0) = Vg(0)
—ì ÌÜÇ ,
where Ì — VI is t h e total m a g n e t i z a t i o n of t h e b o d y . t
Hence,
a
s
0
f
á
H q
G(H )=G(0)-
j
a
MdH .
Mo
a
0
T h e r e are t h r e e distinct regions of integration t o b e considered:
(i) 0 < H
a
< (1 -
n)H
c
t Note that the expression for dG involves dH
Classical Thermodynamics, C.U.P., 1961, p. 26.
af
not dH .
t
See A. B. Pippard, Elements
of
71
THE INTERMEDIATE STATE
FIG . 6.4. Variation of (a) internal magnetic field strength, (b) effective relative
permeability, (c) effective flux density, and (d) intensity of magnetization with applied
magnetic field.
In this range of field strengths the sphere is totally superconducting
and / =
Hence
(ii) (1 - n)H
G(H ) =
a
<H <H
c
a
Vg(0)
s
c
— Â
T h e sphere is in the intermediate state and / =
Therefore 7 = (ç In this range
\)H
< T O = V {0) +
gs
t
= (ç -
C
[ H .(2 -
H (\c
c
s
{
\)H .
(iii) H > H
T h e sphere is completely normal and
1=0.
= Vg (0)H e r e G(HJ = Vg(0) + \Õì&
a
H.
n
»)]
72
INTRODUCTION TO SUPERCONDUCTIVITY
/
/
/
/
v.g
n
G(H )
A
Vg (o )
s
o
H;
H
H
-
A
C
FIG. 6.5. Variation of G i b b s free energy with H
demagnetizing factor.
a
for a body w i t h
non-zero
T h e variation of G w i t h H is s h o w n in Fig. 6.5, w h i c h d e m o n s t r a t e s
clearly t h a t t h e i n t e r m e d i a t e state ( a s represented b y t h e simple m o d e l of
plane laminae) h a s a lower free energy t h a n either t h e purely superconducting or t h e purely n o r m a l state for H' < H < H .
It should b e noted t h a t since t h e c o m p o n e n t of t h e m a g n e t i c field
parallel t o t h e b o u n d a r y b e t w e e n t w o m e d i a is t h e same o n each side of
the b o u n d a r y , and since in addition H is uniform and equal in
m a g n i t u d e t o H for (1 — n) H < H < H , t h e external field at t h e
equator (AA' in F i g . 6.3) is equal t o H w h e n e v e r t h e sphere is in t h e
intermediate state. S o m e a u t h o r s introduce t h e i n t e r m e d i a t e state b y
saying t h a t t h e specimen m u s t leave t h e purely superconducting state
w h e n t h e external field strength at t h e e q u a t o r is equal t o H . T h i s involves a different approach from t h e one w e h a v e used, and w e prefer to
adopt the standpoint of finding t h e state w i t h t h e lowest free energy.
a
c
a
c
t
c
c
a
c
c
c
6.6.
Th e Experimenta l Observatio n of th e Intermediat e Stat e
T h e d o m a i n s t r u c t u r e in t h e i n t e r m e d i a t e state h a s b e e n revealed
experimentally b y m a n y workers. T h e first of t h e s e w e r e M e s h k o v s k y
and Shalnikov, w h o explored t h e m a g n e t i c field in t h e n a r r o w g a p
b e t w e e n t w o h e m i s p h e r e s of tin of d i a m e t e r 4 c m w i t h a very small
b i s m u t h probe. ( T h i s relies on t h e fact t h a t t h e resistivity of b i s m u t h is
very sensitive t o t h e presence of a m a g n e t i c field.) T h e b i s m u t h p r o b e
m e a s u r e s t h e local value of  in t h e region of t h e h e m i s p h e r e i m m e d i a t e -
73
THE INTERMEDIATE STATE
0
1 cm
1 é é éé é é é é éI
FIG. 6.6. Structure of intermediate state in tin sphere, after Meshkovsky ( T = 2· 85TC,
H = 0-7 H ). T h e shaded areas represent normal regions.
a
c
ly adjacent to it, w h i c h is zero in t h e superconducting d o m a i n s and equal
to ì Ç . in t h e n o r m a l regions. A m a p showing their results is shown in
Fig. 6.6, w h e r e t h e normal regions are shown shaded. T h e n o r m a l
regions appear t o consist partly of radial laminae and partly of roughly
cylindrical filaments, showing, as might b e expected, t h a t t h e laminar
model adopted in § 6.3 is too simple.
0
(
O t h e r m e t h o d s of observing t h e intermediate state rely o n the t e n d e n cy of ferromagnetic p o w d e r s t o accumulate in regions of high flux density, of superconducting (diamagnetic) p o w d e r s to accumulate in regions
of low flux density, or on the ability of a p a r a m a g n e t i c glass in a
magnetic field t o rotate the plane of polarization of polarized light
( F a r a d a y effect). M o s t observations of the intermediate state have been
carried out w i t h flat plates which have a value of ç very close to unity
for a perpendicular field. By approximating a circular plate of radius a
and thickness t b y its inscribed ellipsoid, it can be shown that ç = 1 —
(t/2a) so t h a t H (l — n) is a very low field, and almost any value of H is
sufficient t o drive the plate into the intermediate state. A beautiful
photograph of the intermediate state in aluminium plates, obtained b y
Faber, is shown in Fig. 6.7. T h i s w a s taken b y dusting t h e plates w i t h
tin p o w d e r so t h a t the superconducting tin tended to accumulate in
regions of low magnetic field strength, i.e. adjacent to the superconducting regions of t h e aluminium. T h e normal regions are seen to be
laminar, b u t the laminae are not flat and have a characteristic structure.
c
a
74
INTRODUCTION TO SUPERCONDUCTIVITY
F I G . 6.7. Intermediate state in aluminium plate 0 47 cm thick with magnetic field
perpendicular to surface (H = 0-65 H , Ô = 0-92 T ). T h e dark lines are covered with
tin powder and correspond to superconducting regions (after Faberl.
c
6.7.
c
Th e Absolut e S i z e of th e D o m a i n s :
th e Rol e of Surfac e Energ y
T h e simple analysis given in § 6.4 s h o w s h o w t h e r a t i o of t h e w i d t h s
of the superconducting a n d n o r m a l regions is related t o t h e applied
magnetic field, b u t it tells u s n o t h i n g a b o u t their absolute values. O n this
simple model the absolute values are i n d e t e r m i n a t e , since t h e G i b b s
function d e p e n d s only on t h e total a m o u n t of n o r m a l or superconducting
material, i.e. on t h e ratio x /x„. T o discuss t h e factors t h a t d e t e r m i n e the
absolute values of x and x w e need to introduce a n e w concept, t h a t of
a possible surface energy b e t w e e n t h e n o r m a l and superconducting
p h a s e s ; in other w o r d s , w e consider t h e possibility t h a t e x t r a energy m a y
h a v e t o be supplied to form a b o u n d a r y b e t w e e n t h e t w o p h a s e s . S u c h a
concept is q u i t e c o m m o n in p h y s i c s ; t h e surface tension w h i c h exists at
the b o u n d a r y b e t w e e n a liquid and its v a p o u r is an o b v i o u s example, and
the i m p o r t a n c e of surface tension in controlling t h e equilibrium size of
small droplets is well k n o w n . W e shall for t h e m o m e n t a s s u m e t h a t such
a surface energy exists, and defer until § 6.9 t h e q u e s t i o n of h o w it
arises. It is easy t o see qualitatively w h a t t h e role of such a surface
s
s
n
75
THE INTERMEDIATE STATE
energy m a y b e in controlling t h e structure of the intermediate state. T h e
surface energy will give to t h e G i b b s free energy an additional contribution which is proportional to t h e total area of the b o u n d a r y b e t w e e n the
n o r m a l and superconducting phases. If it is positive, then t h e free energy
is minimized b y having as small an area of interphase b o u n d a r y as possible, i.e. b y having a few thick d o m a i n s ; on t h e other h a n d , if the surface
energy is negative, i.e. if energy is released on forming the interphase
b o u n d a r y , it is energetically favourable for t h e b o d y to split u p into a
large n u m b e r of thin d o m a i n s so as to m a k e the area of the interphase
b o u n d a r y as large as possible. I n fact, if the surface energy is negative, a
superconductor in a magnetic field would tend to split u p into normal
and superconducting regions even in the absence of demagnetizing
effects, and not show a t r u e Meissner effect. I t w a s the occurrence of the
Meissner effect in pure superconductors such as lead or tin t h a t led L o n don t o infer t h a t in these superconductors there m u s t b e a positive surface energy.
T h e energy per unit area of b o u n d a r y b e t w e e n the superconducting
and normal regions is usually denoted b y a, and it often simplifies t h e
mathematical analysis if the surface energy is expressed in t e r m s of a
characteristic length Ä such t h a t a = \ì ÇÀÄ. An exact analysis of t h e
d o m a i n structure in the intermediate state is a formidable theoretical
problem, and can only b e a t t e m p t e d if some a s s u m p t i o n about t h e shape
of the d o m a i n s is m a d e . F o r t h e case of a flat plate in a perpendicular
field, t h e problem h a s been analysed b y L a n d a u and b y K u p e r on the
a s s u m p t i o n t h a t t h e d o m a i n s are laminar; b o t h find t h a t t h e ratio of t h e
thickness of t h e superconducting regions x t o t h e thickness of the plate
d is of the order of (A/df/h,
w h e r e h = H /H . I n practice, for supercond u c t o r s such as lead or tin, Ä is found to b e about 5 ÷ 10~ cm, so t h a t
for a plate of 1 c m thickness in a perpendicular magnetic field w h o s e
value is near to t h e critical field, (x /d) ~ 10~ , or x ~ 1 0 " cm. If the
field strength is only about one-tenth of the critical field, then x ~
\0r
cm. Experimental observation of the d o m a i n size in t h e intermediate state provided one of t h e earliest w a y s of measuring the surface
energy, b u t owing t o the uncertainty in the theory the accuracy is not high.
0
s
a
c
5
2
s
2
s
s
l
6.8
Restoratio n of R e s i s t a n c e t o a Wir e in a
Transvers e Magneti c Fiel d
W e have seen in C h a p t e r 4 t h a t t h e restoration of resistance to a long
superconducting cylinder in an axial magnetic field occurs abruptly w h e n
76
INTRODUCTION TO SUPERCONDUCTIVITY
t h e applied field reaches t h e value H
and t h a t in favourable circ u m s t a n c e s t h e t r a n s i t i o n can b e very sharp. If t h e field is applied
perpendicular t o t h e axis, t h e variation of resistance w i t h field is q u i t e
different a n d is typically a s s h o w n in F i g . 6.8.
CJ
H
C
~~
FIG . 6.8. Restoration of resistance to a wire by a transverse magnetic field.
T h e r e are three essential features:
(i) T h e resistance b e g i n s t o r e t u r n w h e n H is little m o r e t h a n 0· 5 H .
T h i s is u n d e r s t a n d a b l e b e c a u s e t h e wire h a s b e e n driven into t h e int e r m e d i a t e state, w h i c h for a cylinder should h a p p e n for fields greater
t h a n 0-5 H . W e h a v e to a s s u m e t h a t t h e d o m a i n s h a v e t h e a p p r o x i m a t e
form of laminae perpendicular t o t h e axis, so t h a t t h e r e is n o c o n t i n u o u s
superconducting p a t h from one end of t h e w i r e t o t h e other. T h e fact
t h a t t h e onset of resistance invariably o c c u r s slightly above t h e value
0-5 H w h i c h w e should expect for a circular cylinder can b e u n d e r s t o o d
in t e r m s of a n i n t e r p h a s e surface energy; t h e analysis of § 6.5, w h i c h
neglects surface energy, p r e d i c t s t h a t t h e i n t e r m e d i a t e state should set in
at H = (1 — n) H . If t h e r e is a positive surface energy, t h e r e will b e an
additional c o n t r i b u t i o n t o t h e G i b b s free energy in t h e i n t e r m e d i a t e
state, and a value of H greater t h a n (1 — n) H m u s t b e reached before
t h e free energy of t h e i n t e r m e d i a t e state is lower t h a n t h a t of t h e p u r e
superconducting s t a t e . t
a
c
c
c
a
c
a
c
t Since, for applied field strengths greater than (1 — n)H , the internal field strength exceeds
H there appears to be a contradiction here with the statement on p. 69 that at a normalsuperconducting boundary the local value of the magnetic field strength is always equal to H .
T h i s contradiction can be resolved by noting that the presence of surface energy modifies the
value which the local magnetic field strength must have at a stationary boundary. T h e effect is
analogous to the increase in the pressure of a vapour in equilibrium with a small drop of liquid
c
c%
c
77
THE INTERMEDIATE STATE
(ii) T h e resistance increases smoothly w i t h H reaching t h e full normal resistance w h e n H attains t h e value H . T h i s again can b e u n derstood if t h e laminae are perpendicular t o the axis, since in t h e intermediate state ç is a linear function of H .
a9
a
c
a
(iii) T h e exact shape of the R versus Ç curve d e p e n d s on the
m a g n i t u d e of the measuring current. T h e implication of this is t h a t t h e
simple laminar model is too c r u d e ; the superconducting regions tend t o
b e linked b y superconducting filaments w h i c h can only carry a small
current. As the current increases, these filaments are driven normal and
the resistance increases.
6.9.
T h e Concep t of Coherenc e an d th e Origi n
of th e Surfac e Energ y
T o explain t h e origin of surface energy at a b o u n d a r y b e t w e e n n o r m a l
and superconducting regions, it is necessary to introduce a very important concept formulated b y P i p p a r d t in 1 9 5 3 — t h a t of the coherence
length. T o explain coherence w e m u s t first r e t u r n t o the idea of t h e
superconducting state as a highly ordered state. W e saw in § 5.2.3 t h a t
w h e n a superconductor is cooled below the transition t e m p e r a t u r e some
extra form of order sets in amongst t h e conduction electrons, and in
§ 1.4 w e introduced the idea t h a t a superconductor can b e regarded as
consisting of t w o interpenetrating electronic fluids, the normal electrons
and the superelectrons. T h e superelectrons in some w a y possess greater
order t h a n the n o r m a l electrons, and we can think of t h e degree of order
of the superconducting phase as being identified w i t h t h e density of
superconducting electrons n . By considering several aspects of t h e
behaviour of superconductors P i p p a r d w a s led to the idea t h a t n cannot
change rapidly w i t h position, b u t can only change appreciably within a
s
s
when there is surface tension at the liquid-vapour boundary. If the surface energy is positive, the
equilibrium value of the magnetic field strength at a superconducting-normal boundary is given
by
and r is the radius of curvature of the boundary. If
where Ä is defined on p. 75 by á = \ì Ç\^
the normal regions were plane laminae, r would be infinite and we should have Ç= H . In practice the domains are never plane laminae but always possess some curvature, so that Ç > / / if Ä
is positive.
0
c
c
t A. B. Pippard, Physica
19, 765 (1953).
78
INTRODUCTION TO
SUPERCONDUCTIVITY
distance w h i c h for p u r e s u p e r c o n d u c t o r s is of t h e order of 10~ c m . T h i s
distance he called the coherence length î. O n e consequence of t h e
existence of t h e coherence length is t h a t the b o u n d a r y b e t w e e n a n o r m a l
and superconducting region c a n n o t b e s h a r p b e c a u s e t h e density of
superelectrons can rise from zero in t h e n o r m a l region to its density n in
the superconducting region only gradually over a distance equal to about
t h e coherence length (see d o t t e d line in Fig. 6.9a).
4
s
Norma l
Superconductin
g
Numbe r o f
superelectron
s
(a ) Penetratio n dept h an d coherenc e rang e a t boundar y
Fre e
energ y
densit y
(c ) Tota l fre e energ y
FIG. 6.9. Origin of positive surface energy.
An i m p o r t a n t p r o p e r t y of t h e coherence length is t h a t it d e p e n d s o n
the purity of the metal, t h e figure of 1 0 " m m w h i c h w e have q u o t e d
being representative of a p u r e s u p e r c o n d u c t o r . If i m p u r i t i e s are present
the coherence length is reduced. T h e coherence length in a perfectly p u r e
superconductor, w h i c h is an intrinsic p r o p e r t y of t h e metal, is usually
denoted b y | , while t h e actual coherence length in an i m p u r e metal or
alloy is w r i t t e n as î. In very i m p u r e specimens, w h i c h are characterized
3
0
79
THE INTERMEDIATE STATE
b y a very short electron m e a n free p a t h / , the coherence length is
reduced t o approximately
0 â
T h e r e are several arguments, mostly of a circumstantial n a t u r e , w h i c h
lead to the concept of coherence. Probably t h e simplest and m o s t direct
arises from the extreme sharpness of the transition in zero field. In a
pure well-annealed specimen the transition may b e less than 10~
degrees wide, and this suggests t h a t the co-operation of a very large
n u m b e r of electrons is involved, as otherwise statistical fluctuations from
point to point would b r o a d e n the transition. T h e idea of coherence as w e
have introduced it here m a y seem r a t h e r v a g u e and ill defined, as indeed
it w a s at its inception, b u t w e shall see w h e n w e discuss t h e microscopic
theory of superconductivity in C h a p t e r 9 t h a t the concept can b e put on
a more precise a n d quantitative basis. It is also in line w i t h the predictions of t h e G i n z b u r g - L a n d a u theory discussed in § 8.5.
Another a r g u m e n t in favour of the idea of coherence is t h a t it allows a
simple explanation of the origin of surface energy, as we shall now show.
Consider a superconducting region adjoining a normal region. As w e saw
in § 6.3, this situation can only arise in the presence of a m a g n e t i c field
of strength H . At the b o u n d a r y , there is not a sudden change from fully
normal behaviour to fully superconducting behaviour; t h e flux density
penetrates a distance ë into the superconducting region, and in accordance w i t h the coherence concept, in the superconducting region t h e
n u m b e r of superelectrons per unit volume n increases slowly over a distance about equal to the coherence length î (Fig. 6.9a).
N o w consider t h e free energy at the b o u n d a r y . If the b o u n d a r y is to b e
stable the superconducting and normal regions m u s t be in equilibrium,
that is to say their free energy per unit volume m u s t be the same. T h e r e
are t w o contributions which change the free energy of the superconducting region relative t o t h a t of the normal region. D u e to the presence of
the ordered superelectrons the free energy density of t h e superconducting state is lowered by an a m o u n t g — g , and, in addition, because the
superconducting region h a s acquired a magnetization w h i c h cancels t h e
flux density inside, there is a positive " m a g n e t i c " contribution \ì^ÇÀ to
its free energy density. F o r equilibrium \ì ÇÀ = g — g , so t h a t well inside the superconducting region the t w o c o n t r i b u t i o n s cancel and the
free energy density is the same as in the neighbouring n o r m a l region. At
the boundary, however, the degree of order (i.e. the n u m b e r of superelect r o n s n ) rises only gradually over a distance determined b y the
coherence length î, so the decrease in free energy d u e to t h e increasing
order of the electrons takes place over the same distance (Fig. 6.9b). O n
(| / )ú
e
5
c
s
n
s
0
s
n
s
80
INTRODUCTION TO SUPERCONDUCTIVITY
the other h a n d t h e " m a g n e t i c " c o n t r i b u t i o n t o t h e free energy rises over
a distance of a b o u t the p e n e t r a t i o n d e p t h ë . In general î and ë are not
the same, so t h e t w o c o n t r i b u t i o n s d o n o t cancel n e a r t h e b o u n d a r y . If,
as in Fig. 6.9, t h e coherence length is longer t h a n t h e p e n e t r a t i o n d e p t h ,
the total free energy density is increased close to t h e b o u n d a r y ; t h a t is t o
say, there is a positive surface energy. I t can b e seen from Fig. 6.9b t h a t ,
roughly speaking, t h e value of t h i s surface energy is approximately
— ë) per unit area of t h e b o u n d a r y . ( T h i s c a n b e seen b y
\ì Ç (î
replacing t h e t w o c u r v e s in F i g . 6.9a b y rectangular s t e p s in w h i c h t h e
changes in flux density and n occur a b r u p t l y at d i s t a n c e s ë and î respectively from t h e edge of the n o r m a l region.) T h e length Ä introduced in
§ 6.7 is therefore to b e identified w i t h î — ë , and since the d o m a i n size
in the i n t e r m e d i a t e state enables Ä to b e m e a s u r e d , t h i s gives t h e value
of î — ë. A s w e saw in § 6.7, t h e value of Ä in a typical p u r e supercond u c t o r such as lead or tin is a b o u t 5 x 1 0 " cm, w h i c h is a b o u t 10 t i m e s
larger t h a n ë, so it is clear t h a t in t h i s case £ ~ Ä ~ 5 ÷ 1 0 ~ cm. T h i s is,
in fact, o n e of t h e a r g u m e n t s w h i c h enabled P i p p a r d to assign an order of
m a g n i t u d e of £ .
2
0
€
s
5
5
0
C o h e r e n c e is a very fundamental p r o p e r t y of s u p e r c o n d u c t o r s ; for
example, w e shall see t h a t t h e coherence length plays an i m p o r t a n t role
in d e t e r m i n i n g t h e properties of t h e t y p e - I I s u p e r c o n d u c t o r s w h i c h w e
consider in t h e second p a r t of t h i s book.
T h e reader should b e w a r n e d t h a t in t h e literature on superconductivity the t e r m "coherence l e n g t h " is used in t w o r a t h e r different senses.
T h e coherence length w h i c h w e h a v e been discussing, and w h i c h is of
concern t o u s in t h i s book, should strictly b e called t h e " t e m p e r a t u r e dependent coherence length", | ( T ) . A s w e h a v e seen, it is related to t h e
fact t h a t t h e superconducting o r d e r p a r a m e t e r c a n n o t v a r y rapidly w i t h
position; i.e. t h e density of superelectrons, or t h e energy g a p or t h e
amplitude of the electron-pair w a v e can only c h a n g e b y a significant
a m o u n t over a distance greater t h a n î(Ô). T h i s a c c o u n t s for t h e lack of
s h a r p n e s s of t h e b o u n d a r y b e t w e e n s u p e r c o n d u c t i n g a n d n o r m a l regions.
T h i s coherence range, w h i c h is a special feature of a superconductor,
varies w i t h t e m p e r a t u r e , increasing at higher t e m p e r a t u r e in m u c h t h e
same w a y as the penetration d e p t h (see § 2.4.1).
T h e o t h e r coherence length, t h e " t e m p e r a t u r e - i n d e p e n d e n t coherence
length", is related to t h e correlation b e t w e e n t h e m o t i o n of electrons.
Because t h e m o t i o n of one electron is correlated w i t h t h e m o t i o n of o t h e r
electrons w h i c h m a y b e some distance away, the c u r r e n t density at any
point is not d e t e r m i n e d j u s t b y t h e fields at t h a t point b u t b y t h e fields
THE INTERMEDIATE STATE
81
averaged over a volume surrounding t h a t point. T h e r a d i u s of this
volume over which the electrons experience the average field is the
temperature-independent coherence range. Strictly speaking, this
coherence range is not peculiar to superconductors; there is some correlation of the electron motion in normal conductors, w h e r e the
coherence length is the electron m e a n free p a t h . In this book, however,
we are only concerned w i t h t h e t e m p e r a t u r e - d e p e n d e n t coherence length
associated w i t h t h e slowness w i t h which superconducting properties can
change with position.
I.T.S.—D
CHAPTE R
TRANSPORT
CURRENTS
IN
7
SUPERCONDUCTORS
7.1. Critica l Current s
T H E EARL Y w o r k e r s in superconductivity soon discovered t h a t there
is an upper limit to t h e a m o u n t of current t h a t can be passed along a
piece of superconductor if it is to r e m a i n resistanceless. W e call this t h e
critical current of that particular piece. If the c u r r e n t exceeds this
critical value, some resistance appears.
W e now show t h a t the critical c u r r e n t is related t o the critical
magnetic field strength H . W e saw in C h a p t e r 3 t h a t all c u r r e n t s in a
superconductor flow at t h e surface w i t h i n t h e p e n e t r a t i o n d e p t h , t h e
current density decreasing rapidly from some value J at the surface. It
w a s pointed out in C h a p t e r 4 t h a t superconductivity b r e a k s d o w n if the
supercurrent density exceeds a certain value w h i c h we call the critical
current density J .
c
a
c
In general there can b e t w o c o n t r i b u t i o n s t o the current flowing on the
surface of a superconductor. Consider, for example, a superconducting
wire along which w e are passing a current from some external source
such as a b a t t e r y . W e call this current the " t r a n s p o r t c u r r e n t " because it
t r a n s p o r t s charge into and out of the wire. If the wire is in an applied
magnetic field, screening c u r r e n t s circulate so as to cancel t h e flux inside
t h e metal. T h e s e screening c u r r e n t s are superimposed on the t r a n s p o r t
current, and at any point the current density J can b e considered to b e
the sum of a c o m p o n e n t J , d u e to the t r a n s p o r t current a n d a c o m p o n e n t
] which arises from the screening c u r r e n t s
H
J=J/+J//W e m a y expect t h a t superconductivity will break d o w n if the m a g n i t u d e
of the total current density J at any point exceeds the critical c u r r e n t
density J .
According to the L o n d o n equation (3.17) there is a relation b e t w e e n
the supercurrent density at any point and t h e m a g n e t i c flux density at
c
82
83
TRANSPORT CURRENTS IN SUPERCONDUCTORS
that point, and this same relation holds w h e t h e r the supercurrent is a
screening current, a t r a n s p o r t current or a combination of b o t h . H e n c e ,
w h e n a current flows on a superconductor, there will at the surface b e a
flux density  and a corresponding field strength H(= Â/ì ) which is
related to the surface current density J .
If the total current flowing on a superconductor is sufficiently large,
the current density at the surface will reach the critical value J and the
associated magnetic field strength at the surface will have a value H .
Conversely, a magnetic field of strength H at the surface is always
associated w i t h a surface supercurrent density J . T h i s leads to the
following general h y p o t h e s i s : a superconductor
loses its zero
resistance
when, at any point on the surface, the total magnetic field strength, due
to transport current and applied magnetic field, exceeds the critical field
strength H . T h e m a x i m u m a m o u n t of transport current w h i c h can b e
passed along a piece of superconductor w i t h o u t resistance appearing is
w h a t we call the critical current of t h a t piece. Clearly the stronger the
applied magnetic field the smaller is this critical current.
If there is no applied magnetic field the only magnetic field will be that
generated b y any t r a n s p o r t current, so in this case, the critical current
will b e t h a t current w h i c h generates the critical magnetic field strength
H at the surface of the conductor. T h i s special case of t h e general rule
stated above is k n o w n as Silsbee's hypothesisf and w a s formulated
before the concept of critical current density w a s appreciated. W e shall
call t h e more general rule for the critical current given in the previous
p a r a g r a p h the "generalized f o r m " of Silsbee's hypothesis.
W e saw in C h a p t e r 4 that the critical magnetic field strength H
d e p e n d s o n the t e m p e r a t u r e , decreasing as the t e m p e r a t u r e is raised and
falling to zero at the transition t e m p e r a t u r e T . T h i s implies that the
critical current density d e p e n d s on t e m p e r a t u r e in a similar manner, the
critical current density decreasing at higher t e m p e r a t u r e s . Conversely, if
a superconductor is carrying a current, its transition t e m p e r a t u r e is
lowered.
0
a
c
c
c
c
c
c
c
c
7.1.1.
Critica l c u r r e n t s o f w i r e s
L e t us consider a cylindrical wire of radius a. If, in the absence of any
externally applied magnetic field, a current i is passed along the wire,
a magnetic field will be generated at the surface whose strength H is
given by
{
t F. B. Silsbee, J. Wash. Acad. Set., 6, 597 (1916).
84
INTRODUCTION TO SUPERCONDUCTIVITY
2ðáÇ
(
= i.
T h e critical current will therefore be
i = 7jiaH .
c
(7.1)
c
T h i s relation for t h e critical c u r r e n t c a n b e tested b y m e a s u r i n g t h e
m a x i m u m current a s u p e r c o n d u c t i n g wire can carry w i t h o u t resistance
appearing, and it is found t h a t , in t h e absence of any externally applied
magnetic field, eqn. (7.1) predicts t h e correct value.
Applie d magneti c fiel d strengt h
Applie d magneti c fiel d strengt h
(a )
(b )
FIG . 7.1. Variation of critical current with applied magnetic field strength, (a)
Longitudinal applied field, (b) T r a n s v e r s e applied field (transport current flowing into
page).
In zero or weak applied m a g n e t i c field s t r e n g t h s t h e critical c u r r e n t s of
s u p e r c o n d u c t o r s can b e quite high. As an example, consider a 1 m m
d i a m e t e r wire of lead cooled to 4 2°K b y i m m e r s i o n in liquid helium. At
(550
this t e m p e r a t u r e the critical field of lead is about 4-4 ÷ 1 0 A m
gauss) so, in the absence of any applied m a g n e t i c field, the wire can c a r r y
u p t o 140 A of resistanceless current.
4
L e t u s n o w consider to w h a t extent the critical c u r r e n t
the presence of an externally applied m a g n e t i c field. F i r s t
an applied magnetic field of flux density B and s t r e n g t h H
in a direction parallel to t h e axis of t h e wire (Fig. 7.1a). If
a
a
- 1
is reduced b y
suppose t h a t
(= Â /ì )
is
a c u r r e n t i is
á
0
85
TRANSPORT CURRENTS IN SUPERCONDUCTORS
passed along t h e wire it generates a field encircling the wire, and at the
surface of t h e wire t h e strength of this field is H = ß/2ðá. T h i s field and
the applied field add vectorially and, because in this case they are at right
angles, the strength Ç of the resultant field at the surface is given b y
(HI+ Ç)Û or
t
H
2
= H
+
2
a
(i/2na) .
2
T h e critical value i of the current occurs w h e n Ç equals
c
H:
c
(7
·
2)
H is a constant, and so this equation, which expresses the variation of i
w i t h i f , is t h e equation of a n ellipse. Consequently, the g r a p h representing the decrease in critical current as the strength of a longitudinal
applied magnetic field is increased h a s the form of a q u a d r a n t of an ellipse
(Fig. 7.1a). In t h i s configuration the magnetic flux density is uniform
over the surface of t h e wire and the flux lines follow helical p a t h s .
Another case of importance occurs w h e n an applied magnetic field is
normal t o the axis of t h e wire (Fig. 7.1b). ( W e assume here t h a t the
applied field is not strong enough to drive t h e superconductor into the
intermediate state.) In this case t h e total flux density is not uniform over
the surface of the w i r e ; t h e flux densities add o n one side of t h e wire and
substract on the other. T h e m a x i m u m field strength occurs along t h e line
L. H e r e , because of demagnetization, as pointed out at the end of § 6.1,
a field 2H is superimposed on t h e field H to give a total field
c
c
fl
a
{
H=2H
a
+
H = 2H
i
a
+
^ .
T h e general form of Silsbee's rule states t h a t resistance first appears
w h e n the total magnetic field strength at any part of the surface equals
H and so t h e critical current in this case is given b y
c9
t = 2ðá(Ç
c
€
-
2H ).
(7.3)
a
I n t h i s case, therefore, the critical current decreases linearly w i t h
increase in applied field strength, falling to zero at jH .
I t should b e emphasized t h a t the critical current of a specimen is
defined as the current at which it ceases to have zero resistance, not as
the current at w h i c h t h e full normal resistance is restored. T h e a m o u n t
of resistance w h i c h a p p e a r s w h e n t h e critical current is exceeded
d e p e n d s on a n u m b e r of circumstances, w h i c h w e examine in the next
section.
c
86
INTRODUCTION TO SUPERCONDUCTIVITY
7.2.
Therma l Propagatio n
T h e variation of critical c u r r e n t w i t h applied m a g n e t i c field predicted
by (7.2) and (7.3) h a s been confirmed b y experiment, t h o u g h m e a s u r e m e n t of critical c u r r e n t s , especially in low m a g n e t i c fields w h e r e t h e
current values can b e high, is n o t always easy. T o see w h y t h e r e m a y b e
a difficulty, w e n o w e x a m i n e t h e processes b y w h i c h resistance r e t u r n s t o
a wire w h e n the critical current is exceeded. Consider, for example, a
cylindrical wire of superconductor. I n practice n o piece of wire can h a v e
perfectly uniform properties along its length; t h e r e m a y b e accidental
variations in composition, thickness, etc., or t h e t e m p e r a t u r e m a y b e
slightly higher at some p o i n t s t h a n others. As a result t h e value of critical
FIG . 7.2. T h e r m a l propagation, (b) shows the temperature variation of the region A
resulting from the current increase shown at (a), (c) shows the return of the wire's
resistance when the normal region spreads from A.
TRANSPORT CURRENTS IN SUPERCONDUCTORS
87
current will vary slightly from point to point, and there will be some
point on the wire w h i c h h a s a lower critical current t h a n the rest of the
wire. In Fig. 7.2 such a region is represented b y the section A w h e r e the
wire is slightly narrower. Suppose w e now p a s s a current along the wire
and increase its m a g n i t u d e , until the current j u s t exceeds the critical
current i (A) of t h e section A (Fig. 7.2a). T h i s small section will become
resistive while the rest of the wire r e m a i n s superconducting, so a very
small resistance r appears in the wire. At A the current i is flowing
t h r o u g h resistive material and at this point heat is generated at a rate
proportional t o i r. Consequently the t e m p e r a t u r e at A rises, and heat
flows away from A along the metal and into t h e surrounding m e d i u m at
a rate which d e p e n d s on the t e m p e r a t u r e increase of A, the thermal conductivity of the metal, the rate of heat loss across t h e surface, etc. T h e
t e m p e r a t u r e of A will rise until t h e rate at w h i c h heat flows away from it
equals the rate i r at w h i c h t h e heat is generated. If the r a t e of heat
generation is low, t h e t e m p e r a t u r e of A rises only a small a m o u n t and the
wire r e m a i n s indefinitely in this condition. If, however, heat is generated
at a high rate, either because t h e resistance of A is high or because the
current i is large, the t e m p e r a t u r e of A m a y rise above the critical
t e m p e r a t u r e of t h e wire (Fig. 7.2b). T h e presence of the current h a s in
fact reduced t h e transition t e m p e r a t u r e of the superconducting wire
from T to a lower value T (i), and if, as a result of the heating of A, the
regions adjacent to A are heated above T (i) they will b e c o m e normal.
T h e current i is n o w flowing t h r o u g h these new normal regions and
generates heat w h i c h drives t h e regions adjacent to t h e m normal.
Consequently, even t h o u g h the current i is held constant, a normal
region spreads out from A until the whole wire is normal and the full
normal resistance R is restored (Fig. 7.2c). T h i s process w h e r e b y a normal region m a y spread out from a resistive nucleus is called thermal
propagation,
and w e see t h a t it is m o r e likely to occur if the critical
current is large and if the resistivity of t h e normal state is high (e.g. in
alloys).
O n account of thermal propagation there can be difficulty in
measuring the critical current of a specimen, especially in low or zero
magnetic fields w h e r e the current value m a y b e high. Consider a superconducting specimen of uniform thickness, as shown in Fig. 7.3a, whose
critical current w e are a t t e m p t i n g to m e a s u r e b y increasing the current
until a voltage is observed. If t h e current is less t h a n t h e critical current
there will b e no voltage d r o p along the specimen and n o heat will be
generated in it. However, the leads carrying t h e current to t h e specimen
c
2
2
c
c
c
n
88
INTRODUCTION TO SUPERCONDUCTIVITY
are usually of ordinary n o n - s u p e r c o n d u c t i n g m e t a l and so heat is
generated in these b y t h e passage of t h e c u r r e n t . C o n s e q u e n t l y t h e e n d s
of the specimen, w h e r e it m a k e s contact w i t h t h e leads, will b e slightly
heated and will h a v e a lower critical c u r r e n t t h a n t h e b o d y of t h e
specimen. As t h e c u r r e n t is increased, therefore, t h e e n d s go n o r m a l at a
current less t h a n t h e t r u e critical c u r r e n t of t h e specimen, and n o r m a l
regions may spread through the wire by thermal propagation.
Consequently a voltage is observed at a c u r r e n t less t h a n t h e t r u e critical
value. T o lessen t h e risk of t h e r m a l p r o p a g a t i o n from t h e c o n t a c t s one
To
voltmete r
To
voltmete r
(a )
(b )
Unsuitabl e
arrangemen t
Suitabl e
arrangemen t
FlG. 7.3. M e a s u r e m e n t of critical current.
should use as thick current leads as possible so t h a t little heat is
produced in t h e m . It is also desirable t o m a k e t h e e n d s of t h e superconducting specimen thicker t h a n t h e section w h o s e critical c u r r e n t
we are measuring, so t h a t t h e critical c u r r e n t of t h e t h i n n e r section will b e reached before t h e r m a l p r o p a g a t i o n s t a r t s from t h e e n d s
(Fig. 7.3b).
A characteristic of t h e r e t u r n of resistance b y t h e r m a l p r o p a g a t i o n is
t h e complete a p p e a r a n c e of t h e full n o r m a l resistance o n c e a certain
current h a s b e e n exceeded, a s a result of t h e n o r m a l region spreading
right t h r o u g h t h e specimen.
89
TRANSPORT CURRENTS IN SUPERCONDUCTORS
7.3.
Intermediat e Stat e I n d u c e d b y a Curren t
If thermal propagation does not occur, t h e full normal resistance does
not appear at a sharply defined value of current b u t over a considerable
current range. Consider a cylindrical wire of superconductor with a
critical field strength H . If t h e radius of the wire is a, a current i
produces a magnetic field strength ß/2ðá at t h e surface. As w e have seen,
the greatest current the wire can carry while remaining wholly superconducting m u s t b e i = 2jiaH , because, if the current were to exceed this,
the magnetic field strength at the surface would b e greater t h a n H .
c
c
c
c
W e might at first suppose that at i an outer cylindrical sheath is
driven normal while the centre r e m a i n s superconducting. However, this
is not possible, as w e shall n o w show. S u p p o s e t h a t an outer sheath
b e c o m e s normal, leaving a cylindrical core of r a d i u s r in the superconducting state. T h e current will now flow entirely in this resistanceless
core, and so the magnetic flux density it p r o d u c e s at the surface of the
core will be H air. Since this is greater t h a n H > the superconducting
core will shrink to a smaller r a d i u s and this process will continue until
the superconducting core c o n t r a c t s to zero radius, i.e. t h e whole wire is
normal. However, it is not possible for t h e wire to b e c o m e completely
normal at a current i because if the wire were normal t h r o u g h o u t , t h e
current would b e uniformly distributed over the cross section and t h e
magnetic field strength inside the wire at a distance r from t h e centre
would be less t h a n the critical field, so t h e material could not b e normal.
c
c
c
c
It therefore appears that, at t h e critical current, t h e wire can b e
neither wholly superconducting nor wholly normal, and t h a t a state in
which a normal sheath s u r r o u n d s a superconducting core is not stable.
In fact, at the critical current, t h e wire goes into an intermediate state of
alternate superconducting and normal regions each of w h i c h occupies
the full cross-section of t h e w i r e . t T h e current passing along t h e wire
n o w h a s t o flow t h r o u g h the normal regions, so at the critical current the
resistance should j u m p from zero to some fraction of the resistance of t h e
completely normal wire. Experiments show that a considerable
resistance does indeed suddenly appear w h e n t h e current is raised to t h e
critical value (Fig. 7.4), t h e resistance j u m p i n g to b e t w e e n 0-6 and 0-8 of
the full normal resistance. T h e exact value depends on factors such as
the t e m p e r a t u r e and purity of the wire.
T h e detailed shapes of the normal and superconducting regions w h i c h
appear w h e n a current exceeding the critical current is passed along a
t F . London, Superfluids,
vol. 1, Dover Publications Inc., New York, 1961.
90
INTRODUCTION TO SUPERCONDUCTIVITY
Current
FIG . 7.4. Restoration of resistance to a wire by a current.
wire h a v e not yet been d e t e r m i n e d experimentally, and it is a c o m plicated problem t o deduce t h e m from theoretical considerations. T h e
configuration s h o w n in Fig. 7.5a is one w h i c h h a s been recently proposed
o n a theoretical basis, and for w h i c h t h e r e is some s u p p o r t i n g
experimental evidence.
It can b e seen from Fig. 7.4 t h a t as t h e c u r r e n t is increased above t h e
critical value i the resistance of the wire gradually increases and a p proaches the full normal resistance asymptotically. L o n d o n suggested
that w h e n the current is increased above t h e critical value i t h e inc
c
(a ) i = i
a
1
c
y
y///\v\\*vmediate
·/
(b ) i> L
FIG . 7.5. Suggested cross-section of cylindrical wire carrying current in excess of its
critical current (based on Baird and Mukherjee, and London).
TRANSPORT CURRENTS IN SUPERCONDUCTORS
91
termediate state c o n t r a c t s into a core surrounded b y a sheath of normal
material w h o s e thickness increases as t h e current increases, so t h a t the
total current is shared between t h e fully resistive sheath and t h e partially
resistive intermediate core (Fig. 7.5b). T h i s model predicts a resistance
increasing smoothly w i t h t h e current in excess of i .
T h e sudden appearance of resistance, either b y thermal propagation or
by the appearance of an intermediate state w h e n t h e critical current is
exceeded, can make t h e m e a s u r e m e n t of t h e critical current of a conducting wire a r a t h e r precarious experiment. A s the m e a s u r i n g current
t h r o u g h the specimen is increased, a resistance R suddenly a p p e a r s w h e n
the critical value i is exceeded. P o w e r
is then generated in t h e
specimen, and if i is large and R is not very small, the heating m a y b e
enough to melt t h e wire, unless the current is reduced very quickly. I n
fact, superconducting wires can act like very efficient fuses w i t h a s h a r p ly defined b u r n - o u t current.
c
c
c
CHAPTE R
THE
8
SUPERCONDUCTING
OF S M A L L
PROPERTIES
SPECIMENS
I T WA S pointed out in C h a p t e r 2 t h a t t h e p e n e t r a t i o n d e p t h ë is very
small, and t h a t m o s t superconducting s p e c i m e n s h a v e d i m e n s i o n s w h i c h
are very m u c h greater t h a n ë. S o m e t i m e s , however, a situation arises, as
for example w i t h t h i n films or fine wires, in w h i c h one or m o r e of the
d i m e n s i o n s of the specimen is c o m p a r a b l e w i t h A. W e shall see in this
chapter t h a t the superconducting p r o p e r t i e s of such specimens are in
some w a y s significantly different from those of large specimens.
8.1.
T h e Effec t o f P e n e t r a t i o n o n t h e C r i t i c a l
Magneti c Fiel d
W e saw in C h a p t e r 4 t h a t if a specimen of s u p e r c o n d u c t i n g metal is in
t h e s u p e r c o n d u c t o r is driven into
a uniform applied m a g n e t i c field H
the normal state w h e n H is increased above a critical value H . F r o m a
t h e r m o d y n a m i c point of view, t h i s is because t h e G i b b s free energy of a
af
a
c
in an
superconducting specimen is changed b y an a m o u n t — j ì ÌÜÇ
ï
applied field H > w h e r e Ì is t h e induced m a g n e t i c m o m e n t .
In the superconducting state Ì is negative, so the free energy is increased, and if this increase is sufficient to m a k e t h e free energy in the
superconducting state exceed t h a t in the n o r m a l state, t h e specimen
b e c o m e s normal. T h e magnetic m o m e n t Ì is equal to jldV, w h e r e V is
the volume of t h e specimen and / is t h e intensity of m a g n e t i z a t i o n given
by
0
á
a
Â=ì Ç
+ ì É.
0
0
(8.1)
In C h a p t e r 4 it is a s s u m e d t h a t  = 0 e v e r y w h e r e inside t h e superconductor, so t h a t / = — Ç and Ì = —HV; in other w o r d s , it w a s a s s u m e d
t h a t t h e magnetic m o m e n t per unit volume is independent of the shape
92
93
THE SUPERCONDUCTING PROPERTIES OF SMALL SPECIMENS
and size of the specimen. I t follows from this t h a t t h e critical magnetic
field is given b y
W 2
= & - &
[see (4.3)],
(8.2)
where g and g are the free energies per unit volume of t h e normal and
superconducting phases in zero magnetic field. H should therefore b e
independent of t h e size of the specimen.
T h i s a r g u m e n t needs modifying w h e n penetration of t h e field is taken
into account. W e saw in C h a p t e r 2 that  does not fall abruptly to zero
at t h e surface of the specimen, b u t decreases w i t h distance (x) into t h e
interior of t h e specimen approximately as e~ '> where ë , t h e " p e n e t r a tion d e p t h " , is about 5 ÷ 10~ cm in most superconductors. It follows
that j u s t inside t h e surface  is not zero, and t h a t in this region the value
of / given b y (8.1) is no longer equal to — H. Instead, the m a g n i t u d e of /
increases from zero at the surface to the value Ç in the interior of t h e
specimen, and as a result the m a g n i t u d e of the magnetic m o m e n t Ì is
less t h a n it would b e if ë w e r e zero. H e n c e , for a given value of H , t h e
n
s
c
x,/
6
a
magnetic contribution t o the free energy, — J
ì ÌÜÇ
or— \ì ÌÇ ,
is
ï
less t h a n it would b e if penetration did not occur, and the applied
magnetic field h a s to b e increased beyond the value given b y (8.2) before
the transition t o the normal state can take place. In other w o r d s , t h e
critical m a g n e t i c field is increased as a result of the penetration of t h e
magnetic flux. T h e m a g n i t u d e of t h e increase d e p e n d s o n t h e reduction
in the magnetic m o m e n t M , which in t u r n d e p e n d s on t h e dimensions of
the specimen relative to the penetration d e p t h ë. T h e effect is only
noticeable if the volume contained within a distance ë of the surface is
comparable w i t h the total volume of the specimen.
8.2.
0
á
0
á
T h e Critica l Fiel d of a Parallel-side d Plat e
T h e effect of penetration o n the critical magnetic field of a specimen is
most easily illustrated b y reference to the case of a parallel-sided plate
w i t h a magnetic field H applied parallel to the surfaces of the plate. T h e
length of the plate in the direction of the field and its w i d t h are b o t h s u p posed m u c h larger than its thickness (Fig. 8.1). T h i s particular geometry
is chosen partly for its practical importance and partly because the flux
distribution within the plate can be easily calculated w i t h t h e aid of t h e
L o n d o n equations.
