GO
a
w
o
3
SUPER
CONDUCTIVITY
w
Ernest A. Lynton
METHUENS MONOGRAPHS ON
PHYSICAL SUBJECTS
Superconductivity
ERNEST A. LYNTON
Although the fascinating phenomenon of
superconductivity has been known for fifty
years, it is largely through the concentrated
experimental and theoretical work of the last
decade that a basic (though at present very
incomplete) understanding of the effect has
been reached. This monograph is a largely
descriptive introduction to superconduc
tivity, requiring little more than an under
graduate physics background. It is written to
serve two functions ; first as a stepping stone
towards more intensive study for those who
intend to work in the field of research and
development of superconductivity and its
applications and, secondly, as a basic refer
ence on the present state of the subject of
superconductivity.
The book contains a description of the
principal characteristics of a superconductor,
together with a detailed discussion of the most
useful phenomenological models which have
been applied to superconductors. The second
part of the monograph describes the funda
mental microscopic properties in terms of the
theory of Bardeen, Cooper and Schrieffer. It
is shown how remarkably successful this
theory has been in explaining the behaviour
of an idealized superconductor. There is a
chapter on superconducting devices, a sub
ject index and a bibliography of more than
330 books and articles.
SECOND EDITION
LONDON :
METHUEN & CO. LTD
NEW YORK
JOHN WILEY & SONS INC
methuen's monographs
on physical subjects
General Editor: B. L. WORSNOP, b.sc, ph.d.
SUPERCONDUCTIVITY
Superconductivity
E. A. Lynton
Professor of Physics
Rutgers, The State University
New Brunswick, N.J., U.S.A.
Powder patterns of the intermediate stale, showing thesbrink
ing of the superconducting (dark) regions as /; takes on the
values (left to right, top to bottom) 0, 008, 027, 053, 079, and
090.
(After Faber, 1958. Reproduced by kind permission of the
Royal Society and the author.) Proc. Roy. Soc. A248 464,
plate 25.
LONDON: METHUEN & CO LTD
NEW YORK: JOHN WILEY & SONS INC
First published in 1962
Second edition 1964
© 1962 and 1964 by E. A. Lynton
Printed in Great Britain by
Spottiswoode, Ballantyne & Co Ltd
London & Colchester
Catalogue No. Methuen 12/4081/66
21
For Carla
CHRJS. :
i
Acknowledgements
This book has grown, beyond recognition, from a set of lecture notes
written and used during my stay at the Institut Fourier of the Univer
sity of Grenoble in 195960. I should like once again to thank my
hosts, Professors Neel and Weil and Dr Goodman, for a stimulating
and pleasant year. I am very grateful to a large number of people who
have helped me with written or oral comments, with news of their un
published work, with preprints, and with copies of graphs. In par
ticular I thank Drs Coles, Collins, Cooper, Douglass, Faber, Gar
funkel, Goodman, Masuda, Olsen, Pippard, Schrieffer, Shapiro,
Swihart, Tinkham, Toxen, and Waldram. My colleagues Lindenfeld,
McLean, and Weiss provided much helpful discussion. Above all my
gratitude is due to Bernard Serin, from whose guidance and friendship
I have profited for many years. He found the time to read the entire
first draft of the manuscript and suggested many improvements, not
all of which I have been wise enough to incorporate.
September 1961
E. A. LYNTON
Preface to the Second Edition
This edition contains revisions and additions which bring the mono
graph essentially up to date, as of the end of June 1964. The treatment
of superconductors of the second kind has been considerably ampli
fied, a discussion of the Josephson effect has been added, and a num
ber of other changes have been made. Many of these were also incor
porated in the excellent French translation of Mme Nozieres, which
was published early this year. I am very grateful to her, as well as to
Dr Nozieres, for his valuable comments and help. A Russian trans
lation, edited with many illuminating footnotes by Dr. Gor'kov,
unfortunately reached me too late for these comments to be included
in the present edition.
August 1964
E. A. LYNTON
vi
Contents
Introduction
page I
n
m
IV
Basic Characteristics
1 . 1 Perfect conductivity and the critical magnetic field
1.2 Superconducting elements and compounds
1.3 The Meissner effect
1.4 The specific heat
1.5 Theoretical treatments
Phenomcnological Thermodynamic Treatment
2.1 The phase transition
2.2 Thermodynamics of mechanical effects
2.3 Interrelation between magnetic and thermal
properties
2.4 The GorterCasimir twofluid model
Static Field Description
3.1 Perfect diamagnetism
3.2 Influence of geometry and the intermediate state
3.3 Trapped flux
3.4 The perfect conductor
3.5 The London equations for a superconductor
3.6 Quantized flux
The Pippard Nonlocal Theory
4.1 The penetration depth, A
4.2 The dependence of A on temperature and field
4.3 The range of coherence
4.4 The Pippard nonlocal relations
The GinzburgLandau Phenomcnological Theory
vii
3
3
5
7
9
11
13
13
16
17
19
22
22
23
26
27
28
32
34
34
36
40
41
48
Vlll
VI
vn
Vffl
IX
Contents
The Surface Energy page 55
6.1 The surface energy and the range of coherence 55
6.2 The surface energy and the intermediate state 59
6.3 Phase nucleation and propagation 61
6.4 Supercooling in ideal specimens 65
6.5 Superconductors of the second kind 67
6.6 The mixed state or Shubnikov phase 71
6.7 Surface Superconductivity 74
Low Frequency Magnetic Behaviour of Small Specimens 75
7.1 Increase in the critical field 75
7.2 High field threads and superconducting magnets 77
7.3 Variation of the order parameter and the energy
gap with magnetic field 78
The Isotope Effect 81
8.1 Discovery and theoretical considerations 81
8.2 Precise threshold field measurements 83
Thermal Conductivity 87
9.1 Low temperature thermal conductivity 87
9.2 Electronic conduction 89
9.3 Lattice conduction 93
9.4 Thermal conductivity in the intermediate state 94
The Energy Gap 95
10.1 Introduction 95
10.2 The specific heat 96
10.3 Electromagnetic absorption in the far infrared 98
10.4 Microwave absorption 100
10.5 Nuclear spin relaxation 104
10.6 The tunnel effect 106
10.7 Far infrared transmission through thin films 109
10.8 The FerrellGlover sum rule 1 14
,.
XII
XIII
Contents
Microscopic Theory of Superconductivity
11.1 Introduction
11.2 The electronphonon interaction
1 1 .3 The Cooper pairs
1 1 .4 The ground state energy
11.5 The energy gap at 0°K
1 1.6 The superconductor at finite temperatures
11.7 Experimental verification of predicted thermal
properties
11.8 The specific heat
1 1.9 Coherence properties and ultrasonic attenuation
11.10 Electromagnetic properties
Superconducting Alloys and Compounds
12.1 Introduction
12.2 Dilute solid solutions with nonmagnetic im
purities
12.3 Compounds with magnetic impurities
1 2.4 Superimposed metals
Superconducting Devices
13.1 Research devices
13.2 Superconducting magnets
1 3.3 Superconducting computer elements
Bibliography
Index
IX
page 116
116
117
118
120
126
127
129
134
136
139
141
141
141
145
150
153
153
154
157
165
183
Introduction
Although the fascinating phenomenon of superconductivity has been
known for fifty years, it is largely through the concentrated experi
mental and theoretical work of the past decade that a basic (though
as yet very incomplete) understanding of the effect has been reached.
Far from being an oddity of little physical interest it has been shown
to be a cooperative phenomenon of basic importance and with close
analogies in a number of fields. At the present time one important
period in the development of the subject has been completed, and the
next is already well under way, with much effort in theory and experi
ment to carry our understanding from the general to the particular,
from the idealized superconductor to the specific metal. Somewhat
coincidentally, there now also is great interest in possible practical
applications of superconductivity.
This monograph is a largely descriptive introduction to super
conductivity, requiring no more than an undergraduate physics back
ground, and written to serve two functions. It can be a first survey and
a stepping stone toward more intensive study for those who intend to
become actively engaged in the further development of superconduc
tivity, be it in basic research or in technical applications. Such readers
will benefit from the extensive bibliography, listing more than 450
books and articles. At the same time the book is sufficiently complete
in its description both of experimental details and of theoretical
approaches to be a basic reference for those who wish to be acquainted
with the present state of superconductivity. It will enable them to
follow further developments as they appear in the scientific and
technical literature.
The contents of the book can be grouped into a number of sections
which treat the subject of superconductivity in successive layers with
increasing resolution of detail. The first three chapters introduce the
reader to the principal characteristics of bulk superconductors, and
treat these in terms of the basic phenomenological models of London
and of GorterCasimir. With this section the reader thus acquires a
broad outline and a general understanding of the thermodynamic and
1
2 Superconductivity
the static electromagnetic behaviour of idealized, bulk superconduc
tors. The treatment of the subject is then pursued in greater detail
along two essentially parallel directions. In the section comprising
Chapters IVVII are discussed those aspects of the behaviour of
superconductors which lead to the nonlocal treatments of Pippard
and of Ginzburg and Landau. These more sophisticated phenomeno
logical models account for an interphase surface energy, in terms of
which the later chapters of this section describe the intermediate state,
phase nucleation, propagation, and supercooling, superconductors
of the second kind, and the magnetic behaviour of specimens of
small dimensions. Chapters VIIIX can be read without a study of
the preceding section (IVVII) and describe in much detail those
characteristics of a superconductor which during the past decade
have indicated the microscopic nature of superconductivity, and
have led to the theory of Bardeen, Cooper, and Schrieffer. The
fundamental aspects of this theory are presented with a minimum of
mathematics.
The book closes with a chapter on the behaviour of alloys and com
pounds, and with one on superconducting devices.
In describing the principal empirical characteristics of supercon
ductors I have tried to include only the key experiments through
which the phenomenon in question was established, as well as more
recent work which gives the most detailed or the most precise informa
tion. It is both unnecessary and impossible in a monograph of this
small size to be encyclopaedic either in the enumeration of all per
tinent experiments, or in the description of superconducting be
haviour in minute detail. My selection of what aspects of the latter
to emphasize may appear arbitrary, especially to those whose work
has been slighted. The choice was not a judgement of the scientific
value of such work, but rather of its didactic usefulness in illuminating
the elementary characteristics of superconductors.
CHAPTER I
Basic Characteristics
1.1. Perfect conductivity and critical magnetic field
The behaviour of electrical resistivity was among the first problems
investigated by Kamerlingh Onnes after he had achieved the lique
faction of helium. In 1911, measuring the resistance of a mercury
sample as a function of temperature, he found that at about 4°K the
resistance falls abruptly to a value which Onnes' best efforts could not
distinguish from zero. This extraordinary phenomemon he called
superconductivity, and the temperature at which it appears the critical
temperature, T c (Kamerlingh Onnes, 1913).
When a metallic ring is exposed to a changing magnetic field, a
current will be induced which attempts to maintain the magnetic flux
through the ring at a constant value. For a body of resistance R and
selfinductance L, this induced current will decay as
/(/) = 7(0)exp(i?//L). (LI)
/(/) can be measured with great precision, for example, by observing
the torque exerted by the ring upon another, concentric one which
carries a known current. This allows the detection of much smaller
resistance than any potentiometric method. A long series of such
measurements on superconducting rings and coils by Kamerlingh
Onnes and Tuyn (1924), Grassman (1936), and others recently cul
minated in an experiment by Collins (1956), in which a superconduct
ing ring carrying an induced current was kept below T c for about two
and a half years. The absence of any detectable decay of the current
during this period allowed Collins to place an upper limit of 10 21
ohmcm on the resistivity of the superconductor.! This can be com
pared to the value of 10 9 ohmcm for the low temperature resistivity
of the purest copper.