If the direction normal to the surfaces of the plate is chosen as the
a
94
INTRODUCTIO N
TO
SUPERCONDUCTIVIT Y
^-direction, the variation of t h e flux density w i t h ÷ as given b y the
L o n d o n e q u a t i o n s (§ 3 . 2 . 1 ) is
FIG . 8.1. Superconducting plate of thickness 2a with a magnetic field parallel to its
surfaces. Its length and height are assumed much greater than 2a. T h e broken lines
show the direction of the screening currents.
where ÷ is m e a s u r e d from t h e mid-plane of t h e film and t h e thickness of
the film in 2a. W e saw in § 3.2 that, for large specimens, t h e variation of
 w i t h ÷ predicted by the L o n d o n e q u a t i o n s is such t h a t A satisfies the
general definition of ë given by (2.2). W e shall therefore neglect t h e distinction between X and ë in w h a t follows, and write the solutions of t h e
L o n d o n e q u a t i o n s in t e r m s of A. T h e dependence of  on ÷ is illustrated
graphically in Fig. 8.2, for the case w h e r e t h e thickness of the film is
m u c h greater t h a n A. Since the value of t h e m a g n e t i c field strength is H
t h r o u g h o u t t h e film,t t h e intensity of m a g n e t i z a t i o n is equal to (Â/ì ) —
H , and the magnetic m o m e n t Ì (given b y j* IdV) and t h e m a g n e t i c conL
L
a
0
a
tribution to the free energy (given b y — j ì ÌÜÇ
or —\ì ÌÇ )
are
ï
b o t h proportional t o the cross-hatched area. F o r the case of Fig. 8.2 this
is only marginally different from t h e value it would have if A w e r e zero.
However, if a ~ A , the situation is as shown in Fig. 8.3, and it is clear
0
+
See Appendix A.
á
0
á
TH E
SUPERCONDUCTIN G
PROPERTIE S
OF SMAL L
95
SPECIMEN S
FIG . 8.2. Variatio n of  wit h distanc e norma l t o th e surfac e for a plat e of thicknes s 2a
(2a ^> ë). Th e cross-hatche d are a is proportiona l to th e magneti c momen t an d to th e
magneti c fre e energy .
ì
0
Ç
3
-a
0
a
FIG . 8.3. Variatio n of  wit h distanc e norma l t o th e surfac e for a plat e of thicknes s 2a
(a
~ ë).
t h a t the reduction in the cross-hatched area relative to t h e value it would
have if ë were zero is now very considerable. At every point t h e
magnetization is I(x) =
B(x)
—H
so the value of M , the magnetic
a9
m o m e n t per unit area of the plate, is given b y
M-
=
-2aH
ë
Ã
n
i1
L
.áú
tanh
a
X.
(8.3a)
96
INTRODUCTION TO
It is convenient to w r i t e Ì
SUPERCONDUCTIVITY
= —2akH
so t h a t
a
1
Á
(8.4)
t a n h ôë.
D u e t o the p e n e t r a t i o n of t h e flux, t h e effective susceptibility of t h e plate
is —k instead of — 1. N o t e t h a t k is positive, and t h a t k = 1 if ë = 0. T h e
magnetic c o n t r i b u t i o n to t h e G i b b s free energy per unit area of t h e plate
is
-\ì ÌÇ
0
=
á
ì á/æÇÀ
0
so t h a t the critical m a g n e t i c field is given b y
ì áÇÇ
0
2
=
€
G
n
-
G
2a(gn
=
s
-
& )
w h e r e G and G refer t o unit area of t h e plate, and g and g t o unit
volume as before. If w e use H' t o denote t h e critical field of a plate of
thickness 2a for a penetration d e p t h ë and H t h e critical field if ë w e r e
zero, then H' is given b y
n
s
n
s
c
c
c
y
c o m p a r e d w i t h \ìïÇ]
Hence
=g
n
k H '
o
~ Ss
H' = k*H
e
e
g*
=
= g
2
c
c
(8.5)
s
b y (8.2).
v e n
#
- g
n
<
1
ë
. a
tanh
á
ë
(8.6)
i.e. the critical field is increased d u e t o t h e p e n e t r a t i o n of t h e flux. W e
have referred to H as t h e critical m a g n e t i c field for t h e case of zero
penetration, b u t since (8.6) involves only t h e r a t i o á/ë, w e could equally
well regard H as t h e critical m a g n e t i c field for an infinitely large
specimen. F o r t h i s reason H is usually referred to as " t h e bulk critical
field".
E q u a t i o n (8.6) can be simplified for t h e cases of á > ë or a < A. If a > ë,
c
c
c
«
1
and
H ^ H
c
ë
, á
tanh ~ 1
á
ë
ë
a
Ë
T
( l - ^ f ^ H
c
( l
+
^ ) ,
(8.7)
which h a s t h e following simple physical interpretation. If a > ë (8.4)
takes t h e form k ~ 1 — (ë/á) , so t h a t Ì ^ —2(á — ë)Ç . T h i s is a s if t h e
intensity of m a g n e t i z a t i o n / h a d r e m a i n e d equal t o — H t h r o u g h o u t t h e
á
a
97
THE SUPERCONDUCTING PROPERTIES OF SMALL SPECIMENS
plate, b u t the thickness of t h e plate had shrunk to 7{a — ë) , i.e. as if each
surface of the plate had receded i n w a r d s a distance ë. T h i s is in accordance w i t h the phenomenological definition of ë given by (2.2). F o r the
other extreme case of a a < ë,
ë
. a
a
1 - - t a n h ~ ~r
á
ë
3ë
1
Ë
T
so t h a t
22
H' ~y](i)^H .
e
(8.8)
e
It is i m p o r t a n t to get some idea of the order of m a g n i t u d e of the
increase in the critical field due to penetration of the flux. W e have seen
(§ 2.4.1) t h a t the penetration d e p t h obeys the relationship
where ë is about 500 A (i.e. 500 x 1 0 ~ m) for most p u r e metals. E q u a tion (8.7) n o w s h o w s that, iur t e m p e r a t u r e s not too near the transition
temperature, the increase in H will only b e significant (say 1 0 % or
more) if t h e thickness of the plate is about 5000 A or less, i.e. if w e are
dealing with a thin film. F o r t h e case of a very thin film, of thickness
100 A or so (8.8) shows t h a t H' may exceed H b y an order of
magnitude, especially if the t e m p e r a t u r e is close to the critical
temperature.
10
0
c
c
8.3.
c
Mor e Complicate d Geometrie s
Although the basic physics is the same as for the rectangular plate, the
effect of penetration on the magnetic properties of a cylinder or sphere is
m u c h more difficult to calculate, the case of the cylinder involving Bessel
functions. W h e n the dimensions of the specimen are m u c h greater t h a n
A, w e may, however, make use of the simple argument discussed i m m e d i ately after (8.7), according t o which the effect of penetration is as if t h e
magnetization remained equal t o —H t h r o u g h o u t t h e specimen b u t t h e
surface of t h e cylinder were to recede i n w a r d s by a distance ë. U s i n g this
argument it m a y easily be shown t h a t the critical magnetic field of a long
cylinder of a r a d i u s a w i t h the field applied parallel to its axis is given b y
a
F o r the case of a < ë, it can b e s h o w n t t h a t H' = / ( 8 ) ( A / a ) / / .
c
t For example, D. Shoenberg, Superconductivity,
v
C.U.P., 1962, p. 234.
c
98
INTRODUCTION TO SUPERCONDUCTIVITY
8.4.
Limitation s of th e Londo n Theor y
It is clear from (8.6) t o (8.8) t h a t , since ë is of t h e order of 1 0 " cm, t h e
effect of p e n e t r a t i o n will b e t o o small t o p r o d u c e a n y appreciable effect
on t h e critical m a g n e t i c field unless o n e of t h e d i m e n s i o n s of t h e sample
at right angles t o t h e field is about 5000 A o r less, a s o c c u r s in t h e case of
a thin film.
5
40 0
r
0
06
10
(T/T )
2
c
FIG . 8.4. Variatio n of paralle l critica l field wit h reduce d temperatur e for ti n films of
variou s thicknesse s ( Ä , 1000A;D , 2000A;V , 5 0 0 è Á ; 0 , lOOOOA) . Also show n ar e
point s calculate d fro m an effective penetratio n dept h a s prescribe d b y Ittne r ( X ,
2000 A). (After Rhoderick. )
T h e d e p e n d e n c e of t h e critical m a g n e t i c field of t h i n films o n their
thickness is q u i t e p r o n o u n c e d , a s c a n b e seen from F i g . 8.4. T h e critical
fields of these films w e r e d e t e r m i n e d b y observing t h e r e s t o r a t i o n of
resistance b y a m a g n e t i c field parallel t o t h e surface of t h e film. I n t h e
case of a film 1000 A thick, t h e critical m a g n e t i c field close t o t h e critical
t e m p e r a t u r e is over an o r d e r of m a g n i t u d e greater t h a n t h a t for t h e bulk
metal. T h e increase relative t o t h e bulk field is m o s t m a r k e d n e a r t h e
critical t e m p e r a t u r e because, a s w a s p o i n t e d o u t in C h a p t e r 2, t h e
p e n e t r a t i o n d e p t h is found experimentally t o vary w i t h t e m p e r a t u r e a p proximately a s [1 — ( T / T ) ] * a n d b e c o m e s infinite a s Ô a p p r o a c h e s T .
4
c
_
c
A c o m p a r i s o n of t h e experimental results w i t h t h e p r e d i c t i o n s of t h e
L o n d o n theory a s expressed b y (8.6) s h o w s g o o d qualitative agreement,
THE SUPERCONDUCTING PROPERTIES OF SMALL SPECIMENS
99
b u t quantitative comparison is not easy. O n e could, of course, assume
the t r u t h of (8.6) and use this equation to calculate an effective value of A
from the m e a s u r e d values of H
H' and a. T h e i m p o r t a n t point,
however, is w h e t h e r t h e value of ë obtained in this way agrees with the
values obtained from the magnetization of bulk specimens and s h o w s the
same t e m p e r a t u r e variation. Only in this w a y is it possible to check the
validity of t h e L o n d o n theory. I t should b e r e m e m b e r e d t h a t the
m e a s u r e m e n t s of ë obtained from the magnetization of bulk specimens,
discussed in C h a p t e r 2, are independent of any particular penetration
law and depend only on t h e definition of ë given by
cf
c
oo
0
where ÷ is m e a s u r e d from t h e surface. As we pointed out on page 97,
(8.7) is consistent with this definition.
E q u a t i o n (8.6), however, is derived from (8.3), which is the penetration law predicted b y the L o n d o n theory. W h e t h e r or not (8.6) correctly
predicts the critical fields for films of any thickness, using the value of ë
determined from bulk m e a s u r e m e n t s is therefore equivalent to verifying
the t r u t h of the penetration law expressed b y (8.3).
A t t e m p t s to correlate theory and experiment in this w a y have been
only partially successful. F i g u r e 8.4 shows points calculated for a film of
thickness 2000 A using for A the value 520 A obtained by magnetization
m e a s u r e m e n t s of bulk specimens and assuming a variation of ë w i t h
t e m p e r a t u r e given by
0
It is seen t h a t the theoretical p o i n t s lie consistently above the experimental ones.
T o account for the discrepancy w e m u s t recall how the L o n d o n theory
originated. I t is essentially a phenomenological
theory, t h a t is to say, it
w a s introduced because it gives a fairly good description of the Meissner
effect. I t is i m p o r t a n t t h a t w e should not accord the L o n d o n theory the
status of, say, M a x w e l l ' s equations, which are believed to b e exact
expressions of inviolable physical laws. O n e obvious limitation is that it
is essentially a classic:1 theory w h i c h t r e a t s the electrons as classical particles, although w e should expect q u a n t u m effects to b e significant. T w o
important a s s u m p t i o n s m a d e in the L o n d o n theory are t h a t the penetra-
100
INTRODUCTION TO SUPERCONDUCTIVITY
tion d e p t h X is i n d e p e n d e n t of t h e s t r e n g t h of t h e applied m a g n e t i c field
and also of the d i m e n s i o n s of t h e specimen. It should not really surprise
us if the first of these a s s u m p t i o n s t u r n s out not t o be strictly true. As
can b e seen from (3.13), it is equivalent to a s s u m i n g t h a t t h e effective
n u m b e r of superconducting electrons is i n d e p e n d e n t of t h e applied field.
H o w e v e r , t h e application of a m a g n e t i c field is k n o w n t o modify t h e
behaviour of electrons in a profound way, so w e m i g h t expect it to
change t h e degree of order, i.e. t h e effective n u m b e r of superelectrons. It
h a s been s h o w n experimentally b y P i p p a r d t h a t t h e p e n e t r a t i o n d e p t h
does in fact increase w i t h applied m a g n e t i c field, although in bulk superc o n d u c t o r s t h e effect is n o t large except near t h e t r a n s i t i o n t e m p e r a t u r e .
T h e L o n d o n theory is therefore essentially a weak field theory. T h e
effect of such a d e p e n d e n c e of ë on the m a g n e t i c field is t h a t t h e
m a g n e t i c m o m e n t Ì of a film is n o longer linearly proportional t o H , as
(8.3a) would predict if ë were c o n s t a n t , b u t increases less rapidly. T h e
g r a p h of Ì against H is therefore non-linear, as s h o w n in F i g . 8.5 and
L
a
a
H
FIG. 8.5. Magnetization curve of
independent of H
a
a
^
a thin superconductor.
assuming ë increases with
assuming ë
H.
a
H h a s t o b e increased further before t h e free energy of t h e s u p e r c o n d u c t ing state b e c o m e s equal t o t h a t of t h e n o r m a l s t a t e ; consequently H is
increased.
a
c
T h e second a s s u m p t i o n m e n t i o n e d above, t h a t t h e p e n e t r a t i o n d e p t h
is independent of t h e d i m e n s i o n s of t h e specimen, seems less open to
objection, and it is difficult t o explain in simple t e r m s w h y it should not
be correct. Suffice it t o say t h a t , as w a s seen in C h a p t e r 6, s u p e r c o n d u c t ing electrons d o not b e h a v e completely independently of each other, b u t
exhibit "long r a n g e o r d e r " extending for a distance k n o w n as t h e
coherence length £, w h i c h in a bulk s u p e r c o n d u c t o r is about 1 0 " cm. If
the d i m e n s i o n s of t h e specimen are less t h a n t h i s bulk coherence range,
4
101
THE SUPERCONDUCTING PROPERTIES OF SMALL SPECIMENS
the value of î is reduced, and various properties of the superconductor
(among t h e m the penetration depth) are modified.
Various a t t e m p t s have been m a d e , b y Ittner, T i n k h a m and others, t o
" p a t c h u p " the L o n d o n theory in so far as its predictions regarding the
critical fields of t h i n films are concerned by incorporating in it the fielddependence, temperature-dependence and size-dependence of the
penetration d e p t h which result from more recent theories, such as the
microscopic theory of Bardeen, Cooper and Schrieffer (see C h a p t e r 9).
T h e success of these a t t e m p t s is only m o d e r a t e , and since they incorporate into the L o n d o n theory n e w features which are in fact foreign to
it, the procedure is not entirely satisfactory. It seems far preferable to
look for a theory, such as the G i n z b u r g - L a n d a u theory, in which the
necessary non-linearity of Ì w i t h respect to Ç and size-dependence of ë
are inherent.
8.5.
Th e Ginzburg-Landa u Theor y
T h e G i n z b u r g - L a n d a u t h e o r y t is an alternative to t h e L o n d o n
theory. T o a certain extent it is a phenomenological theory also, in t h e
sense that it m a k e s certain ad hoc a s s u m p t i o n s whose justification is t h a t
they correctly describe t h e phase transition in zero field, b u t unlike the
L o n d o n theory, w h i c h is purely classical, it uses q u a n t u m mechanics to
predict the effect of a magnetic field. T h e Ginzburg—Landau theory involves a good deal of algebra and a complete description of it would take
us far beyond t h e scope of this book. H o w e v e r , w e will try t o give a brief
description of w h a t it is about, and of some of its m o r e i m p o r t a n t
predictions. X
T h e first a s s u m p t i o n of the G i n z b u r g - L a n d a u theory is t h a t the
behaviour of t h e superconducting electrons m a y b e described by an
"effective wave function" Ø, which h a s the significance t h a t ÉØÉ is equal
to the density of superconducting electrons. It is then assumed t h a t the
free energy of t h e superconducting state differs from t h a t of the normal
state by an a m o u n t which can b e written as a power series in I Ø I . N e a r
the critical t e m p e r a t u r e it is sufficient to retain only the first t w o t e r m s
in this expansion. G i n z b u r g and L a n d a u then point out that if, for any
reason, the wave function Ø is not constant in space b u t h a s a gradient,
this gives rise to kinetic energy whose origin is the same as t h a t of the
2
2
t V. L . Ginzbur g an d L . D. Landau , J.E.T.P.
2 0 , 1064 (1950).
t An introductio n t o th e Ginzburg-Landa u theor y is given by A. D. C. Grassie , The
ducting State, Susse x Universit y Press , 1975.
Supercon-
102
INTRODUCTION TO SUPERCONDUCTIVITY
kinetic energy t e r m ( A / 2 m ) V T w h i c h a p p e a r s in S c h r o d i n g e r ' s e q u a tion for a particle of m a s s m. T o take a c c o u n t of this, an additional t e r m ,
proportional t o t h e square of t h e g r a d i e n t of Ø , is a d d e d t o t h e e x p r e s sion for the free energy of t h e s u p e r c o n d u c t i n g phase. T h e effect of a
m a g n e t i c field is introduced b y resorting t o a t h e o r e m in classical
m e c h a n i c s w h i c h states t h a t t h e effect of t h e L o r e n t z force ( y v x B) o n
t h e motion of a charged particle in a m a g n e t i c field  m a y b e completely
accounted f o r t b y replacing the m o m e n t u m p , wherever it o c c u r s in t h e
expression for the kinetic energy, b y t h e m o r e complicated expression
ñ — j A . H e r e A is t h e m a g n e t i c vector potential defined b y  = curl A. T o
m a k e the t r a n s i t i o n to q u a n t u m mechanics, ñ is replaced b y t h e o p e r a t o r
—ih grad. T h e total m a g n e t i c c o n t r i b u t i o n t o t h e free e n e r g y of t h e
superconducting state is therefore given b y
2
2
the integral being taken over t h e whole v o l u m e V of t h e specimen.
T h e central problem of t h e G i n z b u r g - L a n d a u a p p r o a c h is n o w t o find
functions Ø(÷,ã,æ)
and A(JC , y,z) w h i c h m a k e t h e total free energy of
the specimen a m i n i m u m subject t o a p p r o p r i a t e b o u n d a r y conditions.
F o r weak m a g n e t i c fields t h e problem is easily soluble and r e d u c e s to t h e
same form as t h e L o n d o n e q u a t i o n s . I n a strong m a g n e t i c field t h e
e q u a t i o n s are only soluble by numerical m e a n s . I n t h e case of an infinitely thick plate w i t h t h e applied m a g n e t i c field parallel t o t h e surface, t h e
solution p r e d i c t s t h a t ÉØÉ is c o n s t a n t in t h e interior of t h e plate b u t falls
off t o w a r d s t h e surface b y an a m o u n t w h i c h increases w i t h the applied
2
t T h i s can be seen by the following simple argument. Suppose that a particle of charge q is
moving in a field-free region with velocity v and that a magnetic field is applied at time t = 0.
T h e field can only build u p at a finite rate, and while it is changing there will be an induced electric field which satisfies Maxwell's relation curl Å = — Â. If A is the vector potential, curl Å =
—(d/dt) (curl A), and integration with respect to the space coordinates gives Å = —(dk/dt) apart
from a constant of integration which does not concern us. Hence the m o m e n t u m at time / is
given by
t
dt
— mv, — qA,
so that mv +
= mv,. Hence the vector ñ — ôçí +qA. is conserved during the application of
the field and must be regarded as the effective m o m e n t u m when a magnetic field is present.
However, the kinetic energy å depends only on mv, and if å = f(mv) before the field is applied, we
must write å = / ( ñ — ^A ) in the presence of the field.
2
THE SUPERCONDUCTING PROPERTIES OF SMALL SPECIMENS
103
magnetic field. Since the penetration d e p t h d e p e n d s on the n u m b e r of
superconducting electrons at the surface, as in the L o n d o n theory, and
hence on ÉØÉ, w e at once get a field-dependent penetration depth. In the
case of a thin film, because of t h e b o u n d a r y conditions, t h e variation of
Ø w i t h ÷ d e p e n d s on the thickness of the film, and since ë again d e p e n d s
on | ø | the penetration d e p t h is a function of the thickness of the film.
T h u s the t w o missing elements in the L o n d o n theory are automatically
provided b y G i n z b u r g and L a n d a u . T h e critical magnetic field can b e
calculated b y the usual m e t h o d of equating the free energy of the film in
the superconducting state w i t h t h a t in the normal state. T h e general
expression for H' is complicated, b u t simplifies in t w o special cases:
2
2
c
(i) a > ë
In this case
where 2a is t h e thickness of the film, ë is t h e penetration d e p t h in a weak
magnetic field, and á is a coefficient very close to unity. T h i s is essentially the same as the L o n d o n result.
(ii) a < ë
In this case Ø is approximately constant t h r o u g h o u t the film, and
H' = [^ ^]
6
c
H . T h i s differs from the L o n d o n result (8.8) by a factor of
c
yj2. A surprising consequence of the G i n z b u r g - L a n d a u theory is t h a t in
this case Ø falls gradually
to zero as Ç approaches H' so that the
transition is a second-order one. (It should be r e m e m b e r e d t h a t in bulk
super-conductors the transition is second order in the absence of a
magnetic field b u t first order in the presence of a field.) T h e transition
c
is first- or second-order according as a ^
. T h i s prediction of the
theory h a s been amply confirmed by D o u g l a s s using t h e tunnelling
technique (see C h a p t e r 10).
T h e G i n z b u r g - L a n d a u theory predicts m a g n i t u d e s of the critical
magnetic fields which are not very different from those of the L o n d o n
theory in the limiting cases of very thick and very thin films. In the intermediate case, where solutions have to be obtained numerically, the
104
INTRODUCTION TO SUPERCONDUCTIVITY
G i n z b u r g - L a n d a u theory does not give a significantly b e t t e r fit w i t h
experiment t h a n the L o n d o n theory if values of ë obtained from
m e a s u r e m e n t s on large specimens are used. T h e great success of t h e
G i n z b u r g - L a n d a u theory is t h a t it correctly p r e d i c t s t h e c h a n g e from
first- to second-order t r a n s i t i o n s w i t h decreasing thickness, w h i c h t h e
L o n d o n theory does not.
V a r i o u s refinements h a v e been incorporated into the G i n z b u r g - L a n dau theory to i m p r o v e its q u a n t i t a t i v e predictions. T h e general situation
w i t h respect to t h e theoretical interpretation of critical fields of thin films
is still, however, not really satisfactory.
8.6.
E d g e Effect s
O n e i m p o r t a n t consequence of the d e p e n d e n c e of the critical m a g n e t i c
field of a film on its thickness is t h a t t h e s h a r p n e s s of the m a g n e t i c t r a n sition for a thin film d e p e n d s very m u c h on t h e n a t u r e of its edges. As a
rule films are prepared in the form of a strip b y evaporating t h e superconducting metal o n to an insulating b a s e (or " s u b s t r a t e " ) t h r o u g h a
mask which b e h a v e s like a stencil. Because t h e m a s k is never in precise
contact w i t h t h e substrate, and also b e c a u s e the metal a t o m s are able to
w a n d e r about on t h e s u b s t r a t e before they finally c o m e to rest, t h e edges
of t h e film are never completely sharp b u t always t e n d to b e tapered, as
shown in Fig. 8.6. T h e tapered edges, being t h i n n e r t h a n the rest of t h e
FIG . 8.6. Typical cross-section of evaporated film, showing tapered edges. T h e righthand edge has been trimmed to produce a well defined rectangular geometry.
film, have a higher critical magnetic field and if t h e film is tested for
superconductivity by passing a c u r r e n t t h r o u g h it and seeing w h e t h e r
any voltage difference a p p e a r s across it ends, the edges will r e m a i n
superconducting and give rise to zero resistance even w h e n t h e rest of
the film is normal. T h i s h a s t w o consequences. First, t h e m a g n e t i c field
strength at w h i c h a voltage difference a p p e a r s m a y b e considerably
greater t h a n the t r u e critical m a g n e t i c field of t h e film; and second,
because t h e edges are unlikely t o b e perfectly uniform along t h e length of
the strip, t h e transition from zero to full normal resistance m a y extend
over a considerable range of values of t h e m a g n e t i c field. T o o b t a i n s h a r p
resistance t r a n s i t i o n s at t h e t r u e value of m a g n e t i c field, t h e edges of a
THE SUPERCONDUCTING PROPERTIES OF SMALL SPECIMENS
105
film are usually t r i m m e d as shown on the right-hand side of Fig. 8 . 6 . T h e
effect of t r i m m i n g a film is shown in Fig. 8 . 7 .
é
R
H.
^
FIG. 8.7 . Effect of trimming the edges on resistance transition of thin film.
8.7.
Transition s in Perpendicula r Magneti c Field s
S o far we h a v e limited the discussion to t h e case w h e r e the magnetic
field is applied parallel to the surface of the film. Since films are always
thin compared w i t h their other t w o dimensions, a film is essentially a
"long, thin s p e c i m e n " in the sense used in C h a p t e r 6 , and demagnetizing
effects are quite negligible. However, if a magnetic field is applied
perpendicular to the surface of the film, demagnetizing effects b e c o m e
very important. If the film takes the form of a relatively n a r r o w strip
whose w i d t h w is m u c h less t h a n its length b u t m u c h greater t h a n its
thickness d, t h e n w e m a y approximate the strip b y an elliptical cylinder
whose cross-section h a s axes w and d. T h e demagnetizing coefficient for
such a geometry is given b y ç ~ 1 — (d/w). According to C h a p t e r 6 , w e
expect the film t o enter the intermediate state if exposed t o a perpenTypical values endicular field of m a g n i t u d e H (l — ç) ^ H (d/w).
countered in practice are w ~ 10~ c m and d ^ 1 0 " cm, so t h a t (d/w) ~
1 0 ~ . W e should therefore expect the film to b e driven normal b y an
extremely small perpendicular field—in fact, the e a r t h ' s field should b e
more t h a n sufficient. In practice this is not so, the reason being t h a t t h e
theory of the intermediate state outlined in C h a p t e r 6 is n o longer valid
w h e n the thickness of t h e film is comparable w i t h t h e penetration depth,
mainly because t h e concept of surface energy discussed in C h a p t e r 6 now
requires considerable modification. A complete discussion of the subject
would take u s far beyond t h e scope of this book. According to T i n k h a m ,
the intermediate state in thin films resembles t h e mixed state in type-II
c
c
l
4
5
106
INTRODUCTION TO SUPERCONDUCTIVITY
superconductors (see C h a p t e r 12) and can b e described in t e r m s of
current vortices.
Experimentally it is found t h a t t h e t r a n s i t i o n to t h e normal state of
thin films in perpendicular magnetic fields d o e s occur at a significantly
lower field strength t h a n in the case of parallel fields, b u t n o t as low as
t h e previous p a r a g r a p h would indicate. F o r t h i s reason, in carrying out
studies of t r a n s i t i o n s in parallel m a g n e t i c fields, great care h a s to b e
taken t o ensure t h a t the film is accurately parallel t o t h e applied field so
that there is n o c o m p o n e n t of t h e field perpendicular to t h e surface. I n
the case of resistive transitions t h i s is easily accomplished b y setting t h e
magnetic field so t h a t about half of t h e n o r m a l resistance is restored, and
then r o t a t i n g the m a g n e t (or t h e specimen) until t h e resistance is a
m i n i m u m , w h i c h m e a n s t h a t t h e critical m a g n e t i c field h a s its m a x i m u m
value.
8.8.
Critica l Current s of Thi n S p e c i m e n s
F o r large specimens w e have already seen t h a t t h e critical c u r r e n t can
be calculated from the critical m a g n e t i c field b y m a k i n g use of Silsbee's
criterion. T h i s states t h a t in the absence of an applied m a g n e t i c field t h e
critical current is t h a t current w h i c h p r o d u c e s at the surface of the
specimen a magnetic field equal t o the critical field H . U n f o r t u n a t e l y
this simple rule d o e s not hold for specimens w h i c h have o n e or m o r e
dimensions c o m p a r a b l e w i t h A, t h e m o s t o b v i o u s reason being t h a t even
if some sort of modified Silsbee's rule w e r e to hold w e should not k n o w
w h e t h e r t o insert the bulk critical field H or t h e actual critical field for
the small specimen H' . T h e problem is a complicated one and d e p e n d s ,
a m o n g other things, on t h e current d i s t r i b u t i o n in t h e film. W e therefore
need some w a y of calculating t h e current distribution, and for simplicity
w e shall follow t h e L o n d o n theory, n o t w i t h s t a n d i n g the limitations of
this theory w h i c h w e h a v e already m e n t i o n e d . An entirely a d e q u a t e
theory of critical c u r r e n t s in thin specimens h a s yet to b e formulated, b u t
the predictions of t h e L o n d o n theory are sufficiently correct t o give some
qualitative indication of t h e effects t h a t are to b e expected.
c
c
c
A further difficulty is t h a t w e c a n n o t a d o p t the simple t h e r m o d y n a m i c
approach of e q u a t i n g t h e free energies of the superconducting and normal p h a s e s because, in the presence of a t r a n s p o r t c u r r e n t supplied b y an
external source, the transition is irreversible d u e to t h e fact t h a t energy
is continuously dissipated in the n o r m a l state. F o r t u n a t e l y a w a y r o u n d
this difficulty w a s found b y H . L o n d o n , w h o pointed o u t t h a t t h e
THE SUPERCONDUCTING PROPERTIES OF SMALL SPECIMENS
107
existence of a critical magnetic field H for a bulk superconductor can b e
regarded as resulting from the existence of a critical current density J ,
as we saw in C h a p t e r 4. T o illustrate this principle, consider the special
case of a superconducting plate of thickness 2a w i t h a magnetic field
applied parallel t o its surfaces (see Fig. 8.1). Suppose t h a t the thickness
of the plate lies along the jc-direction, t h a t the applied field is in the zdirection, and t h a t t h e dimension of the plate in the j - d i r e c t i o n is m u c h
greater t h a n 2a. T h e n to exclude flux from the interior of the plate t h e
shielding c u r r e n t s m u s t flow parallel to thejy-axis, as s h o w n in the figure.
As we have already seen, the solution of t h e L o n d o n e q u a t i o n s gives
c
c
D /
B
{
x
x )
=
cosh(*/A)
coA(a/k)^
w h e r e ÷ is m e a s u r e d from t h e midplane of t h e plate. T h e c u r r e n t density
J can be found from Maxwell's e q u a t i o n ! curl  = ì ], which for t h e
geometry shown in Fig. 8.1 simplifies to
y
0
„
ì
J y
1 dB _
ä÷
0
ff sinh(*/A)
ë cosh ( á / ë )
fl
;
 and J are plotted as functions of ÷ in Fig. 8.8.
Notice t h a t although  h a s the same direction t h r o u g h o u t the plate,
the current density h a s opposite directions in the t w o halves. T h e
current density is greatest at t h e surfaces of the plate, w h e r e it h a s the
m a g n i t u d e (Ç /ë) tanh (á/ë). According to L o n d o n ' s postulate, t h e plate
is driven into t h e normal state w h e n the current density at the surface
reaches a critical value J . However, w e k n o w t h a t for thick specimens
(a > ë) the plate is driven normal w h e n H ~ H , and in this case
t a n h (á/ë) -* 1, so t h e relationship b e t w e e n the critical current density J
and the critical magnetic field is
á
c
a
c
c
Jc = ^ -
(8-9)
N o w suppose t h a t instead of applying an external field H to t h e plate
we p a s s in the ^-direction a current which h a s the m a g n i t u d e . / per unit
w i d t h of the plate in the s-direction. Associated w i t h this t r a n s p o r t
current there will b e a flux density  b o t h inside and outside the plate.
Clearly, from the symmetry of the situation, the current density
a
t Not e tha t we hav e writte n thi s equatio n in term s of  rathe r tha n th e usua l H. Thi s is
becaus e J is not th e "free " curren t densit y (see Appendi x A) bu t th e induce d magnetizatio n
current .
108
INTRODUCTION TO
SUPERCONDUCTIVITY
a tan h ·
ë
ë
H
FIG . 8.8. Variatio n of  an d J wit h ÷ for plat e of thicknes s 2a in unifor m applie d field
associated w i t h t h e t r a n s p o r t c u r r e n t m u s t h a v e t h e s a m e direction in
each half of t h e plate, b u t  will h a v e o p p o s i t e directions. W e therefore
need a solution of t h e L o n d o n e q u a t i o n s w h i c h h a s t h e form
B(x) = -B(-x)
and
J(x) =
J(-x).
S u c h a solution is
cosh(*/A)
cosh
 = -ì ë
and
2
0
(á/ë)
curl J = —ì ë
2
0
jj
(8.10)
 a n d J are plotted a s functions of ÷ in F i g . 8.9. J(a)
c u r r e n t per unit w i d t h / b y
Jf=jjix)dx
= 2XJ(a) t a n h
is related t o t h e
(á/ë)
-a
T h e c u r r e n t density is again a m a x i m u m at t h e surface and if w e
a s s u m e t h a t t h e plate will begin t o g o n o r m a l w h e n J(a) h a s t h e value J
— Ç /ë, as before, t h e n t h e critical c u r r e n t per u n i t w i d t h of t h e plate j ^ ^ ,
is given b y
c
€
•f'
e
= 1ë% t a n h
= 2H
C
(á/ë)
t a n h (á/ë).
(8.11)
T H E SUPERCONDUCTIN G PROPERTIE S O F SMAL L
J
109
SPECIMEN S
- _i _coth_ L
2ë
a
ë
I
"ûï_ /
2
FIG. 8.9. Variation of  and J with ÷ for plate of thickness 2a carrying current J
unit width.
per
If a > ë, t a n h (á/ë ) - 1, and t h e critical current per unit w i d t h b e c o m e s
J = 2H . I n t h i s case, from (8.10) the m a g n i t u d e of t h e flux density at
the surface is ì Ç , i.e. the magnetic field strength is H w h i c h is in accordance w i t h Silsbee's rule. I n other words, for plates which are much
thicker than ë , regarding the destruction of superconductivity
as deterto
mined by a critical current density J is in all respects equivalent
thinking of it as associated with a critical magnetic field H . N o t e t h a t
the critical current per unit w i d t h of the plate is independent of its
thickness, which is to b e expected because all the current is concentrated
w i t h i n a penetration d e p t h of t h e t w o surfaces.
c
C
0
€
c9
c
c
I n the case w h e n á is comparable w i t h ë, t h e critical current per unit
w i d t h J ' is, according to (8.11), given b y
c
J'
C
=
2H
C
t a n h (á/ë ) = J
C
t a n h (á/ë) ,
(8.12)
so t h a t the critical current is reduced b y the factor t a n h (á/ë) , in contrast
w i t h the fact t h a t t h e critical magnetic field is increased.
If á <^ ë ,
t a n h (á/ë ) — (á/ë ) a n d J'
is proportional t o the thickness of the film, as
might be expected from the fact t h a t in this case the current is almost
uniformly distributed t h r o u g h o u t the cross-section of the plate.
I t is interesting t o calculate t h e magnetic field at t h e surface of t h e film
w h e n superconductivity is destroyed b y a current. F r o m (8.10) t h e field
c
110
INTRODUCTION TO SUPERCONDUCTIVITY
strength is given b y \H(a)\
J(a)
= J
C
= H /k,
c
= XJ(a) t a n h (á/ë) , and if J(a)
is given b y
then
H(a) = H
c
t a n h (á/ë) .
Since t a n h (á/ë ) < 1, w e see t h a t t h e m a g n e t i c field s t r e n g t h at t h e surface w h e n superconductivity is quenched b y a current, w h i c h w e denote
by Çi, is not only less t h a n t h e m a g n i t u d e of t h e externally applied field
necessary t o drive t h e film n o r m a l in t h e absence of a current H' , b u t is
even less t h a n t h e bulk critical field H , so t h a t Silsbee's rule is not
obeyed in any form. If á <^ ë, H ~ H a/X, and w e have already seen from
(8.8) t h a t for t h i s case H' = / ( 3 ) # ë / á , so t h a t H H' = ( 3 ) / / J . T h e
G i n z b u r g - L a n d a u theory gives essentially t h e same relationship
b e t w e e n H H' and H , b u t w i t h a factor 4/3 instead of 3.
c
c
t
c
h
c
8.9.
c
í
c
0
{
C
V
v
M e a s u r e m e n t s of Critica l Current s
Reliable m e a s u r e m e n t s of t h e critical c u r r e n t s of t h i n films are h a r d t o
m a k e for t w o reasons—first, because t h e a t t a i n m e n t of a suitable
geometry is difficult and, second, b e c a u s e it is not easy to prevent effects
due t o Joule heating confusing t h e issue.
T h e discussion of § 8.2 a s s u m e d t h a t t h e film w a s infinitely w i d e in
the s-direction, so t h a t t h e c u r r e n t density w a s i n d e p e n d e n t of z. F o r a
film of finite w i d t h this is n o longer t r u e . If t h e t h i c k n e s s of the film w e r e
large c o m p a r e d w i t h ë, t h e d i s t r i b u t i o n of c u r r e n t over its surface would
be exactly t h e same as occurs in an analogous situation in electrostatics,
namely t h e distribution of charge on a c o n d u c t o r of the s a m e g e o m e t r y
(see § 2.3). I t is well k n o w n t h a t charge t e n d s to b e c o n c e n t r a t e d w h e r e
the r a d i u s of c u r v a t u r e is greatest, so it follows t h a t in t h e case of a
superconducting plate of rectangular g e o m e t r y t h e c u r r e n t density will
be greatest at t h e edges. If t h e thickness of t h e plate is c o m p a r a b l e w i t h
ë, t h e analogy w i t h electrostatics is n o longer complete, b u t t h e c u r r e n t
density is still greatest at t h e edges of t h e plate (or film). T h e critical
current will therefore b e very sensitive t o t h e geometrical perfection of
these edges and, as w e saw in § 8.6, it is difficult to obtain perfectly
sharp edges unless the films are t r i m m e d .
A further complication w h i c h often arises w h e n one tries to m e a s u r e
the critical c u r r e n t s of t h i n films is t h a t if t h e r e h a p p e n s t o b e a " w e a k
spot", such a small region w h i c h is t h i n n e r or n a r r o w e r t h a n the rest of
the film, or h a s slightly different metallurgical properties, t h e n t h i s weak
spot will b e driven n o r m a l before t h e t r u e critical c u r r e n t of the film is
THE SUPERCONDUCTING PROPERTIES OF SMALL SPECIMENS
111
reached, and the n o r m a l region m a y t h e n spread b y thermal propagation
as discussed in § 7.2. T h i s m a y b e avoided b y using current in the form
of short pulses (say l / 1 0 / / s e c or even less) which d o not allow t i m e for
any appreciable thermal propagation t o take place, or b y taking great
care to see t h a t the film is evaporated on to a substrate w i t h a high
thermal conductivity in good thermal contact w i t h the helium b a t h .
T h e most careful experiments on critical c u r r e n t s of t h i n films to date
seem to b e those of Glover and Coffey, w h o calculated t h e current distribution from t h e L o n d o n equations, and assumed the film to be driven
normal when the current density at the edges of the strip reached J .
T h e y found t h a t their results were consistent w i t h the existence of a
critical current density and t h a t the value of J for tin at 0°K w a s about
2 x 10 A c m " .
c
c
7
2
CHAPTER 9
THE MICROSCOPIC
OF
THEORY
SUPERCONDUCTIVITY
S o FAR, w e h a v e taken a purely macroscopic view of superconductivity;
t h a t is t o say, w e h a v e a s s u m e d t h a t some of t h e electrons b e h a v e as
superelectrons w i t h t h e m y s t e r i o u s p r o p e r t y t h a t , unlike n o r m a l electrons, they can m o v e t h r o u g h t h e metal w i t h o u t h i n d r a n c e of any kind.
W e have t h e n discussed w h a t restrictions are placed o n their collective
behaviour b y t h e l a w s of electromagnetism a n d t h e r m o d y n a m i c s . I n this
c h a p t e r w e delve a little m o r e deeply and take a microscopic view, trying
t o explain from first principles h o w it is t h a t t h i s p r o p e r t y of t h e
superelectrons arises. T h e complete microscopic t h e o r y of superconductivity is extremely complicated, and r e q u i r e s an advanced knowledge of
q u a n t u m mechanics. All w e can d o w i t h i n t h e scope of t h i s book is t o
give a sketch of t h e physical principles involved.
9.1.
S u m m a r y of th e Propertie s of th e Superconductin g Stat e
T o o b t a i n some clues as t o t h e origin of superconductivity it is helpful
t o s u m m a r i z e t h e m o r e i m p o r t a n t p r o p e r t i e s of s u p e r c o n d u c t o r s .
9.1.1.
Zer o resistanc e
A s w e saw in C h a p t e r 1, s u p e r c o n d u c t o r s show zero resistance for
direct c u r r e n t s and not-too-high frequency alternating c u r r e n t s . O n t h e
other h a n d , t h e reflection and a b s o r p t i o n of visible radiation b y a superconductor is t h e same in t h e superconducting a s in t h e n o r m a l state
( s u p e r c o n d u c t o r s d o n o t look any different w h e n cooled below their
critical t e m p e r a t u r e ) , and since optical properties can b e related t h r o u g h
M a x w e l l ' s e q u a t i o n s to resistivity, w e infer t h a t at optical frequencies
the superconducting state exhibits t h e s a m e resistivity as t h e n o r m a l
state.
112
THE MICROSCOPIC THEORY OF SUPERCONDUCTIVITY
113
T h e frequency at w h i c h resistance begins to appear corresponds to the
microwave or long-wave infrared p o r t i o n s of t h e spectrum. T h e
difference b e t w e e n the reflection coefficient of a superconductor in its
superconducting and normal state is shown as a function of frequency
for a few typical metals in Fig. 9.1. T h e curves are all roughly similar in
shape, and show a steep fall to zero commencing at a frequency v w h i c h
varies from metal to metal. T h e value of v for a particular metal d e p e n d s
on t h e t e m p e r a t u r e and t e n d s t o zero as t h e t e m p e r a t u r e rises t o w a r d s
the critical t e m p e r a t u r e , in a w a y reminiscent of the t e m p e r a t u r e dependence of t h e critical magnetic field. Well below t h e critical
t e m p e r a t u r e v is substantially constant and h a s the value 3 ÷ 1 0 H z
for indium.
0
0
11
0
—
Ps
PN
ë (mm )
2
1
0 5
I
I
I
0-3
I
w
A
PN
Ï
5 ÷ 10
11
10
12
Frequenc y (Hz )
FIG . 9.1. Infrared reflection coefficient of various metals at 1 3°K (after Richards and
Tinkham). T h e ordinate is proportional to the difference between the reflection
coefficients in the superconducting and normal states.
9.1.2.
Crysta l structur e
Studies of t h e crystal structure of superconductors by X - r a y
crystallography as the metal is cooled below the critical t e m p e r a t u r e
have revealed n o change in the lattice structure, either as regards t h e
s y m m e t r y of t h e lattice or t h e actual lattice spacing. It h a s also been
found t h a t properties which depend on vibrations of the crystal lattice,
such as t h e D e b y e t e m p e r a t u r e è and t h e lattice contribution to t h e
specific heat, are the same in t h e normal and superconducting phases. It
is fairly clear, therefore, t h a t superconductivity is not associated w i t h
any change in t h e properties of the crystal lattice.
l.T.S.—L
114
INTRODUCTION TO SUPERCONDUCTIVITY
9.1.3·
Electroni c specifi c hea t
A s we h a v e seen in C h a p t e r 5, t h e specific h e a t of a n o r m a l metal at
low t e m p e r a t u r e h a s the form
w h e r e t h e first t e r m refers t o t h e specific heat of the lattice and t h e
second to the c o n d u c t i o n electrons. H o w e v e r , if a superconductor is
cooled below its critical t e m p e r a t u r e in t h e absence of a m a g n e t i c field,
t w o differences occur. First, t h e r e is a very significant j u m p in t h e value
of the specific h e a t w i t h o u t t h e a p p e a r a n c e of any latent heat, w h i c h
s h o w s t h a t t h e transition is a second-order o n e ; and, second, for very
low t e m p e r a t u r e s t h e specific h e a t is given b y an expression of t h e form
T h e value of A is unaltered w h e n t h e s u p e r c o n d u c t o r is driven n o r m a l b y
a m a g n e t i c field, confirming t h a t t h e lattice specific h e a t d o e s not change,
b u t t h e c o n t r i b u t i o n from t h e c o n d u c t i o n electrons is q u i t e different. It is
clear, then, t h a t t h e superconducting state involves some q u i t e drastic
change in t h e behaviour of t h e c o n d u c t i o n electrons.
9.1.4.
Long-rang e orde r
T h e r e is considerable evidence from v a r i o u s sources t h a t t h e superconducting electrons possess some sort of long-range order. F o r example,
the existence of a positive surface energy m e n t i o n e d in C h a p t e r 6 leads
to t h e conclusion t h a t a b o u n d a r y b e t w e e n n o r m a l a n d s u p e r c o n d u c t i n g
regions c a n n o t b e sharp, and t h a t t h e r e is a gradual c h a n g e in p r o p e r t i e s
b e t w e e n t h e m over a distance w h i c h in type-I s u p e r c o n d u c t o r s is a b o u t
1 0 " cm. I n t e r m s of t h e two-fluid model discussed in C h a p t e r 1, w e m a y
say t h a t t h e c o n c e n t r a t i o n of superelectrons c a n n o t fall t o zero abruptly
at a b o u n d a r y b e t w e e n superconducting and n o r m a l regions, b u t
decreases gradually over a distance £, t e r m e d b y P i p p a r d t h e coherence
length, w h i c h in p u r e m e t a l s is of t h e order of 1 0 " cm. Figuratively
speaking, o n e m i g h t say t h a t t h e superelectrons are s o m e h o w a w a r e of
t h e existence of o t h e r electrons w i t h i n a r a n g e of a b o u t 10~ cm, and
modify their behaviour accordingly. F o r t h i s reason superconductivity is
often referred t o as a co-operative
p h e n o m e n o n . A s will b e seen in
4
4
4
THE MICROSCOPIC THEORY OF SUPERCONDUCTIVITY
115
C h a p t e r 11, there are even m o r e striking interference p h e n o m e n a which
show that superconducting electrons m a i n t a i n some sort of phase
coherence over macroscopically large distances, of the order of metres.
9.1.5.
T h e i s o t o p e effect
A n o t h e r experimental result w h i c h m a d e a significant impact on the
theory of superconductivity w a s t h e discovery in 1950 b y Maxwell, and
independently b y Reynolds, Serin, W r i g h t and Nesbitt, that, if
m e a s u r e m e n t s are carried out on specimens m a d e from different isotopes
of a given element, they are found to have different critical t e m p e r a t u r e s .
In m a n y cases t h e critical t e m p e r a t u r e t u r n s out to b e inversely proportional to the square root of the isotopic m a s s . T h u s , although the atomic
lattice does not itself show any change in properties between the normal
and superconducting states, as discussed in § 9.1.2, it m u s t nevertheless
play a very i m p o r t a n t role in determining the change in behaviour of the
conduction electrons.