There is, therefore, little doubt that a superconductor is indeed a
t Quinn and Ittner (1962) have lowered this upper limit to 10" 23 ohmcm
by looking for the time decay of a current circulating in a thin film tube.
3
4 Superconductivity
perfect conductor, in the interior of which any slowly varying electric
field vanishes. A current induced in a superconducting ring will persist
indefinitely without dissipation.
Below T c , the superconducting behaviour can be quenched and
normal conductivity restored by the application of an external mag
netic field. This field, H c , is called the critical or threshold magnetic
field, and, as shown in Figure 1 , it varies approximately as
H c *H [l(lJ],
(1.2)
Normal
T
Temperature T
Fig. 1
where H = H c at T= 0°K. It is convenient to introduce reduced co
ordinates / ■ T/T c , and h(t) = H C (T)/H , in terms of which
ft» l/ 2 . (I.2a)
The actual temperature variation of h is more accurately represented
by a polynomial in which the coefficient of the t 2 term differs from
unity by a few per cent.
The superconductivity of a wire or film carrying a current can be
quenched when this reaches a critical value. For specimens sufficiently
Basic characteristics 5
thick so that surface effects can be ignored, the critical current is that
which creates at the surface of the specimen a field equal to H c .
Smaller samples remain superconducting with much higher currents
than those calculated from this criterion, which is called Silsbee's
rule (Silsbee, 1916).
1.2. Superconducting elements and compounds
Table I lists all presently known superconducting elements and their
characteristic H and T c . In addition there have been found by many
investigators, in particular by Matthias and coworkers, by
Alekseevskii and coworkers, and by Zhdanov and Zhuravlev (see
Table I
Element
T C (°K)
H (gauss)
Aluminium
119
99
Cadmium
056
30
Gallium
109
51
Indium
3407
283
Iridium
014
~20
Lanthanuma
~5
Lanthanum/*
595
1600
Lead
718
803
Mercurya
4153
411
Mercury/?
395
340
Molybdenum
10
—
Niobium
946
1944
Osmium
07
6582
Rhenium
170
201
Ruthenium
049
66
Tantalum
4482
830
Technetium
112
_
Thallium
239
171
Thorium
137
162
Tin
3722
306
Titanium
040
100
Tungsten
~001
Uraniuma
06
~2000
Uraniumy
180
Vanadium
530
1310
Zinc
092
53
Zirconium
075
47
(cf. Roberts (1963) for most references)
6 Superconductivity
Matthias, 1957; Roberts, 1961), a very large number of alloys and
compounds which also become superconducting. Some of these
compounds consist of metals, only one of which by itself becomes
superconducting, some have constituents of which neither by itself is
superconducting, and some even are semiconductors. The possibility
of superconductivity in semiconductors and semimetals has been
discussed by M. L. Cohen (1964), and both GeTe (Hein et al., 1964)
and SrTi0 3 (Schooley et al., 1964) have been found to be supercon
ducting at very low temperatures.
2 4 6 8 10
No. valence electrons/atom
Fio.2
The critical temperatures of superconductors range from very low
values up to 181°K for Nb 3 Sn (Matthias et al., 1954). Matthias
(1957) has pointed out a number of regularities in the appearance of
superconductivity and in the values of T c , the principal of which are
the following :
(1) Superconductivity has been observed only for metallic sub
stances for which the number of valence electrons Z lies between
about 2 and 8.
(2) In all cases involving transition metals, the variation of T c with
the number of valence electrons shows sharp maxima for Z = 3, 5,
and 7, as shown in Figure 2.
(3) For a given value of Z, certain crystal structures seem more
favourable than others, and in addition T c increases with a high power
of the atomic volume and inversely as the atomic mass.
Basic characteristics 7
1.3. The Meissncr effect, and the reversibility of the S.C. transition
If a perfect conductor were placed in an external magnetic field, no
magnetic flux could penetrate the specimen. Induced surface currents
would maintain the internal flux, and would persist indefinitely. By
the same token, if a normal conductor were in an external field before
it became perfectly conducting, the internal flux would be locked in
by induced persistent currents even if the external field were removed.
o o
o
A:H e =0,
B%0.
C:0<H e <H o
B:H e =0
T>T C .
T<T C .
T<T C .
T<T C .
(a.)
(b)
(c)
(d)
Fig. 3
C:0<Hp<H
e n c.
T<T C
B: H e =0,
T<T C .
(d)
Fig. 4
Because of this, the transition of a merely perfectly conducting speci
men from the normal to the superconducting state would not be
reversible, and the final state of the specimen would depend on the
path of the transition.
As an example, Figures 3 and 4 show the flux configuration for a
perfectly conducting sphere taken from point A in Figure 1 to point C
by the different paths ABC and ADC, respectively. The final field
distribution at C, as well as that at B, depends on whether one pro
ceeded via Bov via D, and the irreversibility of the transition is evident.
Careful measurements of the field distribution around a spherical
8 Superconductivity
specimen by Meissner and Ochsenfeld (1933), however, indicated that
regardless of the path of transition the situation at point C is always
that shown in Figure 3c : the magnetic flux is expelled from the interior
of the superconductor and the magnetic induction B vanishes. This is
called the Meissner effect, and shows that the superconducting transi
tion is reversible.
Figure 5 illustrates this by showing B vs. H e curves both for a perfect
conductor and for a superconductor, taking the case of long cylin
drical specimens with axes parallel to the applied field. H e is a uniform,
He
cO
3
"a
c
>'
perfect conductor />
«=
super
conductor
Applied Field He
Fro. 5
external field. In increasing field both specimens have 5=0 until
H e = H c , when they become normal and B = H e . If the field is now
again decreased, the induction inside the perfect conductor is kept at
its threshold value B = H c by surface currents, and in zero field the
specimen is left with a net magnetic moment, as is illustrated in Figure
4d. The superconductor, however, expels the flux at the transition and
returns reversibly to its initial state with B = for < H e < H c .
The vanishing of the magnetic induction, corresponding to the ex
pulsion of the magnetic flux, is the basic characteristic of all ideal
superconducting material with dimensions large compared to a basic
length which will be mentioned later. It is quite independent of the
Basic characteristics 9
connectivity of the body, so that if one has a superconductor with a
hole, the Meissner effect occurs in the metal and only the hole may be
threaded by magnetic flux. The magnetic properties of such a super
conducting ring are thus essentially determined by the relative size of
the diameter of the ring to the diameter of the hole.
1.4. The specific heat
The specific heat of a superconductor consists, like that of a normal
metal, of the contribution of the electrons (C e ) and that of the lattice
(C g ). For a normal metal at low temperatures
*^n Wntt,
(1.3)
, gn = yT+A(T/0) 3 .
y is the Sommerfeld constant, which is proportional to the density of
electronic states at the Fermi surface, is the Debye temperature, and
A a numerical constant for all metals. Experimentally the two contri
butions to C n can be separated by plotting CJTvs. T 2 , so that the
slope of the resulting curve is A/0 3 , and the intercept is y.
In the superconducting phase
C = C es +C gs .
Figure 6 shows values of both C s (H=0) and C„ (H> H c ) for tin as
measured by Corak and Satterthwaite (1954), displaying the charac
teristic features of a sharp discontinuity in C,of the order of 2y7 c at T c ,
and a rapid decrease of C s to values below C n varying about as T 3 .
It is customary to attribute the difference between C s and C„ entirely
to changes in C e , on the assumption that C g is the same in both phases.
This seems reasonable in view of the electronic nature of the super
conducting phenomenon, and is supported by the absence of any
observable change in the lattice parameters (Keesom and Kamerlingh
Onnes, 1924), and by the detection of only minimal changes in the
elastic properties (see, for instance, Alers and Waldorf, 1961). On this
assumption
^s *n — f'es ^e
(1.4)
which allows one to determine C es from measured values of the
specific heat difference after C e „ = y7"has been determined separately.
There has recently been some evidence that the lattice contributions
to the specific heat in the two phases are not quite equal in the case of
10
Superconductivity
Basic characteristics
11
(deg.Kf)
Fig. 6
1.0 1.5 2.0 2.5 3.0 3.5 4.0
1/t
Fig. 7
indium (Bryant and Keesom, 1960; O'Neal et al., 1964), so that 1.4
may not be exact for this element and possibly other superconductors
as well. Ferrell (1961) has suggested that this is due to a shift in the
phonon frequency spectrum. However, the superconducting ele
ments for which reliable values of C„ exist are those with a relatively
high Debye temperature for which C g < C e in both phases down to
very low temperatures. For these elements possible small differences
in C g therefore do not much affect the validity of 1.4.
Figure 7 displays C cs for tin calculated on the basis of 1.4 from the
results in Figure 6, plotted logarithmically in units of l/yT c vs. 1//.
This shows that for l/t>2, one can represent C es by the equation
CJyT c = aexp(b/t). (1.5)
A subsequent chapter will discuss that this is an indication of the
existence of a finite gap in the energy spectrum of the electrons
separating the ground state from the lowest excited state. The number
of electrons thermally excited across this gap varies exponentially
with the reciprocal of the temperature. In recent years it has become
apparent that such an energy gap determines the thermal properties
as well as the high frequency electromagnetic response of all super
conductors, and that it must indeed be one of the principal features
of a microscopic explanation of superconductivity.
1.5. Theoretical treatments
The macroscopic characteristics of a superconductor have been the
subject of a number of phenomenological treatments of which the
principal ones will be discussed in subsequent chapters. F. and H.
London (1935a, b) developed a model for the low frequency electro
magnetic behaviour which is based on a point by point relation
between the current density and the vector potential associated with
a magnetic field. This implies wave functions of the superconducting
electrons which even in the presence of such a field extend rigidly to
the limits of the superconducting material and then vanish abruptly.
A thermodynamic treatment and an associated twofluid model based
on essentially equivalent simplifications were worked out by Gorter
and Casimir (1934a, b). These complementary theories provide highly
successful and useful tools in the semiquantitative analysis of many
problems involving superconductors. Their limitations become
apparent principally in situations in which size and surface effects are
important.
Pippard (1950, 1951) has shown that such effects become tractable
when one takes into account the finite coherence of the superconduct
ing wave functions which is such as to allow them to vary only slowly
over a finite distance. This leads (Pippard, 1953) to a nonlocal
12 Superconductivity
integral relation between the current density at a point and the vector
potential in a region surrounding the point. The equation has only
been solved for a few special cases. In many instances, however, it
reduces to a modified version of the London equation, so that the
much simpler London formalism can then be used with the Pippard
modifications (Tinkham, 1958).
Ginzburg and Landau (1950) have developed on a thermodynamic
basis an alternate method of treating the coherence of the super
conducting wave functions. Their treatment is compatible with
Pippard's electromagnetic approach, and forms a highly useful com
plement to it.
A successful microscopic theory of superconductivity has recently
been developed by Bardeen, Cooper, and Schrieffer (1957). It is based
on the fact, established by Cooper (1956), that in the presence of an
attractive interaction the electrons in the neighbourhood of the Fermi
surface condense into a state of lower energy in which each electron
is paired with one of opposite momentum and spin. Bardeen, Cooper,
and Schrieffer (BCS) have been able to show that a finite energy gap
separates the state with the largest possible number of Cooper pairs
from the state with one pair less. This leads to the correct thermal
and electromagnetic properties to display superconductivity.
The attraction between electrons necessary to form Cooper pairs
can in principle be due to any suitable kind of interaction. The dis
covery (Maxwell, 1950; Reynolds et al., 1950) that for many super
conducting elements the critical temperature depends on the isotopic
mass showed that for these substances the attractive interaction is one
between the electrons and the lattice. The BCS theory and its exten
sions have been worked out on this basis. However, the isotope effect
is apparently absent or considerably reduced in some transition metals
and their compounds (see section 8.1). Furthermore the effect of
pressure in transition metals does not correlate with the Debye
temperatures as it does in nontransition superconductors (Bucher
and Olsen, 1964). Kondo (1962) and Garland (1963a, b) have attri
buted these anomalies to the existence of overlapping bands in the
electronic energy spectrum at the Fermi surface. However, there is
also a hypothesis that in transition metals the attractive interaction
responsible for pairing may be a magnetic one (Matthias, 1960).
CHAPTER II
Phenomenological Thermodynamic Treatment
2.1. The phase transition
Long before the determination of the reversibility of the supercon
ducting transition by the discovery of the Meissner effect, attempts
had been made to apply thermodynamics to it by Keesom (1924), by
Rutgers (Ehrenfest, 1933), and in particular by Gorter (1933), who
virtually predicted the Meissner effect by pointing out that the success
of these early thermodynamic treatments strongly suggested the
reversibility of the transition.
The discovery of the Meissner effect finally enabled Gorter and
Casimir (1934a) to develop a full treatment of the superconducting
phase transition in a manner analogous to that of other phase transi
tions. They start with the fact that two phases are in equilibrium with
one another when their Gibbs free energies (G) are equal. The free
energy of a superconductor is most easily expressed by a diamagnetic
description developed in Chapter III, which attributes to the super
conductor a magnetization M (H e ) in the presence of an external
field H e . Then
V He
G,(H e ) = G/0) j dvj M(H e )dH e . (H.l)
For an ellipsoid, M(H £ ) is uniform, and
He
G s (H e ) = G s (0) VJ M(H e )dH e . (11.10
o
The last term in this expression gives the work done on the specimen
by the magnetic field. As the magnetization is diamagnetic, that is,
negative, the field raises the energy of the superconducting specimen.
It will be shown in Chapter III that only for a quasiinfinite cylinder
parallel to the external field does the superconducting phase change
into the normal one at a sharply defined value of H e . For all other
13
14 Superconductivity
shapes, there is an intermediate state consisting of a mixture of normal
and superconducting regions. Even under these circumstances, how
ever, any magnetic work is done solely on the superconducting por
tions, and for any shape of specimen this always equals, per unit
volume,
He
JM(H e )dH e = H 2 /Stt. (II.2)
o
Thus one can write for any specimen :
G,(/r c ) = (7,(0)+WJ c 2 /87r.
(H.3)
In the normal state the susceptibility is generally vanishingly small, so
that
G n (H c ) = G„(0).
Since the condition of equilibrium defining H C (T) is that
one has
G n (H c ) = G S {H C ),
01.4)
This is the basic equation of the thermodynamic treatment de
veloped by Gorter and Casimir. As S =  (dG/dT) Pi H , differentiation
of 11.4 yields
S n (0)  Sffl =  (VHJAtt) (dHJdT). (II.5)
At T= T c , H c = 0, and S n = S s . At any lower temperature, H c > 0, and
furthermore Figure 1 shows that for < T< T c , dHJdT < 0. Hence
the entropies of the two phases are equal at the critical temperature in
zero field; at any lower, finite temperature the entropy of the super
conducting phase is lower than that of the normal one, indicating that
the former is the state of higher order. This ordering will later be
shown to follow from a condensation of electrons in momentum
space. It follows from Nernst's principle that S n = S s at T= 0, so that
in this limit the slope of the threshold field curve must vanish. As the
entropies of the two phases are also equal at T= T c , their difference
must pass through a maximum at some intermediate temperature.
Phenomenological thermodynamic treatment 15
Equation II.5 also shows that the latent heat Q = T(S n — S s ) is zero
at the transition in zero field, and is positive when H c > 0. Thus there
is an absorption of heat in an isothermal superconductingtonormal
transition, and a corresponding cooling of the specimen when this takes
place adiabatically. The resulting possibility of cooling by adiabatic
magnetization of a superconductor was suggested by Mendelssohn
(Mendelssohn and Moore, 1934) and has been used by Yaqub (1 960)
for low temperature specific heat measurements of tin.
A further differentiation of II.4 yields, upon multiplication by T:
C s C n = (VT/47r)[H c (d 2 H c /dT 2 ) + (dHJdT) 2 ). (II.6)
Atr=r o ^ c = 0,and
C, C n = (*T/4tt) {dHJdTfj^ Te > 0, (H.60
so that the thermodynamic treatment predicts the observed dis
continuity in the specific heat. As the entropy difference between the
tv vo phases passes through an extremum at some temperature below
T c , the specific heats of the two phases at that temperature must be
16 Superconductivity
equal, and at even lower temperatures C s is smaller than C„. Both of
course tend toward zero at T= 0°. The variation of C s  C„ as a func
tion of temperature, as well as that of S s S„, are shown in Figure 8.
2.2. Thermodynamics of mechanical effects
The thermodynamic treatment developed thus far has ignored any
changes in the volume at the transition, as well as any dependence of
H c on pressure as well as on temperature. In taking these into account
one should begin by considering possible magnetostrictive field effects
on the volume in going from II. 1 to II. 1'. Ignoring this, however,
and noting (see Figure 1 1) that for the special case of a quasiinfinite
cylinder parallel to the external field the area under the magnetization
curve up to any field value H e < H c is equal to H 2 j%tt, one can write
QABQGjm = (VJ*ir)Hl 01.7)
Differentiating this with respect top in order to obtain V= (8GI8p) T H
yields
V 5 {H e ) Vjm = (H}l87r)(dV s !8p) T . (n.8)
Similar differentiation of II.3 and II.4 leads to
V n {H c )VM = *mv,EftT,p)l*n\
V n {H c )V s {0) = {H}l%ir)(dVJBp) T +{V s HJAn)(dHcl*P)T 01.9)
Comparing II.9 with II.8 shows that the first term on the righthand
side of the former is just the magnetostriction of the superconductor
upon changing the field from zero to the critical value. It is the second
term which gives the actual volume change at the transition :
V„(H C )V S (H C )  (V s H c /47r)(dH c l8p) T .
(11.10)
This term exceeds the magnetostrictive one by more than an order of
magnitude. The derivatives of 11.10 with respect to T and to p yield
expressions for the changes at the transition of the coefficient of
thermal expansion et=(l/V)(SV/dT) t and of the bulk modulus
K =  V{8pj8V). AtT= T c , H c = 0, this yields
and
«„«, = (U47T)(8H c ldT)(dH c l8p\ (11.11)
*„« = (K 2 /47r)(dH c /8p) 2 . (IU2)
Phenomenological thermodynamic treatment 17
There has been extensive experimental work on pressure effects on
the critical field. This has been reviewed by Swenson (1960) and sum
marized most recently by Olsen and Rohrer (1960). These latter
authors (1957) and, independently, also Cody (1958), have succeeded
in refining earlier work of Lazarev and Sudovstov (1949), and have
obtained for different superconducting elements empirical values of
the length change of a long rod at the transition. (Andres et at., 1 962).
Differences in the behaviour of transition and nontransition metals
have been pointed out by Bucher and Olsen (1964).
The magnitudes of the several mechanical effects are exceedingly
small. Typical values for 8HJ8p are of the order of 10~ 8 10~ 9
gauss/dynecm 2 , and the fractional length change of a long rod
is a few parts in 10 8 . Using the above thermodynamic relations this
yields a difference in the thermal expansion coefficient of about 10 7
per degree, and a fractional change in compressibility of one part
in 10 5 .
2.3. The interrelation between magnetic and thermal properties
One of the most remarkable features of the thermodynamic treatment
outlined in the preceding sections is the manner in which it links the
magnetic and the thermal properties of a superconductor. Equation
II. 5, for example, indicates that quite independently of the detailed
shape of the magnetic threshold field curve, its negative slope indi
cates that the superconducting phase has a lower entropy than the
normal one. The quantitative verification of an equation such as II. 6',
called Rutgers' relation, provides the best available confirmation of
the basic reversibility of the superconducting transition. The following
table, taken from Mapother (1962), compares for a few particularly
favourable elements the specific heat discontinuity measured calori
metrically, with its value calculated with II.6' from measured thres
hold field curves. The agreement is seen to be excellent:
Element
Indium
Tin
Tantalum
(millijoules/°mole)
975 962
106 1056
415 416
18 Superconductivity
The relations between the thermal properties and the threshold field
curve of course also imply that if a specific temperature variation is
either assumed or empirically determined for one of the former, this
uniquely specifies the temperature variation of the latter. Kok (1 934),
for example, showed that if one substitutes into equation II.6 a para
bolic variation of H c , as given by equation 1.2, one obtains a cubic
temperature variation of C cs . It was mentioned in Chapter I that both
of these are only fair approximations to the actual temperature
dependence of these quantities, and that in fact the threshold field can
be represented more accurately by a polynomial which in reduced
coordinates has the form
h(t) = 1 2 a n t\ (11.13)
The first coefficient a, must vanish, as otherwise S s —S„ would not
vanish at T= (see equation II.5), and 2>„ = 1 to make //(l) = 0. If
n
this polynomial is substituted into equation II.6, and one continues
to neglect any changes in the lattice specific heat, it follows that
~^f=(MWHHT^a 2 t 2
...)x
x(2a2) + (2<7 2 /...)}.
(11.14)
Of the two terms on the lefthand side, the second just equals the
Sommerfeld constant y. The first is subject to the following general
argument: As shown by equation II.5, S n > S s , and since S„ varies
linearly with T, S s must approach zero with some power of T greater
than unity. Hence one can write
o: T l+X ,x > 0,
so that
and
C„oc T l+X ,
CJTozT*.
It follows, therefore, that no matter what the precise temperature
dependence of C es is, CJT^Q as T+0°. Applying this limit to
equation 11.14 thus yields
y = (l/27T)a 2 (Hl/Tl). (11.15)
Phenomenological thermodynamic treatment 19
An equivalent expression results from applying the above argument
directly to equation II.6, and recalling that as T+0, dH c /dT+0. One
then obtains
y = (l^Tr^Hl/T^ihd^rldr 2 )^. (11.16)
Both of these last equations are exact expressions which permit the
evaluation of the Sommerfeld constant from a detailed knowledge of
the threshold field curve. Mapother (1959, 1962) has carried out a
searching analysis of the extent to which magnetic and thermal data
can actually be correlated in practice without introducing excessive
errors due to extrapolation; Serin (1955) and Swenson (1962) have
also discussed the relation between the two types of data.
2.4. The GorterCasimir twofluid model
The socalled phenomenological twofluid models of superconduc
tivity have in common two general assumptions:
(1) The system exhibiting superconductivity possesses an ordered
or condensed state, the total energy of which is characterized by an
order parameter. This parameter is generally taken to vary from zero
at T= T c to unity at T= 0°K, and can thus be taken to indicate that
fraction of the total system which finds itself in the superconducting
state.