9.1.6.
T h e M e i s s n e r effect
In m a n y w a y s t h e Meissner effect is the m o s t fundamental property of
superconductors; it incorporates the property of zero resistance inasmuch as the diamagnetic screening c u r r e n t s are constant in t i m e and
d o not die away as long as t h e applied field r e m a i n s unchanged.
However, although an explanation h a s been given within the framework
of the microscopic theory w e are about to develop, it is difficult to interpret on the level of this book, and is not of m u c h assistance for an
elementary u n d e r s t a n d i n g of superconductivity.
9.2.
Th e Concep t of a n Energ y Ga p
T h e reader w h o k n o w s something about semiconductors m a y have
noticed a similarity between Fig. 9.1 and the corresponding curve for t h e
absorption of infrared radiation by a semiconductor. T h e b r o a d features
of the t w o curves are almost identical, b u t w i t h t h e i m p o r t a n t distinction
t h a t in the case of a semiconductor the absorption " e d g e " (i.e. the
sudden increase in absorption) occurs at a frequency w h i c h is about
three orders of m a g n i t u d e greater t h a n in the case of a superconductor.
T h e explanation of the absorption of radiation b y a semiconductor is
116
INTRODUCTION TO SUPERCONDUCTIVITY
now well established. In a s e m i c o n d u c t o r t h e r e is a separation in energy,
or " e n e r g y g a p " , b e t w e e n t h e t o p of t h e (full) valence b a n d of electron
energy levels and t h e b o t t o m of the (empty) c o n d u c t i o n b a n d . If t h e
frequency of t h e incident radiation is sufficiently high for the p h o t o n
energy hv t o exceed the energy gap, t h e p h o t o n s will b e able to excite an
electron from t h e valence b a n d to t h e c o n d u c t i o n b a n d and at t h e s a m e
t i m e b e absorbed in the process. It is natural to p o s t u l a t e t h a t s o m e t h i n g
similar occurs in t h e case of a superconductor, and t h a t r a d i a t i o n is
heavily absorbed w h e n t h e p h o t o n energy is sufficient t o excite electrons
across an energy gap of some sort. Since in a s u p e r c o n d u c t o r a b s o r p t i o n
occurs for frequencies greater t h a n 1 0 H z , t h e energy g a p m u s t b e of
the order of 10~ eV. It m a y b e observed t h a t if w e express t h i s energy
gap in the form kT, then Ô is a b o u t 1°K, w h i c h is of t h e order of
m a g n i t u d e of superconducting critical t e m p e r a t u r e s . T h e significance of
this will a p p e a r later.
11
4
M o r e evidence for the existence of some sort of energy g a p in t h e electron levels c o m e s from specific h e a t data. A s w e h a v e already pointed
out, at very low t e m p e r a t u r e s t h e c o n t r i b u t i o n t o t h e specific heat d u e t o
the conduction electrons in t h e superconducting state is proportional t o
e~ . T h i s is precisely t h e form to b e expected if t h e r e is a g a p in t h e
range of energies available to an electron. A s t h e t e m p e r a t u r e is raised,
electrons are thermally excited across t h e g a p and for each of t h e s e elect r o n s an a m o u n t of energy equal to t h e energy g a p E is absorbed in t h e
process. It follows from a simple application of statistical m e c h a n i c s t h a t
at a t e m p e r a t u r e Ô the n u m b e r of electrons in energy levels above t h e
g a p is proportional t o e~ * > w h e r e k is t h e B o l t z m a n n ' s c o n s t a n t , and
the t h e r m a l energy absorbed in exciting these c o n d u c t i o n electrons is
T h e specific heat associated w i t h
therefore proportional t o E e~ .
this process is proportional t o t h e derivative of t h e energy w i t h respect
T h e T~ t e r m varies m u c h m o r e
to t e m p e r a t u r e , i.e. to (l/T )e~ * .
slowly w i t h Ô t h a n the exponential t e r m , so t h e variation of specific heat
w i t h t e m p e r a t u r e should b e very nearly exponential.
blkT
g
E
/2kT
Eg/2kT
g
2
E
/2kT
2
Additional evidence for t h e existence of an energy g a p in t h e superconducting state is provided b y t h e tunnelling e x p e r i m e n t s t o b e described
in C h a p t e r 10.
It should be stated, however, t h a t u n d e r certain special c i r c u m s t a n c e s
a superconducting metal m a y n o t p o s s e s s a n energy g a p . W e bring u p
this point again in § 9.3.8. T h e s e " g a p l e s s " s u p e r c o n d u c t o r s are not
typical, and u n d e r normal c i r c u m s t a n c e s all elemental s u p e r c o n d u c t o r s
and m o s t alloys exhibit a well-defined energy gap.
THE MICROSCOPIC THEORY OF SUPERCONDUCTIVITY
9.3·
9.3.1.
T h e Bardeen—Cooper-Schrieffe r
117
Theor y
Restatemen t of th e proble m
T o recapitulate, any successful microscopic theory of superconductivity m u s t b e able t o explain the following:
(i) Superconductivity is essentially b o u n d u p w i t h some profound
change in t h e behaviour of the conduction electrons w h i c h is
marked b y the appearance of long range order and a gap in their
energy spectrum of the order of 1 0 eV.
(ii) T h e crystal lattice does not show any change of properties, b u t
m u s t nevertheless play a very i m p o r t a n t part in establishing superconductivity because the critical t e m p e r a t u r e depends on t h e atomic
m a s s (the isotope effect).
(iii) T h e superconducting-to-normal transition is a phase change of the
second order.
- 4
T h e long-range order noted in (i) clearly m e a n s that the electrons m u s t
interact w i t h each other. I t has, of course, been appreciated for a long
time t h a t t h e conduction electrons in a metal interact very strongly
t h r o u g h their coulomb repulsion, and it is surprising t h a t t h e ordinary
free-electron theory of metals and semiconductors, w h i c h neglects this
interaction, w o r k s as well as it does. It is difficult, however, t o believe
t h a t the coloumb repulsion is t h e interaction responsible for superconductivity because there is n o k n o w n w a y in w h i c h a repulsive interaction
can give an energy gap. F u r t h e r m o r e , because the energy g a p is very
small, the interaction responsible for it m u s t b e very weak, m u c h weaker
t h a n the coulomb interaction. T h e apparent lack of any m e c h a n i s m for a
weak attractive interaction w a s for some t i m e the stumbling block in the
way of any microscopic theory of superconductivity.
9.3.2.
T h e electron—lattic e i n t e r a c t i o n
Electrons m a y b e represented b y waves, and in an absolutely perfect
crystal lattice free from thermal vibrations (i.e. cooled to absolute zero),
these w a v e s would propagate freely t h r o u g h the lattice w i t h o u t a t t e n u a tion, r a t h e r in the w a y in which an electric w a v e can p a s s along a lossless
periodic filter w i t h o u t attenuation. However, if the perfect periodicity of
the lattice is destroyed by thermal vibrations, the lattice behaves like a
periodic filter in which the values of some of t h e c o m p o n e n t s (inductances or capacitors) fluctuate in a r a n d o m m a n n e r . T h i s causes a partial
118
INTRODUCTION TO SUPERCONDUCTIVITY
reflection of t h e wave, a n d in t h e same w a y a n electron w h i c h e n c o u n t e r s
any d e p a r t u r e from perfect periodicity of t h e crystal lattice would have a
certain probability of being reflected, or scattered. W e say t h a t t h e elect r o n interacts w i t h t h e lattice a n d speak of t h e e l e c t r o n - l a t t i c e interaction. I t is this e l e c t r o n - l a t t i c e interaction w h i c h d e t e r m i n e s t h e resistivity of pure m e t a l s a n d s e m i c o n d u c t o r s at r o o m t e m p e r a t u r e . Since b o t h
energy a n d m o m e n t u m m u s t b e conserved w h e n an electron is scattered,
one of t h e vibrational m o d e s of t h e lattice m u s t b e excited in t h e
scattering process. T h i s vibrational m o t i o n is quantized, so w e speak of
the emission (or absorption) of a p h o n o n . A p h o n o n b e a r s t h e s a m e
relationship t o a sound w a v e as a p h o n o n d o e s t o a light w a v e .
An early step forward in t h e search for a microscopic theory c a m e in
1950 w h e n Frohlic h pointed o u t t h a t t h e e l e c t r o n - p h o n o n interaction
w a s able t o couple t w o electrons together in such a w a y t h a t they
b e h a v e d a s if t h e r e w a s a direct interaction b e t w e e n t h e m . I n t h e interaction postulated b y Frohlich , o n e electron e m i t s a p h o n o n w h i c h is then
immediately absorbed b y another, a n d h e w a s able t o show t h a t in certain c i r c u m s t a n c e s t h i s emission a n d s u b s e q u e n t a b s o r p t i o n of a p h o n o n
could give rise t o a weak attraction b e t w e e n t h e electrons of t h e sort
w h i c h m i g h t p r o d u c e an energy g a p of t h e right order of m a g n i t u d e . W e
m a y think of t h e interaction b e t w e e n t h e electrons a s being
transmitted
by a p h o n o n . t
W e c a n represent t h e Frohlic h interaction schematically a s in F i g . 9.2,
w h e r e t h e straight lines represent electron p a t h s a n d t h e w a v y line
represents a p h o n o n . D u r i n g t h e process of p h o n o n emission, m o m e n t u m is conserved, so w e m a y w r i t e for t h e electron w h i c h e m i t s t h e
phonon
Pi = Ñß + q ,
w h e r e p is its
scattering, a n d
magnitude by q
the velocity of
x
(9.1)
m o m e n t u m before scattering, p i i t s m o m e n t u m after
q t h e m o m e n t u m of t h e p h o n o n w h i c h is given in
= hv /s, w h e r e v is t h e frequency of t h e p h o n o n a n d s
sound. ( T h i s is analogous t o t h e expression for t h e
q
q
t A macroscopic process in which there is an interaction between t w o particles resulting from
the exchange of a third particle between them may be envisaged as follows. Suppose a skater
throws a ball to a second skater. T h e n , d u e to conservation of m o m e n t u m in t h e acts of throwing
and catching, each skater will receive an impulse which tends to make him recede from his companion. T h e r e will be an apparent repulsion between the skaters despite the fact that there is no
direct interaction between them. W e could convert this into an attraction by replacing the ball by
a boomerang, so that the first skater throws it in a direction away from his partner.
119
THE MICROSCOPIC THEORY OF SUPERCONDUCTIVITY
FIG. 9.2. Schematic representation of electron-electron interaction transmitted by a
phonon.
m o m e n t u m of a photon.) In the same way, w h e n the p h o n o n is absorbed
by the second electron, the m o m e n t u m of t h e latter changes from p to
p2 so t h a t
2
Pi
+
q
=
(9.2)
p i -
F r o m (9.1) and (9.2) w e get
Pi +
Pi
=
Pi +
Pi
(9.3)
showing t h a t m o m e n t u m is conserved b e t w e e n the initial and final
states, as w e should expect. O n the other hand, although energy is
necessarily conserved b e t w e e n the initial and final states, it does not
have to b e conserved b e t w e e n the initial state and the intermediate state
(i.e. the state in w h i c h the first electron h a s emitted a p h o n o n b u t the
second electron h a s not yet absorbed it), or between the intermediate
state and the final state. T h i s is because there is an uncertainty
relationship b e t w e e n energy and time which takes t h e form ÄÅ . Ä* ~ A.
If the lifetime of t h e intermediate state Ä ß, is very short, t h e r e will b e a
large uncertainty, Ä2?, in its energy, so t h a t energy does not have to b e
conserved in the emission and absorption processes. S u c h processes,
which d o not conserve energy, are k n o w n as virtual processes, and virtual emission of a phonon is possible only if there is a second electron
ready to absorb it almost immediately.
It t u r n s out from the detailed q u a n t u m mechanics of t h e process t h a t
if
— e[ < hv w h e r e å and å[ are t h e energies of the first electron
before and after t h e virtual emission of the phonon, then the overall
result of the emission and absorption processes is t h a t there is an a t t r a c q9
ë
120
INTRODUCTION TO SUPERCONDUCTIVITY
tion b e t w e e n t h e t w o electrons. T h e r e is also, of course, t h e c o u l o m b
repulsion b e t w e e n t h e electrons; w h e t h e r t h e net interaction is a t t r a c t i v e
or repulsive d e p e n d s on w h e t h e r t h e p h o n o n - i n d u c e d attraction exceeds
t h e c o u l o m b repulsion o r vice versa.
Frohlich's suggestion t h a t t h e interaction responsible for s u p e r c o n d u c t ivity is one w h i c h involves lattice v i b r a t i o n s (or p h o n o n s ) enabled h i m
to predict t h e isotope effect before it h a d b e e n discovered experimentally.
T h e fact t h a t an e l e c t r o n - p h o n o n interaction is responsible for superconductivity also explains w h y s u p e r c o n d u c t o r s are b a d n o r m a l cond u c t o r s . F o r example, lead, w h i c h h a s o n e of t h e highest critical
t e m p e r a t u r e s , m u s t h a v e a fairly strong e l e c t r o n - p h o n o n interaction and
a s a result is a poor c o n d u c t o r at r o o m t e m p e r a t u r e , w h e r e a s t h e noble
metals, gold and silver, w h i c h are very g o o d c o n d u c t o r s at r o o m
t e m p e r a t u r e , m u s t b e characterized b y a w e a k e l e c t r o n - p h o n o n interaction a n d d o not b e c o m e superconducting even at t h e lowest t e m p e r a t u r e s
yet attained.
9.3.3.
Coope r pair s
It is a p p r o p r i a t e at t h i s point t o s u m m a r i z e t h e b e h a v i o u r of c o n d u c tion electrons in n o r m a l metals. F o r nearly all p u r p o s e s it is permissible
t o neglect interactions b e t w e e n electrons in t h e n o r m a l state a n d t o
describe each electron as h a v i n g individually a n energy å a n d m o m e n t u m
p . Because of t h e r e q u i r e m e n t t h a t t h e wavefunction for an electron shall
satisfy certain b o u n d a r y conditions, t h e r e is a finite n u m b e r of
eigenstates in a given energy range, each eigenstate h a v i n g an energy å
and m o m e n t u m ñ as well defined as t h e u n c e r t a i n t y principle p e r m i t s .
T h e probability t h a t a given eigenstate is occupied b y an electron is then
given b y t h e Fermi—Dirac d i s t r i b u t i o n
•ËÏ
=
e
(e-e )/kT
F
+
é»
w h e r e å is t h e F e r m i energy. At absolute zero, t h e F e r m i - D i r a c function t a k e s t h e form of a step-function, as s h o w n in Fig. 9.3, and the
p o i n t s w h i c h represent t h e m o m e n t a of t h e electrons in threek n o w n as
dimensional m o m e n t u m space occupy a sphere of r a d i u s p
the " F e r m i sea", w h e r e
ñ
F9
PF =
yj(2*ne )F
121
THE MICROSCOPIC THEORY OF SUPERCONDUCTIVITY
10
f(e )
FIG. 9.3. Probability that a quantum state of kinetic energy å is occupied by an
electron, for the case of normal metal at absolute zero.
Following Frohlich's discovery that the electron—electron interaction
can be t r a n s m i t t e d by phonons, the next step t o w a r d s a microscopic
theory of superconductivity w a s taken by Cooper , t w h o discussed w h a t
h a p p e n s w h e n t w o electrons are added to a metal at absolute zero so that
they are forced b y the Pauli principle to occupy states w i t h p > p as
shown in Fig. 9.4. H e w a s able to show t h a t if there is an attraction
between them, however weak, they are able t o form a b o u n d state so t h a t
their total energy is less t h a n 2å . T o see h o w this comes about, w e will
use some elementary ideas from q u a n t u m mechanics w i t h o u t a t t e m p t i n g
a rigorous derivation.
Consider first the case of t w o non-interacting electrons, w i t h m o m e n t a
p ! and p . T h e two-electron wavefunction (p{x y
z p
tf ,j> ,2 , p ) ,
which d e t e r m i n e s t h e probability that an electron w i t h m o m e n t u m p is
while an electron w i t h m o m e n t u m p is at ( # , . y , 2 ) , *
at (X\ yi Zi)
simply the product of t w o single-electron wavefunctions y X ^ u ^ i * * i , P i )
and ø(÷ , yi> %ii Ñ 2 )· W e will take the dependence on the space coordinates as understood and write for brevity ö
F
F
2
li
li
ly
u
2
2
2
2
l
s
9
9
2
2
2
2
2
0(Pi> P 2 ) = V < P i M P 2 ) -
The
will b e simply plane waves, or, m o r e precisely, Bloch wavefunctions. N o w if there is an interaction between pairs of electrons which
causes scattering of the electrons accompanied b y changes in their
momenta, its effect is to " s c r a m b l e " the wavefunctions so that the t w o t L. N. Cooper, Phys. Rev. 104, 1189 (1956).
t T h e reader who knows some quantum mechanics will realize that the right-hand side of this
equation should properly be an antisymmetric sum of products of space- and spin-wave functions, but this is a complication which need not concern us on the level of this book.
122
INTRODUCTION TO SUPERCONDUCTIVITY
FIG . 9.4. Cooper' s problem : tw o electron s interac t abov e a filled Ferm i "sea" . T h e
diagra m is a sectio n of momentu m spac e showin g th e moment a of th e electrons . T h e
conductio n electron s in a norma l meta l hav e moment a whic h ar e uniforml y distribute d
withi n a spher e of radiu s p . T w o additiona l electron s mus t hav e moment a p, an d p
represente d b y point s outsid e th e sphere . Thei r resultan t momentu m Ñ is conserve d in
scatterin g processe s in whic h th e individua l moment a chang e fro m p, , p t o p' , p .
2
F
2
t
2
electron wavefunction is a m i x t u r e of wavefunctions c o m p r i s i n g a w i d e
range of m o m e n t a , and h a s the form
Ö(*1éËé*1é*2éËé*2)
=
2 * < / 0 ( Ñ ß é Ñ ,)/
ij
= ^<>vV<PiMPj)'
(9.4)
W e can interpret t h e wavefunction Ö a s m e a n i n g t h a t t h e t w o electrons
scatter each other repeatedly in such a w a y t h a t their individual m o m e n 2
ta are constantly changing, and t h a t \a (j\ gives t h e probability of finding
t h e electrons at any instant w i t h individual m o m e n t a p , and pj. Since in
each scattering event t h e total m o m e n t u m of the t w o electrons is conserved, p , and pj m u s t satisfy ñ, + pj = c o n s t a n t = P . D u r i n g t h e actual
scattering process t h e electrons are subject t o their m u t u a l interaction,
and if t h i s is an attractive one, t h e potential energy w h i c h results from it
is negative. H e n c e , over a period of t i m e d u r i n g w h i c h t h e r e are m a n y
scattering events, the energy of t h e t w o electrons is decreased b y t h e
time-average of t h i s negative potential e n e r g y , t and t h e a m o u n t of this
t Thi s is a pictoria l descriptio n of a quantum-mechanica l perturbatio n process . Strictl y
speaking , th e perturbe d wavefunctio n Ö is a stationar y state , an d we ough t not t o spea k of time averages . However , th e descriptio n is good enoug h for our presen t purpose .
123
THE MICROSCOPIC THEORY OF SUPERCONDUCTIVITY
decrease is proportional to the n u m b e r of scattering events which take
place, i.e. t o t h e n u m b e r of w a y s in which w e can choose t w o t e r m s from
the wavefunction Ö. It t u r n s out to be a good enough approximation to
assume t h a t each scattering event contributes an equal a m o u n t — V to
the potential energy. (In q u a n t u m mechanical parlance, — V is t h e
" m a t r i x element" of the interaction connecting two-electron states which
have the same total m o m e n t u m , and we are assuming t h a t this is independent of the individual m o m e n t a of t h e electrons.)
W e have said nothing in this section so far about the n a t u r e of the
interaction, apart from requiring that it b e attractive. If the interaction
is of the sort described in § 9.3.2, arising from the virtual emission and
absorption of a phonon, it t u r n s out from the detailed theory that t h e
probability of scattering is appreciable only if the energy deficit b e t w e e n
t h e initial and intermediate states (å[ + hv — å in t h e notation of
§ 9.3.2) is small, i.e. if å — å[ — hv . If we are considering a metal at a b solute zero w i t h t w o electrons added to it, all the eigenstates w i t h kinetic
energies u p to å are occupied, so b o t h å and å[ m u s t b e above å in
order not to violate Pauli's principle.t T h e lowest values of åé and å\
which are above å and at the same time satisfy å — å\ ^ hv lie within
an energy hv of å , w h e r e v is an " a v e r a g e " p h o n o n frequency typical
of t h e lattice, about half the D e b y e frequency. R e m e m b e r i n g that å =
p /2m, this limitation on t h e allowed values of åé and å[ m e a n s that p
and p\ m u s t lie within a range Ä/> = mhv /p
of the F e r m i m o m e n t u m
p . Since all the pairs of values of p and pj w h i c h m a k e u p t h e wavefunction Ö m u s t satisfy the condition p , + pj = P , the allowed values of ñ can
be found b y t h e construction s h o w n in Fig. 9.5. T h e s e m o m e n t a all begin
or end in the ring w h o s e cross-section is shaded. T h e n u m b e r of such
pairs is proportional to the volume of this ring and h a s a very sharp
m a x i m u m w h e n Ñ = 0, in which case the ring b e c o m e s a complete
spherical shell of thickness Äñ. T h u s the largest number of allowed
scattering processes, yielding the maximum lowering of the energy, is obtained by pairing electrons with equal and opposite momenta. I t also
t u r n s out from t h e detailed q u a n t u m mechanical t r e a t m e n t that the
matrix elements V are largest, and the lowering of t h e energy greatest, if
the t w o electrons have opposite spins. T h i s result comes from consideration of t h e spatial s y m m e t r y of t h e wavefunction and is analogous to the
fact t h a t the g r o u n d state of t h e hydrogen molecule also h a s opposite
spins.
q
÷
q
Ã
é
ñ
Ñ
L
÷
Ñ
ß
q
L
2
x
L
F
F
{
t Here and in what follows å is used to denote the kinetic energy (p /2m).
the potential energy which results from the phonon-induced interaction.
2
It does not include
124
INTRODUCTION TO SUPERCONDUCTIVITY
FIG . 9.5. T h e figure shows two shells of radius p and thickness Ä/> =
(mhv /p )
whose centres are separated by the vector P . ( T h e diagram has rotational symmetry
about the vector P. ) All pairs of momenta p , and pj which satisfy the relation p , + pj =
Ñ can be constructed as shown. T h e number of these pairs is proportional to the
volume in p -space of the ring whose cross-section is shown shaded. T h i s volume has a
sharp maximum when Ñ = 0. (After Cooper.)
F
L
F
An essential condition for the wave function Ö to represent t w o elect r o n s w i t h the lowest possible potential energy is therefore t h a t it be
m a d e u p from wavefunctions of the form y(pt)v>(~~pO » w h e r e t h e first
term describes an electron w i t h m o m e n t u m ñ and spin u p , and the
second t e r m an electron w i t h m o m e n t u m —ñ and spin d o w n . E q u a t i o n
(9.4) now b e c o m e s
where
0(p,t, - ñ ì ) = y ( p , t ) y ( - P i + )
and w e have w r i t t e n a instead of a . S u c h a wavefunction describes
w h a t is k n o w n as a Cooper pair. T o o b t a i n t h e total energy of t h e t w o
electrons, w e m u s t add to their potential energy t h e total kinetic energy
associated w i t h their m o m e n t a . T h i s is simply
{
it
W=2^\a \ (p]/2m).
é
2
{
Because the only states available to t h e t w o electrons are those w i t h p >
p
and also because ÓÉá,É = 1 if Ö is normalized, the kinetic energy
m u s t exceed 2p /2m = 2å . T h e m o s t i m p o r t a n t result from C o o p e r ' s
analysis is t h a t in forming a pair with equal and opposite momenta,
the
lowering of the potential
energy due to the interaction
exceeds the
amount by which the kinetic energy is in excess of 2s . H e n c e , if t h e t w o
electrons go into a state represented b y the wavefunction Ö, in w h i c h
they are continually scattered b e t w e e n states w i t h equal and opposite
2
Fy
F
F
F
THE MICROSCOPIC THEORY OF SUPERCONDUCTIVITY
125
m o m e n t a within t h e range Äñ = mhv /e ,
the total energy of t h e system
is less t h a n it would have been had they gone into states infinitesimally
above p b u t w i t h n o interaction b e t w e e n t h e m .
L
F
F
9.3.4.
Th e superconductin g groun d stat e
T h e problem treated b y Cooper is a s o m e w h a t unrealistic one in t h a t
it involves only t w o interacting electrons. I n a metal there are about 1 0
conduction electrons per c m , and while treating interactions b e t w e e n
t w o electrons is clearly an improvement on a theory w h i c h ignores interactions altogether, one might reasonably ask w h y stop at two? Should
we not take interactions b e t w e e n three or m o r e electrons into account?
T h e great step forward t o w a r d s a microscopic theory of superconductivity came in 1957 w h e n Bardeen, Cooper and Schrieffert were able t o
show h o w C o o p e r ' s simple result could b e extended to apply to m a n y
interacting electrons. T h e fundamental a s s u m p t i o n of the Bardeen—
Cooper-Schrieffer theory (usually k n o w n as the B C S theory) is t h a t the
only interactions which m a t t e r in the superconducting state are those
between any t w o electrons which h a p p e n to make u p a Cooper pair, and
t h a t the effect on any one pair of the presence of all the other electrons is
simply to limit, t h r o u g h the Pauli principle, those states into which the
interacting pair m a y b e scattered, since some of the states are already
occupied.
23
3
C o o p e r ' s result described in the previous section refers to w h a t
h a p p e n s when t w o additional electrons are added to a metal at absolute
zero. However, it applied equally well to the situation in which t w o elect r o n s already belonging to the metal, w i t h m o m e n t a infinitesimally
below p , are transformed into a Cooper pair w i t h equal and opposite
m o m e n t a as described b y the wavefunction Ö (9.5). T h e lowering of
their potential energy due to their m u t u a l interaction exceeds the
a m o u n t b y which their kinetic energy exceeds 2e . H e n c e , if w e start
w i t h a metal at absolute zero, so that the distribution of the electrons is
as in Fig. 9.3, w e can form a state w i t h lower energy b y removing t w o
electrons w i t h p very slightly less t h a n p and allowing t h e m to form a
Cooper pair. If w e can d o this for one pair w e can d o it for m a n y pairs
and lower the energy even further. T h i s is possible because m o r e t h a n
one pair of electrons can b e represented b y the same wavefunction Ö
given by (9.5). I n this case all the superconducting electrons can be
F
F
F
t J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957).
126
INTRODUCTION TO SUPERCONDUCTIVITY
represented t o g e t h e r by a many-electron w a v e function Ø
0
w h i c h is a
product of pair w a v e f u n c t i o n s : t
Ø^Ã,, r
2 )
. . . , r„ ) = Ö ( Ã „ r )<D(r , r ) . . . . Ö ( Ã ^ _ , , r„ ),
2
s
3
4
(9.6)
where njl is the total n u m b e r of pairs, r s t a n d s for t h e position coo r d i n a t e s (x , y z ) of t h e n t h electron, a n d t h e O ' s on t h e right h a n d
side are the same for all pairs. [ I t should b e r e m e m b e r e d t h a t although,
for clarity, w e h a v e not included t h e position c o o r d i n a t e s o n t h e right of
(9.5), t h e 0 ' s c o n t a i n t h e position c o o r d i n a t e s implicitly, as m a y b e seen
on reference t o page 121. I n fact, since the y ' s i n t r o d u c e d on page 121
are plane w a v e s proportional t o exp (ip . r /Ë) , each ö o n t h e r i g h t - h a n d
side of (9.5) is a function of ( r _ ! — r„). ] T h e many-electron wavefunction (9.6) gives t h e probability of finding an electron at r while t h e r e is
another at r , and so on, irrespective of their m o m e n t a . T h e fact t h a t w e
can write such a wavefunction, in w h i c h all t h e individual p a i r s are
represented by (9.5), s h o w s t h a t t h e r e is n o limit t o t h e n u m b e r of
Cooper pairs w h i c h m a y b e represented b y wavefunctions of t h e form
(9.5), and t h a t w e can r e g a r d t h e pair as a c o m p o s i t e particle to w h i c h
the Pauli principle in its simplest form d o e s not apply; in o t h e r w o r d s ,
the pair m a y b e regarded as a particle obeying B o s e - E i n s t e i n statistics.
T h i s p r o p e r t y of C o o p e r pairs, t h a t they are all in t h e s a m e q u a n t u m
state w i t h t h e s a m e energy, will prove t o be of great i m p o r t a n c e later on.
n
n
ny
n
n
l
2
O n e might at first think t h a t there is n o limit to t h e n u m b e r of elect r o n s w h i c h m a y b e raised from ñ < p t o form C o o p e r pairs w i t h a
resultant lowering of the total energy, so t h a t w e should end u p w i t h all
the electrons having p > p \ T h i s , however, is clearly absurd, a n d t h e
reason w h y is not h a r d t o find. I n order t h a t a pair of electrons m a y b e
scattered f r o m ( p , t , —p /l) t o ( p t , — Pji) the s t a t e s (p ,t , — p V) m u s t f i r s t b e
occupied and t h e states ( p t , —p/4) m u s t b e e m p t y . As m o r e and m o r e
electrons form C o o p e r p a i r s w i t h / ) > p > t h e c h a n c e of finding t h e s t a t e s
( p t , — P / l ) e m p t y b e c o m e s progressively smaller a n d smaller, so t h e
n u m b e r of scattering processes w h i c h m a y t a k e place is reduced, w i t h a
consequent decrease in t h e m a g n i t u d e of t h e negative potential energy.
Eventually a condition is reached in w h i c h t h e lowering of t h e potential
energy is insufficient to o u t w e i g h t h e increase in t h e kinetic energy, and
it is n o longer possible t o lower t h e total energy of t h e electrons b y forming Cooper pairs. T h e r e will be an o p t i m u m a r r a n g e m e n t w h i c h gives
the lowest overall energy, and this a r r a n g e m e n t can b e described b y
F
F
7
{
7
F
y
+ T h e remarks on page 121 about antisymmetrization apply here also.
THE MICROSCOPIC THEORY OF SUPERCONDUCTIVITY
127
specifying t h e probability h of t h e pair state ( p , t , —ñ, ö) being occupied
in t h e wavefunction Ø . T h i s probability is related to, b u t is not t h e
same as, t h e coefficient a w h i c h occurs in (9.5). T h e Pauli principle as
applied t o p a i r s requires t h a t A, < 1. According to t h e B C S theory, A, is
given b y
t
ó
t
(9.7)
w h e r e e = $/2m
t
and t h e positive square root is taken. T h e q u a n t i t y Ä,
w h i c h h a s t h e d i m e n s i o n s of energy, t u r n s out t o b e of fundamental
i m p o r t a n c e and is given b y
Ä = 2hv
L
exp
(9.7a)
[-U\e )V\-%
F
w h e r e v is t h e average p h o n o n frequency introduced in § 9.3.3, — V is
t h e m a t r i x element of the scattering interaction, and-Ë (e ) is t h e density
of s t a t e s (ignoring spin) for electrons at t h e F e r m i energy of t h e n o r m a l
metal.
L
F
F i g u r e 9.6 s h o w s t h e probability h as given b y t h e B C S theory,
w h e n t h e wavefunction Ø c o r r e s p o n d s t o t h e
plotted as a function oip
state of lowest overall energy, usually referred t o a s t h e g r o u n d state.
Also shown in F i g . 9.6 is t h e probability of t h e single-electron state w i t h
m o m e n t u m p being occupied in a normal metal at 0°K. T h e i m p o r t a n t
feature of t h e figure is t h a t even at the absolute zero t h e m o m e n t u m d i s tribution of t h e electrons in a superconductor does not show an a b r u p t
discontinuity as in the case of a normal metal.
{
i}
ó
{
W e t h u s see t h a t t h e state of lowest energy ( t h e g r o u n d state) occurs
w h e n all t h e electrons w i t h m o m e n t a w i t h i n a range Ap =
mhv /p
about Pf are coupled together in Cooper p a i r s having opposite m o m e n t u m and spin. T h i s state is often referred t o as a condensed state because
the electrons are b o u n d together t o form a state of lower energy, as
h a p p e n s t o t h e a t o m s of a g a s w h e n they condense to form a liquid. I t is
i m p o r t a n t t o stress t h a t t h e paired electrons described b y t h e wavefunction Ø m u s t b e regarded as all belonging t o t h e same q u a n t u m state and
having t h e same energy, because they are all continually being scattered
b e t w e e n single-electron states having m o m e n t a w i t h i n t h e r a n g e Ä/>, so
t h a t their m o d e s of m o t i o n c a n n o t b e distinguished in any way w h a t soever. T h e total energy of t h e interacting electron p a i r s is c o n s t a n t
despite t h e fact t h a t their m o m e n t a are continually changing, so t h e
time-dependent factors b y w h i c h each t e r m on t h e right h a n d side of
L
ó
F
128
INTRODUCTION TO SUPERCONDUCTIVITY
FIG. 9.6.
Probabilit y h tha t th e two-electro n stat e (p ,f, — ñ ,ö ) is occupie d in
th e BC S ground-stat e wavefunction .
Probabilit y tha t a single-electro n stat e wit h momentu m Éñ,É is occupie d in a
norma l meta l at absolut e zero . Not e that , in th e superconductin g groun d state , even at
absolut e zer o ther e ar e vacancie s with/) / < p an d occupie d state s with/) , > p .
x
F
F
(9.5) m u s t b e multiplied t o give a t i m e - d e p e n d e n t w a v e f u n c t i o n t all oscillate w i t h t h e s a m e frequency. Because t h e s e t e r m s h a v e t h e s a m e
frequency, they m u s t h a v e a definite p h a s e relationship w i t h each other,
and so each a in (9.5) is in general complex. T h i s is often expressed by
saying t h a t t h e superconducting g r o u n d state Ø is a coherent mixture of
single-electron wavefunctions ø (/>*).
t
0
9.3.5.
Propertie s of th e BC S groun d stat e
Correlations
W e saw in C h a p t e r 6 t h a t a s u p e r c o n d u c t i n g - t o - n o r m a l b o u n d a r y is
characterized b y a length î of t h e order of 1 0 " cm, called b y P i p p a r d t h e
coherence length, w h i c h is the shortest distance w i t h i n w h i c h t h e r e m a y
be a significant c h a n g e in t h e degree of order (or in t h e c o n c e n t r a t i o n of
superelectrons). I t c a n b e s h o w n t h a t t h e single-pair wavefunction (9.5)
h a s a spatial extent of about 10~ cm, so t h a t in a r a t h e r loose w a y w e
can regard t h e C o o p e r pair as a sort of large molecule having this size. It
is natural, therefore, to identify t h e spatial e x t e n t of t h e pair wavefunction w i t h t h e coherence length î.
4
4
A m o r e precise definition of î can b e given by posing t h e q u e s t i o n :
w h a t is t h e probability of finding an electron w i t h m o m e n t u m —ñ and
spin d o w n in a v o l u m e element dr at a distance r from a volume element
2
t All th e wavefunction s mentione d hithert o hav e bee n time-independent . T o obtai n a time dependen t wavefunctio n eac h time-independen t wavefunctio n mus t b e multiplie d b y an
oscillator y ter m e~ , wher e Å is th e tota l energy .
iEtm
129
THE MICROSCOPIC THEORY OF SUPERCONDUCTIVITY
dx which c o n t a i n s an electron w i t h m o m e n t u m ñ and spin up? In the
normal metal there are no correlations, i.e. t h e probability is independent
of r, and the answer to the question is \n dr dx
where ç is t h e density
of electrons. In t h e superconducting state t h e answer is again
\n dx dx
for very large values of r, b u t for small values of r the probability is
greater, showing t h a t paired electrons are more likely to be close
together t h a n far apart (Fig. 9.7). T h i s increased probability extends
x
2
l
ly
2
l
2
P(r )
FIG . 9.7. T h e probability P(r)dr dr
of finding an electron with m o m e n t u m —ñ and
spin down in a volume dx at a distance r from a volume dx containing an electron
with momentum ñ and spin up. For the normal state (broken line) P{r) is constant and
equal to n / 4 , where ç is the electron concentration. For the superconducting state
P(r) exceeds n / 4 up to a range î ~ 10~ cm.
l
2
2
x
2
2
4
over a distance of about lfr" c m in a pure type-I superconductor, and
this distance is again identified w i t h the coherence length. T h i s confirms
the interpretation of î as the spatial extent of the pair wavefunction
given by (9.5). Notice that w i t h i n a volume î there lie t h e centres of
m a s s of about 10 other pairs, so t h a t the pair wave functions overlap
considerably.
4
3
7
The energy gap
So far we have considered only the ground state of t h e superconductor, b y which we m e a n the state of lowest energy, or the state of the
superconductor at absolute zero. T h e next question we ask is: w h a t
h a p p e n s if the superconductor is excited to a higher state, say b y raising
the t e m p e r a t u r e or by illuminating it w i t h light of an appropriate
wavelength? In § 9.1.1 w e saw t h a t the absorption of infrared radiation
sets in quite abruptly for frequencies above a certain threshold frequency, and in § 9.2 w e suggested t h a t this might be because of the existence
130
INTRODUCTION TO SUPERCONDUCTIVITY
of an energy g a p of some kind. W e n o w show h o w t h i s can b e explained
w i t h i n the framework of t h e B C S theory.
If energy is i m p a r t e d t o a C o o p e r pair, say b y letting light fall o n it, w e
might think t h a t t h i s increase in energy would b e b r o u g h t a b o u t b y increasing t h e m a g n i t u d e s of t h e m o m e n t a w h i c h occur in t h e wavefunction Ö given b y (9.5). H o w e v e r , Ö already c o n t a i n s a m i x t u r e of all
subject only to the
values of m o m e n t a w i t h i n a range Ä/> = tnhv /p ,
restriction t h a t t h e total m o m e n t u m is zero, so w e c a n n o t increase t h e
energy of t h e pair simply b y increasing t h e m o m e n t a of the electrons and
at t h e same t i m e m a i n t a i n i n g t h e condition t h a t their m o m e n t a are equal
and opposite. W h a t can h a p p e n , however, is t h a t a pair m a y b r e a k u p so
that t h e electrons n o longer have equal and o p p o s i t e m o m e n t a . I n t h i s
case they are unable t o t a k e p a r t in such a large n u m b e r of scattering
events as w h e n they formed a C o o p e r pair, and t h e negative potential
energy resulting from their interaction is almost negligible. T h e y b e h a v e
almost like free electrons, and for t h i s reason are referred t o as " q u a s i particles". I t is not meaningful t o talk of t h e m o m e n t a of t h e individual
electrons before t h e pair w a s b r o k e n u p because, as w e h a v e emphasized,
the individual m o m e n t a of t h e electrons represented b y t h e pair
wavefunction Ö c a n n o t b e specified. I t is, however, meaningful t o talk of
the m o m e n t a of t h e electrons after t h e pair is b r o k e n u p , b e c a u s e they
n o w behave almost as free electrons w i t h well specified m o m e n t a . W e
m a y therefore ask h o w m u c h energy is necessary t o break u p a pair so as
to produce t w o electrons, or m o r e properly quasi-particles, w i t h m o m e n ta p , and pj. ( W h e n w e talk of quasi-particles of m o m e n t a p , t a n d p^f it
m u s t be u n d e r s t o o d t h a t t h e c o m p l e m e n t a r y states — ñ,ö and — ñ,ö are
empty, i.e. t h e quasiparticles have n o p a r t n e r s w i t h w h i c h t o form
Cooper pairs.) According to t h e B C S theory, t h e a n s w e r t o t h e q u e s t i o n
is t h a t the a m o u n t of energy required is
L
Å = E + Ej = {(å - e f
(
(
F
F
+ Ä } + {(å, - * ) + Ä }*,
2
1
2
F
2
(9.8)
w h e r e å = p\/2m and t h e positive square r o o t s are taken. T h e q u a n t i t y
Ä is given b y (9.7a). H e n c e t h e m i n i m u m a m o u n t of energy required is
2Ä, w h i c h occurs w h e n pt = pj = p or å = å = å . T h e r e is t h u s an
energy g a p of m a g n i t u d e 2 Ä in t h e excitation s p e c t r u m of t h e superconductor, and radiation of frequency í is absorbed only if hv > 2 Ä.
T h i s energy g a p h a s t w o causes. Firstly, t h e splitting u p of t h e pair so
t h a t t h e electrons n o longer h a v e equal and opposite m o m e n t a results in
t h e disappearance of their b i n d i n g energy, r a t h e r in t h e w a y t h a t energy
is necessary t o split u p a molecule into i t s c o n s t i t u e n t a t o m s . Secondly, if
{
F
ß
ß
ñ
131
THE MICROSCOPIC THEORY OF SUPERCONDUCTIVITY
the state p t is occupied b y an electron b u t t h e state —pi is empty, then
the pair state (pt , — ñö ) is not available to the remaining Cooper pairs, so
that the n u m b e r of scattering events in which they can participate is
reduced, w i t h a corresponding decrease in their binding energy. H e n c e
the total energy of the entire system of electrons, which is w h a t m a t t e r s ,
is increased even further.
It is i m p o r t a n t to emphasize t h a t b y " a quasi-particle w i t h m o m e n t u m
P i " is m e a n t an electron in the state w i t h m o m e n t u m p, t while the c o m plementary state — p j | is e m p t y . T h e m a g n i t u d e of ñ m a y be either
greater t h a n or less t h a n / ) . I n the ground state, where all electrons near
p form Cooper pairs, t h e probability of a state w i t h m o m e n t u m / ) , being
occupied is given b y h as shown in Fig. 9.6. If, as a result of splitting u p
a pair, there is a quasi-particle in the state p „ then this state is now
definitely occupied w h e r e a s previously the probability of its being occupied w a s h . If p > p , Ë, is small, and w e can say t h a t after a pair is
split u p there is definitely an electron in the state p, , w h e r e previously it
w a s m o s t likely t o b e empty. W e can therefore regard t h e quasi-particle
in p , as an electron. O n the other hand, if p < p , h is close to unity, and
the state p , t is definitely occupied (and the state — p , | definitely empty)
w h e r e previously they were b o t h m o s t likely t o be occupied. W e can n o w
regard t h e quasi-particle w i t h m o m e n t u m p , as a vacancy or " h o l e " w i t h
m o m e n t u m — p, . T h i s view of a quasi-particle as an electron if/), > p or
a hole if p < p m a y seem r a t h e r artificial if p ~ p , b u t if p is far
removed from p the description is o b v i o u s . t
F
F
i9
t
t
F
t
F
t
F
t
F
t
F
{
F
9.3.6· M a c r o s c o p i c propertie s of superconductor s accordin g t o
th e BC S theor y
The critical
temperature
T h e previous section w a s concerned w i t h the ground state of the
superconductor, except in so far as w e considered w h a t happened w h e n a
Cooper pair w a s split u p into t w o quasi-particles, e.g. b y radiation. If t h e
t e m p e r a t u r e is raised above absolute zero, pairs are broken u p b y t h e r m al agitation, and at any particular t e m p e r a t u r e the n u m b e r of quasiparticles is given b y the laws of statistical mechanics. T h e r e is one c o m plication, however; the energy gap is not a constant b u t decreases as the
t T h e reader is warned that in the literature of superconductivity the term "hole" is used
simply to denote an unoccupied state. It does not have the additional significance of a vacant
state with negative effective mass near the top of an otherwise full band, as it does in semiconductor usage.
132
INTRODUCTION TO SUPERCONDUCTIVITY
t e m p e r a t u r e rises. I t is easy t o see w h y t h i s h a p p e n s . A s w e saw in
§ 9.3.5, an electron in t h e state ( p t ) w i t h o u t a p a r t n e r in (—ñö) p r e v e n t s
t h e pair state ( p t , —pi) from being available t o C o o p e r pairs, and t h e pair
interaction energy is diminished because t h e n u m b e r of scattering e ve nts
in which they m a y participate is lessened. T h i s decrease in t h e pair interaction energy m e a n s a decrease in t h e energy gap. As t h e t e m p e r a t u r e
rises, t h e n u m b e r of quasi-particles increases and t h e energy g a p cont i n u e s t o fall, until finally a t e m p e r a t u r e is reached at w h i c h t h e energy
gap is zero. T h i s is t h e critical t e m p e r a t u r e T , and above it t h e electrons
cannot b e represented b y a correlated wavefunction of t h e t y p e given b y
(9.5). T h e variation w i t h Ô of t h e energy g a p E = 2 Ä, a s predicted by
t h e B C S theory, is s h o w n in Fig. 9.8; t h e shape of t h i s curve h a s been
c
g
Ä(ï )
Ä (Ô)
ô
—
FIG . 9.8. Variation of Ä with temperature.
well confirmed b y experiment. T h e t h e o r y also p r e d i c t s t h a t t h e critical
t e m p e r a t u r e is simply related t o t h e energy g a p at absolute zero b y
E (0)=
2Ä(0) = 3 5 * T ,
g
(9.9)
C
w h e r e k is B o l t z m a n n ' s constant. An experimental test of t h i s
relationship is given in T a b l e 9.1, t h e values of Ä(0) being o b t a i n e d from
t h e infrared absorption m e a s u r e m e n t s of R i c h a r d s and T i n k h a m . T h e
error is about ± 0 - 2 0 in each case. T h e experimental values are close t o
the theoretical value, b u t t h e deviations are outside t h e experimental
error; t h i s can b e p u t d o w n to t h e simplifications m a d e in t h e theory,
such as t h e a s s u m p t i o n t h a t t h e m a t r i x element V is independent of t h e
change in m o m e n t u m in t h e scattering process.
Inserting t h e expression (9.7a) for Ä into (9.9) gives an explicit e q u a tion for T :
c
3SkT
c
= 4hv
L
exp [ - { · >
(e )V}-].
1
F
(9.10)
133
THE MICROSCOPIC THEORY OF SUPERCONDUCTIVITY
T A B L E 9.1.
RATI O O F SUPERCONDUCTIN G
ENERG Y G A P AT A B S O L U T E ZER O T O
Superconductor
2A(0)/kT
c
kT
c
(Experimental)
Indium
41
Tin
3-6
Mercury
4-6
Vanadium
3-4
Lead
41
Since v is proportional to M * , w h e r e Ì is the isotopic mass, this
explains the origin of the isotope effect. T h e
dependence is frequently found in practice b u t is n o t universal; d e p a r t u r e s from it can b e
explained in t e r m s of the effect of the C o u l o m b interaction b e t w e e n
electrons.