(2) The entire entropy of the system is due to the disorder of non
condensed individual excited particles, the behaviour of which is
taken to be similar to that of the equivalent particles in the normal
state.
In particular, twofluid models make the conceptually useful
assumption that in the superconducting phase a fraction #" of the
conduction electrons are 'superconducting' electrons condensed into
an ordered state, while the remaining fraction 1 — #" remain 'normal'.
The artificiality of this division cannot be overemphasized; its use
fulness will presently appear.
The free energy per unit volume of the ' normal ' electrons continues
to be the same as that of electrons in a normal metal, that is
g n (T) = \yT 2 (11.17)
where y is the Sommerfeld constant. For the 'superconducting' elec
trons g s (T) is taken to be a condensation energy relative to the normal
CHRIST'S COLLEGE
LIBRARY
20 Superconductivity
phase, and the considerations of the first section of this chapter show
this to be
g s (T) = HllSrr. (H.1J
The total free energy per unit volume of the superconducting phase
containing a fraction #" of V electrons and 1  ^ of l n ' electrons is
therefore
G s (ir, T) = a(\  fT)g n {T) + b(iT)8 s (T). (11.19)
The simplest choice of a(liT) = 1iT; b(iT)=ir, makes
G&if, T) a linear function of 1P, so that the equilibrium condition
(3C/air)r = can be satisfied for only one value of Tat which iT can
assume any value between and l . This would mean that for any value
of W the normal and superconducting phases can be in equilibrium
at only that one temperature, which is not the case. Thus it is necessary
to choose a(l  iT) and b(iT) with more care. Gorter and Casimir
(1934b) chose
a(l  #") = (l  ir)\ b{iT) = Hr, (11.20)
so that
G 3 (ir,T) = \{\Hr)*yT 1 inill%iT. (11.21)
Applying the equilibrium condition yields
a(l  *0"~ ' m HllAn yT 2 , (IL22)
which at T c , with 1T  0, reduces to
y = (WttxHHIIT 2 .).
Substituting 11.23 back into 11.22:
(\1T)« 1 = (TJT) 2 = r 2 ,
Phenometwlogical thermodynamic treatment
21
so that
Hr = _^/(i~«0
(11.24)
Oi.:
Putting this back into 11.21 and differentiating to obtain other
thermal quantities yields
and
s s (ir,T) = yT(\iry = y r c r (1+0[)/(1  a) ,
C s (iT,T) = [(l + a)/(la)]yr c / (l+a)/(,  a \
The value of a must be chosen so as to give a reasonable fit to experi
mental data. With a = £ one obtains
QOT.n = 3yT c t\
S S (T) = yT c t\
and
iT(J) = l/ 4 ,
y = (WirKHlfT?).
(11.270
(11.260
(11.250
(11.230
Here again is the cubic temperature variation of the specific heat
which is only an approximation. Clearly no value of a will change
equation 11.27 into an exponential expression. A comparison of
equation 11.23' with equation 11.15 also shows that the choice a = £
makes a 2 =\, and reduces the polynomial representation of h(t) to a
parabolic form. This not only indicates once again the interrelation
between the magnetic and thermal properties, but also points up that
the GorterCasimir model can be used at best only semiquantita
tively. Within this limitation, however, the concept of the two inter
penetrating 'fluids' of condensed and uncondensed electrons is very
useful in obtaining a semiquantitative understanding of many super
conducting phenomena, and will be used repeatedly in subsequent
chapters.
There have been a number of attempts (see [5], p. 280) to improve
the quantitative aspects of the GorterCasimir model so as to yield
more nearly the correct exponential variation of C es and the corre
sponding nonparabolic dependence of h(t). These modifications have
either tried different functional forms for a(l  1T) and b(iT) in
equation TJ.19, or have introduced additional adjustable parameters.
Some of these variations do yield considerably better equations for
the thermal and magnetic superconducting properties. However, the
principal virtue of a twofluid model is to provide a conceptual tool
of primarily qualitative nature, and the various suggested improve
ments rarely add much to the basic physical picture of the two groups
°f electrons.
CHAPTER III
Static Field Description
3.1. Perfect diamagnetism
Even in the absence of a microscopic explanation of the phenomenon
of superconductivity, it is reasonable to assume that the vanishing of
the magnetic induction at the interior of a superconductor is due to
induced surface currents.! In the presence of an external magnetic
field, the magnitude and distribution of this current is just such as to
create an opposing interior field cancelling out the applied one. A
formal description of a macroscopic superconductor in the presence
of an external field H e is, therefore, the following:
in the interior: B, = H, = M, = 0, where M, is the magnetization
per unit volume;
at the surface: 3 S ^ 0, where 3 S is the surface current density; and
outside: B e = U e +H s , where H s is the field due to the sur
face currents.
It is this field which causes the distorted field distribution near a super
conductor as shown in Figure 3c.
Although this description is formally correct, it is much more con
venient to replace it by an equivalent one which treats the supercon
ductor in the presence of an external field as a magnetic body with an
interior field and magnetization such that
in the interior: B, = 0, H, ^ 0, M, ^ 0;
at the surface: 3 S = 0; and
outside: B e = H e + H s , where now H, is the field due to the
magnetization of the sample.
t That electron currents and not, for example, spins are responsible for
the diamagnetism of a superconductor is demonstrated by its gyromagnetic
ratio which is found to have the value of e\2mc (Kikoin and Goobar,
1940; cf. [1], p. 50 and p. 193 ; [2], p. 83).
22
As
Static field description
B = H+4ttM,
23
this description is equivalent to attributing to the superconductor a
magnetization per unit volume
M/= (1/4tt)H,
(HI.1)
which means that the superconductor has the ideal diamagnetic
susceptibility of — 1/47T.
3.2. The influence of geometry and the intermediate state
The great convenience of the diamagnetic mode of description is
illustrated by considering an ellipsoidal superconducting specimen in
an external field H e which is parallel to the major axis. The conven
tional proof, that inside a uniform ellipsoid B, H, and M are all con
stant and parallel to H e , is independent of susceptibility and therefore
applies to the superconductor. Further standard treatments show that
(with vector notation now unnecessary) :
H, = B.AirDMu
(IH.2)
where D is the demagnetization coefficient of the specimen. For an
ellipsoid of revolution this is given by
»(HMS'>
a and b are, respectively, the semimajor and semiminor axes, and
e  (1 b 2 /a 2 ) 112 . For an infinite cylinder with its axis parallel to H e ,
D = 0; for an infinite cylinder transverse to the field, D = \, and for
a sphere, Dl/3.
Combining III.l and III.2 yields :
and
M,= H e /47r(lD)
H, = HJ{lD).
(IU.3)
(IH.4)
In the neighbourhood of the superconductor, the external field is
distorted by the magnetization of the specimen. It follows from the
3
24 Superconductivity
continuity of the normal component of B and of the tangential com
ponent of //that for an ellipsoidal specimen the exterior field distribu
tion is as shown in Figure 9. At the equator of the specimen
Static field description
25
and at the pole
H~ = H,= H e l(lD),
H p = Bi = 0.
(III.5)
(IH.6)
For the longitudinal infinite cylinder with axis parallel to H e ,D =
and H eq = H e . The exterior field at the surface of the specimen is,
therefore, everywhere the same, and the cylinder remains entirely
superconducting until the applied field becomes equal to the critical
Fig. 9
value H c . The entire body then becomes normal. The magnetization
curve for such a specimen is shown in Figure 10, in which for con
venience AnM is plotted against H e .
For all other ellipsoidal shapes, D ^ 0, and the nonuniformity of
the field distribution around the superconductor raises the question
of what happens when H cq = H c > H e . To assume that a portion of
the specimen near the equator then becomes normal, as shown in
Figure 11, would lead to a contradiction: the boundary between the
superconducting and normal regions occurs where H = H C) but in the
now normal region the field would equal H e <H c \ There is, in fact,
no simple, largescale division of such a specimen into normal and
superconducting regions, which allows a field distribution such that
H>H C in the former, H < H c in the latter, and H=H C at the
boundaries.
He
i
(a) transverse
, H cu !i nder (aX
\p) sphere
(c) longitudinal
cylinder
\\
Superconducting state \\
Intermediate state
H c /2 2H c /3
Applied Field H e
Fio. 10
Fig. 11
Instead one must postulate, as was first done by Peierls (1936) and
b y F. London (1936), that once H e > (1  D)H C , the entire specimen
is subdivided into a smallscale arrangement of alternating normal
and superconducting regions, with B = H c in the normal regions, and
** = in the others. The distribution of these regions varies in such a
26 Superconductivity
way that the total magnetization per unit volume changes linearly
from
Mi = HJ47r(l  D) =  BJ4*r at H e = H c {\  D),
to M ( = at H e m H c .
Hence, for (1 D)H c <H e < H c ,
M,= (l/4irZ»(/f c fQ, (III.7)
Hi = H e AnDMi = H c , (UI.8)
Bi = H c (MD)(H C H e ). (III.9)
Magnetization curves for a transverse cylinder (D = I) and for a
sphere (D = 1/3) are also shown in Figure 10. Note that the area under
each of the curves is given by
j MidH e = H?I8tt.
(111.10)
This is just the magnetic work done on the specimen in raising the
field from zero to H c , as cited in equation II.2. In the region
(1 — D) H c =5 H e ^ H c , in which the specimen is neither entirely
normal nor entirely superconducting, it is said to be in the intermediate
state. The detailed structure of this state will be discussed in Chapter
VI ; at this time it is only necessary to emphasize that this intermediate
state exists, in some field interval, for any geometry other than that
of a quasiinfinite cylindrical sample parallel to the external field.
3.3. Trapped flux
It is important to distinguish the reasons and conditions for the inter
mediate state from those giving rise to the phenomenon of trapped
flux or the incomplete Meissner effect. As mentioned earlier, the
magnetic flux threading a multiply connected superconductor is
trapped by an indefinitely persisting current, and cannot change
unless the superconductivity of the specimen is quenched. A similar
situation can arise in a simplyconnected but nonhomogeneous
superconductor. Strains, concentration gradients, and other imper
fections can create inside a superconductor regions with anomalously
Static field description 27
high critical fields. Thus if such a nonideal specimen is placed in a
magnetic field sufficiently high to make it entirely normal, and the
field is then reduced, the anomalous regions will become supercon
ducting before the bulk of the specimen. Should some of these regions
be multiplyconnected, then the flux threading them at the moment
of their transition into superconductivity can no longer escape, and
is trapped even when the external field is reduced to zero, for as long
as the specimen remains superconducting.
Applied Field H e
Fig. 12
As a result, after it has once been normal in an external field, such
an imperfect specimen is less than perfectly diamagnetic in an external
field H< H c , and retains a paramagnetic moment in zero field. This
»s shown in Figure 12 for a long cylinder parallel to the field, using
the same units as in Figure 10. The ratio of m to H c is called the
fraction of trapped flux.
3.4. The perfect conductor
To emphasize once again the difference between a perfect conductor
a nd a superconductor, it is useful to outline an electromagnetic treat
ment of the former, as developed by Becker et al. (1933) just before
the discovery of the Meissner effect.
28 Superconductivity
In a perfect conductor, the equation of motion for an electron of
mass m and charge e in the presence of an electric field E does not
contain a retarding term and would simply be
mv = eE.
(IU.11)
In terms of the current density J = nes, where n is the number density
of the electrons, one can write III.l 1 in the form
E = (47rA 2 /c 2 ) J, 011.12)
where A 2 ■ mc 2 /4wne 2 . (IE. 13)
The parameter A has the dimensions of length, and for a density of
electrons corresponding to one electron per atom it has a value of the
order of 10 6 cm.
Using Maxwell's equation curlE = H/c, one finds that
(4ttA 2 /c) curl J + H = 0, (UI. 14)
and applying another Maxwell equation curlH = 4tt3jc yields for the
perfect conductor the equation
V 2 H = H/A 2 .
(111.15)
Von Laue (1949) showed that the solution of III. 15 for any specimen
geometry yields a value of H which decreases exponentially as one
enters the specimen. For a semiinfinite slab extending in the x
direction from the plane x = 0, the appropriate solution is
H(x) = H(0) exp (  x/X). (UI. 1 6)
Clearly, for x P A, H(x) « 0. Thus equation III. 1 6 confirms that in the
interior of a perfect conductor the magnetic field cannot change in
time from the value it had when the specimen became perfectly
conducting.
3.5. The London equations for a superconductor
The incorrectness of IU.16 was demonstrated by the discovery of
Meissner and Ochsenfeld (1933) that regardless of the magnetic his
tory of the specimen, the field inside a superconductor always
vanishes. F. and H. London (1935a, b; see [2]) therefore proposed to
Static field description 29
add to Maxwell's equations the following two relations in order to
treat the electromagnetic properties of a superconductor:
and
E = (^AVJ),
(47rA 2 /c)curlJ+H = 0.
(A)
(B)
Replacing the field by a vector potential curl A = H and choosing a
gauge such that div A = 0, (B) reduces to
4ttA 2
J + A = 0.
(BO
Note that (A) is identical to UI.2, and thus describes the property of
perfect conductivity, but that the difference between (B) and III.4 is
the important one that application of Maxwell's equations now leads
to
V 2 H = H/A 2 . OH. 17)
Solution of this for any geometry now shows that H, and not only H,
decays exponentially upon penetrating into a superconducting speci
men. For the semiinfinite slab described above, the solution of III. 17
is
H(x) = H(0)exp(x/A), (IU. 18)
which shows that for x > A, H(x) « 0, in accordance with the Meissner
effect.
Clearly the London equations (A) and (B) do not, in fact, yield the
complete exclusion of a magnetic field from the interior of a super
conductor. Instead, the*y predict the penetration of a field such that
it decays to 1 /e of its value at the surface in a distance A. This is called
the London penetration length. Its existence has been fully confirmed
experimentally, although empirical values are consistently higher
than those predicted by the defining equation ni.13, as will be dis
cussed in a later chapter. The existence of this slight penetration of an
exterior field must be taken into consideration in the discussion of
superconducting thin films, wires, or colloidal particles, and in a
detailed treatment of the intermediate state.
30 Superconductivity
Applying curlE = — H/c to equation (B), one obtains
curl[E(47rA 2 /c 2 ) J] = 0,
showing that E— (47rA 2 /c 2 ) J = grad<£,
where <f> is a scalar. In the most general case of a multiply connected
superconductor or a superconducting portion of a currentcarrying
circuit, one cannot prove that <£ vanishes. Hence (A) does not always
follow from (B) and the perfect conductivity implicit in (A) and the
perfect diamagnetism in (B) must be considered as independent
postulates.
In a system of N particles of charge q described by the wave
function
^ / (r l ,r 2 ,...,r N ),
the mean current density at a point R in the presence of a magnetic
field
H(r a ) = curlA(r a ) (III. 19)
is given by
N
1 N
s 2
r U(Rr.
 ^AOrJ V * f 8(R ra )</r, . . .dr N . (111.20)
tnc J
In the absence of a field, A(r a ) ■ 0, W=W Q , and the current density
vanishes, so that
> [■■■ fl^tnv^o^VamscRrjx
Q = l , N
xdr t ...dr N = 0. (IH.21)
If, therefore, one assumes that the wave function W is perfectly rigid
under the application of a magnetic field, that is, that W= X F () always,
then it follows that
N 2
J(R)= 2J ...J ^ c A(r )W*V8Ql.r x )dr l ...dr N . (111.22)
Static field description 3 1
By defining a particle density
/i(R) = S j ...j x P*Y8(R.r a )dr l ...dr N , QSU8)
1 N
equation 111.22 can be written as
J(R) = n(R) — A(R).
mc
(111.24)
But if the particle density n(R) is a sufficiently smooth function so that
one can replace it by a constant n, then in view of the defining equation
III. 13, 111.24 is seen to be identical to (BO
Thus the London equation (B) or (B') implies that the magnetic
properties of a superconductor are due to a complete rigidity of the
wave functions of the superconducting carriers. In F. London's own
words ([2], p. 150): '...superconductivity would result if the eigen
functions of a fraction of the electrons were not disturbed at all when
the system is brought into a magnetic field (H< i/ c ).'
A possible explanation of this is contained in the London equations
themselves. The mean local value of the carriers momentum in the
presence of a field is given by
p = W v + (9/c)A,
which can be rewritten as
P = (^)[(4ttA 2 /c)J + A].
(111.25)
In the same gauge as that leading to (BO, 111.25 for a simply connected
superconductor reduces to
p, = 0. (IIL26)
The London equation thus implies that superconductivity is due to a
condensation of a number of carriers into a lowest momentum state
P s = 0. By the uncertainty principle this requires the essentially un
limited spatial extension of the appropriate wave functions, and
makes it impossible for them to be affected by local field variations.
It also follows from 111.26 that
v* = (qlmc)A,
(IH.27)
32 Superconductivity
showing that in a simply connected superconductor the charge flow
is entirely determined by the externally applied field, and exists only
in its presence.
3.6. Quantized flux
F. London already observed ([2], p. 151] that the unlimited extension
of the wave function of the superconducting charge carriers has a very
fundamental consequence in a multiply connected superconductor.
Consider, for example, a superconductor containing a hole. The wave
functions must then be single valued along any closed path enclosing
the hole. By analogy to the electronic wave functions in an atomic
orbit one can then apply the BohrSommerfeld quantization rules
and require that for the superconducting charge carriers
(J> pdl = nh,
(IU.28)
along any path enclosing the hole. According to 111.25 this then
means that
jMttA 2 m r he
(b Jdl+&>Adl = n (m.29)
Since H = curlA, the contour integral of A is equal to the surface
integral of H over the area enclosed by the contour, and this in turn
equals the magnetic flux <P threading the contour:
Thus
<J)Arfl= f f Hc/S = 0.
r
* he
J<fl+0 = n— •
Q
(111.30)
(111.31)
London called the lefthand side of this equation a. fluxoid, and we see
that according to 111.31 such a fluxoid is quantized in integral
multiples of
he
& »  ' (H1.32)
Note that if the contour is taken at a distance from the hole large
compared to the penetration depth A, the current density vanishes,
Static field description 33
and the fluxoid is just equal to the total flux associated with the hole.
This flux is thus seen to be quantized.
The quantization of flux was verified experimentally by Doll and
Nabauer (1961) and by Deaver and Fairbank (1961). These experi
ments have shown that the quantum of flux is given by
A  kC
2x10 7 gausscm 2 .
This shows that q = 2e, that is, that the superconducting charge
carriers are pairs of electrons. It has already been mentioned that
indeed this is the fundamental premise of the microscopic theory.
A number of authors (Byers and Yang, 1961; Onsager, 1961;
Bardeen, 1961b; Keller and Zumino, 1961; Brenig, 1961) have
extended the London argument for flux quantization in a rigorous
fashion. In particular Byers and Yang as well as Brenig have shown
explicitly that the quantization is due to a periodicity of the free
energy of the superconductor as a function of flux. The free energy of
the normal phase is essentially independent of flux, and there must
therefore occur a corresponding periodic variation of the critical
temperature at which the free energies of the two phases are equal.
This flux periodicity of T c has been observed by Little and Parks
(1961).
CHAPTER IV
The Pippard Nonlocal Theory
4.1. The penetration depth A
The London equations lead to an exponential penetration of an
externally applied magnetic field into a superconductor, so that the
penetration can be characterized by the depth A at which the field has
fallen to 1/e of its value at the surface. Quite in general, and inde
pendently of any particular set of electromagnetic equations for the
superconductor, one can define the penetration depth for an infinitely
thick specimen by
■±J
H{x)dx.
(1V.1)
This would apply equally well to an exponentially decaying field as
to one, improbable though it may be, which remains constant to a
certain depth and then vanishes suddenly.
Shoenberg ([1], p. 140) has pointed out that in this way one can
treat problems involving either very thick specimens (thickness a > A)
or very thin ones {a <^ A) independently of a detailed knowledge of
the appropriate electromagnetic equations. Using IV. 1 to calculate
the ratio of the magnetic susceptibility \ of a sample into which the
applied field has penetrated, to the susceptibility xo of an identical
sample from which the field is entirely excluded, he finds equations
of which the following are applicable to a plate of thickness la in a
uniform field parallel to its surface :
x/Xo = 1  A/a for a > A,
x/xo = aa 2 /A 2 for a < A.
(IV.2)
(IV.3)
The detailed form of the field penetration does not enter at all into
IV.2 and all other equations for large specimens of other shapes, and
does so only through the numerical parameter a in the equations for
34
The Pippard nonlocal theory 35
small specimens. As a consequence it is impossible to test the validity
of any particular penetration law, such as, for example, the London
relation III. 17, by measurements on large specimens; with very small
specimens this can only be done if one can determine absolute values
of a and of A, which is very difficult. On the other hand, one can
measure the variation of the susceptibility of large or of small speci
mens with any parameter affecting only A: temperature, external
field, impurity content, etc. One can then deduce directly the variation
of A with the parameter in question, without having to make any
assumptions about the true penetration law. The results of such
measurements can therefore help to choose between different electro
magnetic theories if these predict different parametric dependences of
the penetration depth.
The oldest method of measuring the penetration depth consists of
determining the magnetic susceptibility of samples with large surface
to volume ratios to make the penetration effects appreciable.
Shoenberg (1940) measured the temperature dependence of the sus
ceptibility of a mercury colloid containing particles of diameter
between 100 and 1000 A. Desirant and Shoenberg (1948) used com
posite specimens consisting of about 100 thin mercury wires of
diameter about 10" 3 cm, and Lock (1951) carried out extensive
measurements of the magnetic behaviour of thin films of tin, indium,
and lead. Casimir (1940) suggested a method using macroscopic
specimens in which the mutual inductance was measured between
two coils closely wound around a cylinder of superconducting
material. It was applied successfully by Laurmann and Shoenberg
(1947, 1949), by Shalnikov and Sharvin (1948), and most recently
with certain refinements by Schawlow and Devlin (1959), and by
McLean (1960).
A superconductor has finite surface impedance at high frequency,
and this impedance is limited in the superconducting phase by the
penetration depth A, as in the normal phase it is limited by the skin
depth S. Pippard (1947a) was the first to use this as a means of
measuring A, and he and his collaborators have carried out a large
number of experiments at different frequencies and varying experi
mental conditions (see Pippard, 1960). Basically all these measure
ments involve observing the change in the resonant frequency of a
36 Superconductivity
cavity containing the specimen when the specimen passes from the
normal to the superconducting phase. At T< T c , where \<8, these
changes are proportional to 8  A. If 8 is independent of temperature,
as is the case for a metal in the residual resistivity range, then any
temperature variation of the observed changes must be due to the
temperature variation of A. Dresselhaus et al. (1964) use instead of a
cavity a rutile resonator to which the specimen is coupled, and also
observe changes in the resonant frequency.