-
L
The latent heat
F i g u r e 9.8 s h o w s t h a t at t e m p e r a t u r e s below about 0 - 6 T , the energy
gap is substantially independent of t e m p e r a t u r e . A constant a m o u n t of
energy, 2Ä(0), is therefore needed t o break u p a Cooper pair and t h e
n u m b e r of pairs broken u p at a t e m p e r a t u r e Ô in this range is p r o p o r T h i s leads t o an electronic specific h e a t which is
tional t o e~ .
as is found experimentally at low t e m p e r a t u r e s
proportional t o e~ ,
(cf. §§ 9.1.3 and 9.2 and C h a p t e r 5).
c
M0)/kT
A{0)/kT
At t e m p e r a t u r e s close to T the specific heat rises m o r e rapidly w i t h
t e m p e r a t u r e because Ä ( Ô) b e c o m e s smaller. Above T , w h e r e the elect r o n s behave as in an ordinary metal, there is no contribution to t h e
specific heat d u e to t h e splitting u p of pairs, so at t h e critical
t e m p e r a t u r e there is an abrupt fall in the specific heat as t h e t e m p e r a t u r e
rises. T h e energy g a p falls smoothly t o zero as Ô approaches T , so the
total energy of the electrons as Ô approaches T from below is exactly
the same as w h e n it approaches T from above. T h e r e is therefore n o
latent heat associated w i t h t h e transition, and as a result n o change in
entropy. T h i s c o m b i n a t i o n of a discontinuity in t h e specific heat w i t h t h e
absence of a latent heat is characteristic of a second order transition; w e
can think of t h e critical t e m p e r a t u r e as the t e m p e r a t u r e at w h i c h t h e internal energy of t h e electrons begins to change (due to the appearance of
c
c
c
c
c
134
INTRODUCTION TO SUPERCONDUCTIVITY
the energy gap) r a t h e r t h a n one at w h i c h it u n d e r g o e s an a b r u p t change.
T h i s is in m a r k e d c o n t r a s t to t h e case of a first-order transition such as
the freezing of a liquid, w h e r e t h e r e is a discontinuity in t h e internal
energy of t h e a t o m s accompanied b y a latent heat.
The critical magnetic field
As we saw in C h a p t e r 4, t h e critical m a g n e t i c field satisfies the
relationship
\ì Ç
0
2
€
= g
n
-
g,
s
where g a n d g are the G i b b s free energy densities of t h e n o r m a l and
superconducting phases. At absolute zero g — g is simply equal to t h e
difference b e t w e e n t h e internal energy densities in t h e n o r m a l and superconducting states (neglecting any difference in v o l u m e b e t w e e n t h e t w o
phases), and according to t h e B C S theory this is simply t h e total b i n d i n g
energy of t h e Cooper pairs as c o m p a r e d w i t h t h e n o r m a l m e t a l in w h i c h
n o pairs are formed. T h i s can b e calculated and t h e result is
n
s
n
W 8
= (gn - A)r=o =
s
ÉÉÌÄ(Ï)]
2
B u t Ä(0) is related to T b y 2Ä(0) = 3-5AT so t h a t
c
c
HI _ 0 - 4 7 y
ÔÉ
ìï
(9.11)
where ã = } ð . 1 \e )k
is the coefficient of Ô in t h e expression for the
is
specific heat in t h e n o r m a l state (see C h a p t e r 5 and § 9.1.3) and Ë(e )
the density of states at t h e F e r m i energy of t h e n o r m a l metal.
T h e r e is t h u s a law of corresponding s t a t e s in t h e sense t h a t if any t w o
out of H , T and ã are k n o w n , t h e third can b e predicted. T h e accuracy
w i t h which relationship (9.11) is satisfied is limited b y t h e accuracy of
(9.9). I t is i m p o r t a n t to n o t e t h a t there are n o disposable p a r a m e t e r s in
(9.11), and it is one of t h e striking successes of t h e B C S theory that,
despite the e n o r m o u s variation in t h e p r o p e r t i e s of m e t a l s in t h e n o r m a l
state, different s u p e r c o n d u c t o r s d o obey q u i t e closely such a law of corresponding states.
2
2
F
F
0
ct
The criterion for the existence of
superconductivity
It is natural to ask w h e t h e r all metals will exhibit superconductivity if
cooled t o low e n o u g h t e m p e r a t u r e s . T h e answer given b y t h e B C S
theory is t h a t t h i s is not necessarily so; m e t a l s will show superconducting behaviour only if t h e net interaction b e t w e e n electrons resulting
THE MICROSCOPIC THEORY OF SUPERCONDUCTIVITY
135
from the combination of t h e phonon-induced and coulomb interactions is
attractive. T h i s is w h y the good normal conductors like silver and
copper, which h a v e a weak electron-phonon interaction, d o not exhibit
superconductivity (at any r a t e d o w n to t h e lowest t e m p e r a t u r e s yet
achieved).
9.3.7·
Th e current-carryin g state s
W e have said nothing so far about the p h e n o m e n o n from which superconductivity acquired its n a m e , viz., the disappearance of resistance.
B o t h the ground state and the excited states which w e h a v e discussed
have a perfecdy isotropic distribution of electrons in m o m e n t u m space,
so t h a t there are as m a n y electrons travelling one way as t h e other, and
no current flows. I t is, however, possible to visualize a situation in which
each Cooper pair, instead of having zero total m o m e n t u m , h a s a resultant m o m e n t u m Ñ which is the same for all pairs. In this case the states
which m a k e u p t h e pair wavefunction have m o m e n t a of t h e form
_(p ' fM-p ' ik
+
+
instead of ( p , t , —p/>0 as in (9.5). T h e entire m o m e n t u m distribution
is shifted bodily in m o m e n t u m space by an a m o u n t P / 2 , as shown
in Fig. 9.9. T h e electron pairs are still able t o take part in a large n u m ber of scattering process which conserve the total m o m e n t u m . T h e s e
can b e described formally as scattering from
[(p,+f)t.(-p,+fk
to
As seen b y an observer moving w i t h velocity
P / 2 m , the situation is indistinguishable from t h a t already discussed in
which the total m o m e n t u m w a s zero, and t h e total energy of t h e elect r o n s is t h e same except t h a t it is increased b y an additional kinetic
energy n^/Sm,
w h e r e n is t h e total n u m b e r of superconducting electrons. ( W e are considering here the situation at absolute zero, w h e r e
there are n o quasi-particles and all the electrons are paired.) T h e
wavefunction of a Cooper pair now b e c o m e s !
s
t T h i s separation of the pair wavefunction into a wavefunction Ö representing the relative
motion of the electrons and a factor representing the motion of the centre of mass cannot always
be done, but it seems to be valid under most conditions of interest.
136
INTRODUCTION TO SUPERCONDUCTIVITY
FIG . 9.9. M o m e n t u m distribution in current-carrying superconductor. T h e momentum vectors are uniformly distributed throughout a sphere of radius p (shown
shaded) whose centre X is displaced by a vector P / 2 from the origin Ï . Ñ is the total
m o m e n t u m of a Cooper pair. In the absence of a current the sphere is centred about
the origin. Above a certain current density it is energetically possible for a pair to split
up into t w o quasi-particles whose momenta are represented by the points A and B.
F
Ö
ñ
=
ö ^ Ñ (r, + r ) / 2 ^
2
(9.12)
w h e r e Ö is t h e pair wavefunction described b y (9.5) and t h e exponential
term represents t h e m o t i o n of t h e c e n t r e of m a s s of the pair w i t h total
m o m e n t u m P . If r d e n o t e s t h e position of t h e centre of m a s s of t h e pair,
r = (r, + r ) / 2 , and (9.12) b e c o m e s
2
Öñ = ÖÝ?' - \
Ñ
r ,
(9.12a)
In t h i s picture t h e current is carried b y p a i r s of electrons which h a v e a
total m o m e n t u m P . W h e n a c u r r e n t is carried b y an o r d i n a r y conductor,
such as a n o r m a l metal or a semiconductor, resistance is inevitably
present because t h e current carriers (either electrons or holes) can b e
scattered w i t h a c h a n g e in m o m e n t u m so t h a t their free acceleration in
the direction of t h e electric field is hindered. T h i s scattering m a y b e d u e
to i m p u r i t y a t o m s , lattice defects or t h e r m a l vibrations. I n t h e case of a
superconductor, t h e electrons w h i c h m a k e u p a C o o p e r pair are constantly scattering each other, b u t since t h e total m o m e n t u m r e m a i n s
constant in such a process there is n o c h a n g e in the current flowing. T h e
only scattering process w h i c h can reduce t h e c u r r e n t flow is o n e in w h i c h
the total m o m e n t u m of a pair in t h e direction of the c u r r e n t changes, and
this can only h a p p e n if t h e pair is b r o k e n u p . H o w e v e r , this de-pairing
requires a m i n i m u m a m o u n t of energy 2 Ä, so t h e scattering can only
h a p p e n if t h i s energy can b e supplied from s o m e w h e r e . F o r low c u r r e n t
densities there is n o w a y in w h i c h t h i s energy can b e i m p a r t e d t o t h e
137
THE MICROSCOPIC THEORY OF SUPERCONDUCTIVITY
pairs, so scattering events which change t h e total m o m e n t u m of a pair
are completely inhibited and there is no
resistance.^
T h e m o m e n t u m of the pairs is related to t h e current density j b y j =
en P/2m, w h e r e n is the total n u m b e r of superconducting electrons and e
the electronic charge. As j increases, the m o m e n t u m distribution s h o w n
in Fig. 9.9 becomes m o r e and m o r e displaced, until finally it b e c o m e s
energetically possible for a Cooper pair t o split u p into t w o q u a s i particles whose m o m e n t a are represented b y p o i n t s such as A and  o n
the surface of the displaced sphere nearest to t h e origin. T h i s can be seen
if w e take as t h e zero of energy the energy of the superconductor w i t h a
full complement of pairs (no quasi-particles) and n o current (P = 0).
Relative to this zero, the energy of the superconductor carrying a
current, b u t still w i t h n o quasi-particles, is
s
s
W =
n P /Sm
2
x
s
which is t h e additional kinetic energy of n electrons each w i t h m o m e n t um P/2.
If the superconductor carries no current b u t h a s one pair split up, t h e
m i n i m u m excitation energy required is 2Ä, and this occurs w h e n t h e
quasi-particles h a v e m o m e n t a p [see (9.8)]. Suppose these quasiparticles have m o m e n t u m vectors pointed t o t h e left (Fig. 9.9) and t h a t
the whole distribution is now displaced an a m o u n t P/2 to t h e right so
t h a t the m o m e n t a of the quasi-particles are represented b y A and B. T h e
energy of the superconductor relative to our zero is now
s
F
w h e r e the first t e r m is the energy required to break u p the pair, the
second t e r m the additional kinetic energy of t h e (n — 2) paired electrons,
and t h e third t e r m t h e change in kinetic energy of t h e quasi-particles. I t
will b e energetically favourable for the pair t o split u p if W > W , i.e. if
s
x
^ > 2 Ä
or
P > ^ .
2
(9.13)
t It is clear that this explanation of zero resistance cannot apply to the "gapless" superconductors referred to on page 116. It is believed that in this special category of superconductors
scattering of pairs is inhibited not by the presence of an energy gap but by the strongly correlated
nature of the pair wavefunction. If this is correct, it seems certain that such a mechanism must
also be important for those superconductors which do exhibit an energy gap.
138
INTRODUCTION TO SUPERCONDUCTIVITY
Since Ñ is proportional t o the c u r r e n t density this m e a n s t h a t t h e r e will
be a critical current density above w h i c h scattering accompanied b y a
change in total m o m e n t u m (i.e. resulting in t h e breaking u p of a pair)
m a y take place. Above this critical current density resistance will appear.
C o m b i n i n g t h e expression for / w i t h t h e condition (9.13) w e find
(9.14)
In C h a p t e r 8 w e stated t h a t for tin, t h e critical c u r r e n t density at a b solute zero is about 2 x 10 A c m . I n (9.14) t h e m o s t u n c e r t a i n q u a n t i ty is n , and if w e substitute for Ä t h e value 1-80AT ( T a b l e 9.1) and for
p t h e value corresponding to v = 6-9 ÷ 1 0 c m s " , w e find n = 8 x
1 0 c m " . T h i s is appreciably less t h a n one electron per a t o m , b u t is not
unreasonable in view of t h e complicated b a n d s t r u c t u r e of tin.
At a non-zero t e m p e r a t u r e , some of t h e p a i r s will b e b r o k e n u p into
quasi-particles even for c u r r e n t s below t h e critical current. T h e q u a s i particles b e h a v e very m u c h like n o r m a l electrons; they can b e scattered
or excited further, a n d if they carry a current, will exhibit resistance. O n
the other hand, t h e r e m a i n i n g p a i r s retain t h e properties of the electrons
at absolute zero, and cannot b e scattered unless an a m o u n t of energy 2 Ä
is available. T h e s e paired electrons are w h a t w e h a v e referred to as
"superelectrons". T h u s w e can identify t w o almost i n d e p e n d e n t fluids of
normal and superelectrons, a concept m a d e u s e of in C h a p t e r 1.
7
- 2
s
C
7
F
1
F
21
9.3.8.
s
3
T h e pai r w a v e f u n c t i o n : l o n g - r a n g e c o h e r e n c e
W e saw in t h e previous section t h a t if a C o o p e r pair h a s a total
m o m e n t u m it can b e presented b y a wavefunction
Ö
ñ
= Öâ*
(9.12a)
à /Ë
w h e r e Ö is t h e wavefunction given b y (9.5). W e also saw in § 9.3.4 t h a t
Ö is characterized b y a coherence length £ ( ~ 1 0 ~ cm), w h i c h is t h e distance w i t h i n w h i c h there is a spatial correlation b e t w e e n electrons w i t h
equal and opposite m o m e n t a . If w e regard t h e pair as being in a r a t h e r
loose sense a sort of b o u n d molecule, t h e n Ö r e p r e s e n t s the internal
motion of t h i s molecule and î its spatial extent. T h e exponential t e r m in
(9.12a) is a travelling w a v e w h i c h r e p r e s e n t s t h e m o t i o n of the c e n t r e of
m a s s of the pair, and the wavelength of this wave is Ë/Ñ, which is t h e
de Broglie wavelength of a particle of m o m e n t u m P . T h e p h a s e coherence
of t h i s travelling w a v e e x t e n d s over indefinitely large distances, very
4
THE MICROSCOPIC THEORY OF SUPERCONDUCTIVITY
139
m u c h greater t h a n | , in fact the wave travels unscattered t h r o u g h the
whole volume of t h e superconducting metal. In the case of a persistent
current in a ring, therefore, the phase coherence extends over centim e t r e s ; in the case of the winding of a superconducting solenoid over
miles I T h e consequences of t h i s very long range coherence are discussed
in C h a p t e r 11.
T h e energy g a p in the electron energy spectrum of a superconductor
can b e reduced b y a n u m b e r of agencies; for example, incorporation of
magnetic impurities. It is found that magnetic impurities lower the t r a n sition t e m p e r a t u r e as well as reducing the energy gap, b u t that t h e t r a n sition t e m p e r a t u r e can still b e above zero at an impurity concentration
which h a s reduced the energy g a p to zero. I n t h i s condition t h e metal is
resistanceless, implying long-range coherence of the electron-pair wave,
even t h o u g h there is no energy gap. T h i s condition is called gapless
superconductivity;
it implies t h a t it is the long-range coherence of the
electron-pair wave, not the energy g a p in the electron energy spectrum,
which is the essential feature of superconductivity. I n gapless superconductors there is at the F e r m i level a m i n i m u m in the density of states,
b u t no actual gap. I n general, if a perturbation causes a superconductor
to p a s s into the n o r m a l state t h r o u g h a second-order transition, the
superconductor goes into t h e gapless state before it b e c o m e s normal.
A discussion of gapless superconductivity is given in the articles b y
Meservy and S c h w a r t z and b y M a k i . t
t T h e s e articles are in Superconductivity,
York).
ed. R. D. Parks, 1969 (Marcel Dekker Inc., New
CHAPTER 10
TUNNELLING
AND T H E
ENERGY
GAP
WE DESCRIBE in t h i s c h a p t e r a t e c h n i q u e introduced b y G i a e v e r in 1960
for t h e direct m e a s u r e m e n t of t h e energy g a p in a s u p e r c o n d u c t i n g metal
b y studying t h e tunnelling of electrons i n t o t h e metal t h r o u g h a very t h i n
insulating film. T h e discovery of t h i s t e c h n i q u e soon after t h e publication of t h e B C S theory m a d e a great c o n t r i b u t i o n t o w a r d s o u r u n d e r s t a n d i n g of superconductivity b e c a u s e of t h e ease w i t h w h i c h it
enabled energy g a p s to b e m e a s u r e d experimentally.
10.1.
Th e Tunnellin g Proces s
T o u n d e r s t a n d t h e m e t h o d , consider first w h a t h a p p e n s if t w o plates
m a d e from a n o r m a l metal are separated b y a very t h i n g a p . T h e energy
b a n d d i a g r a m for this situation is s h o w n in Fig. 10.1. U n d e r o r d i n a r y
Vacuu m
(a )
(b )
FIG. 10.1. Tunnelling between normal metals. T h e single-hatching represents empty
states and the double hatching occupied states. At absolute zero, with no bias between
the plates, as in (a), tunnelling is completely forbidden by the Pauli principle. If a
positive voltage is applied to the right-hand plate, as in (b), so that the Fermi levels no
longer coincide, there are occupied states on the left opposite to unoccupied states on
the right, and tunnelling can take place as shown by the arrows.
140
141
TUNNELLING AND THE ENERGY GAP
conditions an electron in either plate cannot leave t h e metal because its
energy is considerably less t h a n the potential energy of a free electron in
the v a c u u m outside. But if the separation b e t w e e n the metals is extremely small, an electron in one metal can cross to the other b y virtue of a
q u a n t u m - m e c h a n i c a l p h e n o m e n o n k n o w n as tunnelling. I n this process
the electron is represented outside the metal b y an exponentially
attenuated standing wave w h o s e amplitude falls off as e~
instead of
the usual travelling wave which represents t h e electron inside t h e metal.
T h e length æ is typically of t h e order of 10~ cm, so t h e w a v e is
attenuated very quickly indeed. However, if the gap is very thin (of the
order of 10~ cm) there is a small b u t significant chance t h a t an electron
m a y p a s s t h r o u g h the potential barrier separating t h e plates, and
exchange of electrons b e t w e e n the t w o metals b e c o m e s possible. T h i s
" t u n n e l l i n g " process is also possible if t h e t w o electrodes are separated
b y a very thin insulating film, such as the film of oxide w h i c h often forms
on the surface of metals exposed to the atmosphere, and t h i s is the situation which is i m p o r t a n t in practice.
T h e r e are t w o conditions which m u s t b e fulfilled for tunnelling to take
place, apart from t h e obvious one t h a t t h e separation b e t w e e n the m e t a l s
m u s t not b e large compared w i t h the attenuation length æ of the
tunnelling wavefunction. First, energy m u s t b e conserved in the process,
i.e. the total energy of the system, including the metals o n b o t h sides of
the insulating film, m u s t b e the same before and after tunnelling. Second,
tunnelling can take place only if the states to which the electrons tunnel
are empty, otherwise t h e process is forbidden b y the Pauli principle. T o
take as an example the situation shown in Fig. 10.1a, in w h i c h w e h a v e
for simplicity assumed t h a t the metals are separated b y a v a c u u m , at a b solute zero there could be n o tunnelling from one metal t o the other
because all the states which satisfy the first condition are occupied o n
b o t h sides. H o w e v e r , if there is a small voltage difference b e t w e e n t h e
metals (say the left-hand one negative w i t h respect to t h e right), t h e
energy levels will be shifted w i t h respect to each other as in Fig. 10.1b.
T h e r e will now b e e m p t y states on the right opposite to the t o p m o s t
occupied states o n the left, and tunnelling can take place from left t o
right. T h e n u m b e r of states uncovered in this way is proportional to t h e
voltage difference, and if the tunnelling probability is constant, as it is
for very small bias voltages, the resulting current is linearly proportional
to the voltage.
W i t h superconductors, a n u m b e r of situations m a y b e envisaged. W e
m a y have one of t h e electrodes superconducting and the other normal, or
x/:
8
7
142
INTRODUCTION TO SUPERCONDUCTIVITY
b o t h electrodes m a d e of t h e s a m e superconductor, or the electrodes m a d e
from t w o different superconductors. All of these h a v e current—voltage
characteristics w h i c h are peculiar t o t h e particular c o m b i n a t i o n of m e t a l s
involved.
10.2.
T h e E n e r g y L e v e l D i a g r a m for a S u p e r c o n d u c t o r
T h e difference b e t w e e n tunnelling involving s u p e r c o n d u c t o r s and
tunnelling involving normal m e t a l s can b e explained in t e r m s of the
difference b e t w e e n their energy level d i a g r a m s . W h e n w e d r a w a b a n d
d i a g r a m for an ordinary metal, as in Fig. 10.1, we are showing t h e r a n g e
of energies allowed t o an individual electron. T h e electrons are independent of each other, so t h e energy of a particular electron is not affected
b y w h e t h e r or not a n o t h e r level h a p p e n s t o b e occupied. I n the case of a
superconductor t h i s is n o longer t r u e . T h e electrons in t h e condensed
Excite d quasiparticle s
Condense d pair s
FlG. 10.2. Energy level diagram for superconductor.
state are not independent of each other, and their contribution to the
total energy d e p e n d s very m u c h on w h e t h e r or not they h a v e a p a r t n e r
w i t h equal and opposite m o m e n t u m . W e cannot therefore d r a w a b a n d
d i a g r a m in t h e usual way. As w e have already pointed out, all t h e pairs
have the same energy b e c a u s e they are all represented b y t h e s a m e
wavefunction Ö [eqn. (9.5)], and we c a n therefore d r a w a single level as
in Fig. 10.2 to represent t h e average energy per electron (or one half the
energy of a C o o p e r pair) in the condensed state. T h i s level can only contain paired electrons. As w e saw in § 9.3.2, this level can c o n t a i n m a n y
pairs because the Pauli principle in its usual form does not apply to
Cooper pairs. If any of t h e pairs is split u p , the energy of t h e system is
increased in accordance w i t h (9.8), namely
E + Ej=
t
{(å - e f
{
F
+ Ä } * + \{sj - e f
2
F
+ Ä }*,
2
143
TUNNELLING AND THE ENERGY GAP
so t h a t each of the resulting quasi-particles can b e regarded as contributing an a m o u n t of energy {(å — å ) + Ä }*, w h e r e å = p /2m.
Since
the quasi-particles behave almost like independent electrons, their
allowed energy values can b e represented b y a c o n t i n u u m of levels
separated by an interval Ä from t h e level representing the pairs, as in
Fig. 10.2. (Notice that the q u a n t i t y w e have called the energy gap is 2Ä.
T h i s is because if quasi-particles are produced by splitting u p a pair they
are always produced t w o at a time. I t is, however, possible to inject a
single quasi-particle by tunnelling, and if this is done t h e m i n i m u m
energy added to the system is Ä.)
2
2
2
ñ
At absolute zero there are n o quasi-particles and the c o n t i n u u m states
are e m p t y . At t e m p e r a t u r e s above absolute zero, the c o n t i n u u m levels
are partly filled in accordance w i t h the laws of statistical mechanics.
10.3.
Tunnellin g Betwee n a Norma l Meta l an d a
Superconducto r
Consider first tunnelling b e t w e e n a normal metal and a superconductor at absolute zero. T h e energy level d i a g r a m s are as shown in Fig. 10.3.
W i t h the plates at the same potential, the F e r m i level in the normal
metal coincides w i t h the level representing t h e condensed pairs in t h e
superconductor as in Fig. 10.3a. Again tunnelling is impossible if there is
n o potential difference b e t w e e n the plates.
If a positive b i a s of V volts is applied to the superconductor, all its
energy levels are l o w e r e d t relative to those of the normal metal, b u t n o
tunnelling processes which conserve energy are possible until V reaches
the value Ä/â, w h e n the b o t t o m of the c o n t i n u u m of quasi-particle levels
coincides w i t h t h e F e r m i level in t h e normal metal, a s s h o w n in Fig.
10.3b. It now becomes possible for electrons in the normal metal t o
tunnel into t h e quasi-particle states, and t h e n u m b e r of electrons w h i c h
m a y tunnel increases steadily as the bias increases, so t h a t the resulting
current increases monotonically with V as shown in Fig. 10.3d. If t h e
superconductor is given a negative b i a s voltage, so t h a t all its energy
levels are raised relative to those of the normal metal, n o tunnelling can
occur until the voltage b e c o m e s equal to —Ä/e, w h e n a completely new
process b e c o m e s possible which involves the splitting u p of a Cooper
pair. It should b e appreciated t h a t if only one electron is involved in a
t An increase in electrostatic potential lowers the potential energy because the charge on the
electrons is negative.
144
INTRODUCTION TO SUPERCONDUCTIVITY
Super conducto r
Norma l
(c) V = - A / e
Super conducto r
(d )
FIG . 10.3. Tunnelling between a normal metal and a superconductor, (a), (b), and (c)
show the energy levels at 0°K for various differences in voltage V between a superconductor and a normal metal. V is taken as positive if the superconductor is biased
positively relative to the normal metal. T h e singly-hatched areas denote empty states
and the double-hatched areas denote occupied states, (d) Shows the resulting
current—voltage characteristic.
transition the only processes w h i c h c a n conserve energy are those in
which the electron m o v e s horizontally on t h e energy b a n d d i a g r a m
b e t w e e n states of t h e same energy. H o w e v e r , if t w o electrons are involved it b e c o m e s possible for one t o gain energy and the other t o lose it
as long as t h e total energy is conserved. S u c h a process is shown in F i g .
10.3c. I n this, a C o o p e r pair splits u p , one of the electrons tunnelling to
an e m p t y state j u s t above t h e F e r m i level of t h e n o r m a l metal w i t h a loss
of energy Ä, while the second electron, h a v i n g lost its partner, is converted into a quasi-particle and occupies t h e lowest excited state in t h e
superconductor w i t h a gain in energy Ä. T h u s the total energy of t h e
complete system before t h e t r a n s i t i o n s indicated b y a r r o w s h a v e taken
place is equal t o t h e total energy after, and t h e process is allowed. T h e
n u m b e r of pairs t h a t m a y split u p in t h i s w a y increases w i t h t h e b i a s
TUNNELLING AND THE ENERGY GAP
145
voltage, because m o r e quasi-particle states and states in t h e normal
metal become accessible, and t h e resulting negative current increases
with — V as shown in Fig. 10.3d. Ä is t h u s given directly b y the voltage
at which the tunnelling current between t h e superconductor and any
normal metal s h o w s a sudden increase. At t e m p e r a t u r e s above absolute
zero, a very small current m a y flow at voltages between the values +A/e,
because there are a few electrons excited to states above t h e F e r m i level
in the normal metal which may, if there is positive bias, tunnel into the
quasi-particle states, and there are a few e m p t y states below the F e r m i
level of the normal metal to w h i c h one of the m e m b e r s of a pair m a y
tunnel if there is negative bias. However, there will still b e a sharp rise in
tunnelling current w h e n V= ± Ä / â . Therefore, from the tunnelling I— V
characteristic we can directly determine t h e energy g a p ( = 2Ä) of the
superconductor. Tunnelling is a relatively easy experiment and is a most
useful way of measuring the energy gap.
10.4. T u n n e l l i n g
Betwee n
Tw o
Identica l
Superconductor s
T h e energy level diagram for t w o identical superconductors w i t h no
applied bias is as shown in Fig. 10.4, where it is assumed that the
t e m p e r a t u r e is above absolute zero so that t h e quasi-particle states are
partially occupied. It is possible for quasi-particles to tunnel in either
direction, because the states to which they m a y tunnel are not completely full, b u t at zero b i a s the current due to tunnelling from left to right will
be the same as t h a t from right t o left, so t h a t no net current flows.
N o w suppose t h e left-hand electrode is biased positively b y a voltage
V relative to the right-hand electrode. T h i s m e a n s t h a t t h e energy level
diagram on the left will b e shifted d o w n w a r d s relative to t h a t on the
right by an a m o u n t eV, as shown in Fig. 10.4b. T h e r e will now b e a net
flow of electrons from right to left, because t h e lowest quasi-particles on
the left have n o states on the right to tunnel into, while all t h e quasiparticles on the right are able to tunnel. T h e current increases w i t h V
until the occupied quasi-particle states on the left are below the b o t t o m
of the c o n t i n u u m states on the right, so t h a t quasi-particles can no
longer tunnel from left to right. Since the quasi-particles will all lie
within about kT of the lowest level, this stage is reached w h e n eV~
kT
or V~ 10~ V. If Vis increased further the current r e m a i n s m o r e or less
constant because none of the quasi-particles o n the left can tunnel, and
those on the right, which are able to tunnel, are c o n s t a n t in n u m b e r .
However, w h e n V reaches the m a g n i t u d e 2A/e an additional process
4
y
I.T.S. — ¥
146
INTRODUCTION TO SUPERCONDUCTIVITY
-e a
ÈÈ -
-È È
-È È
ÈÈ —
-è è
(a )
V=0
is
ÈÈ -
èè (b )
V~10" \
2Ä
2Ä/â
(c )
V = 2A/e
(d)
FIG . 10.4. Tunnelling between identical superconductors, (a), (b), and (c) energy
levels.-ÈÈ-denote s a Cooper pair, · denotes a quasi-particle. (d) C u r r e n t - v o l t a g e
characteristic.
b e c o m e s possible w h i c h involves t h e splitting u p of a C o o p e r pair, as
shown in F i g . 10.4c. O n e of t h e electrons t u n n e l s into t h e left-hand
superconductor so as t o occupy t h e lowest quasi-particle state. T h i s
process is accompanied b y a loss of energy Ä. T h e second electron,
having lost its partner, is converted into a quasi-particle and occupies
t h e lowest excited state on t h e right w i t h a gain in energy Ä. T h u s t h e
total energy before t h e t r a n s i t i o n s indicated b y t h e a r r o w s h a v e t a k e n
place is identical w i t h t h e total energy after, and t h e process is allowed.
As a result of t h i s process, extra electrons flow from right to left and t h e
current increases. If V is increased b e y o n d liSje, a process of t h i s sort
continues t o b e possible except t h a t t h e quasi-particles d o n o t go into t h e
lowest excited states on each side. T h e r e is n o w a large n u m b e r of c o m b i n a t i o n s of excited s t a t e s w h i c h can serve as final states, and t h e
tunnelling c u r r e n t increases rapidly, as s h o w n in Fig. 10.4d. I n t h i s case
also w e can d e t e r m i n e t h e energy g a p from t h e voltage at w h i c h t h e
tunnelling c u r r e n t increases rapidly.
TUNNELLING AND THE ENERGY GAP
10.5.
147
Th e Semiconducto r Representatio n
T h e reader is w a r n e d t h a t there is another w a y of representing energy
levels in a superconductor which is widely used in the literature. In this
model the energy levels are represented by t w o b a n d s , one of which is
completely full at absolute zero and the other completely empty, as in a
semiconductor. T h e w i d t h of t h e gap b e t w e e n the b a n d s is 2Ä, not Ä as
in Fig. 10.2, t h o u g h , as w e shall see, t h e t w o representations are
equivalent.
T o see how t h i s model arises, consider a superconductor at a t e m perature above absolute zero, so t h a t there are quasi-particles present,
and suppose a single electron is added to it. T h e n one of t w o things m a y
happen. T h e added electron m a y go into a m o m e n t u m state p i whose
p a r t n e r —p i is e m p t y . In t h i s case it behaves as an excited quasi-particle.
Alternatively, it m a y join u p w i t h another unpaired electron and form a
Cooper pair. C o m p a r e d w i t h the first case, the second gives rise to a
total energy w h i c h is lower b y at least 2Ä. Since the superconductor is in
the same condition before the electron is added in b o t h instances, the
energy of the extra electron before it enters t h e superconductor m u s t be
lower by at least 2Ä in t h e second case if energy is to b e conserved
overall. W e may therefore regard the added electron as belonging to one
of t w o b a n d s w h i c h are separated in energy b y 2Ä. An electron can only
enter the superconductor w i t h overall conservation of energy if, before
entering, its energy lies above the b o t t o m of t h e upper b a n d or below the
top of the lower b a n d . T h e s e b a n d s therefore define the r a n g e s of energy
which an electron m u s t have if it is to tunnel into the superconductor. It
is i m p o r t a n t to realize t h a t in this model b o t h the b a n d s are single electron b a n d s . It should not be t h o u g h t t h a t the levels in t h e u p p e r b a n d
correspond to quasi-particles w i t h p > p and the lower to quasiparticles w i t h / ) < p ; a full range of m o m e n t u m values is present in each
case.
F
F
T h e semiconductor representation is most easily illustrated b y considering tunnelling b e t w e e n a normal metal and a superconductor. I n
Fig. 10.5a,b w e describe this process in t e r m s of the excited q u a s i particle representation, as w e did in § 10.3 and Fig. 10.3. In Fig. 10.5c,d
a description is given in t e r m s of the semiconductor representation. T h e
physics of the process is m u c h m o r e clearly brought out b y the quasiparticle representation. T h e advantage of the semiconductor representation is t h a t the allowed transitions always correspond to horizontal
a r r o w s which represent transitions m a d e by a single electron, and this
148
INTRODUCTION TO SUPERCONDUCTIVITY
Super conducto r
Norma l
-ÈÈ -
(a ) V = A / e
(c ) V = A / e
(b )
(d )
V=-A/e
Ä/e
V= - Ä / e
Ä/e
(e )
FIG . 10.5. Tunnellin g betwee n a superconducto r an d a norma l meta l at 0 ° K . (a) an d
(b ) Excite d quasi-particl e representation , (c) an d (d ) Semiconducto r representation , (e)
Current-voltag e characteristic .
m a k e s the detailed interpretation of tunnelling p h e n o m e n a s o m e w h a t
easier; probably for this reason, it h a s been m u c h m o r e c o m m o n l y used
in t h e literature of tunnelling. I t will b e n o t e d t h a t an electron can only
be injected into t h e lower b a n d if t h e r e is a quasi-particle w i t h w h i c h it
can c o m b i n e t o form a C o o p e r pair, so t h e n u m b e r of e m p t y s t a t e s in t h e
lower b a n d is equal to t h e n u m b e r of quasi-particles in t h e u p p e r b a n d ,
r a t h e r like an intrinsic semiconductor. T h e m e c h a n i s m b y w h i c h t h e t w o
b a n d s c o m e a b o u t is, however, totally different from t h a t responsible for
the b a n d s t r u c t u r e of a semiconductor and great care m u s t b e used in
applying t h i s r e p r e s e n t a t i o n in t h e case of a superconductor. T h e
149
TUNNELLING AND THE ENERGY GAP
difference b e t w e e n the t w o representations h a s been discussed
Schrieffert and Adkins. ö
10.6.
by
Othe r Type s of Tunnellin g
T h e r e are other, m o r e complicated, types of tunnelling. F o r example,
if t h e t w o m e t a l s are dissimilar superconductors, a characteristic as
shown in Fig. 10.6 is obtained. T h e r e is a negative resistance region
between V — ( Ä — Ä)/â and V = ( Ä + A )/e. T h e explanation of this
negative resistance region d e p e n d s on the way in which the density of
states in the quasi-particle b a n d varies with energy, and w e shall not discuss it here. F o r further details t h e reader is referred to the original
paper by Giaever and Megerle.§
2
÷
2
(Ä -Ä.)/â
V
2
x
-
2Ä./â
FlG . 10.6. Tunnellin g betwee n tw o superconductor s wit h energ y gap s Äé an d Ä
( Ä > Ä,).
2
2
T h e r e is also the possibility, in the case of t w o superconductors, of
t w o electrons w h i c h form a pair tunnelling as a pair, so t h a t they m a i n tain their m o m e n t u m pairing after crossing the gap. T h i s type of
tunnelling, k n o w n as Josephson tunnelling, is possible because t h e superconducting g r o u n d state can contain m a n y pairs. It only a p p e a r s under
very special circumstances, namely exceptionally t h i n insulating layers
( < 1 0 " e m ) . Josephson tunnelling, the consequences of which are discussed at length in the following chapter, takes place w h e n there is no
7
t J. R. Schreiffer , Rev. Mod. Phys. 3 6 , 200 (1964).
J C . J. Adkins , Rev. Mod. Phys. 3 6 , 211 (1964).
§ I. Giaeve r an d K. Megerle , Phys. Rev., 122, 1101 (1961).
150
INTRODUCTION TO SUPERCONDUCTIVITY
difference in voltage b e t w e e n t h e s u p e r c o n d u c t o r s , so t h a t a c u r r e n t m a y
flow w i t h o u t any accompanying voltage d r o p . W e m i g h t call it a
tunnelling supercurrent. T h i s supercurrent h a s a critical c u r r e n t density
j w h i c h is characteristic of t h e j u n c t i o n .
If t h e J o s e p h s o n tunnelling c u r r e n t density exceeds t h e value j , a
voltage difference V a p p e a r s across t h e j u n c t i o n a n d t w o processes
occur. S o m e electrons tunnel individually, a s in § 10.4, w i t h an I—V
characteristic a s s h o w n in F i g . 10.4d or F i g . 10.6 (depending o n w h e t h e r
the s u p e r c o n d u c t o r s are identical or n o t ) and, at t h e same time, some
electrons continue t o tunnel in t h e form of electron-pairs. H o w e v e r , t h e
condensed states are n o longer opposite each o t h e r o n an energy-level
diagram, so pairs c a n n o t tunnel from o n e condensed state t o t h e other
w i t h conservation of energy if t h e energy of t h e electron p a i r s alone is
considered. T h e energy balance is m a d e u p b y t h e emission of a p h o t o n
of electromagnetic radiation of frequency í such t h a t
c
c
hv=2eV.
(10.1)
T h e factor 2 c a n b e considered as arising either because t h e pair c a n b e
considered a s a particle w i t h charge 2e, or because t w o electrons each
with charge e are involved in t h e transition. W h i c h e v e r w a y o n e looks at
it, t h e energy required t o m a k e u p t h e balance is 2eV. T h i s process,
which involves t h e emission of radiation, is k n o w n a s t h e a.c. Josephson
effect. Since V is normally of t h e order of 1 0 " V t h e radiation is in t h e
short wavelength m i c r o w a v e p a r t of t h e s p e c t r u m . T h e emission of such
radiation from very thin tunnel j u n c t i o n s h a s been detected b y L a n g e n burg, Scalapino, a n d T a y l o r .
3
y
Josephson tunnelling will b e considered m o r e fully in t h e next c h a p t e r
(§ 11.3.1 et seq.).
10.7.
Practica l Detail s
T u n n e l l i n g is a very useful p h e n o m e n o n , because w e c a n u s e it to
m e a s u r e simply a n d directly t h e energy g a p of a superconductor. T h e
majority of superconducting tunnelling e x p e r i m e n t s have been carried
out using evaporated films of t h e t w o metals. I n a typical e x p e r i m e n t a
thin film of o n e of t h e metal s is evaporate d o n t o a glass plat e (such as a
microscope slide) in t h e shape of a strip a millimetre or so wide. T h i s
film is then oxidized by exposure t o air or oxygen until a layer of oxide a
few ten s of Angstro m unit s thick is built u p o n t h e surface . A film of t h e
second metal is t h e n evaporated, usually as a strip w h i c h crosses t h e first
151
TUNNELLING AND THE ENERGY GAP
one at right angles, so t h a t t h e area t h r o u g h w h i c h tunnelling can take
place is a few square millimetres. Electrical contact is then m a d e t o the
films, as a rule w i t h indium solder, so t h a t t h e c u r r e n t - v o l t a g e
characteristic can b e observed. O n c e t h e I-V characteristic h a s been
observed, t h e energy gap can b e determined from t h e voltage at w h i c h
the curve s h o w s a pronounced change in slope. T h i s occurs at a voltage
equal t o Ä/e for s u p e r c o n d u c t o r - n o r m a l tunnelling and t o 2Ä/â for
s u p e r c o n d u c t o r - s u p e r c o n d u c t o r tunnelling (see Figs. 10.3d and 10.4d).
T u n n e l l i n g c u r r e n t s are usually less t h a n 1 0 " A for applied voltages
around 1 0 " V, so care m u s t be taken w i t h the electrical m e a s u r e m e n t s . It
is c o m m o n practice to superpose a small alternating c o m p o n e n t on t h e
steady voltage V applied t o t h e s a n d w i c h ; t h e current t h e n c o n t a i n s an
alternating c o m p o n e n t which is proportional t o t h e value of t h e differential conductance dl/dV. T h i s a.c. c o m p o n e n t can be amplified, using a
tuned amplifier. I n this w a y a very sensitive direct m e a s u r e m e n t can b e
m a d e of t h e slope of t h e I—V curve.
F i g u r e 10.7 s h o w s some m e a s u r e m e n t s of t h e t e m p e r a t u r e dependence
of t h e energy gap of indium, tin and lead, measured b y Giaever and
3
3
10
Ä-^^Ï-¼-Ï^Ï
0-8
· 0%0 Ï
BC S Theory -
Æ
06
<
0-4 |
ï Indiu m
ä Tin
• Lea d
^ Ä
\
Ï
Ä
\ è Ä
02
h
\ Ä
\
02
04
0-6
0-8
10
T/Tc
FIG . 10.7. T h e energy gap of lead, tin, and indium versus temperature, as determined
by tunnelling experiments. (After Giaever and Megerle.)
Megerle w h o studied tunnelling from these m e t a l s into a n o r m a l metal.
T h e fit w i t h t h e t e m p e r a t u r e dependence predicted b y the B C S theory is
shown t o b e q u i t e good. W e should not expect t h e B C S theory to m a k e
absolute predictions of t h e energy gap and transition t e m p e r a t u r e of a
superconductor w i t h any great accuracy, so t h e figure s h o w s t h e ratio of
t h e energy gap at t e m p e r a t u r e Ô to t h e energy g a p at 0°K, plotted
152
INTRODUCTION TO SUPERCONDUCTIVITY
against Ô expressed as a fraction of t h e transition t e m p e r a t u r e T . T h e
curve obtained in this w a y should b e t h e s a m e for all s u p e r c o n d u c t o r s .
O n e of t h e m o s t impressive examples of t h e p o w e r of the tunnelling
technique in m e a s u r i n g energy g a p s is its use to m e a s u r e t h e d e p e n d e n c e
of t h e energy g a p on an applied m a g n e t i c field. A s w e saw in C h a p t e r 5,
the superconducting to normal transition in the presence of a m a g n e t i c
field is a first-order transition, w h i c h implies t h a t the energy g a p should
go abruptly t o zero as t h e specimen is driven n o r m a l b y t h e field. T h e
data represented b y circles in Fig. 10.8, w h i c h show the energy g a p of an
c
\
Ä
\
\
I
Aluminu m
Approx . Reduce d
2 |- Symbol Thicknes s
Temp .
ä
3000 A
-745
4000 A
-774
•1 ì
ï
0
d.
* \
ë
1
1-9
3 5
I
I
I
I
I
.
1
1
1
-1
2
-3
-4
-5
6
-7
8
-9
(Magneti c field) ( ^ )
2
IJo— I
1 0
2
FIG. 10.8. Energy gap of aluminium as a function of magnetic field for films of
thickness 3000 A and 4000 A. (After Douglass.)
aluminium film 4000 A thick, obtained from tunnelling into superconducting lead, show the expected behaviour. T h e small decrease of energy
gap w i t h increasing field before H is reached can be explained in the
same way as the field-dependence of t h e p e n e t r a t i o n d e p t h discussed in
§8.4. H o w e v e r , for a film w h i c h is 3000 A thick, represented b y triangles
in Fig. 10.8, t h e energy g a p falls linearly to zero as Ç increases to H .
T h i s gradual disappearance of t h e energy g a p is indicative of a secondorder transition, and is effective confirmation of the prediction b y G i n z b u r g and L a n d a u , m e n t i o n e d in § 8.5, t h a t below a certain film thickness
the transition in a m a g n e t i c field should b e c o m e of second order.
c
c
CHAPTE R
COHERENCE
OF T H E
QUANTUM
11.1.
11
ELECTRON-PAIR WAVE;
INTERFERENCE
Electron-pai r W a v e s
WE HAVE seen that in a superconductor the resistanceless current
involves the motion of pairs of electrons. W h e n considering the current,
each pair may be treated as a single " p a r t i c l e " of m a s s 2m and charge 2e
whose velocity is t h a t of the centre of m a s s of the pair. As w i t h ordinary
particles, these current carriers m a y b e described by m e a n s of a wave. In
a normal metal t h e conduction electrons suffer frequent scattering accompanied b y violent changes of phase and so their electron waves are
coherent only over very short distances. T h e Cooper pairs in a superconductor are not, however, randomly scattered and their w a v e s remain
coherent over indefinitely long distances. W e saw in C h a p t e r 9 [eqn.
(9.12a)] that each pair m a y be represented by a wavefunction which w e
can write as
Ö
Ñ
=
ö * · í*
ñ
Ã
â
(11.1)
where Ñ is the net m o m e n t u m of the pair whose centre of m a s s is at r .
T h e term e - * has the form of a travelling wave and describes the
motion of the centre of m a s s of the pair.
It w a s shown in § 9.3.7 that, if the current density is uniform, all the
electron-pairs in a superconductor have the same m o m e n t u m and
therefore have w a v e s of the same wavelength. T h e superposition of a
n u m b e r of coherent waves of equal wavelength simply results in another
wave with the same wavelength, so all the electron-pairs in a superconductor can together be described by a single wave of a form similar to
(11.1), i.e.
/ ( P
r)/
Ø
Ñ
=
øâ^-'í *
(11.2)
where É Ø É is t h e density of electron-pairs and Ñ is the m o m e n t u m per
pair. W e shall call this wave, describing the motion of all the electron2
Ñ
153
154
INTRODUCTION TO SUPERCONDUCTIVITY
pairs, the electron-pair
wave. T h e electron-pair w a v e r e t a i n s its p h a s e
coherence over indefinitely long distances. T h i s c h a p t e r is concerned
w i t h some of t h e p h e n o m e n a w h i c h occur as t h e result of t h i s long-range
coherence. T h e s e p h e n o m e n a are analogous to t h e interference and
diffraction effects observed w i t h ordinary electromagnetic waves, and,
because they are manifestations on a macroscopic scale of q u a n t u m
behaviour, t h e p h e n o m e n a are often referred to u n d e r t h e collective title
of " q u a n t u m interference".
11.1.1.
Phas e of th e electron-pai r w a v e
It is i m p o r t a n t t o u n d e r s t a n d w h a t t h e coherence of t h e electron-pair
wave implies. Coherence of a w a v e travelling t h r o u g h a region m e a n s
t h a t if w e k n o w t h e p h a s e and a m p l i t u d e at any point w e can, from a
knowledge of t h e wavelength and frequency, calculate t h e p h a s e and
amplitude at any o t h e r point. I n o t h e r w o r d s , because the w a v e travels
undisturbed, t h e a m p l i t u d e s and relative p h a s e s at all p o i n t s in t h e
region are uniquely related b y t h e w a v e equation.