Schawlow (1958), Jaggi and Sommerhalder (1959, 1960), and most
recently Erlbach et al. (1960) have measured the penetration of a
magnetic field through a thin cylindrical film of thickness less than
the penetration depth.
4.2. The dependence of A on temperature and field
According to the London theory, an external magnetic field pene
trates into a superconductor to a depth characterized by (see equa
tion HI. 13)
A = (mc 2 l4irn s e 2 ) 112 ,
where n s is the number density of the superconducting electrons. It is
reasonable to expect this to be the only temperaturedependent factor
in this defining equation, and in fact the GorterCasimir twofluid
model assumes that
n s (t) = #xq*m av.4)
where i^it) is the order parameter.
The temperature dependence of W is given by 11.25', so that sub
stituting this and IV.4 into the defining equation for the penetration
depth one obtains
A(/) = A(0)/(l/ 4 ) ,/2 , (IV.5)
where A(0) = (mc 2 /47r/;/0)e 2 ) ,/2 (IV.6)
is the penetration depth at T= 0°K. Very near T c , IV.5 can be written
as
*»£Ww.
(IV.7)
The Pippard nonlocal theory 37
sented to a very high degree of approximation by IV.5. This is a
striking success of the phenomenological theories discussed in the
preceding chapters. Close inspection of the recent very precise
measurements, however, shows a small deviation from IV.5 at
/ < 08, which becomes particularly pronounced at low temperatures.
This deviation is barely discernible in the normal plot of A(r) vs. y(t),
where y(.t) = (1f 4 ) 1 ' 2 , but is displayed strikingly in Figure 13,
which shows for Schawlow's results (1958) the variation of the slope,
1400p
1200 >■
1000
d y 600
400
200
a
BCS
THEORY
1
Daunt et al. (1948) were the first to point out that the empirical
temperature variation of the penetration depth can indeed be repre
1.0 1.5 2.0 25 3.0 3.5 4.0 4.5 5.0
Fig. 13
dXjdy, with y(t). The solid line indicates the values of dX/dy calculated
by Miller (1960) on the basis of the BCS theory; the experimental
results appear to deviate less from IV.5 than is predicted by theory.
Furthermore it appears that in impure specimens no deviation from
IV.5 can be found at all (Waldram, 1961).
The slope of A(r) plotted as a function of y(t), as well as the inter
cept, yield values for A(0) if one ignores the small deviations from
IV.5. Appropriate empirical values for pure bulk samples are shown
in Table II. They exceed by a factor of about five what one would
expect from the London definition IV.6, unless one makes rather
38 Superconductivity
unlikely assumptions of low densities of superconducting electrons
or of a large effective mass. Experiments on very small samples, and
measurements on impure metals, yield even higher values of A(0),
although none of the factors in IV.6 appear to depend on size or
purity. This failure of the London theory will be discussed in the
following sections of this chapter.
Table II
lement
A(0)(A)
Reference
Al
500
Faber and Pippard, 1955a
Cd
1300
Khaikin, 1958
Hg
380450*
Laurmann and Shoenberg, 1949
In
640
Lock, 1951
Nb
440
McLean and Maxfield (1964)
Pb
390
Lock, 1951
Sn
510
Pippard, 1947; Laurmann an
Shoenberg, 1949; Lock, 1951
470600*
Schawlow and Devlin, 1959
Tl
920
Zavaritskii, 1952
* Anisotropy.
Pippard ( 1 950) investigated the change of the penetration of a small,
r.f. field (94 kMc/s) at a given temperature as an external d.c. field
H e is raised from zero to the critical value. This change, divided by
the penetration depth in zero field, is plotted against temperature in
Figure 14. There are clearly two effects: one at low temperatures
(which Bardeen (1952, 1954) has shown to follow from an extension
of the London equations to include nonlinear terms), and one near
T c . This latter involves a change in A with H in just that region in
which A varies appreciably with temperature. By a thermodynamic
derivation Pippard has shown that this temperature variation of A
leads to a dependence of the superconducting entropy on field. He
finds that
S(H e )S(0) = ^^V/U ~/ 4 )] av>8)
for a superconductor of total surface A, assuming IV.5 to hold. Near
T c this change contributes as much as onefourth of the total entropy
The Pippard nonlocal theory
39
difference between the normal and superconducting phases, which is
quite considerable.
Pippard pointed out that to assume that the entire entropy change
takes place in the thin layer into which the field penetrates would,
therefore, result in an unreasonably high entropy density in this layer.
Yet this is just what one is led to believe by the London model, accord
ing to which the superconducting wave functions or, in twofluid
language, the corresponding order parameter #", remains rigidly
unchanged by the application of an external field. Any change in the
Z0 25 3.0
T(°K)
Fig. 14
35 T C
thermodynamic functions with field must therefore be confined to the
thin layer into which the field penetrates.
Recent measurements of the field dependence of the penetration
depth at 1 and 3 kMc/s (Spicwak, 1959; Richards, 1960, 1962; Pip
pard, 1960; Dresselhaus et al., 1964) have shown that this effect has
certain unexpectedly complicated features. Not only is the magnitude
of the change frequency dependent, but even its sign can change under
certain conditions. In particular there can be an increase in the pene
tration depth when the applied field is parallel to the specimen surface,
and at the same frequency a decrease when the field is perpendicular.
Bardeen (1958) and Pippard (1960) have suggested that these com
plexities may be due to field induced deviations of the superconducting
and normal electron densities from their equilibrium values.
4
40
Superconductivity
4.3. The range of coherence
The unreasonably high entropy density in the surface layer led
Pippard (1950) to propose a basic modification of the London model,
according to which the order parameter changes gradually over a
certain length f , which he calls the range of coherence of the super
conducting wave functions. In terms of the microscopic theory this
distance can be considered as the typical size of the Cooper pairs. Any
change in the thermodynamic functions of course extends over as
wide a region as the change in the order parameter, and thus a value
£ > A would correspond to a more reasonably small value of the field
induced entropy density.
Pippard (1950) obtained an estimate of the range of coherence of
the order parameter by minimizing the Gibbs free energy of the super
conductor in the presence of an external field. The resulting relation
between the fractional change of the penetration depth and the ratio
A(0)/£ allows him to estimate from his experimental data on the field
effect on A that £ M 20A(0) « 10 4 cm. Such a distance is much larger
than the smallest colloidal specimen or the thinnest films in which
superconductivity is still known to exist, and it is therefore of great
importance to realize that, to quote Pippard (1950, p. 220) : ' the range
of order must therefore not be regarded as a minimum range necessary
for the setting up of an ordered state, but rather as the range to which
order will extend in the bulk material'.
Strong support is given to the existence of this range of coherence
by the extreme sharpness of the superconducting transition under
suitable conditions. De Haas and Voogd (1931) have observed
resistive transitions in single crystals of tin taking place within a range
of one millidegree, and a sharpness approaching this value has come
to be the criterion for the quality of a specimen of suitable shape and
orientation. Applying a simple statistical argument, Pippard shows
that fluctuations would create a broader transition unless the super
conductivity of a bulk sample can be created or destroyed only over
an entire domain of diameter M 10A(0).
In the next section as well as in 6.5 it will be shown that the range of
coherence is much smaller than 10 ~ 4 cm in low mean free path alloys
as well as in certain pure metals. For such materials one would expect
The Pippard nonlocal theory 41
a broadened transition even for ideally homogeneous samples.
Goodman (1962c) has discussed this in some detail.
4.4. The Pippard nonlocal relations
In 1953 Pippard measured the penetration depth in a series of dilute
alloys of indium in tin, and found that the decrease in the normal
electronic mean free path of the metal was accompanied by an
10
E
o
.L
10
20
)0 6 4
40
30.
(cm)
Fig. 15
50
appreciable rise in the value of A(0). This has been confirmed by
Chambers (1956), and by Waldram (1961), whose results are shown
in Figure 15. Such a dependence of A(0) on the mean free path is quite
incompatible with the London model, for clearly none of the para
meters in the defining equation IV.6 varies appreciably with the
electronic mean free path.
This experimental result, added to the previous questions which had
been raised about the correctness of the London phenomenological
treatment, led Pippard (1953) to develop a fundamental modification
of this model, based on the concept of the range of coherence of the
42 Superconductivity
superconducting phase. The basic London equations, it will be
remembered, lead to the relation
J(R)= 
4nXl
A(R),
where A^, = mc 2 /4ime 2 , so that one can also write this as
„2
IW~
J(R) = A(R).
(IV. 10)
(IV.ll)
One way to introduce a dependence of the penetration depth on the
electronic mean free path is to write
(IV.P1)
where £ is a constant of the superconductor in question, and £(/) a
parameter depending on the mean free path /. It is evident from the
analysis in Chapter HI that IV.P1 leads to an expression for the field
penetration into a semiinfinite slab which has the London form:
but where now
H{x) = H e exp(x/X),
X = X L
(TV. 12)
As experimentally A is found to increase with decreasing /, it is clear
that £(/) must decrease as / decreases.
As a first step toward a modification of the London model, Pippard
identifies £ with the range of coherence of the pure superconductor,
and assumes that £(/) tends toward this value as />• a>, but that
£(/) >/as />0. This is the case if, for example,
1 1 1
(IV. 13)
where a is a constant of order unity. £(/) is thus an effective range of
coherence which has a size (w 10 " 4 cm) characteristic of the metal in
a pure superconductor, but which becomes limited by the normal
electronic mean free path as the latter becomes much smaller than
10 4 cm.
The Pippard nonlocal theory 43
These equations satisfactorily explain the onset of a mean free path
effect on the penetration depth at a critical value of /, as found by
Pippard and others, as well as the very large penetration depth values
obtained from experiments where / is limited by boundary scattering.
They do not, however, satisfactorily explain the finding that A(0) in
pure, bulk superconductors exceeds the London value IV.6 by a
factor of four to five. According to Pippard this is because IV.P1 does
not correctly describe the relation between current density and the
vector potential in such a case. PI still implies, as does equation
III.B', the basic London idea of a wave function which is completely
rigid under the application of an external field because the electronic
momenta are ordered or correlated over an infinite distance. Thus the
distance over which H and A vary is quite immaterial ; the same kind of
relation would hold if the field varied very slowly as if it varied very
rapidly. But according to Pippard the range of momentum coherence
is not infinite but only about 10~ 4 cm, so that the electromagnetic
response of the superconductor should be affected profoundly if the
field varies rapidly over this distance. A relation like PI could apply
only if the field varied slowly over a distance of the order off.
The situation is somewhat analogous to the problem of electrical
conductivity in a normal conductor, for which the relation
J(R) = a(/)E(R)
(IV. 14)
is valid only if E(R) varies slowly over a distance of the order of/. An
applied alternating field penetrates only a finite distance, S, which
varies inversely as the square root of the frequency. At sufficiently low
temperatures and high frequencies, the electronic mean free path in
the normal metal may be longer than this skin depth, so that electrons
may spend only part of the time between collisions in the field pene
trated region. Pippard (1947a) showed how this makes the electrons
less effective as carriers of current and leads to a higher surface
resistance, as observed by H. London (1940) and Chambers (1952).
Under these conditions Ohm's law (IV. 14) can no longer be a valid
approximation; the current at a point must be determined by the
integrated effect of the field over distances of the order of the mean
free path (see Pippard, 1954). The details of this socalled anomalous
44 Superconductivity
skin effect were worked out by Reuter and Sondheimer (1948), who
derived that
3a CR(RE)e R "dT
JCR)_ 4ir/J F
(IV. 15)
where a is the d.c. conductivity and / the mean free path. The form
of this equation ensures that in the case of a rapidly varying field the
current density at a point R is determined by the integral of the field
over a distance comparable to the mean free path /.
In a superconductor of range of coherence £, the current density
at a point in the case of a rapidly varying field should also be deter
mined by an integral of the field over a distance of the order of g, and
not, as is implicit in the London equation as well as in PI, by the field
variation over a quasiinfinite distance. Because of this analogy to the
anomalous conduction in a normal metal, and because some special
solutions of equation IV. 15 were already known, Pippard (1953)
proposed as the basic relation for the electromagnetic response of a
pure superconductor the equation
J(R)= 
3ne 2 C R(RA)e~ R ltdr
4n$ m
S
R*
(IV.P2)
Somewhat misleadingly, as this erroneously implies that the basic
London equation is a truly local relation, P2 is called the Pippard
nonlocal relation.
This relation leads to a reversal of the phase of the magnetic field
penetrating into a superconductor (Pippard, 1953). Drangeid and
Sommerhalder (1962) have observed this effect.
The validity of P2 is strongly supported by Bardeen's proof ([5],
pp. 303 ff.) that an energy gap in the single electron spectrum requires
a nonlocal relation between current density and vector potential. In
fact the BCS theory leads to a relation entirely equivalent to P2 if one
assumes
& = nv /7re(0), (IV. 16)
where 2e(0) is the energy gap at 0°K. Substituting the BCS value
2e(0) = 352k B T c , IV.16 becomes
g, = 0l8hv /k B T c .
(IV.17)
The Pippard nonlocal theory 45
This is just the expression IV.9 for the range of coherence derived
from an uncertainty principle argument, with a = 018.
From P2, the penetration depth A as defined by IV. 1 can be
evaluated explicity in two limiting cases:
A = V(Zolt)*L for 5< A, (London limit), (IV. 18a)
A. feM
1/3
for g> A, (Pippard limit). (IV. 1 8b)
The second of these is the one applicable to the case of an infinite
mean free path, and correctly predicts a penetration depth into very
pure superconductors which is much larger than the London value.
IV. 18a is identical to IV. 12 obtained directly from PI. This is of
course a reflection of the fact that PI is the limiting form for £ <^ A of
the more general equation P2.
Equations IV. 18 show that the range of coherence of a super
conductor can be calculated from absolute values of the penetration
depth. Faber and Pippard (1955a) have in this way obtained values
of 21 x 10 5 cm for tin, and 123 x 10 5 cm for aluminium. These
values differ very much, but when substituted into equation IV.9
together with known values of T c and v (from anomalous skin effect
data [Chambers, 1952]), both correspond to a = 015. This is in
striking agreement with the BCS value of 018, as cited in IV.17. A
later chapter will mention how measurements of the transmission of
infrared radiation through thin superconducting films lends further
strong support to this value.
Peter (1959) has solved the Pippard nonlocal relation P2 for the
case of cylindrical superconducting films of thickness d < A and
radius r. He finds that an external field H e penetrates through the
film to a value Hi such that
HJH^Vrd^faFitld).
(IV. 19)
£o 's the range of coherence in a specimen of unlimited mean free path,
and can be calculated from IV.17; £ is the actual range of coherence
in the film, and A should be the London penetration depth as calcu
lated from IV.5 and IV.6. Schawlow (1958), however, has shown that
46 Superconductivity
good agreement with his measurement on tin films can be found by
substituting for A the empirical value for bulk samples (510 A) and
considering £ as being determined by the sizelimited mean free path
of the electrons in the films. A similar analysis has been used by
Sommerhalder (1960).
It is now generally accepted that whenever one applies the equation
of the Pippard theory (or those of the GinzburgLandau treatment
to be discussed presently) to the case of small or impure specimens,
one obtains good agreement by using for the ideal penetration depth
in a bulk sample, not the London value A^. but rather the depth deter
mined experimentally. For example, the results of Whitehead (1956)
on the magnetic properties of mercury colloids were shown by
Tinkham (1958) to be in excellent agreement with the prediction of
the London limit of the Pippard theory if one modifies equation
IV.18b and writes
■\W
(IV.20)
where X b is now the empirical penetration depth for a bulk sample
and takes the place of the London value A L . The mean free path / is
limited by boundary as well as by impurity scattering. Ittner (1960a)
has similarly found that such a modification of the Pippard equations
adequately predicts the results of the observations by Blumberg (1 962)
of the critical field of moderately thin films. In analysing the magnetic
behaviour of small (or very impure) specimens, for which £ * / <^ A, it
is thus in general possible to obtain adequate precision without
attempting to solve the difficult relation IV. P2. Instead one can use
IV.20 to calculate the penetration depth, and then substitute this value
of A into the London equation IV.10.
In discussing the mean free path dependence of the coherence
length one must remember that it is related to the behaviour of a
superconductor in two subtly different ways. One of these, as men
tioned in Section 4.3, is the distance over which the order parameter
of the superconducting phase varies. It is this aspect which, for
example, in Chapter VI will enter into the discussions of the width
of a boundary between the normal and superconducting phases.
It follows from Gor'kov's analysis of the influence of impurities
The Pippard nonlocal theory 47
(Gor'kov 1959b) that the mean free path dependence of this aspect of
the range of coherence is given by
£ = £oX 1/2 (0 (IV.21)
x(/) is a function of the mean free path shown graphically by Gor'kov
and approximated to within about 20 per cent by the simple expression
(Douglass and Falicov, 1964)
x(/)
(s4
In the limit / <^ £ , IV.21 thus reduces to
i = V(t o 0
(IV.22)
(IV.23)
The relatively slow variation of this aspect of the coherence length
with mean free path is essentially due to the fact that not every
electronic collision destroys the superconducting coherence.
The other aspect of the range of coherence is that it determines the
distance over which the magnetic field or the vector potential at a
given point influences the current density. This is expressed by the
Pippard equation IV.P2. What is important in this application is the
actual mean distance between electron collision, so that now equa
tion IV. 13 applies. This means that
$ « / (IV.24)
for / <g £ . It is this mean free path dependence which enters, for
example, into equation IV.20.
CHAPTER V
The GinzburgLandau
Phenomenological Theory
In 1950 Ginzburg and Landau (GL) introduced a phenomenological
approach to superconductivity which, like that of Pippard, modifies
the absolute rigidity of the superconducting order parameter or wave
function which is implicit in the London model. Although the theory
was originally formulated so as to reduce always to the 'local'
London equations in zero field, Bardeen (1954) has shown that it can
be modified so as to be compatible with a nonlocal equation of the
Pippard type. Furthermore, Gorkov (1959, 1960) has derived the GL
equations, under certain conditions, from his formulation of the BCS
theory.
GL introduce an order parameter >p which they normalize so as to
make \*fi\ 2 = n s , where n s is the density of the superconducting elec
trons, ip is thus a kind of 'effective' wave function of the supercon
ducting electrons. According to the general LandauLifshitz theory
of phase transitions (1958), the free energy of the superconductor
depends only on \ifj\ 2 and can be expanded in series form for tem
peratures near T c . In the absence of an external field, the supercon
ducting free energy (per unit volume, as are all equations listed) is then
(7,(0) = G B (O) + o# 2 +(8/2)0 4 . (V.l)
Minimizing the free energy with respect to \<p\ 2 yields the zero field
equilibrium value
M>l 2 =«/A (V.2)
from which (7,(0)  (7„(0) = <x 2 /20. (V.3)
In the immediate vicinity of T c one can assume that the coefficients a
and /? have the simple form
*(j) = (r c r)(sa/ar) r=re
and jB(D = p(T c )  fi e
48
(V.4)
The GinzburgLandau phenomenological theory 49
With these one then finds from V.3, remembering that the free energy
difference between the phases equals the magnetic energy, that
47ra 2 47r(r c r) 2/fl  x2
2 _
Ht =
\8T/ T=Te
(V.5)
P Pc
Near T c , H c indeed is known to vary linearly with (T c T), so that
the correctness of equation V.5 justifies the assumptions V.4. All
further thermodynamic manipulations are now possible, but they
and all other conclusions drawn using V.4 are restricted to tempera
tures very near T c . Both Bardeen (1954) and Ginzburg (1956a) have
considered extensions of the model to the full superconducting range
by introducing different forms for a(T) and @{T), the former using
expressions based on the GorterCasimir twofluid model.
The outstanding contribution of the GL model in any temperature
range arises from its ability to treat the superconductor in an external
field H e « H c . The free energy G s (H e ) is now increased not only by the
usual volume term H*/9v, but also by an extra term connected with
the appearance of a gradient of i/j, as ifj is not completely rigid in the
presence of H e . Such a gradient would contribute to the energy in
analogy to the kinetic energy density in quantum mechanics which
depends on the square of the gradient of the wave function. Intro
ducing this extra energy is equivalent to requiring that */> not change
too abruptly. One is thus led to a concept of gradual, extended varia
tions of the superconducting order parameter quite analogous to
Pippard's model of the range of coherence.
In order to preserve gaugeinvariance, GL assume the extra energy
term to be
JL/^^aJ , (V.6)
where A is the vector potential of the applied field, and e* a charge
which, as stated in the original version of the theory, 'there is no
reason to consider as different from the electronic charge'. Modifica
tions of this view will be discussed presently.
GL thus write
t2 1 r * 12
(V.7)
G s (H e ) = GM + fr+^A W*~A
Uc m ~M 
50 Superconductivity
One must now minimize this with respect to both tp and to A, which
leads to the two equilibrium equations:
2m\ c J r dtfj*
V 2 A = ^J, = t^'w^W^
c mc
(V.GL1)
+ •
Aire*"'
mc
I0I 2 A.
(V.GL2)
In a very weak field, HxO, the function iff remains practically con
stant (that is, rigid), V0 = 0, 4> » 0o> and GL2 reduces to
V 2 A
Ane* 2
mc'
l^ol 2 A =
Aire* 1
« C A.
(V.8)
</ 2
dz 2
tN**w^*t^* H
^4
A 2
mc 2
(V.10)
Here, as in V.2, the subscript denotes the zero field value. This of
course is just London's expression (B). Nonlocal versions of the GL
treatment are obtained by substituting an integral expression for the
second term in V.6. In its present local form the GL treatment is
restricted to temperatures near T c for two reasons: in the first place
because of the simple forms assumed for the functions a(T) and fi(T),
and secondly because only near T c is A §> £ , and can the nonlocal
electromagnetic character of superconductivity be ignored.