L e t u s r e w r i t e t h e electron-pair w a v e (11.2) in t h e f o r m t
(for simplicity w e consider a one-dimensional wave), and suppose t h a t
the w a v e frequency í is related t o the total kinetic energy Å of a C o o p e r
pair b y the familiar relation Å = hv, a n d t h a t t h e wavelength ë is related
to t h e m o m e n t u m Ñ of t h e p a i r ' s centre of m a s s b y t h e d e B r o g u e relation ëÑ = h.
Consider a length of s u p e r c o n d u c t o r joining t w o p o i n t s X and Y. If n o
current is flowing b e t w e e n X and F , t h e m o m e n t u m Ñ of t h e electron
pairs is zero and the wavelength ë is infinite. Consequently, the p h a s e of
an electron-pair w a v e is t h e s a m e at X as at Y.
S u p p o s e n o w t h a t a resistanceless c u r r e n t flows from X to Y. T h e
electron pairs n o w h a v e a m o m e n t u m Ñ and t h e electron-pair w a v e a
finite wavelength ë = h/P. T h e r e will therefore b e a p h a s e difference
(Äö)
b e t w e e n t h e p o i n t s X and Y and t h i s p h a s e difference r e m a i n s
c o n s t a n t in t i m e . T h e p h a s e difference b e t w e e n t w o p o i n t s past w h i c h a
plane w a v e is travelling i s :
֋
t In accordance with the usual convention, the time-dependent oscillatory factor e"' ' '
been omitted from the expression for the wavefunction in (11.1).
2 7 1
has
COHERENCE OF THE ELECTRON-PAIR WAVE; QUANTUM INTERFERENCE
155
Y
(Äö)
֋
= ö -öã=
jj-dl,
2ð
÷
÷
where ÷ is a unit vector in the direction of the wave propagation, and dl
is an element of a line joining X to Y. ( T h o u g h ÷ is, by definition, parallel
to t h e direction of propagation, the wave itself need not p r o p a g a t e in a
straight line. L a t e r in this chapter we shall need to consider waves
travelling r o u n d a closed circular path.) N o w for the electron-pair wave
ë = h/P and the pair m o m e n t u m is Ñ = 2mv> where í is the velocity of the
pairs due to t h e current. T h e relation of í to the supercurrent density J
is J = \n . 2.5. v, where n is the density of superelectrons, and \n is
t h e density of electron pairs. T h e wavelength is therefore
s
s
s
s
s
2 _
h
n
*
e
so t h e phase difference b e t w e e n X and Y d u e t o the current is
ã
(A<P) =^jj .dl,
XY
s
(11.3)
X
since ÷ is necessarily parallel to J .
Because t h e electron-pair w a v e retains its coherence t h r o u g h o u t a
superconductor it should b e able to produce long-range interference
phenomena, and it should be possible to observe effects analogous to
familiar optical interference, e.g. Fraunhofer diffraction a n d diffraction
by a grating. As w e shall see, such effects in superconductors can b e
d e m o n s t r a t e d experimentally, and such d e m o n s t r a t i o n s provide a convincing justification for treating superelectrons as w a v e s w i t h long-range
coherence.
s
11.1.2.
Effec t o f a m a g n e t i c
field
T h e phase of t h e electron-pair wave m a y b e strongly affected b y the
presence of an applied magnetic field. W e saw in § 8.5 that in the
presence of a magnetic field the m o m e n t u m ñ of particles w i t h charge q
takes the form m\ + j A , where A is the magnetic vector potential
defined b y curl A = B . In the case of electron pairs, Ñ = 2mv + 2eA. W e
can still derive t h e wavelength ë from the m o m e n t u m Ñ by ë = h/P, and
so, b y the same a r g u m e n t which led to (11.3) w e find t h a t in the presence
of a magnetic field, the phase difference b e t w e e n t w o p o i n t s X and Y is
156
INTRODUCTION TO SUPERCONDUCTIVITY
<
^
W
-
g
|
5
l
.
.
«
^
+
÷
j
A
.
A
(Ð.4)
÷
W e interpret t h e first t e r m on t h e r i g h t - h a n d side as being t h e p h a s e
difference d u e to t h e current, and t h e second t e r m as an additional p h a s e
difference d u e to the m a g n e t i c field. W e can therefore w r i t e
(Äø)÷ã
= [(Äö)÷ãÉ
+
[(ÄÖ)×Õ]Â>
where [ ( Ä ^ ) ^ ]/ is the p h a s e difference due to any current, and
[(ÄÖ)×Õ\Â
is the phase difference d u e to any m a g n e t i c field. W e have, therefore,
ã
[{W)xrl= jj^jh.dl
S
(11.5)
X
and
ã
[ ( Ä 0 ) , Ë = ^ Ã / Á . < / 1.
(11.6)
÷
T h e phase difference w h i c h can be produced b y t h e presence of a
magnetic field plays a very i m p o r t a n t p a r t in t h e p h e n o m e n a w h i c h are
n o w to b e described.
11.2.
TheFluxoi d
W e n o w consider t h e case of a supercurrent circulating a r o u n d a
closed p a t h . T h e results of this section will b e of i m p o r t a n c e in a later
chapter.
F i g u r e 11.1 s h o w s a superconductor enclosing a n o n - s u p e r c o n d u c t i n g
region N. S u p p o s e t h a t in Í there is a flux density  d u e to superc u r r e n t s flowing a r o u n d it (see § 2.3.1). S u c h a situation m i g h t arise
where Í is a hole t h r o u g h the material or, in t h e case of a solid piece of
superconductor, w h e r e t h e magnetic field g e n e r a t e d b y t h e encircling
current m a i n t a i n s the region Í in the n o r m a l state.
Consider a closed p a t h , such as t h e d o t t e d curve in Fig. 11.1, which
encircles t h e n o r m a l region. T h e r e will b e a p h a s e difference of t h e
electron-pair w a v e b e t w e e n any t w o p o i n t s on this curve, d u e b o t h to t h e
presence of the magnetic field and to t h e circulating current. As w e h a v e
seen, the p h a s e difference b e t w e e n the p o i n t s X and Y is given by (11.4):
÷
÷
COHERENCE OF THE ELECTRON-PAIR WAVE; QUANTUM INTERFERENCE
157
FIG. 11.1. Superconductor enclosing a non-superconducting region.
N o w consider t h e phase change occurring around a closed path, say
XYZX. T h e total phase change will b e
N o w by Stokes* theorem, <f A . dl = jj
curl A . J S , where dS is an ele-
s
m e n t of area, and furthermore curl A = B, so w e can rewrite (j>A . dl a s
J*J *B . dS, where S is t h e area enclosed by XYZX,
and t h e phase change
s
r o u n d t h e closed p a t h m a y b e rewritten
W h a t follows in this chapter d e p e n d s o n t h e realization that, if t h e
superelectrons are t o b e represented b y a w a v e , t h e w a v e at any point
can, at any instant, have only one value of phase and amplitude.
Consequently, t h e phase change Ä 0 around a closed p a t h m u s t equal
2ðç where ç is any integer. W e shall refer to this as t h e " p h a s e cond i t i o n " or " q u a n t u m c o n d i t i o n " . ! W e have therefore
^
•
"
•
ô
/
ß
·
·
"
-
2
*
»
<
s
which can be r e w r i t t e n
s
t T h i s condition is also responsible for the quantized electron orbits in the Bohr atom.
m
)
158
INTRODUCTION TO SUPERCONDUCTIVITY
F . and H . L o n d o n n a m e d t h e q u a n t i t y o n t h e left-hand side of t h i s e q u a tion t h e fluxoid enclosed b y t h e curve XYZX. I t is given t h e symbol Ö '
to distinguish it from t h e flux Ö :
Ö' = ^ É , . Ë + { [ Â . < * 8 ,
(11.8)
s
5
and w e have s h o w n t h a t , because of t h e single-valuedness of t h e w a v e ,
the fluxoid c a n only exist in integral multiples of t h e unit h/2e:
Ö' = « £ .
(11.9)
N o t e t h a t it is s o m e t i m e s convenient to w r i t e t h e
equivalent form
^
=
^ j h '
d
l
+
j '
A
d l
-
fluxoid
in t h e
( °)
1L1
T h e fluxoid w i t h i n a closed curve is closely related t o , t h o u g h not
identical with, t h e m a g n e t i c flux w i t h i n t h e curve. T h e first t e r m o n t h e
right-hand side of (11.8) c o n t a i n s t h e line integral of t h e current density
a r o u n d t h e curve XYZX, b u t b e c a u s e t h e p e n e t r a t i o n d e p t h is small,
nearly all t h e circulating c u r r e n t will in fact b e c o n c e n t r a t e d very close t o
the b o u n d a r y of t h e normal region N a n d t h i s t e r m will b e negligible u n less w e are considering a curve of w h i c h a considerable p a r t lies very
close t o N. T h e second t e r m on t h e r i g h t - h a n d side of (11.8) is j u s t t h e
magnetic flux contained w i t h i n Í and t h e p e n e t r a t i o n d e p t h a r o u n d it.
So unless w e are considering a closed curve w h i c h lies very close to t h e
b o u n d a r y of N t h e fluxoid w h i c h a curve encloses is practically t h e same
as t h e flux it encloses.
9
9
W e see, therefore, t h a t any flux (strictly fluxoid) contained w i t h i n a
superconductor should only exist as multiples of a q u a n t u m , t h e fluxon,
Ö , given b y
0
<t> = h/2e
(11.11)
Q
= 2 07 ÷ 1 0 ~
15
weber.
It can b e seen t h a t this predicted value of t h e fluxon is extremely small.
T h e value of t h e fluxon h a s b e e n m e a s u r e d experimentally a n d t h e fact
t h a t its value is found to be, as predicted, Planck's c o n s t a n t divided b y
twice t h e electronic charge, is strong evidence t h a t t h e supercurrent is
carried b y pairs of electrons.
COHERENCE OF THE ELECTRON-PAIR WAVE; QUANTUM INTERFERENCE
159
W h a t is measured in experiments is the flux, not the fluxoid, b u t as w e
have seen, in m o s t cases these are indistinguishable, and the expression
"quantized flux" is often used instead of the strictly accurate "quantized
fluxoid". M e a s u r e m e n t s have, for example, been m a d e of t h e magnetic
m o m e n t of a long, hollow cylinder of superconductor which w a s
repeatedly cooled below its transition t e m p e r a t u r e in very weak axial
magnetic fields, À T h e thickness of t h e wall w a s small c o m p a r e d to the
diameter of the central hole ( t h o u g h large compared t o t h e penetration
depth) and consequently the magnetic m o m e n t of the cylinder w a s
proportional t o t h e flux t r a p p e d in the hole. T h e m e a s u r e m e n t s showed
t h a t the cylinder could only t r a p an a m o u n t of flux corresponding to an
integral n u m b e r of fluxons.
S u p p o s e a ring or hollow cylinder b e c o m e s superconducting while in
an applied magnetic field. W e have seen t h a t once t h e material h a s
become superconducting the flux threading the hole can only b e an integral n u m b e r of fluxons ç Ö . However, in general, the strength of the
applied field will not b e such t h a t t h e flux from it which t h r e a d s the hole
is an exact n u m b e r of fluxons, t h o u g h the difference will b e small
because t h e fluxon is small. W h e n the ring b e c o m e s superconducting the
strength of the circulating current which arises on the inner surface to
maintain flux in the hole is j u s t t h a t which, together w i t h the applied
field, produces t h e nearest integral n u m b e r of fluxons. F o r example, if the
applied field h a p p e n e d to produce a flux (w + | ) Ö in the hole, then in t h e
superconducting state t h e flux in the hole will b e c o m e ç Ö . If, however,
the applied field strength h a d been such as to produce a flux (ç + | ) Ö in
t h e hole, the m a g n i t u d e of t h e circulating current will b e such as to
produce a flux of (ç + 1)Ö (for an explanation of this see § 11.4).
F l u x quantization is a special property of superconductors and does
not occur in normal metals for the reason discussed in § 11.1. W e shall
see t h a t fluxons play an i m p o r t a n t role in the properties of type-II superconductors w h i c h are discussed in t h e second part of this book.
0
0
0
0
0
11.2.1.
Fluxoi d withi n a superconductin g meta l
I n the previous section w e have been considering a closed curve w h i c h
s u r r o u n d s a non-superconducting region, e.g. a hole. M a g n e t i c flux can
thread this hole accompanied b y a current flowing r o u n d the hole
t For example, Doll and Nabauer, Phys. Rev. Letters 7, 51 (1961). Deaver and Fairbank, ibid.
7, 43 (1961).
160
INTRODUCTION TO SUPERCONDUCTIVITY
(§ 2.3.1). T h e fluxoid enclosed w i t h i n t h e curve will b e an integral
n u m b e r ç of fluxons, b u t t h e value of ç will b e zero if n o flux t h r e a d s t h e
hole. If, however, w e consider a closed curve w h i c h does not encircle a
non-superconducting region, so t h a t t h e area enclosed b y t h e curve is entirely superconducting, t h e n ç is always zero. T h i s will b e so even if t h e
curve passes close to a b o u n d a r y of t h e superconductor w h e r e , w i t h i n
t h e penetration d e p t h , neither t h e flux density  nor the current density
J n e e d b e z e r o . T h e m a g n i t u d e s a n d d i r e c t i o n s of  a n d J.. are
everywhere such t h a t , w h e n integrated a r o u n d t h e curve, t h e t w o t e r m s
contributing to t h e fluxoid [right-hand of (11.8)] cancel to give zero.
s
T h e s t a t e m e n t t h a t t h e fluxoid w i t h i n any closed curve not s u r r o u n d ing a non-superconducting region is zero is a m o r e precise w a y of
expressing t h e perfect d i a m a g n e t i s m of a superconducting material,
since it is valid everywhere in t h e metal, including t h e p e n e t r a t i o n d e p t h .
11.3.
11.3.1.
Wea k Link s
Josephso n tunnellin g
Consider t w o s u p e r c o n d u c t o r s Ñ and Q completely isolated from each
o t h e r ; the p h a s e of the electron-pair w a v e in Ñ will b e unrelated to the
phase of t h e w a v e in Q. S u p p o s e n o w t h a t t h e g a p separating t h e t w o
pieces is gradually reduced to zero. W h e n t h e separation b e c o m e s very
small, electron pairs can tunnel across t h e g a p and t h e electron-pair
w a v e s in Ñ and Q t e n d t o b e c o m e coupled together. A s Ñ and Q
approach each other there is an increase in t h e interaction b e t w e e n their
electrons d u e t o t h e tunnelling, a n d t h e p h a s e s of t h e w a v e s in t h e t w o
pieces b e c o m e progressively m o r e tightly locked together. Eventually
w h e n Ñ and Q c o m e into contact, they form o n e single piece of metal and
there must, u n d e r a given set of external conditions, b e a definite relation
b e t w e e n t h e p h a s e s t h r o u g h o u t Ñ and Q.
Before contact is m a d e , the interaction occurs as a result of the spread
of the electron-pair w a v e t h r o u g h t h e gap, i.e. a tunnelling of electron
pairs from Ñ and Q and vice versa. T u n n e l l i n g of an electron pair m e a n s
t h a t the t w o electrons m a i n t a i n their m o m e n t u m pairing after crossing
the g a p . W e have already m e t this t y p e of tunnelling in t h e previous
chapter, w h e r e it w a s referred t o as " J o s e p h s o n tunnelling". If t h e g a p is
thin the tunnelling across it of electron pairs is relatively probable and so
an appreciable resistanceless c u r r e n t can flow t h r o u g h it. A gap,
COHERENCE OF THE ELECTRON-PAIR WAVE; QUANTUM INTERFERENCE
161
however, h a s a critical current as does an ordinary superconductor.
W h e n resistanceless current flows across such a gap by electron-pair
tunnelling there is a phase difference b e t w e e n the electron-pair w a v e s on
each side of t h e gap. It can b e shown t h a t if i is the supercurrent
crossing the gap from Ñ to Q
s
i = i sin ( 0
s
c
Q
-
(11.12)
ö)
Ñ
where 0 and ö are the p h a s e s on each side of the gap. T h e m a x i m u m
value of the supercurrent occurs w h e n there is a phase difference of ð / 2
across the gap, and i then equals the critical current t of the gap.
Q
Ñ
s
c
It is i m p o r t a n t to realize that, because the phase difference in (11.12)
is not restricted to the range 0 to ð / 2 , the phase difference is not u n i q u e ly determined for a given value of i b u t m a y take one of t w o alternative
values, Ä 0 or ð — Ä 0 . |
E q u a t i o n (11.12) relates the electron-pair current tunnelling across a
Josephson j u n c t i o n to the phase difference of the electron-pair wave on
the t w o sides. W e saw in §§ 1.4 and 3.1 that if the supercurrent in a
superconductor is varying w i t h time, a voltage is developed across the
superconductor. T h e same is true of a Josephson tunnelling junction. If
the current t h r o u g h the junction is varying w i t h time, the phase
difference Ä 0 across it m u s t also be changing with time, and it can b e
shown t h a t a voltage V is developed across it, related t o the rate of
change of the phase difference by
s
2 Ý ? Ê = Ë^ Ä 0 .
(11.13)
It can b e seen t h a t this equation is consistent with equation (10.1).
E q u a t i o n s (11.12) and (11.13) show h o w the current t h r o u g h a Josephson
tunnelling j u n c t i o n and any voltage which m a y be developed across it are
related t o the phase difference of t h e electron-pair w a v e s on t h e t w o
sides:
i = i sin Áö
s
e
2eV=h^-A(p
(11.14.i)
(ll.14.ii)
+ T h e phase difference may also take on values 2ðç + Ä 0 or 2ðç + (ð — Áö) but these are not
physically distinguishable from Ä 0 and ð — Ä 0 .
162
INTRODUCTION TO SUPERCONDUCTIVITY
T h e s e t w o e q u a t i o n s are t h e basic e q u a t i o n s of a J o s e p h s o n tunnelling
junction, and from t h e m we can derive nearly all its properties. T h e y
follow from the coherent n a t u r e of t h e electron-pair w a v e in a superconductor. W e d o not derive t h e m in t h i s book b u t a fairly simple derivation
is given b y F e y n m a n . f
11.3.2.
Pendulu m analogu e
W e shall n o w show t h a t t h e r e is a close analogy b e t w e e n a J o s e p h s o n
tunnelling j u n c t i o n and a simple p e n d u l u m . T h i s analogy is very useful
because t h e m a t h e m a t i c s of J o s e p h s o n j u n c t i o n s can b e r a t h e r c o m plicated, and t h e results difficult t o interpret, b u t it is often possible t o
visualize intuitively h o w a simple p e n d u l u m will b e h a v e and d r a w conclusions t h r o u g h t h e analogy a b o u t t h e b e h a v i o u r of a J o s e p h s o n j u n c tion. F u r t h e r m o r e , even if one c a n n o t predict h o w a p e n d u l u m will
behave u n d e r a certain set of circumstances, it is relatively easy t o perform e x p e r i m e n t s on a p e n d u l u m a n d o b t a i n results w h i c h c a n b e
transferred t o a J o s e p h s o n j u n c t i o n .
T o establish t h e analogy, consider t h e v a r i o u s m e c h a n i s m s b y w h i c h current m a y flow t h r o u g h a J o s e p h s o n j u n c t i o n . F i r s t t h e r e is t h e electron-pair
tunnelling current i,. W e consider t h e general case in w h i c h t h i s c u r r e n t
varies w i t h t i m e producing, as p o i n t e d o u t on p . 161, a voltage F across t h e
junction. Because there is a voltage across t h e j u n c t i o n , a c u r r e n t of n o r m a l
(unpaired) electrons will also flow across t h e j u n c t i o n b y t h e n o r m a l
tunnelling process. T h i s process is resistive a n d c a n b e r e p r e s e n t e d b y a
resistance/? across t h e j u n c t i o n [Fig. 11.2(a)]. Because t h e j u n c t i o n c o n s i s t s
of t w o metal surfaces very close together, there is also a capacitance C across
the j u n c t i o n . T h e t h r e e parallel b r a n c h e s in Fig. 11.2(a) therefore form t h e
equivalent circuit of a Josephson tunnelling j u n c t i o n . If w e p a s s a c u r r e n t /
t h r o u g h t h e j u n c t i o n from an external source, t h i s c u r r e n t m u s t equal t h e
sum of the c u r r e n t s flowing d o w n each of t h e t h r e e b r a n c h e s of t h e
equivalent circuit:
T
dV
n
V
.
.
A
.
T o find h o w t h e total c u r r e n t is related t o t h e p h a s e difference Ä 0 across
the junction, w e use eqn. ( l l . 1 4 . i i ) to replace Vby (Ë/2Ý?×Ý//Ë )Ä0 w i t h
t h e result t h a t
t R. P. Feynman, Lectures on Physics, Vol. I l l , pp. 2 1 - 1 6 (Addison-Wesley).
COHERENCE OF THE ELECTRON-PAIR WAVE; QUANTUM INTERFERENCE
163
I
i -i
s
c
R
ä ß ç Äö 1|—é
(a) E q u i v a l e n t c i r c u i t of J o s e p h s o n
junctio n
pendulu m
FIG . 11.2. Pendulum analogue of a Josephson tunnelling junction.
r
CH d
2
A
,
¹
d
A
,
.
.
A
.
(11.15)
which is t h e equation relating t h e total current t h r o u g h a Josephson
tunnelling j u n c t i o n to the p h a s e difference of t h e electron-pair w a v e s on
each side.
L e t u s n o w consider t h e simple rigid p e n d u l u m of Fig. 11.2(b) w h i c h
consists of a light stiff rod of length / w i t h a bob of m a s s m at its lower
end. T h e p e n d u l u m can r o t a t e freely about the pivot P . If an external
t o r q u e Ô is applied t h e p e n d u l u m will swing out of the vertical. L e t t h e
angle of deflection at any instant be È. By analogy w i t h N e w t o n ' s second
law, t o r q u e p r o d u c e s a r a t e of change of angular m o m e n t u m , so if Ì is
the m o m e n t of inertia of t h e p e n d u l u m about P ,
M-rpi
(11.16)
= total torque.
N o w the total t o r q u e acting on the p e n d u l u m consists of several p a r t s :
there is the applied t o r q u e Ô w h i c h deflects the p e n d u l u m ; t h i s is o p posed by the weight of the b o b which exerts a t o r q u e — mgl sin è and, if
the pendulum is not in a v a c u u m and r o t a t e s w i t h velocity dd/dt> t h e
viscosity of t h e air will exert an opposing t o r q u e —çÜè/dt. So (11.16) can
be written
M-j-ú =
T-mglsmd-ç-j-.
R e a r r a n g e m e n t t o bring all t e r m s in è together gives
^
.,Üè
T—M-i-y
2
dd
+ ç -j-
+ mgl
. .
sin
a
È.
(11.17)
164
INTRODUCTIO N
TO
SUPERCONDUCTIVIT Y
T h i s is the e q u a t i o n of rotational m o t i o n of a simple p e n d u l u m . W e can
see t h a t t h i s e q u a t i o n h a s exactly t h e s a m e form as e q u a t i o n (11.15)
which relates the total c u r r e n t t h r o u g h a J o s e p h s o n j u n c t i o n to t h e p h a s e
difference of the electron-pair w a v e on each side. A rigid p e n d u l u m is
therefore an analogue of a J o s e p h s o n tunnelling j u n c t i o n . C o m p a r i s o n of
the t w o e q u a t i o n s t e r m b y t e r m ,
(junction) / = ^
^
Ì^- È
( p e n d u l u m ) T=
2
Ä0 + ^
+
^ Ä 0 + i sin Ä 0,
c
ç-^è
+ mgl sind
(11.18)
(11.19)
gives the following correspondence b e t w e e n the m e c h a n i c s of t h e pendulum and t h e electrical properties of t h e j u n c t i o n :
Junction
P h a s e difference, Ä 0
T o t a l current across j u n c t i o n , J
Capacitance, C
N o r m a l tunnelling conductance, l/R
Electron-pair tunnelling current,
i = i sin Ä 0
Voltage across junction,
s
c
Pendulum
Deflection, è
Applied t o r q u e , Ô
M o m e n t of inertia, Ì
Viscous damping, ç
H o r i z o n t a l displacement of b o b ,
÷ = / sin è
Angular velocity,
Üè
T h e s e analogues are s u m m a r i z e d in F i g . 11.3.
As pointed o u t earlier, this analogue is very useful because b y
visualizing or experimenting on the m o t i o n of a p e n d u l u m w e can deduce
the electrical behaviour of a J o s e p h s o n tunnelling j u n c t i o n . As an e x a m ple, w e n o w investigate h o w t h e voltage across a j u n c t i o n is related t o
the current t h r o u g h it. W e represent a gradual increase of c u r r e n t
t h r o u g h the junction b y a gradual increase of t o r q u e applied t o t h e p e n d ulum. W e can imagine t h e t o r q u e t o be provided b y w e i g h t s h a n g i n g
from a d r u m attached to the pivot of the p e n d u l u m , as in Fig. 11.3.
W h e n a small t o r q u e is applied (i.e. a small c u r r e n t passed t h r o u g h t h e
junction) t h e p e n d u l u m finally settles d o w n at a c o n s t a n t angle of deflection È. T h e r e is t h e n no angular velocity, so t h i s implies t h a t t h e r e is n o
voltage across a j u n c t i o n w h e n a small c u r r e n t is passing t h r o u g h it, i.e.
the j u n c t i o n is superconducting. If the t o r q u e is gradually increased, t h e
pendulum deflects to a greater b u t steady angle, i.e. w e can p a s s m o r e
COHERENCE OF THE ELECTRON-PAIR WAVE; QUANTUM INTERFERENCE
165
A p p l i e d torqu e — ^ t o t a l c u r r e n t
FlG. 11.3. Analogy between pendulum and Josephson tunnelling junction.
current t h r o u g h a junction w i t h o u t any voltage appearing. T h e r e is,
however, a m a x i m u m torque w h i c h can be applied to the p e n d u l u m and
which still p r o d u c e s a stationary deflection. T h i s is the t o r q u e which
deflects the p e n d u l u m t h r o u g h a right angle so that it is horizontal. If w e
apply any greater t o r q u e t h e p e n d u l u m rises, accelerates u p w a r d s , passes
t h r o u g h t h e vertically " u p " position, and thereafter continues to r o t a t e
continuously a r o u n d its axis so long as t h e t o r q u e continues to b e
applied. W e see, therefore, t h a t if m o r e t h a n a certain "critical t o r q u e " is
applied the p e n d u l u m cannot remain at rest b u t r o t a t e s continuously.
Because angular velocity is t h e analogue of voltage across a junction and
the angular velocity is always in the same direction, this unstable
behaviour of t h e p e n d u l u m shows t h a t a d.c. voltage will appear across a
j u n c t i o n if the current passed t h r o u g h it exceeds a critical value, i.e. a
junction h a s a critical current.
11.3.3.
a . c . J o s e p h s o n effect
L e t u s consider further w h a t can b e deduced from the p e n d u l u m
analogue of a Josephson tunnelling junction. S u p p o s e t h a t a pendulum is
rotating continuously because a t o r q u e greater t h a n the critical torque is
being applied t o it. As the pendulum rotates, the horizontal deflection ÷
of t h e b o b (Fig. 11.3) oscillates, changing from right to left and back
166
INTRODUCTION TO SUPERCONDUCTIVITY
again. We have seen that the horizontal deflection of the bob corr e s p o n d s to the electron-pair c u r r e n t tunnelling across t h e j u n c t i o n . So
t h e analogue tells u s t h a t w h e n t h e r e is a d.c. voltage across a J o s e p h s o n
tunnelling j u n c t i o n an a.c. electron-pair c u r r e n t t u n n e l s back and forth
across it. T h i s is t h e a.c. Josephson effect m e n t i o n e d in § 10.6. W h a t is
the frequency of t h e a.c. tunnelling c u r r e n t ? T o a n s w e r t h i s w e r e t u r n to
our p e n d u l u m analogue. W h e n a c o n s t a n t t o r q u e T, greater t h a n the
critical t o r q u e , is applied to a p e n d u l u m , t h e p e n d u l u m r o t a t e s , a n d its
rotation accelerates until t h e energy lost d u e t o viscous d a m p i n g d u r i n g
each rotation equals t h e work d o n e on t h e p e n d u l u m b y t h e c o n s t a n t
applied t o r q u e . W h e n t h i s state is reached t h e period of each revolution
is t h e same. T h e angular velocity, however, is not c o n s t a n t b u t varies
d u r i n g each revolution (on t h e half-cycle d u r i n g w h i c h t h e b o b is rising
the r o t a t i o n decelerates b e c a u s e gravity o p p o s e s t h e applied t o r q u e , b u t
on t h e following half-cycle t h e b o b is accelerated b y gravity as it falls).
T h e frequency of t h e r o t a t i o n is(l/2n)(dd/dt)
w h e r e (dd/dt) is t h e t i m e
average of t h e angular velocity over one cycle. W e deduce, therefore, t h a t
t h e electron-pair tunnelling current, w h i c h is t h e analogue of t h e
horizontal displacement of t h e p e n d u l u m ' s b o b , oscillates back and forth
across t h e j u n c t i o n w i t h frequency í equal ×ï (\/2ð) {(d/ dt)/S>(p). N o w w e
have seen t h a t w h e n the p h a s e difference across a j u n c t i o n is varying at a
r a t e (d/dt)N(p a voltage V a p p e a r s across t h e j u n c t i o n w h o s e i n s t a n t [eqn. (11.14.ii)]. By analogy w i t h the p e n d a n e o u s value is (h/2e)(d/dt)&(f)
ulum (d/dt)A(p always h a s t h e same sign b u t is not c o n s t a n t , so t h a t t h e
voltage V c o n t a i n s b o t h d.c. and a.c. c o m p o n e n t s , i.e. it is a d.c. voltage
w i t h an a.c. ripple o n it. T h e average value {(Ü/Üß)Äö) of t h e r a t e of
c h a n g e of p h a s e across t h e j u n c t i o n is given b y
because t h e d.c. c o m p o n e n t is t h e t i m e average of t h e voltage.
C o n s e q u e n t l y t h e frequency of oscillation is related t o t h e d.c. voltage
across the j u n c t i o n by
T h e a . c J o s e p h s o n effect can b e observed t h r o u g h the a.c. c o m p o n e n t
of t h e voltage w h i c h a p p e a r s across t h e j u n c t i o n and w h i c h c a n be
detected b y suitable i n s t r u m e n t a t i o n . T h i s alternating c o m p o n e n t of t h e
voltage causes emission of radiation w i t h frequency í given by hv =
COHERENCE OF THE ELECTRON-PAIR WAVE; QUANTUM INTERFERENCE
167
2eVd . T h i s is in agreement w i t h eqn. (10.1) derived from consideration
of the energy balance of the tunnelling electron pairs.
W e m a y sum u p the situation as follows: w h e n a steady direct current,
greater t h a n t h e critical current, is fed t h r o u g h a Josephson j u n c t i o n
from a constant current source a d.c. voltage V^ a p p e a r s across the
junction. T h e electron-pair tunnelling current has, however, a large
oscillatory c o m p o n e n t which tunnels back and forth w i t h frequency
across t h e junction. T h i s a.c. Josephson tunnelling can b e
(2e/h)Vd
observed as a voltage ripple of frequency (2e/h)V^ superimposed on t h e
d.c. voltage.
C
c
c
c
T h e a.c. Josephson effect forms t h e b a s i s of a n u m b e r of useful
devices. An oscillating Josephson junction can b e used as a tunable
oscillator whose frequency í is given b y í = 2eV/h, w h e r e V is the
d.c. voltage across the junction. T h e p o w e r o u t p u t is very small b u t its
usefulness lies in t h e fact t h a t t h e frequency can b e very accurately controlled simply by adjusting t h e voltage across the junction. Josephson
j u n c t i o n s are also used as very sensitive detectors of radiation; because
an alternating electric field acting on a biased oscillating junction can
drive a d.c. current t h r o u g h an external load.
11.3.4.
Couplin g energ y
W e have seen t h a t the supercurrent i tunnelling t h r o u g h a Josephson
junction is sinusoidally related to the phase difference of the electronpair waves on t h e t w o sides [eqns. (11.14.i) and (11.14.U)]. If, therefore, a
certain current, i say, is passed t h r o u g h the junction there are t w o
possible phase differences across it, Äö and ð — Äö (Fig. 11.4). W e
naturally ask w h i c h of these t w o possible phase differences will in fact
occur. T h e answer is determined by the fact that, as w e shall now show,
the energy of a Josephson tunnelling j u n c t i o n d e p e n d s on the phase
difference across it. Consider a j u n c t i o n t h r o u g h which a constant
current i is being passed, the current having been raised from zero to
this value over a t i m e t. D u r i n g the time the current is increasing there
m u s t b e a rate of change of current di/dt, b u t eqn. ( l l . 1 4 . i i ) tells u s t h a t
if the current t h r o u g h a j u n c t i o n is changing a voltage will appear across
it, so during the t i m e t t h a t the current is being established there is a
s
x
é
÷
s
voltage V across t h e junction, and power iV is being delivered to it. S o
t
work, W = j iVdty is performed in setting u p the current and the cono
sequent phase difference across it, and W m u s t be the increase in energy
168
INTRODUCTION TO SUPERCONDUCTIVITY
is
FIG . 11.4. Josephso n junction ; relationshi p of tunnellin g curren t i, an d energ y W to
phas e differenc e Ä0.
of t h e j u n c t i o n d u e to t h e passage of a c u r r e n t t h r o u g h it. E q u a t i o n s
(11.14.i) and ( l l . 1 4 . i i ) relate t h e voltage and current to the p h a s e
difference, and substitution of these gives t h e increase i n j u n c t i o n energy
as
t
ï
W=^[l-cosA0].
(11.20)
T h i s relationship is shown in Fig. 11.4. W e see t h a t the energy W increases w i t h Äö for p h a s e differences u p t o ð and so, if a c u r r e n t i is
passed t h r o u g h the j u n c t i o n it is energetically favourable for the smaller
phase difference Äö to establish itself across t h e j u n c t i o n r a t h e r t h a n
the larger difference ð — Äö . W h e n there is n o current t h r o u g h the
junction, the phase difference across it will be zero not ð ; in t e r m s of our
pendulum analogue this is equivalent to saying that, for energetic
reasons, if no t o r q u e is applied the p e n d u l u m h a n g s vertically
d o w n w a r d s not vertically u p w a r d s .
x
÷
÷
COHERENCE OF THE ELECTRON-PAIR WAVE; QUANTUM INTERFERENCE
11.3.5.
169
Weak-link s
Josephson j u n c t i o n s are a special case of a m o r e general kind of weak
contact b e t w e e n t w o superconductors. T h e s e other forms have properties similar to t h e Josephson tunnelling j u n c t i o n s w e have j u s t described.
A resistanceless current can b e passed t h r o u g h a very n a r r o w constriction or a point contact b e t w e e n t w o superconductors (Fig. 11.5) while a
phase difference is maintained between t h e superconductors on each
side. W e shall include all these configurations under t h e general n a m e of
"weak-links". A weak-link is a region which h a s a m u c h lower critical
current than t h e superconductors it joins and into w h i c h an applied
magnetic field can penetrate.
(a )
(b ) C o n s t r i c t i o n
Tunnellin g
junctio n
( c ) Poin t contac t
FIG. 11.5. Superconducting weak-links.
F o r all weak-links, including Josephson tunnelling junctions, the
resistanceless current flowing t h r o u g h the link increases as the p h a s e
difference across the link increases, and the critical current is t h e current
at which the phase difference reaches ð / 2 . H o w e v e r , the current is not in
general proportional to the sine of the p h a s e difference; in this respect
the Josephson junction is a special case.
T h o u g h the behaviour of weak-links which are n o t Josephson
tunnelling j u n c t i o n s m a y differ in detail from t h a t of a Josephson
tunnelling j u n c t i o n their general behaviour is very similar, and t h e effects
described in this chapter for Josephson tunnelling j u n c t i o n s also occur
for the other forms of weak-link.
Weak-links in the form of constrictions or point c o n t a c t s are usually
better t h a n tunnelling j u n c t i o n s as radiation sources or detectors because
radiation can m o r e easily b e coupled into or out of them.
170
11.4.
INTRODUCTION TO SUPERCONDUCTIVITY
Superconductin g Quantu m Interferenc e D e v i c e (SQUID )
W e n o w consider a superconducting device w h o s e p r o p e r t i e s well
illustrate t h e coherence of the electron-pair wave t h r o u g h o u t a superconductor and the effect of weak-links on this coherent wave. T h e device is
called a superconducting
quantum
interference
device, usually a b breviated to " S Q U I D " . S Q U I D s can b e used t o detect a n d m e a s u r e
extremely weak magnetic fields and are t h e basic e l e m e n t s of a r a n g e of
very sensitive and useful i n s t r u m e n t s . T h e basic element of a S Q U I D is
a ring of superconducting metal containing o n e or m o r e weak-links. W e
shall consider here t h e particular form w h i c h includes t w o weak-links, as
shown in Fig. 11.6. T h e ring of s u p e r c o n d u c t i n g material h a s t w o similar
weak-links X and W w h o s e critical c u r r e n t i is very m u c h less t h a n the
c
®
B
a
FIG . 11.6. Superconductin g ring wit h tw o weak-links .
critical current of t h e rest of t h e ring. Because s u p e r c u r r e n t s flowing
r o u n d such circuits cannot exceed the critical c u r r e n t of t h e weak-links,
resistanceless c u r r e n t s m u s t b e very m u c h less t h a n t h e critical c u r r e n t s
of the s u p e r c o n d u c t o r s joined b y t h e weak-links. Because of t h e low
current density in t h e s u p e r c o n d u c t o r s , t h e m o m e n t u m of t h e C o o p e r
pairs in t h e m is small and t h e corresponding electron-pair w a v e s h a v e a
very long wavelength. T h e r e is, consequently, very little p h a s e difference
b e t w e e n different p a r t s of any o n e superconductor, and w e shall a s s u m e
t h a t t h i s difference is negligible, so that, if t h e r e is n o applied m a g n e t i c
field, t h e phase is t h e s a m e t h r o u g h o u t any o n e piece of s u p e r c o n d u c t o r
which does n o t include a weak link. A s a result, a p h a s e difference
a p p e a r s across the s u p e r c o n d u c t o r s only w h e n a m a g n e t i c field is
applied.
W e n o w e x a m i n e t h e effect of t h e application of a m a g n e t i c field t o a
S Q U I D . In studying t h e behaviour of t h i s circuit, w e shall apply t w o of
the results discussed in t h e previous sections. First, t h a t if a m a g n e t i c
COHERENCE OF THE ELECTRON-PAIR WAVE; QUANTUM INTERFERENCE
171
field is applied perpendicular t o t h e plane of t h e ring, it produces a
difference in p h a s e of the electron-pair w a v e along t h e p a t h XYW and
along the p a t h WZX (Fig. 11.6); we assume t h a t the weak-links X and
W are so short t h a t the phase difference across t h e m produced b y t h e
applied magnetic field is negligible. Second, that if a small current flows
round the ring, it produces a phase difference across the weak-links, b u t
negligible phase difference across the thick segments XYW and WZX.
S u p p o s e t h a t the ring h a s been cooled below its transition t e m p e r a t u r e
in the absence of an applied magnetic field so that there is n o magnetic
flux threading the hole. N o w a magnetic field of gradually increasing flux
density B is applied perpendicular to the plane of the ring. If the ring
had n o weak-links the application of t h e magnetic field would induce a
current i (Fig. 11.6) which would circulate to cancel the flux in the hole.
W e shall use the convention t h a t the clockwise direction in Fig. 11.6 h a s
a positive sign. F o r the cancellation to be complete the m a g n i t u d e Ii\ of
the circulating current would have to b e such that
a
L I /I = Ö
(11.21)
á
w h e r e L is the inductance of the ring and Ö = ?/B ,
of the hole enclosed by the ring.
T h e presence of the weak-links h a s t w o effects:
á
a
sJ being the area
(i) T h e y have a very small critical current * , so any circulating
current m u s t be less t h a n this value. Consequently, unless B is
very small, not enough current can circulate to cancel t h e flux in
the hole, and this flux can n o longer b e maintained at zero.
(ii) Even t h o u g h any circulating current is limited b y t h e weak-links
to a m a x i m u m value of i , it can nevertheless introduce a significant phase change across each of t h e m .
c
a
c
T h e particular form of S Q U I D w e are about to consider is constructed so t h a t b o t h the critical current i of the weak-links and the
inductance L of t h e ring are extremely small, so t h a t
c
iL < Ö .
c
0
(11.22)
N o w iL is the flux generated in the hole by a circulating current, so the
presence of the weak-links m e a n s t h a t an induced circulating current
cannot generate even one fluxon, and the net flux Ö in the hole is scarcely
different from the flux Ö of the applied magnetic field;
á
172
INTRODUCTION TO SUPERCONDUCTIVITY
T h e presence of t h e weak-links also m e a n s t h a t t h e flux within t h e hole
is no longer necessarily an integral n u m b e r of fluxons.t H o w e v e r , the
q u a n t u m condition t h a t t h e total p h a s e c h a n g e a r o u n d any closed p a t h
m u s t equal nln can still b e satisfied b e c a u s e the circulating current,
t h o u g h unable to generate even one fluxon, can produce considerable
phase differences across each weak-link.
E q u a t i o n (11.6) tells u s t h a t t h e application of a m a g n e t i c
produces a p h a s e difference Ä 0( â ) a r o u n d t h e ring
field
A0(fi)=^pj>A.rfl .
N o w I A . dl is the flux Ö produced in t h e ring b y t h e applied magnetic
field (we have seen t h a t t h e flux Li p r o d u c e d by t h e circulating c u r r e n t is
negligible, eqn. (11.22)), and h/2e e q u a l s the fluxon Ö , so
á
0
Ä 0( â ) = 2 ô Ã ^.
(11.23)
So, w h e n a m a g n e t i c field is applied to the superconducting ring, its
effect on t h e electron-pair w a v e is to p r o d u c e a p h a s e c h a n g e a r o u n d the
ring which is proportional to the flux of t h e applied field, as s h o w n in
Fig. 11.7(a). If, for example, t h e flux density B of the applied field is
such t h a t the flux t h r e a d i n g t h e ring is Ö (Fig. 11.7(a)), t h e p h a s e
change it p r o d u c e s around the ring is Äö(Â).
I n general t h e s t r e n g t h of
the applied magnetic field will not b e such as to p r o d u c e an integral
n u m b e r of fluxons in the hole and t h e resulting p h a s e c h a n g e Äö{Â) will
not equal a multiple of 2 ð . H o w e v e r , the p h a s e change r o u n d any closed
superconducting circuit must equal an integral multiple of 2ð; a c u r r e n t i
therefore circulates r o u n d t h e ring of such a strength t h a t t h e additional
phase different 2 Ä 0 ( é) it p r o d u c e s t h e t w o weak-links b r i n g s the total
phase c h a n g e r o u n d the circuit t o a multiple of 2ð (Fig. 11.6):
x
÷
÷
÷
Ä 0 ( â ) + 2 Ä 0 ( ß) = ç . 2 ð .
(11.24)
But in an applied m a g n e t i c field producing a flux Ö, (Fig. 11.7(a)), there
are t w o possible circulating c u r r e n t s w h i c h could m a k e t h e total phase
change r o u n d the ring an integral multiple of 2 ð . A current of m a g n i t u d e
t In a different form of S Q U I D , which we shall not discuss, the critical current of the weaklinks and the inductance are not so small. In this form of S Q U I D the flux in the hole is quantized
and different from the flux of the applied field. T h i s form of S Q U I D operates in a different
manner from the one discussed in this book.
COHERENC E O F T H E ELECTRON-PAI R
WAVE ; QUANTU M
INTERFERENC E
173
FIG . 11.7. Effect of magneti c field on SQUI D wit h tw o weak-link s (Li < Ö ).
(a) Phas e chang e Äö(Â ) roun d rin g du e to magneti c field, (b) Circulatin g curren t i in
absenc e of measurin g current . (c) Critica l measurin g curren t I .
c
0
á
c
if could circulate in the clockwise direction so that the phase difference
2Ä0(*Õ) (Fig. 11.7(a)) it produces across the t w o weak-links a d d s on to
the phase difference Äö(Â) produced in the t w o halves of the ring b y
the magnetic field ( n = l ) , or a smaller anticlockwise current of
m a g n i t u d e i\ could circulate to produce t h e smaller negative phase
different — 2 A 0 ( / f ) which would exactly cancel the phase change due to
the magnetic field (n = 0). In fact the current will circulate in the anticlockwise direction because, as we saw in § 11.3.4, the energy of weaklinks depends on the phase difference across them and, for phase
differences less t h a n ð , increases as the phase difference increases.
Therefore in o u r S Q U I D t h e smaller, anticlockwise, current is
energetically favourable (because the circulating current flows t h r o u g h
b o t h junctions, the phase differences across them add and the phase
difference across each m u s t be less t h a n ð ) .
÷
W e can find h o w the m a g n i t u d e of the circulating current depends on
174
INTRODUCTION TO SUPERCONDUCTIVITY
the strength of the applied m a g n e t i c field b y m e a n s of (11.14.i) which
tells u s that a current i passing t h r o u g h a J o s e p h s o n tunnelling j u n c t i o n t
produces a p h a s e different Ä 0( é) given b y
â ß ç Ä 0( é) = i/i .
(11.25)
e
F o r the phase difference across the weak-links to cancel the p h a s e
difference d u e to the m a g n e t i c field
2 Ä 0 ( * -) -
-Ä0(â).
S u b s t i t u t i n g the values of Äö{Â) and Äö(é~) from (11.23) and (11.25)
we obtain t h e strength of the circulating current
Ö
If I = L sin ð -=4.
Öï
If the magnetic field strength is increased so t h a t t h e flux t h r o u g h the
ring approaches | Ö the m a g n i t u d e of t h e anticlockwise circulating
current increases as s h o w n b e t w e e n Ï and A in Fig. 11.7(b). It can b e
seen that w h e n t h e applied field is such t h a t the flux t h r o u g h the ring
equals \Ö the circulating current reaches t h e critical current i of the
weak-links, so at this field strength t h e links go n o r m a l and the circulating current dies away.
S u p p o s e the strength of the applied m a g n e t i c field is still further increased so t h a t t h e flux Ö t h r o u g h t h e ring n o w h a s a value Ö greater
it p r o d u c e s across the t w o
t h a n | Ö , and t h e phase change Áö(Â)
halves of t h e ring exceeds ð (Fig. 11.7(a)). Clearly it is n o w energetically
favourable for the total phase change to b e b r o u g h t u p to 2ð b y a current
f J circulating in t h e clockwise direction to p r o d u c e p h a s e c h a n g e s across
the weak-links w h i c h add on to those d u e t o t h e m a g n e t i c field. W h e n Ö
is j u s t greater t h a n | Ö t h e circulating current is relatively large b u t as
the strength of t h e magnetic field is increased the current b e c o m e s
smaller, until at Ö it h a s fallen to zero (Fig. 11.7(b)). It can b e seen that
the circulating c u r r e n t i varies periodically w i t h the s t r e n g t h of the
applied field, switching from —i to + i w h e n the flux t h r o u g h t h e ring is
an odd multiple of | Ö . T h i s periodic dependence of t h e circulating
current on the strength of an applied m a g n e t i c field is the basis of all
SQUIDs.
0
0
c
á
2
0
2
á
0
0
c
c
0
T In this analysis we shall assume the weak-links to be Josephson tunnelling junctions. For
other weak-links, such as constrictions or point contacts, where the current is periodic in Ä0 but
not sinusoidally related, a similar analysis can be carried out and leads to a qualitatively similar
result.
COHERENCE OF THE ELECTRON-PAIR WAVE; QUANTUM INTERFERENCE
175
T h e period of variation of t h e circulating current corresponds to a flux
change of one fluxon, Ö , a very small a m o u n t of magnetic flux, so the
value of the circulating current is affected b y very small changes in
applied magnetic field strength. If w e could detect changes in the circulating current w e should, therefore, b e able to detect very small
changes in the strength of t h e applied magnetic field, and our S Q U I D
could b e used as a sensitive m a g n e t o m e t e r . W e can, as w e shall now see,
detect changes in the circulating current b y passing a current / across
the ring b e t w e e n t w o side-arms Y and Z , as s h o w n in Fig. 1 1 . 8 . W e shall
0
i
I
FIG. 11.8. Superconductin g quantu m interferenc e devic e (SQUID) .
call / the " m e a s u r i n g c u r r e n t " . If the ring is symmetrical, / divides
equally, { / f l o w i n g t h r o u g h each weak-link; b u t so long as t h e S Q U I D is
everywhere superconducting n o voltage will b e detected b y the voltmeter
V connected across it. However, if w e increase the measuring current /
enough, a voltage will b e developed. T h e value of m e a s u r i n g current / at
which the voltage appears is called the critical measuring current I of
the ring. It can be seen t h a t current i — \l flows across one weak-link
and a larger current i + {/ across the other, and it might be t h o u g h t t h a t
the critical m e a s u r i n g current must be t h a t which raises i + {/ t o the
critical value i of the weak-link t h r o u g h which it flows, i.e. t h a t t h e
critical measuring current would be 2{t — t). However, this is not
necessarily t r u e ; in general it b e c o m e s impossible, as w e shall now see,
for t h e phase c h a n g e of t h e electron-pair w a v e r o u n d the ring to equal an
integral multiple of 2ð (a necessary condition for there to b e a superconc
c
c
176
INTRODUCTION TO SUPERCONDUCTIVITY
ducting p a t h right r o u n d t h e ring) even t h o u g h t h e c u r r e n t t h r o u g h
neither weak-link h a s reached i . T h a t is to say, a voltage a p p e a r s across
the ring at a m e a s u r i n g current I less t h a n 2(z — i). T h i s is an excellent
example of t h e fundamental i m p o r t a n c e of t h e coherence of t h e electronpair w a v e in a superconductor.
W e now find t h e value of the critical m e a s u r i n g current and h o w it
d e p e n d s on t h e strength of t h e m a g n e t i c field. L e t a and â b e t h e phase
changes produced b y c u r r e n t s flowing across t h e t w o weak-links ( w e
m u s t r e m e m b e r o u r sign c o n v e n t i o n : t h e clockwise direction is positive,
so a positive p h a s e difference is an increase in p h a s e angle ö in a
clockwise direction r o u n d t h e ring), and let Äö(Â) b e t h e total phase
change produced b y t h e applied m a g n e t i c field a r o u n d the t o p and b o t t o m halves of t h e ring. If the ring is superconducting,
a + â + Ä 0( Â ) = ç. 2 ð .
c
c
c
E q u a t i o n (11.23) gives Ä 0( Â ) in t e r m s of t h e flux Ö
á
enclosed b y the
r i n g ; so
i.e.
a + â + 2ð^
= ç.2ð.
(11.26)
T h i s relation m u s t always b e satisfied if t h e r e is t o b e superconductivity
everywhere r o u n d t h e ring. T h e m a g n i t u d e s of á a n d â d e p e n d on the
total c u r r e n t passing t h r o u g h t h e weak-links so, w h e n t h e current / is fed
t h r o u g h the ring t h e circulating c u r r e n t i adjusts itself so t h a t t h e phase
condition (11.26) is still satisfied.
I n t h e absence of the m e a s u r i n g c u r r e n t / t h e p h a s e differences across
t h e t w o weak links are equal b e c a u s e t h e s a m e c u r r e n t i flows t h r o u g h
each. F r o m (11.26) w e see t h a t in t h i s case á = â = ð[ç — ( Ö / Ö ) ] . B u t
w h e n we p a s s t h e current / , a and â are n o longer equal, b e c a u s e the
current t h r o u g h X is n o w i — \l and t h a t t h r o u g h W is now i + \l. Since
a + â m u s t r e m a i n constant, t h e decrease in a m u s t equal t h e increase in
â. L e t u s w r i t e t h a t w h e n t h e m e a s u r i n g c u r r e n t / is passed from Y to Æ
á
a = [ç
ð
- (Ö /Ö )] â
0
0
ü,
(11.27)
â = ð[ç - ( Ö / Ö ) ] + ä
á
0
w h e r e ä d e p e n d s on t h e current / . If the weak-links are J o s e p h s o n
tunnelling j u n c t i o n s the p h a s e difference across t h e m d u e to current
t h r o u g h t h e m is given b y (11.14.i), so for t h e weak-links X and W w e
have
COHERENCE OF THE ELECTRON-PAIR WAVE; QUANTUM INTERFERENCE
i— jl—
i + \I=
t sin
;
c
("-|Ç·
177
(11.28)
i sin
c
W e are interested in h o w the S Q U I D behaves as w e increase / , so w e
eliminate i b y subtraction,
/ = i { ß ç [ð (ç - | ? ) + ü ] c
8
= 2i cos
c
8
ß ç [ ð{ç - ^ -
4-ì
sino .
ü] )
(11.29)
T h e right-hand side of this equation could have either positive or
negative values, depending on w h e t h e r the cos and sin t e r m s have the
same or opposite signs. But w e are considering w h a t h a p p e n s w h e n the
measuring current / is fed t h r o u g h from Y to Æ in Fig. 11.8, and we have
taken / in this direction to b e positive. If / t u r n s out to b e negative, this
is equivalent to turning the figure upside down. T h i s inversion does not,
however, represent a physically different situation, so w e m a y write
(11.29) as
I=2L
Isin ä
cos
I.E.
I=2L
cos ð
Ö
-=r ^
· sin ä
(11.30)
However, sin ä cannot have a m a g n i t u d e greater t h a n unity, so (11.30)
can only be satisfied and all of the ring r e m a i n superconducting if
/<2/
c
cos ð
Ö
a
Öç
Therefore the critical measuring current is
cos ð
Öá
Öç
(11.31)
I t can b e seen from this that, w h e n a magnetic field is applied, the
critical measuring current of t h e S Q U I D d e p e n d s in a periodic m a n n e r
on t h e strength of the field, being a m a x i m u m w h e n the field is such that
178
INTRODUCTION TO SUPERCONDUCTIVITY
the flux Ö t h r o u g h the ring is an integral n u m b e r of fluxons (Fig.
11.7(c)).
A g r a p h such as Fig. 11.7(c), w h i c h s h o w s t h e periodic variation of
critical c u r r e n t w i t h applied m a g n e t i c field strength, is referred t o as t h e
"interference p a t t e r n " of the superconducting q u a n t u m interferometer.
In order t o o b t a i n a p a t t e r n such as Fig. 11.7(c) t h e critical c u r r e n t of
the links and t h e area of t h e hole m u s t b e small, so t h a t Li < Ö and t h e
induced current c a n n o t appreciably screen t h e hole from the applied
field. Often, however, t h e critical c u r r e n t s of t h e links and t h e size of t h e
hole m a y b e large e n o u g h so t h a t Li > ã Ö , and there is appreciable
screening of t h e hole. I t can b e s h o w n t h a t in t h i s case t h e d e p t h of
modulation (Fig. 11.9) of t h e critical c u r r e n t is reduced t o ÄÉ = Öï/L.
á
c
c
0
0
€
_L
ìÃ
Ì·ï Á
Á
2<D
0
Ho
A
Applie d magheti c fiel d strength , H
a
FIG. 1 1 . 9 . Interference pattern of superconducting q u a n t u m inteferometer (Li
11.4.1.
"Diffraction "
c
> Ö ).
0
effect s
W e have seen t h a t the critical c u r r e n t of a S Q U I D is m o d u l a t e d b y an
applied magnetic field. I t also h a p p e n s , however, t h a t t h e critical c u r r e n t
of a single weak-link is m o d u l a t e d by a m a g n e t i c field. Although, as w e
showed in the last section, a m a g n e t i c field p r o d u c e s negligible phase
difference across t h e weak-links in a S Q U I D , it nevertheless h a s a considerable effect o n their critical current. I t can b e s h o w n t h a t t h e critical
current i of a J o s e p h s o n j u n c t i o n in a m a g n e t i c field parallel t o t h e plane
of t h e j u n c t i o n is itself a periodic function of t h e s t r e n g t h of t h e m a g n e t i c
field, being a m i n i m u m w h e n t h e m a g n e t i c flux passing t h r o u g h t h e j u n c tion equals an integral n u m b e r of fluxons.
c
COHERENCE OF THE ELECTRON-PAIR WAVE; QUANTUM INTERFERENCE
179
E q u a t i o n (11.31) gives t h e critical current of a S Q U I D , i.e. t w o weaklinks in parallel. T h i s equation h a s exactly the same form as the
Fraunhofer formula for the optical interference pattern from t w o parallel
slits ( Y o u n g ' s fringes) if w e take I as equivalent t o the resultant
amplitude A of t h e optical oscillation, i as the amplitude a of each of the
t w o interfering b e a m s and ð Ö / Ö as half t h e phase difference b e t w e e n
the t w o interfering b e a m s . T h e amplitude of Y o u n g ' s fringes is,
however, modulated by the diffraction p a t t e r n of t h e individual slits. T h e
Fraunhofer formula for optical diffraction from a single slit is
c
c
á
á =
0
á
sin â
0â
where â is one-half of the phase difference of r a y s from the t w o opposite
edges of the slit. By analogy, therefore, w e might expect the critical
current i of a single weak-link to depend on t h e magnetic field in t h e
following w a y :
c
i =
c
i (0)
c
sin ( ð Ö , / Ö )
0
ðÖ ,,/Ö ç
(11.32)
where Ö , is t h e flux of t h e applied magnetic field passing t h r o u g h the
area of the link. I n fact, a direct calculation, similar to that in the
previous section, of the phase of t h e electron-pair w a v e in the
neighbourhood of a weak-link leads to exactly this r e s u l t . t
Just as optical interference p a t t e r n s which are formed b y a n u m b e r of
slits are modulated in amplitude b y the diffraction which occurs at each
individual slit, t h e interference pattern from a superconducting
interferometer is m o d u l a t e d b y t h e "diffraction" occurring at t h e weaklinks. W e neglected this diffraction effect in d r a w i n g Figs. 11.7(c) and
11.9. An actual interference pattern obtained from a q u a n t u m
interferometer is shown in Fig. 11.10. T h e modulation of the interference
pattern b y the diffraction p a t t e r n of the weak-links is evident.
It is very difficult to m a k e a superconducting q u a n t u m interferometer
in which b o t h weak-links have the same cross-sectional area. C o n s e q u e n t l y t h e i n t e r f e r e n c e p a t t e r n is u s u a l l y m o d u l a t e d b y t w o
periodicities and a complicated multiply-periodic p a t t e r n results.
W e have seen t h a t there is a close analogy b e t w e e n q u a n t u m interference effects in superconductors and optical interference. Nevertheless, the reader should b e w a r n e d against d r a w i n g false conclusions
t T h e detailed calculation may be found in Jaklevic, Lambe, Mercereau and Silver, Phys. Rev.
140A, 1628 (1965).
180
INTRODUCTIO N TO SUPERCONDUCTIVIT Y
Critica l
curren t
-500
- 400
-300
- 200
-100
0
100
200
300
400
500
Magneti c field (Milligauss )
FIG . 11.10. Trac e of interferenc e patter n fro m a quantu m interferometer , showin g
modulatio n of interferenc e b y diffraction . (Reproduce d b y kin d permissio n of R. C.
Jaklevic , J. Lambe , J . E . Mercereau , an d A. H . Silver , Scientifi c Laboratory , For d
Moto r Company. )
f r o m t h i s a n a l o g y a b o u t t h e n a t u r e of s u p e r c o n d u c t i n g q u a n t u m
interference. It m u s t be r e m e m b e r e d t h a t t h e physical situations are q u i t e
different. T h e essential feature of optical interference is t h a t interference
b e t w e e n t w o coherent light w a v e s p r o d u c e s at any point a resultant o s cillation w h o s e a m p l i t u d e d e p e n d s o n t h e relative p h a s e and a m p l i t u d e
of t h e t w o interfering w a v e s at t h a t point. T h i s leads t o a p a t t e r n in
space w h o s e intensity varies from place t o place. I n any uniform piece of
superconducting metal, however, t h e intensity (i.e. s q u a r e of amplitude)
of t h e coherent electron-pair w a v e is e v e r y w h e r e t h e same. T h i s is
because t h e intensity is proportional t o t h e density of superelectrons
w h i c h h a s negligible variation from place t o place. E q u a t i o n (11.32),
t h o u g h of t h e same m a t h e m a t i c a l form as t h e e q u a t i o n of optical diffraction, does not c o n t a i n any position variables and d o e s not describe any
variation of intensity in space. T h e S Q U I D described in § 11.4 is n o t t o
be r e g a r d e d as an analogue of Y o u n g ' s slits in w h i c h t h e weak-links correspond to t w o slits w h o s e diffracted w a v e s interfere at Z .
S u p e r c o n d u c t i n g q u a n t u m interference can b e p u t t o practical use in a
n u m b e r of w a y s . M o s t of t h e applications m a k e u s e of t h e fact t h a t a
superconducting interferometer can b e used t o detect very small c h a n g e s
of m a g n e t i c field strength. W e h a v e seen t h a t t h e critical c u r r e n t of t h e
device u n d e r g o e s a complete cycle for a c h a n g e of m a g n e t i c flux of o n e
fluxon t h r o u g h t h e enclosed hole, a n d b e c a u s e t h e fluxon is only a b o u t
2 ÷ 1 0 " weber (2 ÷ 1 0 " g a u s s c m ) a very small c h a n g e in t h e applied
magnetic field s t r e n g t h can b e observed as a c h a n g e in critical c u r r e n t .
As well as being used directly as m a g n e t o m e t e r s , S Q U I D S can b e used
as very sensitive galvanometers, b e c a u s e any electric c u r r e n t m u s t
generate a m a g n e t i c field and t h i s field can b e detected b y a S Q U I D .
1 5
7
2
CHAPTER 1 2
T H E MIXED S T A T E
FOR MANY years it w a s supposed t h a t the behaviour which we have
described in P a r t I of this book w a s characteristic of all superconductors.
It had, indeed, been noticed t h a t certain superconductors, especially
alloys and i m p u r e metals, did not behave quite in the expected way, b u t
this anomalous behaviour w a s usually ascribed t o i m p u r i t y effects, not
considered to b e of great scientific interest, and consequently little effort
w a s m a d e to u n d e r s t a n d it. However, in 1957 Abrikosov published a
theoretical paper pointing out t h a t there might b e another class of superconductors w i t h s o m e w h a t different properties, and it is n o w realized
t h a t the apparently anomalous properties of certain superconductors are
not merely trivial impurity effects b u t are t h e inherent features of this
other class of superconductor n o w k n o w n as " t y p e - I I " .
O n e of t h e characteristic features of t h e type-I superconductors w e
considered in t h e first part of this book is the Meissner effect, the
cancellation w i t h i n the metal of t h e flux d u e t o an applied magnetic field.
W e mentioned in § 6.7 t h a t the occurrence of this perfect d i a m a g n e t i s m
implies the existence of a surface energy at the b o u n d a r y b e t w e e n any
normal and superconducting regions in t h e metal. T h i s surface energy
plays a very i m p o r t a n t role in determining t h e behaviour of a supercond u c t o r ; for example, as w e shall now see, it d e t e r m i n e s w h e t h e r t h e
material is a type-I or a type-II superconductor.
Consider a superconducting b o d y in an applied magnetic field of
strength less t h a n t h e critical value H , and suppose t h a t within the
material a normal region were t o appear w i t h b o u n d a r i e s lying parallel to
the direction of the applied magnetic field. T h e appearance of such a normal region would change t h e free energy of t h e superconductor, and w e
m a y consider t w o contributions to this free energy c h a n g e : a contribution arising from t h e bulk of the normal region and a contribution due to
its surface. As w e saw in C h a p t e r 4, in an applied magnetic field of
the free energy per unit volume of the normal state is
strength H
c
a9
183
184
INTRODUCTION TO SUPERCONDUCTIVITY
greater t h a n t h a t of the superconducting, perfectly d i a m a g n e t i c , state b y
— Ç ). F u r t h e r m o r e , as s h o w n in C h a p t e r 6, t h e r e is
an a m o u n t \ì (Ç
a surface energy associated w i t h t h e b o u n d a r y b e t w e e n a n o r m a l and a
superconducting region. F o r t h e type-I s u p e r c o n d u c t o r s w e considered
in t h e first part of this book t h i s surface energy is positive. H e n c e , if a
n o r m a l region w e r e to form in t h e superconducting material, t h e r e would
b e an increase in free energy d u e b o t h to t h e bulk and to t h e surface of
the n o r m a l region. F o r t h i s reason, t h e a p p e a r a n c e of normal regions is
energetically unfavourable, and a type-I superconductor r e m a i n s superconducting t h r o u g h o u t w h e n a m a g n e t i c field of strength less t h a n H is
applied.
2
2
0
c
S u p p o s e , however, t h a t in certain m e t a l s t h e surface energy b e t w e e n
normal and superconducting regions w e r e negative instead of positive
(i.e. energy is released w h e n t h e interface is formed). I n t h i s case t h e
appearance of a n o r m a l region would reduce t h e free energy, if the increase in energy d u e to t h e bulk of t h e region w e r e o u t w e i g h e d b y t h e
decrease d u e to its surface. A material a s s u m e s t h a t condition w h i c h h a s
t h e lowest total free energy, so in t h e case of a sufficiently negative surface energy we would expect t h a t , in order t o p r o d u c e the m i n i m u m free
energy, a large n u m b e r of n o r m a l regions would form in t h e superconducting material w h e n a m a g n e t i c field is applied. T h e material would
split into some fine-scale m i x t u r e of superconducting and n o r m a l regions
w h o s e b o u n d a r i e s lie parallel t o t h e applied field, t h e a r r a n g e m e n t being
such as to give t h e m a x i m u m b o u n d a r y area relative t o v o l u m e of n o r m a l
material. W e shall call t h i s the mixed state. I n the next section it will b e
shown t h a t t h e conditions in some s u p e r c o n d u c t o r s are such t h a t t h e
surface energy is indeed negative. T h e s e m e t a l s are therefore able to g o
into the mixed state, and these are t h e t y p e - I I superconductors.
It is i m p o r t a n t to distinguish clearly b e t w e e n t h e mixed state w h i c h
o c c u r s in type-II s u p e r c o n d u c t o r s and the intermediate
state which occurs in type-I superconductors. T h e i n t e r m e d i a t e state occurs in t h o s e
type-I superconducting b o d i e s w h i c h h a v e a non-zero d e m a g n e t i z i n g
factor, and its appearance d e p e n d s on t h e shape of t h e b o d y . T h e mixed
state, however, is an intrinsic feature of t y p e - I I superconducting
material and a p p e a r s even if t h e b o d y h a s zero demagnetizing factor (e.g.
a long rod in a parallel field). In addition, t h e s t r u c t u r e of t h e int e r m e d i a t e state is relatively coarse and the g r o s s features can b e m a d e
visible to the naked eye [Figs. 6.6 and 6.7 (pp. 73 and 74)]. T h e s t r u c t u r e
of the mixed state is, as w e shall see, o n a m u c h finer scale w i t h a
frontispiece).
periodicity less t h a n 10~ cm (see
5
185
THE MIXED STATE
12.1.
Negativ e Surfac e Energ y
I n § 6.9 w e showed that, as a result of t h e existence of t h e penetration
d e p t h and coherence length, there is a surface energy associated w i t h the
b o u n d a r y b e t w e e n a normal and superconducting region. I t w a s shown
that, if the coherence range is longer t h a n the penetration depth, as it is
in most p u r e metallic elements, the total free energy is increased close to
t h e b o u n d a r y [Fig. 6.9 (p. 78)], t h a t is t o say there is a positive
surface
energy.
Norma l
Superconductin
g
Numbe r o f
superelectron
s
(a ) Penetratio n dept h a n d coherenc e rang e
Fre e
energ y
densit y
(c ) Tota l fre e energ y
FIG . 12.1. Negative surface energy; coherence range less than penetration depth.
(Compare this with Fig. 6.9.)
T h e relative values of t h e coherence length î and t h e penetration
d e p t h ë vary for different materials. I n m a n y alloys and a few p u r e
metals the coherence range is greatly reduced, as w a s pointed out in
C h a p t e r 6. A similar a r g u m e n t t o t h a t used in § 6.9 shows that, if t h e
186
INTRODUCTION TO SUPERCONDUCTIVITY
coherence length is shorter t h a n t h e penetration d e p t h , t h e surface
energy is negative, as Fig. 12.1 illustrates, and therefore such a supercond u c t o r will b e t y p e - I I . I n m o s t p u r e m e t a l s t h e coherence length h a s a
value £ of a b o u t 10~ c m . T h i s is considerably greater t h a n the p e n e t r a tion depth, which is about 5 x 10~ cm, so in such m e t a l s t h e surface
energy is positive and they are t y p e - I . A reduction in t h e electron m e a n
free p a t h , however, reduces t h e coherence length and increases t h e
penetration d e p t h (§ 6.9, § 2.4.1). I m p u r i t i e s in a metal reduce t h e electron m e a n free p a t h , and in an i m p u r e metal or alloy the coherence range
can easily b e less t h a n t h e p e n e t r a t i o n d e p t h . Alloys or sufficiently impure metals are, therefore, usually t y p e - I I superconductors.
4
0
6
12.2.
Th e Mixe d Stat e
W e have seen t h a t it m a y be energetically favourable for superconductors w i t h a negative surface energy b e t w e e n n o r m a l a n d superconducting
regions t o g o into a mixed state w h e n a m a g n e t i c field is applied. T h e
configuration of the normal regions t h r e a d i n g t h e superconducting
material should b e such t h a t t h e ratio of surface t o volume of normal
material is a m a x i m u m . It t u r n s o u t t h a t a favourable configuration is
one in which the superconductor is t h r e a d e d b y cylinders of normal
material lying parallel to t h e applied m a g n e t i c field (Fig. 12.2). W e shall
refer t o these cylinders as normal cores. T h e s e cores a r r a n g e themselves
in a regular p a t t e r n , in fact a triangular close-packed lattice (Fig. 12.2
and
frontispiece).
FIG . 12.2. T h e mixed state.
W e might expect the n o r m a l cores t o h a v e a very small r a d i u s because
the smaller t h e r a d i u s of a cylinder t h e larger t h e r a t i o of its surface area
to its volume. T h e picture of t h e mixed state w h i c h e m e r g e s from these
THE MIXED STATE
187
considerations is as follows. T h e bulk of t h e material is diamagnetic, the
flux due to the applied field being opposed b y a diamagnetic surface
current which circulates around the perimeter of the specimen. T h i s
diamagnetic material is threaded b y normal cores lying parallel to t h e
applied magnetic field, and within each core is magnetic flux having t h e
same direction as t h a t of the applied magnetic field. T h e flux within each
core is generated b y a vortex of persistent current t h a t circulates around
the core w i t h a sense of rotation opposite to t h a t of the diamagnetic surface current. ( W e saw in § 2.3.1 t h a t any normal region containing
magnetic flux and enclosed b y superconducting material m u s t be encircled b y such a current.) T h e p a t t e r n of c u r r e n t s and t h e resulting flux
are illustrated in Fig. 12.3.
FlG. 12.3. T h e mixed state, showing normal cores and encircling supercurrent
vortices. T h e vertical lines represent the flux threading the cores. T h e surface current
maintains the bulk diamagnetism.
T h e vortex current encircling a normal core interacts w i t h t h e
magnetic field produced b y t h e vortex current encircling any other core
and, as a result, any t w o cores repel each other. T h i s is s o m e w h a t
similar to the repulsion b e t w e e n t w o parallel solenoids or b a r m a g n e t s .
Because of this m u t u a l interaction the cores threading a superconductor
in the mixed state do not lie at r a n d o m b u t arrange themselves into a
regular periodic hexagonal a r r a y t as shown in Fig. 12.3. T h i s array is
usually k n o w n as the fluxon lattice. T h e existence of t h e normal cores
and their arrangement in a periodic lattice h a s been revealed b y t w o
experimental techniques. T h e decoration technique of E s s m a n n and
t Occasionally a square lattice may be formed, but this is very uncommon and only occurs
under special circumstances.
188
INTRODUCTION TO SUPERCONDUCTIVITY
T r a u b l e reveals t h e p a t t e r n of t h e n o r m a l cores b y allowing very small
(500 A) ferromagnetic particles t o settle on t h e surface of a t y p e - I I
superconductor in the mixed state. T h e particles locate themselves
w h e r e t h e m a g n e t i c flux is strongest, i.e. w h e r e t h e n o r m a l c o r e s intersect t h e surface. T h e resulting p a t t e r n can t h e n b e e x a m i n e d b y elect r o n microscopy. T h e frontispiece s h o w s the p a t t e r n of n o r m a l cores in
t h e mixed state revealed b y t h i s m e t h o d . An alternative m e t h o d ,
developed b y Cribier, Jacrot, R a o and F a r n o u x , m a k e s use of n e u t r o n
diffraction. N e u t r o n s , because of their m a g n e t i c m o m e n t , interact w i t h
magnetic fields. T h e regular a r r a n g e m e n t of c u r r e n t vortices in t h e
mixed state p r o d u c e s a periodic m a g n e t i c field w h i c h a c t s as diffraction
grating, and scatters a b e a m of n e u t r o n s shone t h r o u g h t h e specimen
into preferential directions given b y t h e B r a g g law. O b s e r v a t i o n of the
directions into w h i c h t h e b e a m of n e u t r o n s is scattered s h o w s t h a t the
cores are arranged in a hexagonal periodic array.
12.2.1.
Detail s of th e m i x e d stat e
T h e picture of t h e m i x e d s t a t e w e h a v e j u s t given, w i t h t h i n cylindrical normal cores t h r e a d i n g t h e superconducting material, is a good
enough a p p r o x i m a t i o n for m a n y purposes, b u t it d o e s n o t accurately
describe t h e details of t h e structure. F o r o n e thing, t h e cores are not
sharply defined. W e saw in § 6.9 t h a t t h e r e c a n n o t b e a s h a r p b o u n d a r y
between a superconducting and a n o r m a l region; t h e t r a n s i t i o n is spread
out over a distance w h i c h is roughly equal t o t h e coherence r a n g e î.
F u r t h e r m o r e , t h e m a g n e t i c flux associated w i t h each core s p r e a d s into
t h e surrounding material over a distance a b o u t equal t o t h e p e n e t r a t i o n
d e p t h ë.
A detailed analysis of t h e free energy of t h e mixed state, w h i c h w e
shall not a t t e m p t here, s h o w s t h a t t h e n o r m a l cores should h a v e an
exceedingly small radius. H o w e v e r , as t h e size of a n o r m a l cylinder is
reduced so t h a t it a p p r o a c h e s î it b e c o m e s progressively m o r e difficult to
define its r a d i u s exactly o n account of t h e diffuseness of t h e b o u n d a r y .
Because it is not possible t o define exactly t h e v o l u m e and surface area of
such a small core, w e c a n n o t properly divide its free energy into distinct
volume and surface c o n t r i b u t i o n s , and w e m u s t consider i t s free energy
as a whole. A detailed consideration of free energy gives t h e following
structure for t h e mixed state.
189
THE MIXED STATE
(a )
(B)
Â
(c )
X
FIG . 12.4. Mixed state in applied magnetic field of strength just greater than H .
(a) Lattice of cores and associated vortices, (b) Variation with position of concentration of superelectrons. (c) Variation of flux density.
el
T h e properties of the material vary w i t h position in a periodic
manner. T o w a r d s the centre of each vortex the concentration n of
superelectrons falls to zero, so along the centre of each vortex is a very
thin core (strictly a line) of normal material (Fig. 12.4b). T h e d i p s in t h e
superelectron concentration are about t w o coherence-lengths wide. T h e
flux density d u e to the applied magnetic field is not cancelled in the normal cores and falls to a small value over a distance about ë away from
the cores (Fig. 12.4c). T h e total flux generated at each core by the encircling current vortex is j u s t one fluxon (see § 11.2).
s
W e shall n o w confirm that, w h e n a magnetic field is applied to a t y p e II superconductor, the appearance of cores of the form w e have j u s t
described does result in a lowering of the free energy. At each core t h e
n u m b e r n of superelectrons decreases and energy m u s t b e provided t o
split u p the pairs. As an approximation we m a y think of each core as
equivalent to a cylinder of normal material w i t h r a d i u s î. T h e
appearance of a normal core will therefore result in a local increase in
per unit length of core due t o the decrease in
free energy of ðî .\ì ÇÀ
electron order. However, over a radius of about ë the material is not
s
2
0
190
INTRODUCTION TO SUPERCONDUCTIVITY
diamagnetic so there is a local decrease in magnetic energy a p p r o x i m a t e per unit length, w h e r e H is the strength of the
ly equal to ðë .\ì ÇÀ
applied field. If there is t o b e a net reduction in free energy b y t h e formation of such cores, w e m u s t have
2
0
a
ðî\\ ÇÀ
ìï
< ðë ë ì Ç .
2
2
2
0
(12.1)
According to this relation, if the mixed state is to appear in applied fields
less t h a n H (a necessary condition, o t h e r w i s e an applied field would
drive the whole superconductor into t h e n o r m a l state before t h e mixed
state could establish itself), w e m u s t have î < ë. T h i s is the same condition as t h a t w e derived for negative surface energy. So, as predicted by
the simple a r g u m e n t s on page 184, the mixed state is p r o d u c e d b y the
application of a m a g n e t i c field to s u p e r c o n d u c t o r s w h i c h would h a v e
negative surface energy b e t w e e n superconducting and n o r m a l regions.
c
12.3.
Ginzburg-Landa u Constan t of Metal s an d Alloys
L e t u s w r i t e t h e ratio of the penetration d e p t h ë to t h e coherence
length î as a p a r a m e t e r ê:
ê =
ë/î.
ê varies for different superconductors and is k n o w n as the G i n z b u r g - L a n d a u c o n s t a n t of the m a t e r i a l . t It is an i m p o r t a n t p a r a m e t e r
because its value d e t e r m i n e s several p r o p e r t i e s of t h e s u p e r c o n d u c t o r ;
for example, according to the considerations of t h e previous sections, a
superconductor is type-I or type-II d e p e n d i n g on w h e t h e r its value of ê
is less or greater t h a n unity.
A m o r e detailed t r e a t m e n t t h a n w e have given s h o w s t h a t t h e sign of
the surface energy and t h e possibility of t h e formation of a mixed state
depends, strictly, not on w h e t h e r the ê of the material is less or greater
t h a n unity, b u t on w h e t h e r ê is less or greater t h a n 1/^2:
ê < 0-71 surface energy positive (type-I),
ê > 0-71 surface energy negative (type-II).
+ T h e constant ê appears in the theoretical treatment of superconductivity by Ginzburg and
Landau, which is an extension of the London treatment and explicitly includes the surface energy
between normal and superconducting regions. In the G i n z b u r g - L a n d a u treatment ê is defined by
where Ö is the q u a n t u m of magnetic flux (see § 11.2). If the electron mean
ê = (^2)2ééëì Ç ./Ö
free path is very short, ë increases and so ê is large in alloys. For our purposes, however, we can
consider ê to be the ratio of the penetration depth to the coherence range.
2
0
(
0
0
191
THE MIXED STATE
T h i s correct critical value of ê is, however, not very different from t h e
value of unity w e obtained by simple considerations.
It w a s pointed out in § 12.1 that in alloys and impure metals the
coherence range is shorter t h a n in pure metals; consequently ê can have
a large value and these superconductors are usually type-II. It is,
however, possible for even pure metals to be type-II superconductors. It
can be shown t h a t superconductors with high transition t e m p e r a t u r e s
can be expected to have relatively short coherence ranges and, in fact,
three superconducting metals (niobium, vanadium, and technetium) h a v e
ê greater t h a n 0-71 even in the absence of impurities. T h e s e are called
intrinsic
type-II superconductors. P u r e niobium, v a n a d i u m , and
technetium have ê values of 0-78, 0-82 and 0-92, respectively.t
However, pure metals are usually type-I and alloys are usually t y p e - I I .
T h e G i n z b u r g - L a n d a u constant ê of a superconductor which contains
impurities is related to its resistivity in the normal state because the
scattering of electrons by the impurities shortens the coherence range î
and also increases the normal resistivity p. F o r a given metal, therefore,
ê increases with the normal state resistivity.
12.4.
12.4.1.
L o w e r a n d U p p e r Critica l F i e l d s
L o w e r critica l
field,
H
cl
W e have seen t h a t when a magnetic field is applied to a type-II superconductor it m a y be energetically favourable for it to go into the mixed
state whose configuration h a s been described in previous sections.
However, a certain m i n i m u m strength of applied field is required to
drive a type-II superconductor into the mixed state. T h i s can be seen by
examination of (12.1), which gives the condition for the free energy to b e
lowered by the appearance of the mixed state. F o r a given value of î
relative to A (remembering that, in a type-II superconductor, î < ë) , w e
see that H m u s t be greater t h a n a certain fraction of H . Therefore a
certain m i n i m u m strength of applied field is required to drive a type-II
superconductor into the mixed state, and this is k n o w n as the lower
critical field, H . W e can get an approximate value for H from eqn.
a
c
cl
cl
+ In fact, for intrinsic type-II superconductors and dilute alloys ê varies slightly with
temperature, increasing as the temperature falls. In vanadium, for example, ê at the transition
temperature (5-4°K) is 0 82 but rises to 1 - 5 at 0°K. Similarly the alloy P b . T l . has a ê-valu e of
0-58 at its transition temperature, 7 2°K, and so is type-I, but on cooling to 4 3°K the ê-valu e
rises to 0 71, so below this temperature the alloy is type-II.
99
01
192
INTRODUCTIO N
TO
SUPERCONDUCTIVIT Y
(12.1) ( a p p r o x i m a t e because eqn. (12.1) w a s based on a simplified model
of a core). F r o m this equation we see that the mixed state will b e
energetically favoured if the strength of the applied field exceeds Ç î/ë,
i.e.
€
H
Clearly the value of H
H /K.
~
CL
C
relative to H
cl
c
decreases as t h e value of ê
increases.
12.4.2.
Uppe r critica l
field,
H
c2
In the previous section w e h a v e shown t h a t if a gradually increasing
magnetic field is applied to a type-II superconductor it goes into the
mixed state at a "lower critical field" H w h i c h is less t h a n H . N o w , in
a type-I superconductor, H is the field strength at which the magnetic
free energy of the superconductor has been raised to such an extent t h a t
it b e c o m e s energetically favourable for it to go into the n o r m a l state. A
type-II superconductor in the mixed state has, however, in an applied
field, a lower free energy t h a n if it w e r e type-I and perfectly d i a m a g n e t i c .
Consequently we m a y expect t h a t a magnetic field stronger t h a n H m u s t
be applied to drive a type-II superconductor normal. ( T h i s is similar to
the a r g u m e n t used in § 8.1 to show t h a t the critical field of a thin superconductor is greater t h a n t h e critical field of t h e bulk material.) F u r t h e r more, we m a y note that an a r g u m e n t similar t o that on p. 189 s h o w s that
in fields above H the mixed state can have a lower free energy t h a n the
completely normal state. T h e h i g h magnetic field s t r e n g t h u p t o w h i c h
the mixed state can persist is called the upper critical field, H .
cl
c
c
c
c
c2
At the lower critical field strength H a type-II s u p e r c o n d u c t o r goes
from the completely superconducting state into the mixed state and a
lattice of parallel cores is formed. As the strength of the applied magnetic
field is increased above H the cores pack closer together and, because
each core is associated w i t h a fixed a m o u n t of flux, the average flux density  in the superconductor increases. At a sufficiently high value of
applied magnetic field the cores merge together and the m e a n flux density in the material d u e to the cores and the diamagnetic surface current
of the applied m a g n e t i c field (Fig.
approaches the flux density ì Ç
12.5). At the u p p e r critical field H t h e flux density b e c o m e s equal to
ì Ç
and the material goes into the normal state.
cl
cl
0
á
c2
0
á
TH E MIXE D
193
STAT E
Â
÷
FIG. 12.5. Mixed state at applied magnetic field strength just below H .
C2
W e have n o w seen that, whereas type-I superconductors can exist in
one of t w o states, superconducting or normal, type-II superconductors
can be in one of three states, superconducting, mixed or normal. T h e
phase d i a g r a m s of the t w o types are compared in Fig. 12.6. In a type-II
superconductor, the larger the value of ê the smaller will b e H b u t the
larger will be H relative to the critical field H .
cX
c2
12.4.3.
c
T h e r m o d y n a m i c c r i t i c a l field , H
c
In C h a p t e r 4 w e saw that for a type-I superconductor the critical field
h a s a value given by
(12.2)
where (g — g ) is the difference in free energy densities of the normal
and superconducting states in the absence of an applied magnetic field.
W e may define the critical field for all types of superconductor b y m e a n s
of (12.2), which can apply equally to type-I and type-II, since for each
superconductor there must be, in the absence of an applied magnetic
field, a characteristic energy difference (g — g ) between the completely
superconducting and completely normal states. H is a m e a s u r e of this
energy difference and, to distinguish it from the upper and lower critical
fields, it may be called the thermodynamic
critical field. Only in a type-I
n
s
n
s
c
194
INTRODUCTIO N T O
SUPERCONDUCTIVIT Y
Norma l
Ç
Ï
Ô
Typ e
Norma l
Ç
Ô
ô
Ï
1
ô
Typ e Ð
F i g . 12.6 . Phase diagrams of type-I and type-II
superconductors.
superconductor does the material b e c o m e normal in a field strength
equal to H .
c
W h e n , in describing a type-II superconductor, w e say t h a t the upper
or lower critical fields have certain values relative to the t h e r m o d y n a m i c
critical field H we may loosely think of H as being a b o u t the critical
field that would be characteristic of an equivalent type-I superconductor,
i.e. one w i t h the same transition t e m p e r a t u r e . t A s will b e seen in
§ 12.5.1, the value of H for a type-II superconductor can nevertheless be
determined indirectly from its experimentally m e a s u r e d m a g n e t i z a t i o n
curve.
C9
c
c
12.4.4.
Valu e of th e uppe r critica l
field
It can be s h o w n that for a t y p e - I I superconductor the u p p e r critical
field h a s a value
H
c2
(12.3)
= ( /2)êÇ ,
í
so materials with a high value of ê r e m a i n in t h e mixed state and d o not
go normal until strong m a g n e t i c fields are applied.
T h e ability of type-II s u p e r c o n d u c t o r s w i t h high values of ê t o resist
being driven normal until strong magnetic fields are applied is of considerable technical importance, especially in the construction of super+ We saw in Chapter 9 that the BCS theory of superconductivity predicts a law of corresponding states for different superconductors, from which it follows that superconductors with the
same transition temperature should have the same value of H at any temperature. T h i s law of
corresponding states is fairly well obeyed in practice.
C
195
THE MIXED STATE
conducting solenoids to generate strong magnetic fields. W h e r e a s at
4-2°K type-I superconductors have critical fields of only a few t i m e s
10 A m " (i.e. a few h u n d r e d gauss), type-II superconductors can have
upper critical field strengths exceeding a million A m " . S o m e typical
examples are s h o w n in T a b l e 12.1.
4
1
1
T A B L E 12.1.
UPPE R CRITICA L F I E L D H
C2
AT 4 2 ° K , ê-VALUE
AND
T R A N S I T I O N TEMPERATUR E O F SOM E Ô Õ É ¸ - I I ALLOY S COMPARE D
T O LEA D (TYPE-I )
i/ (4-2°K )
c2
ê
A rrr
(Mo -R e
Ti -Nb
3
Type-I I
2
I Nb Sn
3
6-7 ÷ 10
- 8 X 10
~ 1 · 6 ÷ 10
6
7
c
12.4.5.
4-4 ÷ 10
Pb
8,400
-100,000
-200,000
5
H (4
Type- I
T (°K )
C
Gaus s
1
4
20
34
10
9
18
2°K)
550
4
0-4
7-2
Paramagneti c limi t
T h e question m a y b e asked, if ê is m a d e indefinitely large, is there any
limit t o the strength of magnetic field required to drive a type-II superconductor normal? T o answer this question, consider a material with a
high transition t e m p e r a t u r e and a large value of ê . At t e m p e r a t u r e s well
below the transition t e m p e r a t u r e the t h e r m o d y n a m i c critical field will b e
fairly high and so at these temperatures, according to (12.3), w e should
have a very large value of H . F o r example, at 1 2°K an alloy of 60
atomic per cent t i t a n i u m and 40 atomic per cent niobium is predicted to
have an u p p e r critical field strength H of about 20 ÷ 10 A m " . Experim e n t shows, however, t h a t for such materials with a very high predicted
value of H > resistance in fact r e t u r n s at a considerably w e a k e r field. I n
the case of the titanium—niobium alloy, the normal state is restored by a
magnetic field of strength 10 x 10 A m , about half the predicted value
oiH .
cl
6
1
c2
cl
6
- 1
cl
T h i s reduced critical field h a s been ascribed to p a r a m a g n e t i s m arising
from the spins of t h e conduction electrons. A magnetic field applied to a
normal metal t e n d s to align parallel to itself the spins of the electrons
196
INTRODUCTIO N T O
SUPERCONDUCTIVIT Y
near the F e r m i level (Pauli p a r a m a g n e t i s m ) . F o r m o d e r a t e m a g n e t i c field
strengths the degree of alignment and the resulting lowering of the free
energy is small, and until now w e have neglected it, considering a normal
metal to be non-magnetic. But in very strong m a g n e t i c fields there m a y
be a considerable reduction of m a g n e t i c free energy if t h e spins of the
electrons align parallel to the applied field. S u c h an alignment is,
however, incompatible w i t h superconductivity, w h i c h requires t h a t in
each Cooper pair the spins of t h e t w o electrons shall be anti-parallel.
Consequently, in a sufficiently strong m a g n e t i c field it m a y be
energetically favourable for the metal to go into the normal p a r a m a g n e t i c
state with electrons near to the F e r m i level aligned parallel to the field,
rather than to remain superconducting w i t h electrons in anti-parallel
pairs.
A calculation b y Clogston suggests that as a result of the electron
p a r a m a g n e t i s m a superconductor m u s t go into the normal state if the
applied field strength exceeds a strength H
equal to about 1-4 X
10 T A NRR . Consequently the mixed state of a type-II superconductor
cannot persist in fields above this value, no m a t t e r h o w large the value of
ê.
T h e same limitation should apply to the increased critical field of a
very thin specimen of type-I superconductor. E q u a t i o n (8.8) implies t h a t
the critical field of a thin film could be m a d e indefinitely high if the film
were thin enough. In fact, however, the critical field will not increase
above the value H .
It m u s t be r e m e m b e r e d t h a t this effect of the normal electron
p a r a m a g n e t i s m is only i m p o r t a n t in s u p e r c o n d u c t o r s w i t h a very high
or more. T h e effect of electron
upper critical field, say 5 ÷ 1 0 A m
p a r a m a g n e t i s m is negligible in s u p e r c o n d u c t o r s to which w e only apply
a relatively weak magnetic field.
P
6
Y
1
c
P
6
- 1
T o s u m m a r i z e : subject to the considerations which have been discussed
in the previous p a r a g r a p h s , the values of the lower and u p p e r critical
fields for a type-II superconductor w i t h G i n z b u r g - L a n d a u constant ê
are given approximately by
H
C2
-
ê
KH .
C
TH E MIXE D
12.5.
197
STAT E
Magnetizatio n of Type-I I Superconductor s
W e n o w examine the magnetic properties of type-II superconductors.
At applied magnetic field strengths H below H , a type-II superconductor behaves exactly like a type-I superconductor, exhibiting perfect
diamagnetism and a magnetization equal to —H (Fig. 12.7). W h e n t h e
a
cl
a
I
Â
.
Applie d magneti c fiel d
Applie d magneti c fiel d
*-
+
»-
FiG. 12.7. Magnetization of a type-II superconductor.
applied field strength reaches H
normal cores with their associated
vortices form at the surface and pass into the material. T h e flux
threading the vortices is in the same direction as that d u e to the applied
magnetic field, so t h e flux in t h e material is no longer equal to zero and
the m a g n i t u d e of the magnetization suddenly decreases (Fig. 12.7). I n
fields between H a n d H the n u m b e r of vortices which occupy the sample
is governed by the fact t h a t vortices repel each other. T h e n u m b e r of normal cores per unit area for a given strength of applied magnetic field is
such that there is equilibrium between the reduction in free energy of t h e
material d u e t o t h e presence of each non-diamagnetic core and t h e
existence of t h e m u t u a l repulsion between t h e vortices. As the strength
of t h e applied field is increased, the normal cores pack closer together, so
the average flux density in the material increases and the m a g n i t u d e of
the magnetization decreases smoothly with increasing H . N e a r to the
upper critical value H the flux density and magnetization change linearly with applied field strength. At H there is a discontinuous change in
the slope of t h e flux density and magnetization curves, and above H the
material is in t h e normal state with flux density equal to ì Ç
and zero
magnetization.
cl9
cl
c2
a
c2
c2
c2
0
á
198
INTRODUCTIO N T O
SUPERCONDUCTIVIT Y
W e pointed out in § 4.1 that t h e total area enclosed b y t h e m a g n e t i z a tion curve is always equal to the difference b e t w e e n G a n d G , i.e. to
\ì ÇÀí,
and this r e m a i n s true for a type-II superconductor. In Fig. 12.8
n
s
0
I
FIG . 12 .8. Illustration of thermodynamic critical field H of type-II superconductor.
T h e dotted right-angled triangle is drawn to have an area equal to the shaded area
within the magnetization curve.
C
the relationship b e t w e e n H and the t h e r m o d y n a m i c critical field is illustrated. T h e ratio b e t w e e n H and H is such t h a t t h e area enclosed b y
the dashed curve equals t h e area enclosed by the m a g n e t i z a t i o n curve of
the type-II superconductor.
c2
c7
12.5.1.
c
Determinatio n of ê
T h e value of ê of a type-II superconductor can b e d e t e r m i n e d if a
magnetization curve has been obtained. If the area u n d e r t h e m a g n e t i z a tion curve is measured, the ratio of H to H can b e deduced b y use of
the construction shown in Fig. 12.8. F o r m u l a (12.3) then gives ê =
F u r t h e r m o r e , it h a s been s h o w n t h a t t h e slope of t h e
(\/^2)(H /H ).
magnetization curve w h e r e it c u t s the applied field axis at H is given b y
cl
c2
c
c
c2
dl
dH.
-1
1 16(2ê — 1)'
2
so we can also d e t e r m i n e ê from the slope of the m e a s u r e d m a g n e t i z a tion curve. Note, however, that these procedures to d e t e r m i n e ê are only
valid if the magnetization is reversible (i.e. the s a m e curve is traced in
b o t h increasing and decreasing fields). As w e shall see in t h e next section, the magnetization is often not reversible, and the greater t h e degree
of hysteresis the less accurate are the values of ê d e t e r m i n e d from the
magnetization curve.
199
T H E M I X ED S T A T E
12.5.2.
Irreversibl e magnetizatio n
If a type-II superconductor is perfectly h o m o g e n e o u s in composition,
its magnetization is reversible, i.e. t h e curves in Fig. 12.7 are t h e s a m e
whether t h e applied field H is increased from zero or decreased from
some value greater than H . Real samples, however, usually show some
irreversibility in their magnetic characteristics (Fig. 12.9). Irreversibility
a
c2
Applie d magneti c fiel d
—
Applie d magneti c fiel d
FIG. 12.9. Irreversible type-II magnetization curves.
is attributed t o t h e fact t h a t t h e normal cores which thread t h e supercond u c t o r in t h e mixed state, can b e " p i n n e d " t o imperfections in t h e
material and so are prevented from being able t o m o v e freely.
Consequently, o n increasing t h e applied field strength from zero there is
no sudden entry of flux at H because t h e cores formed at t h e surface are
hindered from moving into t h e interior. Similarly, on reducing the
applied field strength from a value greater t h a n H , there is a hysteresis,
and flux m a y b e left permanently t r a p p e d in t h e sample, because some of
the normal cores are pinned and cannot escape. I t a p p e a r s t h a t almost
any kind of imperfection whose dimensions are as large or larger than
the coherence length c a n pin normal cores. F o r example, b o t h the long
chains of lattice faults, called dislocations, a n d particles of chemical i m purities, such a s oxides, c a n give rise t o magnetic irreversibility. B u t
type-II materials which are very carefully prepared a n d purified so as t o
be free of such defects can show very little irreversibility a n d c a n exhibit
magnetization curves almost as " i d e a l " as those in Fig. 12.7. I n general,
however, specimens contain a n u m b e r of imperfections and their
magnetization curves exhibit some irreversibility and permanently
trapped flux. W e shall see in t h e next chapter that t h e pinning of normal
cores b y imperfections plays a very i m p o r t a n t part in determining t h e
critical c u r r e n t s of type-II superconductors.
cl
c2
INTRODUCTION TO SUPERCONDUCTIVITY
200
12.6.
Specifi c Hea t of Type-I I Superconductor s
In § 5.2 w e saw t h a t if a type-I s u p e r c o n d u c t o r is h e a t e d t h e r e is, in
general, a sudden c h a n g e in the specific heat as the metal goes from the
superconducting into t h e normal s t a t e ; the m a g n i t u d e of this c h a n g e
being given b y (5.4). If the heating is d o n e in a c o n s t a n t applied
the transition is of t h e first order * and latent h e a t is
magnetic field H
absorbed.
and
A type-II superconductor, however, h a s t w o critical fields, H
H . O n heating a sample in a c o n s t a n t applied m a g n e t i c field H (e.g.
from A t o  in F i g . 12.10) w e m a y expect to observe t w o c h a n g e s in t h e
specific heat, first at T and t h e n at T .
ay
cX
c2
a
2
fiel d strengt h
x
Magn e
ï
H
a
A
iN.
\
Nj
ï
ô,
Â
\
iSl
ô
Temperatur e
2
*-
FIG . 12.10. T y p e - I I superconductor heated in a magnetic field.
At t e m p e r a t u r e T the metal passes from the superconducting to
mixed state. It can b e seen from Fig. 12.11 t h a t this transition is
associated w i t h a large n a r r o w peak in the specific heat curve. According
to the Abrikosov model of a type-II superconductor, the m a g n e t i z a t i o n
curve at H h a s an infinite slope (see Fig. 12.7), from which it can b e
shown that at T the e n t r o p y is c o n t i n u o u s b u t h a s an infinite
t e m p e r a t u r e derivative in the mixed state. T h e lack of discontinuity in
the e n t r o p y leads to a second-order transition and t h e infinite
t e m p e r a t u r e derivative should p r o d u c e a " ë - t y p e " specific heat anomaly.
T h e sharp peak observed at T in Fig. 12.11 is consistent w i t h such a
x
cX
x
x
201
THE MIXED STATE
ë-typ e specific heat anomaly. At T the specimen passes from the mixed
to normal state. According t o t h e description of the mixed state given in
§ 12.4.2 w e would expect that at T the n a t u r e of the mixed state a p proaches t h a t of the normal state. W e d o not therefore expect any
sudden increase in entropy as t h e metal is heated t h r o u g h T b u t expect
t h a t the entropy of the mixed state will rise t o w a r d s t h a t of the normal
state as the t e m p e r a t u r e is raised t o w a r d s T . H e n c e t h i s transition
should also b e of second order, and w e expect at T a sudden d r o p in t h e
specific heat similar to t h a t observed in type-I superconductors in t h e
absence of a magnetic field. It can b e seen in F i g . 12.11 t h a t at T such a
specific heat d r o p is indeed observed.
2
2
2
2
2
2
Temperatur e ( K )
e
FIG. 1 2 . 1 1 . Specific hea t of type-I I superconducto r (niobium ) measure d in a constan t
applie d magneti c field. (Base d on McConvill e an d Serin. )
CHAPTER
CRITICAL
OF TYPE-II
13
CURRENTS
SUPERCONDUCTORS
THE CRITICAL c u r r e n t s of t y p e - I I s u p e r c o n d u c t o r s are of considerable
practical interest. W e have m e n t i o n e d previously t h a t electromagnets
capable of generating strong m a g n e t i c fields c a n b e w o u n d from w i r e s of
type-II superconductors, and clearly t h e m o r e c u r r e n t t h a t can b e passed
t h r o u g h t h e w i n d i n g s of such an electromagnet w i t h o u t resistance
appearing t h e stronger will b e t h e m a g n e t i c field t h a t can b e generated
w i t h o u t heat being produced.
In C h a p t e r 7 w e saw that, provided the specimen is considerably
larger t h a n t h e p e n e t r a t i o n depth, t h e critical c u r r e n t of a type-I superconductor is successfully predicted b y Silsbee's hypothesis, i.e. if t h e
resistance is to r e m a i n zero, t h e total m a g n e t i c field s t r e n g t h at t h e surface, due t o t h e current and applied m a g n e t i c field together, m u s t not
exceed H . T h e situation in t y p e - I I s u p e r c o n d u c t o r s is, however, m o r e
complicated, because t h e state of t h e material c h a n g e s at t w o field
strengths, H and H , not at a single field s t r e n g t h H .
c
cl
c2
c
It should be pointed out t h a t at present (1977) t h e b e h a v i o u r of
c u r r e n t s in t y p e - I I s u p e r c o n d u c t o r s is b y n o m e a n s fully u n d e r s t o o d .
Consequently w e shall only discuss r a t h e r general aspects of t h e c u r r e n t carrying capacity and w e shall not t r y t o present any detailed t r e a t m e n t ,
because present ideas are almost b o u n d t o b e modified b y future
developments.
13.1.
Critica l Current s
In a magnetic field w h o s e strength is less t h a n H a t y p e - I I superconductor is in t h e completely superconducting state and b e h a v e s like a
type-I superconductor, w h e r e a s in field s t r e n g t h s greater t h a n H it goes
into the mixed state. W e shall see later that, c o n t r a r y to w h a t one m i g h t
expect, t h e mixed state does not necessarily have zero resistance. W e
might guess, therefore, t h a t , so long as t h e applied m a g n e t i c field is not
cl
cl
202
203
CRITICAL CURRENTS OF TYPE-II SUPERCONDUCTORS
by itself strong enough to drive the material into the mixed state, t h e
critical current should b e determined b y a criterion like Silsbee's rule
(§ 7.1) for a type-I superconductor, b u t w i t h H substituted for H \ i.e.
the metal will b e resistanceless so long as t h e magnetic field generated b y
the transport current does not bring the total field at t h e surface to a
value above H .
E x p e r i m e n t s show t h a t for weak applied magnetic field strengths this
modified Silsbee's rule is obeyed, b u t only in t h e case of extremely
perfect samples, i.e. those w i t h reversible magnetization curves. In the
case of a field applied at right angles to a wire of pure type-II superconductor, the critical current falls linearly w i t h increasing field strength, as
would be expected (compare curve a Fig. 13.1 and Fig. 7Ab, p. 84).
b u t there
However, the critical current does not fall to zero at \H
remains a small critical current; and even above H , w h e r e t h e applied
field is b y itself strong enough to drive the metal into the mixed state, a
type-II superconductor can still carry some resistanceless current. C u r v e
a on Fig. 13.1 s h o w s this small critical current extending u p to about
cl
c
cl
cly
cl
H
c 1
——Mixe d stat e
·-
H
c ;
Applie d transvers e magneti c field ,
FIG.
H —» a
13.1. Typical variation of critical currents of wires of (a) highly perfect,
(b) imperfect type-I I superconductors in transverse applied magnetic field.
H . M o s t samples are not, however, extremely perfect, and for such imperfect samples t h e critical current is increased considerably b o t h above
cl
204
INTRODUCTION TO SUPERCONDUCTIVITY
and below H (curve b, Fig. 13.1).
In t h i s c h a p t e r w e shall b e chiefly concerned w i t h t h e critical c u r r e n t s
w h e n t h e applied m a g n e t i c field is perpendicular t o t h e c u r r e n t flow.
T h e r e is particular interest in t h i s configuration b e c a u s e it is t h e o n e w h i c h
occurs in electromagnets. I n a solenoid, for example, t h e field g e n e r a t e d
is perpendicular to t h e coil windings. T h e critical c u r r e n t curves of F i g .
13.1 refer t o t h i s situation. T h e shape of c u r v e b w i t h a plateau e x t e n d ing u p t o about H , is characteristic of t y p e - I I s u p e r c o n d u c t o r s w h i c h
are imperfect a and is q u i t e unlike t h a t w h i c h would b e predicted b y any
form of Silsbee's rule. It is found t h a t w h e n a s u p e r c o n d u c t o r is in t h e
mixed state its critical c u r r e n t is almost completely controlled b y t h e
perfection of t h e m a t e r i a l ; t h e m o r e imperfect t h e material t h e greater is
the critical current, i.e. t h e higher and m o r e p r o n o u n c e d is t h e plateau
(Fig. 13.1). A highly imperfect wire m a y b e able t o carry a b o u t
10 A m m " of its cross-section. Conversely, a r a t h e r perfect specimen
h a s a very small critical current, p e r h a p s a few t e n s of m i c r o a m p s per
m m , w h e n it is in t h e mixed state. I t is extremely i m p o r t a n t to u n d e r stand that t h e critical current of a t y p e - I I s u p e r c o n d u c t o r in t h e mixed
state is entirely d e t e r m i n e d b y imperfections and i m p u r i t i e s and not b y
any form of Silsbee's rule. T h i s d e p e n d e n c e of t h e critical c u r r e n t on t h e
perfection of t h e material is of considerable technical i m p o r t a n c e b e c a u s e
superconducting electromagnets require resistanceless w i r e of h i g h
as t h e a p current-carrying capacity. If Silsbee's rule applied, w i t h H
propriate magnetic field, t h e critical c u r r e n t s w o u l d b e o r d e r s of
m a g n i t u d e greater t h a n those w h i c h are actually found.
cl
}
c2
3
2
2
c2
It s o m e t i m e s h a p p e n s t h a t at h i g h applied m a g n e t i c field s t r e n g t h s t h e
critical current increases w i t h increasing field strength, rising t o a
m a x i m u m near H . T h i s is k n o w n as t h e " p e a k effect". H o w e v e r , t h e
reason for t h e occasional a p p e a r a n c e of t h i s effect is not yet u n d e r s t o o d ,
and w e shall not consider it further in t h i s book.
c2
13.2.
Flo w Resistanc e
Before a t t e m p t i n g an explanation of w h a t d e t e r m i n e s h o w m u c h
resistanceless c u r r e n t can flow t h r o u g h a t y p e - I I s u p e r c o n d u c t o r w h e n it
is in t h e mixed state, w e m u s t d r a w a t t e n t i o n to an i m p o r t a n t
experimental observation. S u p p o s e w e take a length of w i r e of t y p e - I I
t W h e n speaking of the perfection of a material we mean the lack of both "chemical" impurities (i.e. foreign atoms) and "physical" impurities (i.e. faults in the periodic arrangement of
the atoms in the crystal lattice).
205
CRITICA L C U R R E N T S O F TYPE-I I S U P E R C O N D U C T O R S
superconductor and apply perpendicular t o it a magnetic field H of
sufficient strength to drive t h e material into the mixed state. W e n o w
p a s s a current along the wire and observe h o w the voltage V developed
between t h e e n d s varies as the m a g n i t u d e i of the current is altered (Fig.
13.2). So long as t h e current is less t h a n the critical value i n o voltage is
a
c
Current ,
i
FIG . 13.2. Voltage—current characteristic of type-I I superconductor
< H < H ).
magnetic field ( H
C L
A
in
transverse
CL
observed along t h e wire, b u t w h e n the current is increased above i 2L
voltage appears which, at c u r r e n t s s o m e w h a t greater t h a n i approaches
a linear increase w i t h increasing current. It should b e noted t h a t t h e
voltage developed is considerably less t h a n t h a t w h i c h would b e
observed if the wire were in t h e normal state. F i g u r e 13.3 s h o w s t h e
c
cf
>
0
i (A)
c
i (B )
c
i (C )
c
Curren t
FIG . 13.3. Voltage-current characteristics of three wires of the same type-I I
superconductor in the mixed state in the same transverse applied magnetic field.
Curves A, B, and C refer to specimens which are progressively less perfect.
206
INTRODUCTION TO
SUPERCONDUCTIVITY
v o l t a g e - c u r r e n t characteristics, m e a s u r e d at t h e s a m e s t r e n g t h of
applied magnetic field, of three w i r e s of equal d i a m e t e r of t h e s a m e t y p e II superconductor b u t of different degrees of perfection. T h e critical
current is different for each wire, the p u r e r or m o r e perfect wires having
lower critical c u r r e n t s ; b u t the slope of t h e characteristic is the s a m e for
all three specimens. W e see, therefore, that, t h o u g h t h e value of t h e
critical current of a specimen d e p e n d s o n the perfection of the material,
t h e r a t e at which voltage a p p e a r s w h e n t h e critical current is exceeded is
an innate characteristic of t h e particular material and d o e s not d e p e n d
on h o w perfect it is.
T h e value of the slope dV/di which the characteristic a p p r o a c h e s at
c u r r e n t s well above i is k n o w n , for r e a s o n s w h i c h will b e c o m e a p p a r e n t
later, as t h e flow resistance R of the specimen. T h e flow resistivity p' of
c
>
>
'c 3
'c 2
'd
Current, i
FIG . 1 3 . 4 . Effect of applied magnetic field strength on V-i characteristic of a type-I I
superconductor in mixed state in a transverse magnetic field (H < H < H < # < H ).
cX
{
2
3
cl
the material from w h i c h t h e specimen is m a d e m a y b e defined b y R —
p'l/s/,
w h e r e / is t h e length of t h e specimen and si its cross-sectional
area. I t is found t h a t , for a given s t r e n g t h of applied m a g n e t i c field, t h e
flow resistivity is proportional to t h e n o r m a l resistivity of the metal.
F u r t h e r m o r e , t h e flow resistivity increases w i t h increasing s t r e n g t h of
applied magnetic field (Fig. 13.4), approaching t h e normal resistivity as
the applied field strength a p p r o a c h e s i / .
1
c 2
13.3.
13.3.1.
Flu x Flo w
Lorent z forc e an d critica l curren t
W e h a v e seen t h a t a type-II s u p e r c o n d u c t o r in t h e mixed state is able
CRITICAL CURRENTS OF TYPE-II SUPERCONDUCTORS
207
to carry some resistanceless current and t h a t the critical current cannot
be determined b y any modification of Silsbee's rule. F u r t h e r m o r e , t h e
m a n n e r in which voltage appears w h e n the critical current is exceeded is
quite different from the case of a type-I superconductor. W e now ask
w h a t determines t h e m a g n i t u d e of the critical current of a type-II superconductor in t h e mixed state, and w h a t is the source of t h e voltage w h i c h
appears at c u r r e n t s greater t h a n the critical current.
T h e current-carrying properties of t y p e - I I superconductors can b e
qualitatively explained if it is supposed t h a t w h e n a current is passed
along a type-II superconductor, which h a s been driven into the mixed
state by an applied magnetic field, t h e current flows not j u s t at the surface, as in a type-I superconductor, b u t throughout the whole body of the
metal
Consider a length of type-II superconductor in an applied transverse
magnetic field of strength greater t h a n H
(Fig. 13.5). If a current is
cl
FIG. 13.5. T y p e - I I superconductor carrying current through the mixed state. For
stationary cores the Lorentz force F is perpendicular both to the axes of the cores and
to the current density J.
L
passed t h r o u g h this specimen, there will b e at every point a certain
t r a n s p o r t current density J. ( T h e " t r a n s p o r t c u r r e n t " is the current
flowing along a specimen. W e use the t e r m t o distinguish t h i s m o t i o n of
the electrons from the circulating vortex c u r r e n t s around the cores.)
208
INTRODUCTION
TO
SUPERCONDUCTIVITY
H o w e v e r , because the metal is in t h e mixed state, it is t h r e a d e d b y the
magnetic flux associated w i t h the normal cores. T h e r e will therefore b e
an electromagnetic force ( L o r e n t z force) b e t w e e n t h i s flux and t h e
current. W h e n speaking of a L o r e n t z force b e t w e e n a m o v i n g charge and
a magnetic field o n e is in fact referring t o t h e m u t u a l force w h i c h exists
b e t w e e n t h e moving charge and t h e source of the m a g n e t i c field. I n t h i s
case, therefore, t h e force acts b e t w e e n t h e electrons carrying t h e
t r a n s p o r t c u r r e n t and the vortices generating t h e flux in t h e cores. H e n c e
there will b e o n each vortex a L o r e n t z force Fi acting at right angles t o
b o t h the direction of t h e t r a n s p o r t c u r r e n t and to t h e direction of t h e
flux.
S u p p o s e t h e specimen is of length /, cross-sectional area sJ and carries
a current i in an applied m a g n e t i c field which is at an angle è to t h e
direction of the c u r r e n t . If the flux density of t h e field is  t h e L o r e n t z
force on t h e specimen is UB sin È. H o w e v e r , since each vortex encloses
an a m o u n t of flux Ö , t h e m e a n flux density is  = ç Ö , w h e r e ç is t h e
n u m b e r of vortices per unit area perpendicular to B and t h e L o r e n t z
force is therefore lin Ö sin È. T h e total length of all t h e vortices
threading t h e specimen is nl V , so t h e m e a n force per unit length of
vortex is (é"/</) Ö sin È. T h o u g h the c u r r e n t density varies b e t w e e n t h e
cores, the average current density J equals i/s/, so the L o r e n t z force on
unit length of each vortex can b e seen to b e
0
0
y
0
0
F = JQ> sin È.
L
0
(13.1)
In the special case w h e r e t h e applied m a g n e t i c field is perpendicular to
the direction of the current, è = 90° and the force is j u s t
FL=J*O-
(13.2)
In t h e previous c h a p t e r w e have seen t h a t t h e cores tend to b e pinned at
imperfections in the material. Consequently, if the L o r e n t z force is not
too great, t h e cores r e m a i n stationary and d o not m o v e u n d e r its action.
( T h e electrons w h i c h carry the t r a n s p o r t c u r r e n t c a n n o t m o v e s i d e w a y s
in t h e opposite direction, b e c a u s e t h e r e can b e n o c o m p o n e n t of c u r r e n t
across the specimen.) N o t every individual core is directly pinned to t h e
material, b u t the interaction b e t w e e n t h e vortex c u r r e n t s is sufficient to
give the lattice of cores a certain rigidity, so t h a t if only a few cores are
pinned the whole p a t t e r n is immobilized. W h a t m a t t e r s , therefore, is t h e
average pinning force per core. L e t t h e average pinning force per unit
length of core b e F . So long as t h e t r a n s p o r t c u r r e n t density J p r o d u c e s
p
CRITICAL CURRENTS OF TYPE-II SUPERCONDUCTORS
a L o r e n t z force per unit length of core which is less t h a n F
tice will n o t move, i.e. there will b e a stable situation if
py
209
t h e core lat-
If, however, t h e t r a n s p o r t current is increased, so t h a t t h e L o r e n t z force
exceeds F , t h e core lattice is n o longer prevented from moving t h r o u g h
the specimen. If t h e cores are set in m o t i o n , ! and if there is some viscous
force opposing their motion t h r o u g h t h e metal, work m u s t b e d o n e in
maintaining this motion. T h i s work can only b e supplied b y t h e
transport current, and so energy m u s t b e expended in driving this
current t h r o u g h t h e material. I n other w o r d s , if t h e current sets t h e cores
into motion, and if their motion is impeded, there will be a voltage drop
along the material. T h i s motion of t h e cores (and t h e fluxons they contain) t h r o u g h t h e material is k n o w n as "flux flow" and is t h e source of
the flow resistivity observed at c u r r e n t s greater than t h e critical current.
p
T h i s motion of t h e cores w h e n t h e current exceeds t h e critical current
h a s been observed directly b y t h e effect it h a s on n e u t r o n diffraction
from t h e mixed stated (see p . 188).
T h e m e c h a n i s m s producing t h e viscous force that opposed t h e m o t i o n
of t h e cores t h r o u g h t h e metal are complicated and w e shall n o t discuss
t h e m here. O n e contribution t o this viscous drag arises because t h e cores
contain magnetic flux and, as each core moves, this flux m o v e s w i t h it
t h r o u g h t h e metal. T h i s flux m o t i o n induces an e.m.f. w h i c h drives a
current across t h e core, t h e current returning via t h e surrounding superconducting material. T h e s e c u r r e n t s m a y b e t h o u g h t of as eddy c u r r e n t s
set u p b y t h e flux motion. Since t h e core is normal, work is dissipated in
driving t h e current across it, and this is one reason w h y energy m u s t b e
provided t o keep t h e cores in motion.
It should b e stressed t h a t t h e situation w i t h regard t o t h e voltage is
different in type-I and type-II superconductors. I n a type-I superconductor, if t h e critical current is exceeded, t h e voltage is d u e to t h e t r a n s p o r t
current flowing t h r o u g h normal regions which span t h e whole specimen.
W h e n flux flow is occurring in a type-II superconductor, t h e material is
still in t h e mixed state and there are still c o n t i n u o u s superconducting
p a t h s threading t h e whole specimen.
T h e critical current will b e t h a t which creates j u s t e n o u g h Lorentz
t When the cores are moving the forces acting on them are different. T h e velocity and direction of the core motion will be discussed in § 13.3.2.
ö Schelten, Ullmaier and Lippmann, Phys. Rev. Â 1 2, 1772 (1975).
I.T.S.—Ç
210
INTRODUCTION TO SUPERCONDUCTIVIT Y
force t o d e t a c h t h e cores from t h e pinning centres, i.e. t h e critical c u r r e n t
density J
c
will b e given b y
7c%
=
FP-
W e can n o w see w h y t h e m o r e imperfect s p e c i m e n s have higher critical
c u r r e n t s : if there are m a n y imperfections, a g r e a t e r fraction of cores will
be pinned t o t h e material and t h e m e a n pinning force per core will b e
greater.
I n t h e previous c h a p t e r w e saw t h a t t h e presence of pinning c e n t r e s
gives rise t o irreversible m a g n e t i z a t i o n of imperfect t y p e - I I supercond u c t o r s . If t h e above explanation of t h e critical c u r r e n t is correct, t h e
critical current in t h e mixed state should b e greater in t h o s e m a t e r i a l s
w h i c h have m o r e irreversible m a g n e t i z a t i o n curves. T h i s indeed is found
t o b e t h e case. F i g u r e 13.6 s h o w s m a g n e t i z a t i o n and critical c u r r e n t
curves of a wire of t y p e - I I s u p e r c o n d u c t o r (an alloy of t a n t a l u m and
Ï
0
IXIO
2XIQ
3XI0
Strengt h of opplie d moqneH c field (A/m)
5
5
5
IXIO
2XIQ
3XI0
Strengt h of applie d magneti c field (A/m)
5
5
FIG . 1 3 . 6 . Magnetization and critical current of imperfect (a) and nearly perfect
(b) type-II superconductor (tantalum-niobium alloy measured at 4 2 ° K ; Heaton and
Rose-Innes).
5
CRITICAL CURRENTS OF TYPE-II
211
SUPERCONDUCTORS
niobium) before and after it h a d been purified.! T h e t o p t w o curves
show the magnetization and critical current of the wire after it had been
d r a w n from an ingot and so contained m a n y imperfections due to the
drawing process. T h e magnetization curve is very irreversible and there
is a plateau of high critical current extending to H . T h e lower curves
show the properties of the same piece of wire after it had been carefully
purified and the imperfections eliminated by heating for several d a y s in a
very good v a c u u m . It can be seen t h a t the magnetization h a s b e c o m e
almost perfectly reversible and, t h o u g h the mixed state still persists u p
t h e high current-carrying capacity h a s been lost.
to H
cl
c2l
_
10
H = 7 2 ÷ 10 A/ry
6
X
a
å
<
å
<
/
05
J
-20 - ÉÏ
0
é
"Ñ- - 9 - -Q—at- -r H - o ^ - o - l w i
10 2 0
Orientatio n
ot
30 40
50
60
70
80
90
I00
applie d magneti c fiel d
0
(0)
(a )
F l G . 13.7.
Variatio n of critica l curren t of N b S n stri p in mixed stat e wit h orientatio n
3
of applie d magneti c field. (After G. D. Cod y an d G. W . Cullen , RC A
Laboratories. )
If the critical current is t h a t current which produces a L o r e n t z force
strong enough to m o v e the vortices off any pinning centres, w e should
expect it to depend markedly on the angle b e t w e e n the magnetic flux and
the current. According to (13.1), if è is the angle the applied magnetic
field m a k e s w i t h t h e current, t h e critical current should b e inversely
proportional to sin È. Figure 13.7 shows the results of an experiment to
test this relationship. I t can b e seen that, except w h e n t h e applied field is
nearly parallel to the current, the inverse of the critical current does vary
t It ma y be notice d tha t man y of th e experiment s used t o illustrat e th e propertie s of type-I I
superconductor s hav e bee n performe d on tantalum-niobiu m or niobium-molybdenu m alloys.
Thi s is becaus e thes e alloys ar e tw o of th e few type-I I superconductor s whic h can be prepare d in
a reall y pur e stat e so tha t the y show reversibl e magnetization . Thoug h ther e ar e man y type-I I
superconductin g alloys, mos t of thes e cannot , for metallurgica l reasons , be produce d in ver y
perfec t form .
212
INTRODUCTION TO SUPERCONDUCTIVITY
as sin è in accordance w i t h t h e L o r e n t z force model. If t h e inverse of t h e
critical current were to vary as sin è for all values of 0, the critical
current would t e n d to infinity as t h e direction of the applied m a g n e t i c
field is r o t a t e d t o lie parallel t o t h e c u r r e n t . Clearly the critical c u r r e n t
cannot b e infinite, and w h e n t h e field is parallel t o t h e wire t h e critical
current h a s a certain m a x i m u m value. T h e factors w h i c h limit t h e
critical c u r r e n t is parallel applied m a g n e t i c fields are not yet, however,
properly u n d e r s t o o d .
13.3.2·
Flu x
flow
W e have associated t h e voltage, w h i c h a p p e a r s w h e n t h e t r a n s p o r t
c u r r e n t is increased above t h e critical value, w i t h t h e w o r k required t o
drive the cores t h r o u g h the metal. F o r a given c u r r e n t t h e voltage is independent of time, from w h i c h it m a y b e deduced t h a t t h e core lattice is
not accelerating u n d e r t h e forces w h i c h act o n it b u t m o v e s w i t h constant velocity. T h i s implies t h a t t h e metal b e h a v e s as t h o u g h it w e r e a
viscous m e d i u m as far as m o t i o n of t h e cores is concerned. W e h a v e seen
- . 0
(b )
FIG . 1 3 . 8 . Velocity and force diagrams for vortex motion through mixed state. v is
the velocity of the electrons carrying the transport current i, and í is the velocity of a
vortex through the material. T h e applied magnetic field is directed into the plane of
the page. (Based on Volger, Stass, and van Vijfeiken.)
t
CRITICAL CURRENTS OF TYPE-II SUPERCONDUCTORS
213
t h a t t h e electrical flow resistivity does not depend on the purity of the
superconductor, so to each material w e can ascribe a "viscosity cons t a n t " ç such t h a t , if there is a net force F per unit length acting o n a
core, the core will acquire a velocity õ = F/ç.
W e now consider w h a t m o t i o n t h e vortices take u p w h e n the t r a n s p o r t
current is raised above the critical value, so t h a t t h e vortices b e c o m e
detached from t h e pinning centres. It is i m p o r t a n t to realize that w h e n
t h e cores are in motion, t h e m a g n i t u d e and direction of t h e forces acting
on t h e m are different from w h e n they are held stationary in t h e material.
Consider the special case w h e n t h e applied magnetic field is perpendicular t o t h e direction of t r a n s p o r t current flow. If t h e cores are held
stationary in t h e metal, the relative velocity b e t w e e n the cores and t h e
electrons carrying the t r a n s p o r t current h a s a certain value, and, as w e
have seen, there is a L o r e n t z force tending t o detach t h e vortices from
their pinning sites. T h i s force is perpendicular b o t h to t h e t r a n s p o r t
current and t o t h e axis of the cores. W h e n , however, t h e cores h a v e been
set in motion, t h e r e is a different relative velocity b e t w e e n t h e m and t h e
electrons carrying t h e current, so t h e force acting o n t h e m is changed.
W e m u s t r e m e m b e r t h a t the vortices encircling the cores represent a circulatory motion of t h e superelectrons. T h i s m o t i o n is superimposed o n
the linear motion produced b y t h e t r a n s p o r t current, and in the absence
of any other forces the vortices would b e carried along b y the t r a n s p o r t
current w i t h t h e velocity v of its electrons (Fig. 13.8a). If this w e r e to
h a p p e n there would b e n o relative m o t i o n of t h e vortices and t r a n s p o r t
current electrons, and so there would b e n o L o r e n t z force o n t h e vortices.
However, as w e h a v e seen, t h e appearance of a voltage suggests t h a t
there is opposition t o t h e m o t i o n of t h e vortices t h r o u g h t h e metal, and
w e n o w show t h a t , as a result, the vortices acquire a c o m p o n e n t of
velocity at right angles to the t r a n s p o r t current.
t
Because of t h e viscous drag exerted b y t h e material of t h e metal on t h e
vortices they will not m o v e as fast as t h e t r a n s p o r t current electrons, b u t
only w i t h some lower velocity v . In other w o r d s , in t h e direction of the
t r a n s p o r t current there will b e a relative velocity b e t w e e n t h e vortices
and t h e electrons carrying t h e t r a n s p o r t current. Consequently there will
now b e a L o r e n t z force on t h e flux threading the vortices, and this gives
t h e m a c o m p o n e n t of velocity v sideways across t h e conductor. T h e
resultant velocity í of the vortices is therefore at an angle á to the direction of t h e t r a n s p o r t current (Fig. 13.8a).
W e can find t h e direction of t h e vortex m o t i o n b y the following selfconsistent a r g u m e n t . If in t h e steady state t h e vortices m o v e t h r o u g h the
t
t
214
INTRODUCTION TO SUPERCONDUCTIVITY
metal w i t h a velocity í w h i c h is c o n s t a n t , t h e forces o n t h e m m u s t
balance. T h e s e forces are illustrated in Fig. 13.8b. T h e metal e x e r t s a
viscous d r a g o n vortices m o v i n g t h r o u g h it, so t h e r e will b e a force F
acting in a direction opposite t o t h e vortex velocity v :
v
F = -çí.
(13.3)
v
N o w the electrons of t h e t r a n s p o r t c u r r e n t have a velocity (v, — v )
relative t o t h e core of t h e vortex and t h i s p r o d u c e s , at right angles t o
( /~~ )> t h e L o r e n t z force F w h i c h drives the vortices in t h e direction í
(see Fig. 13.8b). T h e m a g n i t u d e of F is given b y
v
v
L
L
F=
L
l
V /
- íÉç^Ö ,
(13.4)
0
w h e r e n is t h e n u m b e r of superelectrons per u n i t volume. F o r t h e velocity of t h e vortices to b e c o n s t a n t , F m u s t equal —F , w h i c h from (13.3)
and (13.4) gives
s
L
lv/ — vl =
v
•
and t h e angle á w h i c h t h e vortex m o t i o n m a k e s w i t h t h e t r a n s p o r t
c u r r e n t is
á = tan
- 1
T h i s s h o w s t h a t the greater the viscous d r a g t h a t t h e metal exerts o n
t h e vortices (i.e. t h e bigger t h e value of ç) t h e m o r e nearly will t h e vortices move at right angles t o t h e t r a n s p o r t current. M e a s u r e m e n t s o n
type-II s u p e r c o n d u c t o r s h a v e s h o w n t h a t á is close t o 90°. ( T h i s result
h a s been obtained from Hall effect m e a s u r e m e n t s . T h e Hall angle equals
90° — a.) T h e fact t h a t á is nearly 90° implies t h a t t h e viscous d r a g is
large so t h a t w h e n flux flow o c c u r s t h e vortices m o v e virtually at right
angles t o t h e direction of t h e t r a n s p o r t c u r r e n t .
13.3*3.
E.M.F . d u e t o cor e m o t i o n
In t h e previous sections w e h a v e seen t h a t a sufficiently strong c u r r e n t
é passed t h r o u g h a superconductor in t h e mixed state sets t h e cores into
motion, and w e h a v e supposed t h a t t h e metal e x e r t s a viscous d r a g on
cores moving t h r o u g h it. T h e energy required t o keep t h e cores moving
can only c o m e from t h e c u r r e n t a n d t h i s m e a n s t h a t , irrespective
215
CRITICAL CURRENTS OF TYPE-II SUPERCONDUCTORS
of t h e detailed m e c h a n i s m of t h e process, w o r k is required t o m a i n t a i n
Vbetween
t h e c u r r e n t ; in other w o r d s , t h e r e will b e a voltage difference
the e n d s of t h e specimen, and t h e specimen s h o w s resistance. If Ñ is t h e
power required t o keep t h e cores in motion, i.e. t h e power dissipated in
the specimen, t h e voltage will b e simply V = P/i.
T h e above a r g u m e n t a s t o w h y a voltage a p p e a r s at c u r r e n t s greater
t h a n t h e critical current is q u i t e general b u t gives n o insight into t h e
m e c h a n i s m b y w h i c h t h e voltage is generated. I n fact, t h e voltage m a y
be ascribed t o an induced e.m.f. generated b y t h e m o t i o n of t h e magnetic
flux in t h e moving cores. Consider, for example, t h e circuit s h o w n in Fig.
13.9, w h e r e Ì is a piece of type-II superconductor driven into t h e mixed
state b y a magnetic field applied perpendicular t o t h e plane of t h e page.
If t h e cores m o v e across t h e sample, a s t h e result of a L o r e n t z force d u e
t o a current passed t h r o u g h t h e sample or for any other reason, a voltage
V will b e recorded o n t h e voltmeter. T h i s voltage is induced b y t h e
m o t i o n of t h e magnetic flux contained in t h e cores. T h e r a t e at w h i c h
flux crosses b e t w e e n t h e c o n t a c t s t o t h e voltmeter is n<t> v d w h e r e ç is
the n u m b e r of cores per unit area, Ö is t h e magnetic flux w i t h i n each
core (i.e. t h e fluxon), v t h e transverse velocity of core motion, and d t h e
separation of t h e contacts. T h e voltage measured is
0
t
0
t
V = n<I> v d.
0
t
T h i s "induced v o l t a g e " is t h e s a m e voltage as t h e "resistive v o l t a g e " d u e
to t h e passage of t h e current u
Ì
FIG . 13.9. Generation of e.m.f. by flux flow in mixed state.
If it is t h e m o t i o n of t h e fluxons across a type-II superconductor
w h i c h produces t h e e.m.f. in t h e mixed state, an e.m.f. should appear
whenever t h e cores are set in m o t i o n , a n d t h e appearance of t h e e.m.f.
should n o t depend o n t h e cause of t h e motion. T h a t this is so h a s been
216
INTRODUCTION TO SUPERCONDUCTIVITY
d e m o n s t r a t e d b y Lowell, M u n o z , and Sousa, w h o w e r e able t o set t h e
fluxons in m o t i o n t h r o u g h a specimen of niobium—molybdenum alloy, in
w h i c h n o t r a n s p o r t c u r r e n t w a s flowing, b y h e a t i n g o n e e n d of t h e
specimen. If w e consider t h e v a r i a t i o n w i t h t e m p e r a t u r e of t h e
magnetization curve of a t y p e - I I s u p e r c o n d u c t o r w e see t h a t , in a u n i form applied m a g n e t i c field, t h e r e will b e a greater density of fluxons in
h o t t e r regions t h a n in colder ones. C o n s e q u e n t l y in a t e m p e r a t u r e
gradient, because of the m u t u a l repulsion of t h e fluxons, t h e r e will b e a
force driving t h e fluxons from t h e h o t t e r to t h e colder p a r t s . F i g u r e 13.10
s h o w s t h e a r r a n g e m e n t . W h e n a t e m p e r a t u r e difference w a s established
b e t w e e n t h e e n d s of t h e specimen, a voltage difference V appeared
b e t w e e n t h e t w o edges. Since n o t r a n s p o r t c u r r e n t w a s present, t h e
observed voltage c a n n o t h a v e any " o h m i c " source. T h e voltage changed
sign w h e n t h e direction of t h e applied m a g n e t i c field H w a s reversed.
T h i s experiment is t h e m o s t convincing d e m o n s t r a t i o n t h a t m o t i o n of
fluxons t h r o u g h t h e mixed state can g e n e r a t e an e.m.f. T h e e x p e r i m e n t
can only b e performed o n very perfect specimens, o t h e r w i s e t h e m o t i o n
of t h e fluxon p a t t e r n is prevented b y t h e pinning d u e t o imperfections.
a
W e can s u m u p t h e current-carrying b e h a v i o u r of t h e mixed state as
follows: it is supposed t h a t w h e n a c u r r e n t is passed t h r o u g h a t y p e - I I
superconductor in the mixed state t h i s c u r r e n t flows t h r o u g h o u t t h e
metal. T h i s c u r r e n t exerts a L o r e n t z force o n t h e cores w h i c h t h r e a d t h e
mixed state. T h e s e cores m a y b e anchored t o imperfections b u t if t h e
current exceeds a certain value, t h e critical c u r r e n t , t h e cores m a y b e
driven across t h e material. W h e n t h i s "flux flow" occurs a voltage
a p p e a r s perpendicular to t h e direction of flux m o t i o n and heat is
generated in the material.
Ç
Hot
FIG . 13.10. E . M . F . generated by fluxon motion due to a temperature gradient.
217
CRITICAL CURRENTS OF TYPE-II SUPERCONDUCTORS
It should b e m a d e clear t h a t at present (1977) t h e details
processes are b y no m e a n s fully understood. I n particular, it is
w h y imperfections pin the core lattice against motion, nor can
of the variation of critical current w i t h applied magnetic field
(e.g. Fig. 13.1) b e properly explained.
13.4.
of these
not clear
the form
strength
Surfac e Superconductivit y
It can be seen from Fig. 13.1 t h a t t h o u g h t h e critical current of a typeII superconductor falls rapidly w h e n t h e applied magnetic field strength
exceeds H , t h e material is nevertheless able to carry a small, t h o u g h
rapidly decreasing, resistanceless current in fields greater t h a n H . T h i s
is surprising, because above H the metal should b e in t h e normal state
and not able to carry any resistanceless current at all. F o r a n u m b e r of
years, a t t e m p t s w e r e m a d e t o explain this and similar anomalies by supposing that the material w a s inhomogeneous. I t ' m i g h t , for example, be
threaded by a network of regions w h o s e critical field w a s higher t h a n
that of the bulk material. It h a s recendy been realized, however, that
these " a n o m a l i e s " are a manifestation of a property w h i c h is possessed
even by perfectly p u r e and homogeneous superconductors. T h i s property
is t h a t of surface
superconductivity.
c2
c2
cl
3*H
J
H
C
2*H
C
c3
AC ÇÓ
Ï
ÉÏ
20
30
40
50
60
70
80
90
Orientation(^)o f applie d magneti c field (deg )
FIG. 13.11. Variation of H
E3
with angle of applied magnetic field to surface.
In 1963 St. J a m e s and D e G e n n e s deduced from theoretical considerations t h a t superconductivity can persist close to t h e surface of a
superconductor in contact with an insulator (including vacuum), even in
a magnetic field w h o s e strength is sufficient to drive the bulk material
normal. T h i s superconducting surface layer c a n occur in materials whose
218
INTRODUCTION TO SUPERCONDUCTIVITY
G i n z b u r g - L a n d a u p a r a m e t e r ê exceeds 0-42. Surface superconductivity
is not a special property of type-II s u p e r c o n d u c t o r s (ê > 0-71) b u t can
occur in any superconductor w h o s e ê -value exceeds 0-42. H o w e v e r ,
because its effects are usually observed in t y p e - I I superconductors, w e
have delayed its discussion till t h i s chapter.
T h e surface superconducting l a y e r t can persist in applied magnetic
fields u p t o a certain m a x i m u m s t r e n g t h w h i c h w e call H . T h e value of
H
d e p e n d s on t h e angle the applied m a g n e t i c field m a k e s w i t h t h e surface, and H
is a m a x i m u m w h e n t h e applied field is parallel to t h e surface. In t h i s case H
= 2 - 4 K H (i.e. 1-7 H
for a t y p e - I I s u p e r c o n d u c tor). If t h e angle t h e applied field m a k e s w i t h t h e surface is increased, t h e
decreases (Fig. 13.11) reaching a m i n i m u m value of ^/2KH
value of H
(i.e. H
in t h e case of a type-II superconductor) w h e n t h e field is perpendicular to t h e surface.
C L
C 3
C 3
C 3
C
C 2
C 3
C
C 2
Typ e I
Typ e Ð
1
Ln Ç
ê = 0 42
Ln ic
* = 0 71
*
FIG . 1 3 . 1 2 . Dependence of characteristics of superconductors on value of
G i n z b u r g - L a n d a u constant ê. H \ \ is value of limiting field strength for surface
superconductivity when field is parallel to surface.
E 3
W e are n o w in a position t o p r o d u c e a d i a g r a m (Fig. 13.12) which
s h o w s h o w t h e n a t u r e of s u p e r c o n d u c t o r s d e p e n d s o n t h e value of their
G i n z b u r g - L a n d a u p a r a m e t e r ê . F o r ê < 0-42 t h e superconductor is
type-I and can exist in o n e of t w o states, superconducting or normal,
t T h i s surface layer is often referred to as the superconducting surface "sheath". W e , however,
prefer to speak of the superconducting surface "layer" to emphasize that, as will be soon shown,
this layer may appear only on parts of the surface and does not necessarily enclose the specimen.
219
CRITICAL CURRENTS OF TYPE-II SUPERCONDUCTORS
depending on w h e t h e r the magnetic field strength is below or above the
t h e r m o d y n a m i c critical field H . However, if ê exceeds 0-42 a thin
superconducting layer can exist on the surface in fields u p to i / . W h e n
ê is greater t h a n 0-71 a superconductor is type-II and can exist in four
possible conditions—superconducting, mixed, normal w i t h surface
superconductivity, and completely normal.
It w a s pointed out in the previous chapter t h a n the ê of pure metals
varies slightly w i t h temperature, increasing as the t e m p e r a t u r e falls.
H e n c e it is possible for a metal to change its type of superconductivity if
the t e m p e r a t u r e is changed. Lead, for example, h a s a ê-valu e of about
0· 37 at 7· 2°K, b u t its ê-valu e increases to 0· 58 on cooling to 1 · 4°K. T h e
value of ê becomes equal to 0-42 at 5-8°K, so at t e m p e r a t u r e s below this
a superconducting surface layer can appear on a lead specimen.
c
c 3
Surface superconductivity occurs only at an interface b e t w e e n a superconductor and a dielectric (including vacuum), and does not occur at an
FIG . 13.13. Bands of surface superconductivity along a cylindrical specimen in a
transverse magnetic field.
interface between a superconductor and a normal metal. H e n c e , surface
superconductivity can b e prevented by coating the surface of a specimen
w i t h normal metal. W e can, for example, test if an observed behaviour is
due to surface superconductivity b y seeing w h e t h e r the behaviour
persists after the specimen h a s been copper-plated.
In general only part of the surface of a specimen is parallel to t h e
applied magnetic field, and consequently, in a magnetic field of strength
between H and H , the superconducting layer will only cover this part
of t h e surface. Figure 13.13 illustrates the i m p o r t a n t special case of a
cylindrical rod or wire in a transverse magnetic field. At an applied field
strength equal to H the superconducting layer extends right round the
c2
c 3
c2
220
INTRODUCTION TO SUPERCONDUCTIVITY
surface; b u t as the field s t r e n g t h is increased, this layer c o n t r a c t s into
t w o b a n d s A and A' r u n n i n g along t h e sides of t h e specimen. As t h e field
strength a p p r o a c h e s H these b a n d s contract t o zero along t h e t w o lines
LL and L'L' w h e r e t h e surface is parallel to t h e applied field. It is these
superconducting b a n d s w h i c h account for t h e small resistanceless
current t h a t a t y p e - I I superconductor can carry in applied field s t r e n g t h s
exceeding H .
c3
cl
CHAPTE R
HIGH-TEMPERATUR E
14
SUPERCONDUCTOR S
I N 1 9 8 7 it was discovered t h a t some m e t a l oxide ceramics w h i c h are
s e m i c o n d u c t o r s at r o o m t e m p e r a t u r e b e c o m e s u p e r c o n d u c t i n g at a
relatively h i g h t e m p e r a t u r e , a r o u n d 100°K. T h o u g h t h e t r a n s i t i o n
t e m p e r a t u r e of these ceramics is nearly 200°K below r o o m t e m p e r a t u r e
t h e y h a v e b e c o m e k n o w n as ' h i g h - t e m p e r a t u r e ' s u p e r c o n d u c t o r s
b e c a u s e t h e i r t r a n s i t i o n t e m p e r a t u r e s are m u c h h i g h e r t h a n t h e
' t r a d i t i o n a l ' m e t a l l i c s u p e r c o n d u c t o r s w i t h critical t e m p e r a t u r e s of
23°K or less.
E x a m p l e s of such h i g h t e m p e r a t u r e s u p e r c o n d u c t i n g ceramics are:
Y B a C u 0 (often called ' Y B C O ' ) w i t h T = 93°K, B i ( S r C a ) C u O
(often called ' B S C C O ' ) w i t h T = 110°K, and T l B a C a C u O (which
has the highest transition temperature, 125°K). T o show superconductivity the oxygen content of these ceramics m u s t be slightly less t h a n 7, 8 or
10 respectively (i.e. t h e r e m u s t be a small oxygen deficiency). T h e s e
superconducting ceramics are of extreme interest, b o t h from a scientific
p o i n t of view because there is as yet no acceptable explanation of h o w
superconductivity can occur at such high temperatures, and from a p r a c tical point of view because these temperatures can be achieved w i t h o u t
difficulty by t h e use of easily-available liquid nitrogen. A t the present
t i m e (1993), however, our u n d e r s t a n d i n g of t h e superconductivity of
these materials has b e e n h i n d e r e d because it has n o t b e e n possible to
p r o d u c e reasonably large single crystals w h i c h have s u p e r c o n d u c t i n g
properties. T h e fact t h a t the individual crystallites in a polycrystalline
sample have extremely anisotropic superconducting properties adds to
the difficulty in understanding t h e m .
2
3
7
c
c
2
2
2
2
2
3
2
s
1 0
T h e h i g h - t e m p e r a t u r e superconducting ceramics have a layered crystal
structure. I n general there are several neighbouring layers of copper oxide
separated from the next g r o u p of copper oxide layers by several layers
of o t h e r metal oxides ('isolation planes', see Fig. 14.1). T h e electrical
c o n d u c t i v i t y a n d s u p e r c o n d u c t i v i t y are a s s o c i a t e d w i t h t h e c o p p e r
oxide planes. For a given basic c o m p o u n d , t h e h i g h e r t h e n u m b e r of
221
222
INTRODUCTION TO SUPERCONDUCTIVITY
YI BA CU 0
2
3
7
FlG. 14.1. Structure of typical high-temperature superconducting ceramic.
neighbouring copper oxide planes (up to a m a x i m u m of 3 or 4), the higher
is the transition temperature.
T h e association of the superconductivity w i t h the copper oxide planes
is illustrated by t h e fact t h a t t h e s u p e r c o n d u c t i n g p r o p e r t i e s are little
altered by c h a n g i n g t h e c o m p o s i t i o n of t h e i n t e r m e d i a t e p l a n e s . For
example, if in Y B a C u 0 t h e y t t r i u m , w h i c h lies b e t w e e n t h e c o p p e r
oxide planes, is replaced by a n o t h e r rare earth t h e properties are scarcely
affected. I n particular, if the yttrium, w h i c h has n o magnetic m o m e n t , is
replaced by gadolinium, w h i c h has a large magnetic m o m e n t , the t r a n sition temperature is scarcely altered, although t h e introduction of m a g netic atoms into a superconductor normally causes a dramatic decrease in
the transition temperature (see § 9.3.8).
2
3
7
As w o u l d be expected from their h i g h transition t e m p e r a t u r e , these
ceramic superconductors have a very short coherence range, only a few
atomic spacings long, and a very deep penetration d e p t h . T h e y are, c o n sequently, extreme examples of t y p e - I I s u p e r c o n d u c t o r s . F u r t h e r m o r e ,
because the atoms are arranged in parallel planes, their superconducting
(and normal state) properties are very anisotropic, t h e superconducting
p r o p e r t i e s b e i n g s t r o n g e r in d i r e c t i o n s parallel t o t h e a t o m i c p l a n e s
Cafe -planes') t h a n in t h e p e r p e n d i c u l a r d i r e c t i o n ( t h e c - a x i s ) . F o r
example, it seems t h a t the coherence range parallel to the c-axis is about
0*2 n m b u t parallel to the á/3-plane s it is about 1*5 n m . A s a result the
G i n z b u r g - L a n d a u constant, ê , is highly anisotropic varying in m a g n i -
HIGH-TEMPERATURE SUPERCONDUCTORS
223
tude from a few tens parallel to the c-axis to a few h u n d r e d parallel to the
atomic planes. Properties such as critical magnetic fields a n d critical currents can be an order of t h e m a g n i t u d e greater w h e n measured parallel to
the atomic planes t h a n in the perpendicular direction.
T h e u p p e r critical m a g n e t i c field, H , can be extremely h i g h , as w e
would expect from the high value of ê. Just below the transition temperature the rate at which H increases w i t h decreasing temperature is about
1 0 A m " p e r degree w h e n t h e field is applied parallel t o t h e a t o m i c
planes, t h o u g h considerably less w h e n it is parallel to the perpendicular
direction. T h i s leads us to expect very high values of the upper critical field,
H at low temperatures. For Y B a C u 0 , the upper critical field at liquid
helium temperature (4*2°K) is estimated to be about 5 × 10 A m " w h e n
the magnetic field is applied parallel to the copper oxide planes and about
1 × 1 0 A m " w h e n t h e field is in t h e perpendicular direction. T h e s e
magnetic field strengths are too high to be achievable with existing m a g nets but pulsed magnetic field experiments have shown that H is indeed
greater t h a n 8 × 10 A m " .
A p a r t from any scientific interest, t h e value of t h e critical current of
superconductors is i m p o r t a n t for practical applications. A t low, liquid
helium, temperature the critical current density of superconducting ceramic
is less t h a n that of the best conventional superconductors if any applied
magnetic field is weak. However, the resistanceless current-carrying capacity
of t h e ceramics scarcely decreases w h e n t h e m a g n e t i c field s t r e n g t h is
increased, so t h a t in h i g h magnetic fields their critical current is m u c h
greater than that of conventional superconductors (Fig. 14.2).
c2
c2
7
1
c2y
2
3
7
8
8
1
1
c2
7
1
Magneti c fiel d (Teslas )
FlG . 14.2. Effect , at low temperature , of magneti c field on critica l curren t of super conductin g cerami c (Bi Sr Ca Cu O ) compare d wit h bes t 'conventional ' supercon ducto r (Nb Sn ) (afte r Sat o eta/.).
2
2
2
3
10
3
224
INTRODUCTIO N T O SUPERCONDUCTIVIT Y
A t 'high' temperatures (i.e. at a b o u t 77°K, the t e m p e r a t u r e of liquid
nitrogen, where conventional superconductors are n o t superconducting)
the ceramic superconductors have disappointingly low critical currents.
T h e r e appear to be several reasons for this: in the bulk material composed
of many crystallites the current must cross m a n y boundaries between the
crystallites a n d each of these acts as a w e a k link; f u r t h e r m o r e in some
crystallites the current may be constrained to flow perpendicularly to the
copper oxide planes. T h e critical current of long lengths can be considerably increased by treatments w h i c h align the crystallites so that the planes
of copper oxide lie parallel to the axis of the wire or tape. Indeed, in tapes
of B S C C O which have been processed so t h a t the crystallites are aligned
with their copper oxide planes roughly parallel to the plane of the tape, the
interfaces between crystallites no longer act as weak links, and the critical
current is determined by fluxon p i n n i n g within the crystallites. However,
even in single crystals the critical current is low, because thermal excitation
enables t h e fluxons to d e t a c h t h e m s e l v e s from t h e p i n n i n g sites (see
§ 13.3). Furthermore, it seems that at these relatively high temperatures
the fluxon lattice loses its rigidity so that those fluxons which are not t h e m selves p i n n e d can move u n d e r t h e L o r e n t z force exerted by a transport
current.
T h e ability of t h e fluxons in h i g h - t e m p e r a t u r e s u p e r c o n d u c t o r s to
Because of t h e p i n move is revealed in the p h e n o m e n o n of
flux creep.
c
ï
X 0
c
CO
Applied
magnetic
field
Ideal type-II
superconductor
FlG. 14.3. Flux creep. If the applied magnetic field is increased from zero to a strength
H' and then held steady at that value, the negative magnetization slowly decreases, as
shown by the vertical line, showing that fluxons are entering into the sample. Similarly, if
the magnetic field is decreased from a high value to some lower value H" the magnetization slowly decreases as fluxons leave the sample.
HIGH-TEMPERATURE SUPERCONDUCTORS
225
ning centres, these type-II superconductors have irreversible m a g n e t i z ation (see § 12.5.2), b u t if the applied magnetic field is changed the n e w
m a g n i t u d e of t h e magnetization slowly decays towards its equilibrium
value (see Fig. 14.3), this flux-creep m o t i o n decaying logarithmically
w i t h time. T h i s shows t h a t fluxons can enter into or leave the specimen
in spite of the pinning.
W h y superconductivity appears in these materials at such a high t e m p e r a t u r e is n o t at p r e s e n t (1993) u n d e r s t o o d . Q u a n t u m interference
e x p e r i m e n t s (see C h a p t e r 11) s h o w t h a t t h e s u p e r c u r r e n t is c a r r i e d
by electron pairs, as in conventional metallic superconductors, b u t there
is as yet n o general agreement as to the m e c h a n i s m w h i c h causes this
pairing. T h e r e is some experimental evidence for an energy gap, b u t it is
n o t yet clear w h e t h e r a B C S - l i k e m e c h a n i s m is r e s p o n s i b l e for t h e
superconductivity.
APPENDI X A
TH E
DENSIT Y
SIGNIFICANC E
 AND
TH E
OF TH E
MAGNETI C
MAGNETI C
FIEL D
FLU X
STRENGT H
Ç
IT IS i m p o r t a n t for a p r o p e r appreciation of t h e m a g n e t i c properties of
superconductors t h a t t h e reader should h a v e a sound u n d e r s t a n d i n g of
the significance of  and H. T h i s is especially so in view of t h e fact t h a t
the M K S system involves a r a t h e r different a p p r o a c h to m a g n e t i s m t h a n
the mixed e.s.u.—e.m.u. system.
A . 1.
Definition of Â
T h e m o d e r n a p p r o a c h is to a b a n d o n t h e concept of free m a g n e t i c
poles and instead t o discuss m a g n e t i s m entirely in t e r m s of the interaction b e t w e e n c u r r e n t s . F r o m t h i s point of view, t h e m a g n e t i c flux density
vector  is t h e basic m a g n e t i c q u a n t i t y , in t h e sense t h a t in electrostatics
the fundamental q u a n t i t y is t h e electric field vector E . If a c o n d u c t o r
carrying a current i is placed in a magnetic field a force will act on the
conductor d u e to the presence of the field. T h e m a g n e t i c flux density Â
of t h e field is defined b y the relationship:
d¥ = tdl x Â,
(A.l)
where d¥ is the force on a current element tdl. T h i s is analogous to the
definition of Šas the force on unit charge, and establishes  as t h e
fundamental property.
M a g n e t i c fields are generated by currents, and the flux density
resulting from a given geometrical a r r a n g e m e n t of c u r r e n t s in free space
can b e calculated from t h e B i o t - S a v a r t l a w :
d B = ^ ± ,
(A.2)
w h e r e dB is the c o n t r i b u t i o n t o t h e flux density at a point Ñ d u e to a
current element idl, r is t h e distance of Ñ from t h e element dl t t h e unit
vector in t h e direction of r, and ì t h e permeability of free space. T h i s
9
0
226
APPENDI X A
227
expression for  corresponds to the expression
for the electric field in free space due to a charge dq å being the permittivity of free space. It can be shown from the B i o t - S a v a r t law that in
free space  satisfies the Ampere circuital law
9
)Â.Ë
0
// Ë
=
(A.4)
0
where the line integral j> Â . dl is taken round any closed p a t h and J
the net current linking t h a t p a t h .
is
It can easily b e shown by applying the Amp£re circuital law to an infinitely long solenoid t h a t the flux density inside it is uniform and given
by
 = ì ôçß,
(A.5)
0
where m is the n u m b e r of t u r n s per unit length and i the current through
each of them. F u r t h e r m o r e , the flux density outside the solenoid is zero.
A.2.
T h e Effec t o f M a g n e t i c M a t e r i a l
All the foregoing equations w i t h the exception of (A.l) apply only in
free space. T o introduce the effect of magnetic material, consider a long
cylinder of paramagnetic material inside an infinitely long solenoid, as
shown in Fig. A . l . P a r a m a g n e t i s m arises because the material contains
Solenoi d (m i Am" )
/
1
Imaginar y surfac e current s
(I A m " )
1
FIG . A.l. Rod of magnetic material in infinite solenoid.
integration (A.7).
ABCD
path of
within it elementary atomic dipoles, and these dipoles tend to be aligned
by the field of t h e solenoid, so t h a t they point predominantly in the direction of the field. R e m e m b e r i n g t h a t the atomic dipoles are due not to free
228
INTRODUCTIO N T O SUPERCONDUCTIVIT Y
magnetic poles, b u t to small circulating c u r r e n t s w h i c h arise either from
electron spin or from t h e orbital m o t i o n of electrons w i t h i n the a t o m s , it
will b e seen t h a t these circulating c u r r e n t s are as s h o w n in Fig. A.2a
w h e n viewed parallel to t h e axis of the solenoid. Because of the aligning
influence of t h e field, all these c u r r e n t s circulate in the s a m e sense. T h e
degree of magnetization of t h e material can be described b y specifying
its intensity of magnetization
(usually called simply its " m a g n e t i z a t i o n " )
I, which is a vector pointing in t h e direction of magnetization and
having a m a g n i t u d e equal to t h e resultant m a g n e t i c dipole m o m e n t per
unit volume.
T h e total flux density w i t h i n t h e cylinder is n o w the resultant of t h e
flux density due to the solenoid and t h a t d u e to t h e atomic c u r r e n t s .
(a )
(b )
FlG. A.2. Equivalence of aligned current dipoles and surface current (viewed in
direction of field). T h e atomic current loops which generate the magnetic dipoles all
circulate in the same direction due to the aligning action of the field, as in (a). T h e
average flux density produced by these current loops within the material is the same
as would be produced by imaginary surface currents of density / A m " , as in (b).
1
Clearly  will not b e uniform w i t h i n t h e m a g n e t i c material b u t will fluct u a t e from point t o point w i t h t h e periodicity of t h e atomic lattice. B u t
there will b e an average value of B , and it can b e s h o w n t t h a t if t h e
magnetic material h a s a m a g n e t i z a t i o n I this average value is exactly t h e
same as would b e produced b y fictitious c u r r e n t s flowing a r o u n d the
periphery of t h e cylinder in planes perpendicular t o the axis, as in Fig.
A.2b, and having a surface density of IA m . T h e m a g n e t i z a t i o n of t h e
p a r a m a g n e t i c material is therefore equivalent to an imaginary solenoid
carrying a current of / A m and t h e additional flux density produced b y
this solenoid is, in accordance w i t h (A.5), given b y
_ 1
- 1
B
m
=
t See, for example, A. F. Kip, Fundamentals
1962.
ì É.
0
of Electricity and Magnetism, McGraw-Hill,
229
APPENDI X A
T h e flux density due to this imaginary solenoid simply a d d s to the flux
density produced by the real solenoid, so the m a g n i t u d e of the total flux
density within the material is given b y
 = ì ôçß + ì É.
0
(A.6)
0
T h e r e are t w o conventions in the literature about the dimensions of / .
M o s t books on electromagnetic theory give / the dimensions of amperes
per metre, as w e have done here. Books dealing with the magnetic
properties of solids, however, often describe their m a g n e t i s m in t e r m s of
a magnetic polarization
/ ' , a magnetic flux density w h i c h the sample
adds onto the flux density of t h e applied field. So à h a s t h e same d i m e n sions as B and (A.6) becomes
f
 = ì ôçé + / ' .
(A.6a)
0
T h e r e are a r g u m e n t s in favour of each convention, b u t the definition of /
embodied in (A.6) as a magnetic field strength contributed by the sample
seems to be m o r e in keeping with the parallelism between m a g n e t i s m
and electrostatics, and we have adopted it in this book.
A.3.
T h e Magnetic Field Strength
If we take the line-integral of  around the p a t h ABCD in Fig. A . l ,
where AB = CD = x, w e find from (A.6), remembering t h a t 5 = 0 outside the solenoid,
 . dl = (ja mi + ì É)÷.
0
(A.7)
0
ABCD
W e can write this as
= ì (./ +
§Â.ÜÉ
0
J ),
/
(A.8)
m
where J — xmi is the total current t h r o u g h the t u r n s of the solenoid
linking ABCD, and J
= Ix is the total effective surface current which
is equivalent to the magnetization J. J is often referred to as the "free"
current. H e n c e (A.4) only r e m a i n s valid if we identify . / w i t h J +
J.
T h i s is not a useful relationship, however, because although we know J
w e d o not in general know « / . It is therefore convenient to introduce a
new vector, called the magnetic field strength H , which is defined by
f
r
m
f
f
m
f
m
 = ì Ç + ì1
0
so that (A.6) gives
H=
0
mi
(A.9)
(A.10)
230
INTRODUCTIO N T O SUPERCONDUCTIVIT Y
and (A.7) b e c o m e s
<J)H . dl — xmi =
(A.l 1)
H e n c e A m p e r e ' s circuital law for Ç involves only the real or "free"
current J and is independent of the presence of t h e magnetic material.
It is this i m p o r t a n t result w h i c h m a k e s Ç a useful q u a n t i t y . It is an u n fortunate feature of the M K S system t h a t t h e fundamental distinction
between  and H as exemplified b y (A.8) and ( A . l l ) , t e n d s to b e
obscured b y the presence of a dimensional factor ì in (A.8) w h i c h is
absent in ( A . l l ) .
F o r m a n y materials (but not iron) it is found t h a t the m a g n e t i z a t i o n /
is proportional to t h e m a g n e t i c field intensity within the specimen, so
that inside t h e material I = ÷¢ w h e r e ÷ is t h e susceptibility. H e n c e
f
f
0
 = ^ ( Ç + É) =
0
( 1 + / Ê Ç
where ì,. = 1 + ÷ is the relative permeability, w h i c h is a p u r e n u m b e r .
F o r ferromagnetic or p a r a m a g n e t i c m a t e r i a l s ÷ is positive and ì > 1,
b u t for diamagnetic materials ^ is negative and u < 1.
T o sum up, for the case of a long t h i n rod, t h e flux density  w i t h i n
the rod includes a contribution from t h e m a g n e t i z a t i o n of the material,
while the magnetic field s t r e n g t h Ç does not. In this case the magnetic
field strength inside the rod is t h e same as if the rod w e r e not there.
Ã
r
A.4.
Th e C a s e of a Superconducto r
T h e foregoing discussion applies also to t h e case of a type-I superconductor (or a type-II superconductor below H ), b u t w i t h an i m p o r t a n t
distinction. T h e flux density w i t h i n t h e superconductor is zero if w e ignore penetration effects, and this perfect d i a m a g n e t i s m is b r o u g h t about
by real c u r r e n t s w h i c h circulate around t h e periphery of the superconductor; they are in fact the screening c u r r e n t s discussed in C h a p t e r 2.
W h e n it c o m e s to the concept of  and Ç in a superconductor, there
are t w o w a y s in which w e can proceed. W e can focus attention on the
screening c u r r e n t s and regard t h e m as real c u r r e n t s n o t different in
n a t u r e from the current in the w i n d i n g s of the solenoid. O n this view,
which r e g a r d s t h e bulk of t h e superconductor as n o n - m a g n e t i c , it is a p propriate t o rewrite (A.6) in t h e form
cl
 = ì (ôçß + Ë ) ,
0
where j
s
(A. 12)
is t h e surface density of the screening c u r r e n t s per unit length
231
APPENDI X A
parallel to the axis. T h e vanishing of  within a superconductor is due to
the fact that the t e r m s in brackets in (A. 12) are equal and opposite. In
this case we have for any closed path
§Â.ÜÉ
= ì (./
0
/
+ Ë ë
(A.13)
where
is the total current t h r o u g h the solenoid t u r n s linking the path,
and •'/ the total diamagnetic screening current linking the same path.
However, although the screening currents are certainly real, we cannot
measure t h e m with an ammeter, and we shall continue to define the
magnetic field strength Ç so that
. dl is always equal to t h e "free" or
measurable current / . T h i s m e a n s that w e retain (A. 10), which states
that
f
s
f
H=mi,
(A. 10)
just as in the case of a paramagnetic material, and the screening
currents
affect  but not Ç inside the
material.
Alternatively, it is possible to invert t h e argument summarized in
§ A.2 and regard the real c u r r e n t s flowing on the surface of the supercond u c t o r as e q u i v a l e n t t o fictitious d i p o l e s u n i f o r m l y d i s t r i b u t e d
throughout the body of the superconductor. (Since the superconductor is
diamagnetic, these imaginary dipoles would point the opposite way to
the real dipoles in a ferromagnetic or paramagnetic specimen.) F r o m this
point of view, w e can talk about the intensity of magnetization / of a
superconductor and regard this either as the magnetic m o m e n t per unit
volume of the equivalent dipoles or the surface density of the screening
currents in a m p s per metre. A practical definition of / is that it is the
total magnetic m o m e n t of the specimen arising from the surface c u r r e n t s
divided by the volume of the specimen. / is equal to and has the same
dimensions as the q u a n t i t y ^ occurring in (A. 12). In t e r m s of /, w e may
write, in accordance with (A.6),
 = ti (mi
0
+ /),
(A.6)
where the term u mi represents the flux density due to the solenoid and
ì É is the flux density resulting from the fictitious equivalent dipoles. As
before, we write
y
0
0
 = ìÇ
0
+ ì 1,
(A.9)
0
and the vanishing of  implies that within the superconductor the field
strength Ç m u s t b e equal to —/, so that w e have ÷ = — 1 or ì = 0, i.e.
the bulk of the superconductor is perfectly diamagnetic.
Ã
232
INTRODUCTIO N T O SUPERCONDUCTIVIT Y
W h i c h e v e r w a y w e choose to look at it, w e have
H.dl
= · / Ö—
(
ô
<f B . r f l ,
(A.l la)
and the value of Ç inside a long thin superconductor is the same as if t h e
superconductor were n o t there. (H is in fact t h e q u a n t i t y w e h a v e called
the applied field H ). W h i c h e v e r view w e adopt, we say t h a t deep w i t h i n
a superconductor  = 0, b u t in t h e presence of an applied field Ç d o e s
not v a n i s h . !
F o r most p u r p o s e s t h e second approach is m o r e convenient, since it
allows u s to apply to a superconductor c o n c e p t s such as energy of
magnetization and demagnetizing field w h i c h w e r e first developed for
the case of ordinary magnetic materials. As an example, t h e t r e a t m e n t of
the intermediate state given in C h a p t e r 6 would b e very difficult if w e
had to take t h e screening c u r r e n t s into account explicitly. T h e r e are,
however, cases w h e r e w e are particularly interested in t h e spatial d i s tribution of flux density (and of screening currents) near t o t h e surface of
a superconductor. An example is t h e development of t h e L o n d o n
equations in C h a p t e r 3. I n this case, it is a d v a n t a g e o u s to adopt t h e first
approach and recognize the existence of the real surface c u r r e n t s w h i c h
circulate around t h e specimen w h o s e bulk consists of n o n - m a g n e t i c
material. T h e distinction b e t w e e n  and Ç is n o w revealed b y (A.l l a )
;nd (A. 13), w h i c h in point form b e c o m e
a
and
curl Ç = ]
f
curl  = ì (É/
0
+ D-
ì ].
W i t h i n t h e s u p e r c o n d u c t o r , ] = 0, so t h a t curl Ç = 0 and curl  =
B o t h approaches are encountered in t h e literature, and the reader
should familiarize himself w i t h each of t h e m .
f
A.5.
0
5
D e m a g n e t i z i n g Effect s
So far w e h a v e limited our discussion t o the case of a long thin
specimen in w h i c h end effects are u n i m p o r t a n t . W e n o w show t h a t the
t T h i s standpoint is not invariably taken in the literature. It is quite common to find authors
who write  = ì Ç everywhere within the superconductor. In this case the vanishing of  implies the vanishing of Ç also. But there is really no point in introducing  and Ç if they are
everywhere simply related by a universal constant of proportionality. T h i s way of looking at it
causes difficulties in the explanation of demagnetizing effects, and also means that the usual
boundary condition involving the continuity of the tangential component of Ç at the boundary
between two media, which we have made use of in Chapter 6, is no longer valid at the interface
between the superconducting and normal phases.
0
233
APPENDI X A
relationship  = ì (Ç + I) h a s to b e interpreted s o m e w h a t differently if
the specimen is not long in comparison w i t h its width. T o b e specific,
consider a sphere of paramagnetic material which is placed in a uniform
applied field, say within a long solenoid. It is possible to show that the
magnetization I , defined as the magnetic m o m e n t per unit volume, is
uniform within the sphere and is parallel to the axis of the solenoid. It
can also be s h o w n t that the flux density due to the magnetization is the
same as would b e produced b y surface c u r r e n t s having a constant surface density / per unit length measured parallel to the axis. T h e flux
density produced b y such a distribution of surface c u r r e n t s around the
surface of the sphere is equal t o %ì É, so within the sphere the flux
density is uniform and given by
0
0
= ì^çé
(A.13)
+ |ì 7.
0
C o m p a r i n g this w i t h (A.6) and (A.9), which state that for a long thin
specimen
 = ì^ß
+ ìÉ = ìÇ
0
+
0
ì É,
0
w e might be t e m p t e d to define the magnetic field strength b y
 = ìÇ
0
+
áì É,
0
w h e r e á is a numerical factor depending on the geometry of the
specimen, equal to unity for a long thin specimen and to 2/3 for a sphere.
If w e were to do this, the field strength inside the body would be equal to
mi b o t h for a long thin specimen and for a sphere. However, unlike the
case of a long thin rod, in the case of the sphere the surface c u r r e n t s alter
outside
the flux density (and therefore the field strength, since  = ì Ç)
the sphere, and an argument identical w i t h t h a t given on page 65
(Chapter 6), s h o w s t h a t we should no longer have <j>H . dl = .9 . It can
be shown that if we wish to retain <|>H · dl = .//
for any closed curve,
where
is t h e total current in those t u r n s of the solenoid which link
the curve, then w e m u s t also retain (A.9) as the definition of the
magnetic field strength H , inside the sphere, i.e.
0
f
Â, = ì Ç , + ì É·
0
0
(Á.9')
But, according t o (A.13),
 = ì^çé
{
t See, for example, A. F. Kip, Fundamentals
ed., 1969, p. 370.
+ |ì 7·
0
(Á.13)
of Electricity and Magnetism, McGraw-Hill, 2nd
234
INTRODUCTION TO SUPERCONDUCTIVITY
T h e s e e q u a t i o n s can b o t h b e satisfied only if
H, = mi-
j/ ,
i.e. t h e magnetic field strength inside t h e sphere is different from t h e
value mi w h i c h it would h a v e if t h e sphere w e r e absent and w h i c h w e
have called t h e applied field H . H e n c e for a sphere
a
H = H
t
—} / ,
a
and for a b o d y of arbitrary shape
H, = H
f l
-nI,
(A.14)
w h e r e ç is t h e " d e m a g n e t i z i n g factor", equal t o 1/3 for a sphere. F o r a
superconductor, if w e adopt t h e a p p r o a c h w h i c h ignores t h e screening
c u r r e n t s b u t r e g a r d s t h e bulk of t h e superconductor as perfectly
diamagnetic, t h e n
JB, = ì Ç
0
+ ì É = 0,
ß
0
so t h a t
I =
and from (A.14) H
specimen is increased
write
{
 = ì
ß
0
-H,
— H + nH , i.e. the field strength inside the
to H = H /(l — Þ). F o r the general case w e m a y
a
{
t
(
Ç
a
á
-
nl
applied d e m a g n e t i z i n g
field field
internal field, H ,
+
I
)
magnetization
APPENDIX Â
FREE ENERGY OF A M A G N E T I C
BODY
SUPPOSE t h a t a b o d y of volume V is uniformly magnetized by an applied
field H , so t h a t its magnetic m o m e n t is M. T h e field is n o w increased to
H + dH and produces a change of magnetic m o m e n t dM. T h e total
energy which m u s t be supplied (e.g. b y a b a t t e r y driving a coil which
generates the magnetic field) to make these increments in field and
magnetic m o m e n t at constant t e m p e r a t u r e i s t
a
a
a
dW
tot
= V. Ü(\ Ç )
+
2
ìï
á
ì Ç ÜÌ.
0
á
T h e first t e r m on the right-hand side represents the work d o n e in
increasing the strength of the applied field over the volume occupied b y
the b o d y ; this work m u s t b e done to increase the field w h e t h e r the body
is present or not. T h e second term, ì Ç ÜÌ,
is the energy which m u s t
be supplied to increase the magnetic m o m e n t of the body. L e t u s call this
dW
0
á
M9
dW
=
M
ì Ç ÜÌ.
0
á
C o m p a r i s o n of this with the work done b y an external pressure p in
producing an infinitesimal change in volume dV of a body,
dW
v
=
-pdV
f
shows t h a t t h e expression for the work to magnetize the b o d y h a s a
corresponds top and Ì to — V.t
similar form, if w e suppose that ìïÇ
N o w the G i b b s free energy for a body in the absence of a magnetic
field is
á
G=U-TS
+
pV
y
t See, for example , A. B. Pippard , Classical Thermodynamics Cambridg e Universit y Press ,
1957, p. 26.
X Th e sign s ar e differen t becaus e energ y mus t be given to a bod y t o increas e it s magnetization ,
wherea s wor k is don e by a bod y whe n it increase s in volum e agains t an externa l pressure .
y
235
236
INTRODUCTION TO SUPERCONDUCTIVITY
where U and S are its internal energy and entropy. As w e might expect
from the above considerations, t h e effect of m a g n e t i z a t i o n is t o add a
term —ì Ç Ì
analogous to the t e r m +pV:
0
á
G=
U-TS
+
ñí - Ç Ì.
ìï
á
Small c h a n g e s in the conditions will p r o d u c e a change in G given by
dG = dU-
TdS - SdT + pdV + Vdp - ì Ç ÜÌ
0
-
á
ì ÌÜÇ .
0
á
If the applied field H and the m a g n e t i c m o m e n t Ì are changed while
the t e m p e r a t u r e and pressure are kept c o n s t a n t (dT = dp = 0), w e have
a
dG = dU-
TdS + pdV - ì Ç ÜÌ
0
-
á
ì ÌÜÇ .
0
á
But for a magnetic body u n d e r t h e s a m e conditions of constant Ô a n d / ) ,
dU = TdS - pdV +
ì Ç ÜÌ.
0
á
work d o n e o n body
Therefore dG = —ì ÌÜÇ ,
and t h e change in free energy of a body
w h e n it is magnetized to a magnetic m o m e n t Ì b y a field of strength H
is
0
á
a
G(H )-G(0)
a
=
- JMdH .
Mo
0
a
INDE X
Coope r pair s see Electro n pair s
Core s
186
motio n 209,212-216
pinnin g 199,208
Correlatio n betwee n electron s 80, 129
Correspondin g state s 134
Coulom b repulsio n betwee n electron s 117,
120,133
Couplin g energ y 167
Criterio n for superconductivity , BC S
theor y 134
Critica l curren t 40, 82-91, 202-204, 206-212,
223
densit y see Critica l curren t densit y
effect of magneti c field 82, 84-85, 202-206,
210-212
an d irreversibl e magnetizatio n 210
Josephso n junctio n 150, 161
measuremen t of 87, 91
quantu m interferomete r
173,175,178,179
Silsbee' s hypothesi s 83
thi n specimen s 106-111
type-I I superconductor s 202-204, 206-212,
217
weak link 1 6 9 , 1 7 8 , 2 2 4
wire s 83
Critica l curren t densit y 40, 82,107, 111, 138
BC S theor y 138
an d critica l magneti c field 83, 107
Critica l magneti c field 40-53, 223
BC S theor y 134
of cylinde r 97
effect of penetratio n 92
element s 6
H
218
191, 196
lower critica l field H
parallel-side d plat e 93
paramagneti c limit 195
temperatur e dependenc e 43
thermodynami c valu e 43, 193, 198
Adiabati c magnetizatio n 61
A. C . Josephso n effect 150, 165
Alloys xvii, 6, 185
therma l propagatio n 87
type-I I superconductivit y xvii, 186, 211
uppe r critica l fields 195
Bardeen-Cooper-Schrieffe r theor y
(BC S theory ) 117-139
criterio n for superconductivit y 134
critica l curren t densit y 138
critica l magneti c field 134
current-carryin g state s 135-139
energ y gap 129-133, 142,151
fundamenta l assumption s 125
groun d stat e 125
laten t hea t 133
law of correspondin g state s 134
macroscopi c propertie s 131-135
second-orde r transitio n 133
transitio n temperatur e 131
two-flui d mode l 139
zer o resistenc e 137
Boundar y betwee n superconductin g an d norma l
region s 68, 80
boundar y condition s for  an d Ç 69
Cerami c superconductor s 7, 221
Circuits , resistanceles s 8
Coherenc e 77, 153
BC S theor y 128
of electro n pair s 128, 138, 153
an d surfac e energ y 77,185
Coherenc e lengt h 77, 114, 128
an d electro n mea n fre e pat h
79
high temperatur e superconductor s
orde r of magnitud e 78
an d purit y 78
an d surfac e energ y 80, 185
temperature-dependen t 80
temperature-independen t 80
222
c3
cl
237
238
INDEX
Critica l magneti c field—continued
thi n film 97-99
thi n wir e 97
type-I I superconductor s 191-196
192, 194, 196, 223
uppe r critica l field H
Critica l temperatur e see Transitio n
tempertur e
Cryotro n 45
Crysta l structur e an d superconductivit y 113
Current-carryin g states , BC S theor y 135
Current s in superconductor s 11, 22, 26, 36,
202
analog y with electrostati c charg e 23
critica l see Critica l curren t and Critica l cur ren t densit y
resistanceles s network s 11
transpor t 23, 82, 192
c2
Defects see Imperfection s
Demagnetizatio n
66,227-229
Demagnetizin g facto r 64, 6 6 , 1 0 5 , 229
Demagnetizin g field 66, 229
Densit y of state s at Ferm i surfac e 127,134
Diamagnetism , perfec t 17, 21, 160
Dislocation s 199
Edg e effects in magneti c transition s 104
Effectiv e momentu m in magneti c field 102
Electri c field in superconducto r 13, 14, 32, 35
Electrica l resistanc e
maximu m valu e in superconducto r 8
metal s an d alloys 3, 112
at optica l frequencie s 14, 112
residua l 4 , 5
restoratio n by curren t 89
zer o resistanc e 3, 8, 31, 83, 137, 202
Electrica l resistivit y
a.c. 12
residua l 4
of a superconducto r 10
Electron-electro n interactio n 117
coulom b repulsio n 117, 120, 133
by electron-phono n interactio n 118-120
Electron-lattic e interactio n 118-120
conservatio n of energ y an d
momentu m
119-120
by phono n emission 118
Electro n pair s (Coope r pairs ) 120, 158
Bose-Einstei n statistic s 126
energ y to brea k up 130
formatio n of groun d stat e 121, 123
long-rang e coherenc e 138, 153
momentu m pairin g conditio n 123
motio n of centr e of mas s 136
an d Paul i principl e 126, 127
phas e coherenc e 128, 139, 154
spatia l exten t 129
tota l momentu m 135
wave functio n 124, 136
Electron-pai r waves 124,136, 153, 154,157
coherenc e 153, 154
diffractio n 178
effect of magneti c field 155
frequenc y 154
interferenc e 170, 178, 179
momentu m 154
phas e
154,155,157,160
wavelengt h
153,154,155
Electron-phono n interactio n see Electron lattic e interactio n
Electron s
Ferm i "sea " 120
norma l 12, 138
orderin g of 60
superelectron s 12, 138
in vacuu m 14
Element s
critica l fields 6
superconductin g xvii, 6
transitio n temperature s 6
Energ y conservatio n in phono n emission or
absorptio n 119
Energ y gap 61, 115, 129-132, 140
absorptio n of electromagneti c radiatio n 115
BC S theor y 130-132, 142, 151
definitio n of Ä 127
measuremen t of 132, 145, 150
an d specific hea t 61, 116
temperatur e dependenc e 132
Energ y level diagra m 142
semiconducto r representatio n 147
Entrop y 54
type-I I superconductor s 201
Ferm i "sea " 120
Ferromagnetic s 5
Flo w resistanc e 204
Flu x see Magneti c flux
Flu x cree p 224
Flu x flow 2 0 6 , 2 1 2
du e to temperatur e gradien t
e.m.f. 214
Flu x lines see Fluxon s
Fluxoi d 156-160
216
INDEX
Fluxo n lattic e 187
Fluxon s 1 5 8 , 1 7 1 , 1 8 9
arrangemen t in mixed stat e
Frontispiece,
187,197
Fre e energ y see Gibb s fre e energ y
Gaples s superconductor s 116,137, 139
Gibb s free energ y 4 1 - 4 3 , 70, 230-231
an d critica l magneti c field 41-44, 93
effect of magneti c field 4 1 , 1 9 1 , 231
in intermediat e stat e 70
Ginzburg-Landa u constan t ê 190,198, 218
dependenc e on resistivit y 191
H
218
surfac e superconductivit y 217
temperatur e variatio n 191,219
an d typ e of superconducto r 190, 218
Ginzburg-Landa u theor y 101
critica l magneti c field of thi n films 103
effect of magneti c field 102
second-orde r transitio n in thi n film 103
Groun d state , BC S theor y 125
c3
Hal l effect in mixed stat e 214
Hig h frequencie s 14,113
High-temperatur e superconductor s
Hollo w superconducto r 24, 159
Hysteresi s 48,199, 210
221
Idea l superconducto r 47
Imperfection s 48,199, 211
Inductanc e of a superconducto r 13-14
Intensit y of magnetizatio n see Magnetizatio n
Interferomete r 170
Intermediat e stat e 66-81
domai n structur e 72
experimenta l observatio n of 72
induce d by curren t 89
lamina r structur e 76
magneti c propertie s 69
size of domain s 74
thi n films 105
Interna l magneti c field 64, 69, 228-229
Isotop e effect 115,133
Josephso n tunnellin g 149, 160
a.c. effect 150, 165
couplin g energ y 167
junctio n 1 4 9 , 1 6 0 , 1 6 2
pendulu m analogu e 161
Kappa , ê see Ginzburg-Landa u constan t ê
239
Lamina r structur e of intermediat e stat e 68-70,
76
Laten t hea t 58, 200
BC S theor y 133
La w of correspondin g state s 134
Londo n equation s 34, 37
Londo n theor y 33-39
limitation s of 98
penetratio n dept h 35
Long-rang e orde r 77, 114
an d coherenc e lengt h 7 7 , 1 1 4 , 1 2 8 - 1 2 9
Lorent z forc e on fluxons 208
Magneti c field
"applied " 16
critica l see Critica l magneti c field
effect on critica l curren t 84, 203, 206
effect on electron-pai r wave 155
effect on free energ y 4 1 - 4 3 , 1 9 1 , 231
effect on Josephso n junctio n 178
interna l 66, 69, 228-229
quantu m interferomete r 170-180
Magneti c field strength , definitio n 224
Magneti c flux
Frontispiece,
arrangemen t in mixed stat e
186-189
distributio n insid e type- I superconduc tor 36, 38
distributio n in parallel-side d plat e 39, 93
fluxoid
156-160
penetratio n 26
in perfec t conducto r 16
quantu m of see Fluxon s
in resistanceles s circui t 8-11
trappe d 48, 199
Magneti c flux densit y 46
definitio n of 221
measuremen t of 49
zer o in superconducto r 19
Magneti c momen t
effect of penetratio n 95
parallel-side d plat e 95
Magneti c shieldin g by superconductor s 11
Magnetizatio n 46
adiabati c 59
definitio n of 223, 224, 226
hysteresi s 4 8 , 1 9 8 , 2 1 1
irreversible
48, 199, 210
measuremen t of 50-53
parallel-side d plat e 95
"perfect " conducto r 16
superconducto r 19
240
INDEX
Magnetization— continue d
type-I I superconducto r 197-199, 210, 222
Matri x elemen t of scatterin g interactio n 123,
127
Maxwell' s equatio n 32
Meissne r effect 19, 34, 115
see also Perfec t diamagnetis m
Mercur y xvi
Microscopi c theor y of superconductivit y
112-139
Mixe d stat e 1 8 4 , 1 8 6 - 1 9 0 , 1 9 1 - 1 9 3 , 1 9 7 - 1 9 8
critica l curren t 202-212
current-carryin g propertie s 192-220
e.m.f. du e to cor e motio n 214
flux flow 206-217
188, 197
fluxon patter n Frontispiece,
fre e energ y 189, 192
Hal l effect 214
lower critica l field H
191, 196
uppe r critica l field H
192, 194-196
Modulu s of elasticity , chang e at transitio n 62
Momentu m
(l
(2
conservatio n in phono n emission or
absorptio n 118
of electro n pai r 135-137, 153
Network , resistanceles s 11
Neutro n diffractio n fro m mixed stat e 188, 209
Niobiu m
transitio n temperatur e 6, 7
type-I I superconductivit y 6, 191
Non-idea l superconducto r 47, 199, 200
Norma l electron s 12, 138
Order , of electron s
60
Paramagnetis m 195
Peltie r coefficient 63
Pendulu m analogu e 162
Penetratio n dept h 26-30, 98, 160
dependenc e on magneti c field 100
dependenc e on specime n size 100
effect on critica l magneti c field 92
effect on magnetizatio n 93
high-temperatur e superconductor s 222
Londo n theor y 35, 37, 99
an d surfac e energ y 80, 185
temperatur e variatio n 28, 36
"Perfect " conducto r 16, 20
magneti c behaviou r 16
Perfec t diamagnetis m 17, 21, 26, 28, 34, 160
see also Meissne r effect
Permeabilit y of superconducto r
Persisten t curren t 8, 10
Phas e see Electron-pai r wave
Phas e coherenc e of Coope r pair s
Phas e diagra m 43, 194
Phas e transitio n
21
128,138,153
first orde r 59
second orde r 57, 200
Phonon s
averag e frequenc y 123,127
electron-lattic e interactio n 117
virtual emission
119,123
Polymer , superconductin g 6
Quantize d magneti c flux see Fluxo n
Quantu m interferenc e 153, 170, 180
Quantu m interferomete r 170, 180
diffractio n effects 179
interferenc e patter n 173, 178, 180
Quasiparticle s 130, 131
energ y of 143
Resistanc e see Electrica l resistanc e
Resistanc e transition s 4, 8, 83-91, 205
rol e of surfac e energ y 76
temperatur e rang e 8
in transvers e magneti c field 76
zer o 83, 138, 202
Resistanceles s conducto r see Perfec t conducto r
Resistivit y see Electrica l resistivit y
Rutger' s formul a 58
Screenin g current s 17, 21, 25
Second-orde r transitio n 57, 200
BC S theor y 133
of thi n film in magneti c field 103, 152
Silsbee' s hypothesi s 83-86
thi n specimen s 106,110
type-I I superconducto r 203, 204
Smal l specimen s 92
Solenoid , superconductin g 10, 195, 202
Sommerfel d specific hea t constan t 60, 134
Specific hea t 5 7 - 6 1 , 1 1 4
electroni c 59, 114
an d energ y gap 61, 116
lattic e 59, 114
Rutgers ' formul a 58
Sommerfel d constan t 60, 134
type-I I superconductor s 200
S Q U I D see Quantu m interferomete r
Superconductin g stat e
summar y of propertie s
112
241
INDEX
Superelectron s 12, 138
Surfac e energ y 74, 183
coherenc e rang e 77, 185
an d domai n size in intermediat e stat e 74
effect on magneti c field at boundar y 76
measuremen t of 76, 80
negativ e 184, 185-186
Surfac e superconductivit y 217
Susceptibilit y of superconducto r 21
Switch , therma l 2
Symbol s xii
Technetium , type-I I superconducto r 6, 191
Therma l conductivit y 62
Therma l expansion , chang e at transitio n 62
Therma l propagatio n 8 6 - 8 8 , 1 1 1
Therma l switch 62
Thermodynami c variable s 41
Thermodynamic s 5 4 - 6 3 , 2 3 0
Thermoelectricit y 6 3 , 2 1 6
Thi n films 28, 9 3 - 1 0 0 , 1 0 3 - 1 1 1
critica l magneti c field 93-98, 196
curren t distributio n 107-109,110
intermediat e stat e 106
therma l propagatio n 111
transitio n in perpendicula r magneti c field
105
Thomso n coefficient 63
Transitio n
first-orde r 59
laten t hea t 58
reversibl e 4 0 , 4 1
second-orde r 57, 103, 152, 200
sharpnes s of 8, 79
Transitio n temperatur e
alloys an d compound s 7
BC S theor y 131
element s 6
an d energ y gap 132
high-temperatur e superconductor s 221, 222
isotop e effect 115,133
niobiu m 7
rang e 8
ver y low 5
Transpor t curren t see Current s in
superconuctor s
Tunnellin g 140-152, 160, 167
a.c. Josephso n effect 150, 165
attenuatio n lengt h 141
betwee n dissimila r superconductor s 149
betwee n identica l superconductor s 145
betwee n norma l metal s 140
betwee n norma l meta l an d super conducto r 143
condition s for 141
experimenta l detail s 150
Josephso n 149, 160,167
measuremen t of energ y gap 145
wavefunctio n 141
Two-flui d mode l 13, 7 7 , 1 1 4
BC S theor y 138
Type- I superconductor s xvii, 3-180, 184, 190,
218
Type-I I superconductor s xvii, 183-225
alloys 186
critica l curren t
202-220
current-carryin g propertie s 207
e.m.f. du e to cor e motio n 209, 214
flow resistanc e 206, 209
Ginzburg-Landa u paramete r 190
Hal l effect 214
high temperatur e 222
intrinsi c 191
191, 196
lower critica l field H
magnetizatio n 197, 199
mixed stat e 183, 186
specific hea t 200
thermodynami c critica l field 193, 198
thermoelectri c effects 63, 216
1 9 2 , 1 9 4 , 1 9 6 , 223
uppe r critica l field H
cl
c2
Vanadium , type-I I superconducto r 6, 191
Virtua l emission of phono n 119,123
Voltag e du e to flux flow 214
Volum e chang e at transitio n 62
Vortice s see Core s
Wavefunctio n
BC S groun d stat e 126
electro n pair s 124,135
Wavelengt h of electron-pai r wave 153, 155
Waves , electron-pai r 153
Wea k link 160,169
critica l curren t 161, 169, 178
diffractio n 178
high-temperatur e superconductor s 224