The set GLl and GL2 of coupled nonlinear equations in ifj and
A have been solved for essentially onedimensional problems. Taking
the zaxis to be normal to the infinite superconducting boundary, the
field H along the>axis, and the current J s and potential A along the
xaxis, one obtains (using V.l)
The GinzburgLandau phenomenological theory 5 1
Note that with this geometry
H = curlA = ~
dz
The meaning of these equations becomes clearer by introducing a
dimensionless parameter k defined by
2e* 2
KT := ., .,/ZcAq,
h 2 <
where
Ag
m
c 2
4ne* 2 ifa
(V.ll)
(V.12)
The subscript again denotes zero field, k, A , and H c are the three
parameters of the GL theory which are to be determined experi
mentally, and in terms of which various field and size effects can
be expressed. H c is the bulk critical field. A is the empirical penetra
tion depth of a superconductor in the weak field limit, and is the
quantity which through equation V.12 determines the zero field
equilibrium value of the order parameter i/jq. For a bulk sample con
taining impurities A increases, as was discussed in the previous
chapter, and this in turn affects both O and k.
k can be determined in a number of ways, two of which follow
directly from the defining equation V. 1 1 . In the immediate vicinity of
T ct the experimental variation of A (/) can be expressed by IV.7:
Also one can write
H c =
dH r
so that
8/jV
dT
dH r
xAT,
T=T e
dT
T=T„
xT 2 xX 4 (0).
(V.l 3)
Thus k is seen to be temperature independent, at least for T& T c .
52 Superconductivity
One can also use the expression for the penetration depth derived
from the BCS theory to be, very near T c :
m = x ^ (l _„./! _y?/ik
J/2
V2
V2 \ATj
so that
,2 _
,*2
4/i 2 ,
dH„
dT
t=t c
x7?xAf(0).
(V.14)
where now A L (0) is the London penetration depth calculated from
111.13 using the actual free electron density n. This can be calculated
from the value of the normal state anomalous skin resistance
(Chambers, 1952).
Another method of calculating k for a given superconductor is to
use results on supercooling, as will be discussed in a subsequent sec
tion. Ginzburg (1955) pointed out already before the formulation of
the BCS theory that values as calculated from V.13 and V.14 could
be made to agree very well with those obtained from supercooling
data by taking e* = 2 or 3e. More recently Gor'kov (1958) has formu
lated the electromagnetic equation of the BCS microscopic theory in
terms of Green's functions, and was able to show (1959, 1960) that
the GL equations GLl and GL2 are identical to his expressions
near T c when ip is taken to be proportional to the energy gap, and
when one takes e* = 2e. This again is an indication that the current
carriers in superconductivity are the doubly charged Cooper pairs.
With this value of e*, V.13 and V.14 yield
and
*c= 108 xlO 7
k = 216xl0 7
dH r
dT
dH c
T=T r
tMo),
dT
T=T e
T c X 2 L (0).
(V.130
(V.140
For tin, the first of these yields k = 0158, the second 0149, two
values which are in excellent agreement. For indium, however, the
respective values are 01 12 and 0051 (Davies, 1960; Faber, 1961).
For aluminium, the equations yield 005 and 001 (Davies, 1960).
This lack of agreement may be in part due to errors in anomalous
The GinzburgLandau phenomenological theory 53
skineffect measurements used to evaluate A L (0), and in part, particu
larly in the case of aluminium, due to the large value of £ > because of
which nonlocal conditions set in very close to T c . The values of k
calculated from supercooling are probably the most reliable.
In terms of the parameters k, Aq, and H c , equation (V.9) reduces
Far from the phase boundary, for z> °o, tp 2 = ipl, and
dz
At the boundary, z = 0, V.9' is satisfied in the absence of an external
field {A = 0) by ip 2 = ipl; difi/dz = 0. In other words, the presence of
the phase boundary as such has no influence on the function tp, which
has the same value tp everywhere. In the presence of an external field
H e , however, this solution no longer applies, and one must integrate
V.9' and V.10 with the boundary condition tp 2 = ipl for z»co, and
the condition H = dA/dz = H e , and difi/dz = for z = 0. This integra
tion cannot be carried out exactly. Neglecting higher order terms,
however, one finds equations for i/j and for A as functions of z. At
z = 0, the value of tp is
^o 4(«+V2) CV ' 13)
With values of k m 0 1 , this equation predicts a decrease of ip by only
about 23 per cent when H e = H c . It is not surprising, therefore, that
the change in penetration depth with field is also very small. This can
be calculated formally by using the defining equation IV. 1 from which
one finds that, with a weak measuring field normal to H e :
1 +
An 1 +
8(k+V2) 2 # ( 2
k H 2
For a measuring field parallel to H e , the effect is tripled.
(V.16)
54 Superconductivity
It is evident that in the limit k*0, the effect of the external field on
ifj and on A vanishes, so that one returns to a situation formally
equivalent to the London picture. It must however be noted that even
for k = 0, j/tq is deduced from the empirical value of A . As a result
one can in certain cases, such as, for example, the treatment of very
thin films, allow k to vanish without necessarily reducing the GL
treatment to the London one.
CHAPTER VI
The Surface Energy
6.1. The surface energy and the range of coherence
Closely tied to the range of coherence of the superconducting wave
functions is the existence of an appreciable surface energy on a
boundary between the superconducting and normal phases. H.
London (1935) already pointed out that the total exclusion of an
external field does not lead to a state of lowest energy for a super
conductor unless such a boundary energy exists. In the presence of
an excluded external field, H e , the energy of a superconductor in
creases by Hg/Sir per unit volume. It would, therefore, be energetically
more favourable for a suitably shaped superconductor to divide up
into a very large number of alternately normal and superconducting
layers such that the width of the latter is less than A, and that of the
former very much smaller than that. The resulting penetration of the
external magnetic field into the superconducting layers much reduces
the magnetic energy of the sample, while the extreme narrowness of
the normal layers keeps negligible their contribution to the total free
energy. This situation is made energetically unfavourable by the
existence of a surface energy. To make each superconducting layer
narrower than A, a slab of thickness c/must have d/X such layers. This
is avoided by an interphase surface energy cc„, per unit surface whose
contribution exceeds the gain in magnetic energy, that is:
2d
Hid
8tt '
(VI.l)
where the energies have been calculated for a volume of slab of unit
surface area. Hence
XH[
> 28tt
55
(WW)
56 Superconductivity
It is convenient and customary to express the surface energy in terms
of a parameter A ' of dimensions of length, such that
Thus one sees that
8tt
A'>
(VI.2)
(VI.3)
Position
Fio. 16
is the condition for the diamagnetic behaviour of superconductors.!
Empirical values of A ' for pure superconductors turn out to be an
order of magnitude larger than the penetration depth.
The surface energy is intimately related to the Pippard range of co
herence. Figure 16 shows the variation of the order parameter iV and
of the externally applied field H e along a direction perpendicular to
the sn interphase boundary. One can define two effective bound
aries, indicated by M and C. M is the magnetic boundary defined so
that if inside the superconductor B = H c up to M and then dropped
off sharply to zero, the total magnetic energy would equal the actual
value, given by the integral of BH/Stt over the entire superconductor.
t F. London ([2], pp. 125130) has shown that taking into account the
detailed field penetration leads to the condition A' > A.
The surface energy 57
Similarly C is the configurational boundary such that if #" dropped
sharply to zero at C after being constant up to that point, one would
have the same superconducting free energy as the actual amount. The
free energy per unit volume of the superconductor is lower than that
in the normal state by an amount Hc/Stt. A configuration boundary
as shown on the inside of the magnetic boundary is essentially equiva
lent to a reduction of the superconducting volume and hence an
increase in the total free energy by an amount equal to HcI%t times
the distance CM per unit area of interphase boundary. The intro
duction of the Pippard range of coherence thus leads to a configur
ational boundary surface energy A ' w £. From this one must subtract
the decrease in energy due to the penetration of the field. Figure 16
indicates that the distance CM corresponds to the resulting net
surface energy parameter
A m £A. (VIA)
The condition for the Meissner effect is that f > A, i.e. that A > 0.
The GinzburgLandau theory was formulated so as to lead ex
plicitly to the existence of a surface energy, which arises as in the
Pippard approach from the gradual variation of the order parameter
*p over a finite distance, from the zero value in the normal region to
its full equilibrium value in the superconducting domain. Again the
surface energy is that amount which is needed to equate the energies
of the two phases in equilibrium, with H e = H c . In the supercon
ducting phase the increase in the free energy in the region where
is changing is given by V.6; in addition there is a reduction of the
energy due to the penetration of the field equal to
H(z)H c
MH„ = 
4tt
If,
(VI.5)
where H(z) is the value of the penetrated field at any point inside the
superconducting region. Thus the surface energy is given by the
integral of the difference between the superconducting and normal
free energies over the entire superconducting halfspace :
00
G s {H,z)
H(z)H c . H\
H:
4tt
+PG /I (0)^
(VI.6)
58
which gives for A :
Superconductivity
— CO
where A is the empirical penetration depth into a bulk supercon
ductor, and k the dimensionless parameter defined in the previous
chapter. This equation requires numerical integration. For k <^ 1 it
reduces to
A = 189° ■
K
(VI.8)
The thickness of the transition layer is thus, according to the GL
theory, of the order of A /k « 10A for most pure elements. The
intimate relation between the GL model and Pippard's range of
coherence is shown by Gorkov's derivation of GLl and GL2 from
first principles. He finds an expression for the GL parameter k in
terms of the critical temperature and the Fermi momentum and
velocity of the metal. Using equations IV. 17 and 111.13, this simplifies
to
096 A ?
50
(VI.9)
Comparing VI.8 and VI.9 shows that, as expected, the Pippard range
of coherence g and the surface energy parameter A as derived from
the GL theory are of comparable size. In short, both approaches
necessarily lead to a positive surface energy because both require that
the characteristic superconducting order parameter vary over a finite
distance. Both, therefore, obtain a net surface energy parameter of
length comparable to the difference between this distance and the
penetration depth of an external magnetic field.
It therefore also follows from both theories that the surface energy
must decrease and may even become negative when the range of co
herence decreases and the penetration depth increases. Equations
IV.12 and IV.21 show that this is just what happens to A and to £ when
the mean free path of the superconductor decreases. In alloys one
would, therefore, expect A to decrease with increasing impurity con
The surface energy 59
tent, and ultimately to become negative. This has indeed been inferred
by Pippard (1955) and by Doidge (1956) from their studies of flux
trapping and the superconducting transition in dilute solid solutions
of indium in tin. Direct measurements of J in such alloys by Davies
(1960) has demonstrated its decrease with shortening /, and Wipf
(1961) has traced this decrease to actual negative values. All the work
cited indicates that A becomes negative at a critical concentration of
approximately 25 atomic per cent of indium in tin.
Changes of A with decreasing mean free path also follow from the
numerical integration of VI.7, which yields that A < for
k > 1/V2.
(VI. 10)
This prediction is in good agreement with the work on tinindium
alloys just cited. Chambers (1956) found that the addition of 25
atomic per cent of indium to tin about doubles the penetration depth
as compared to pure tin, so that according to the defining equation k
should be increased by a factor of approximately four. This would
make k « 06, which is close to the theoretical value of 0707.
6.2. The surface energy and the intermediate state
Chapter II mentioned that a superconducting specimen with a demag
netization coefficient D is in the intermediate state when the external
magnetic field H e satisfies the inequality (1 D)H C <H C < H c . All
experiments on the detailed structure of this state have generally sub
stantiated the suggestion of Landau (1937, 1943) of a laminar struc
ture of alternating normal and superconducting layers. The thickness
of the normal layer grows at the expense of the superconducting one
as the external field approaches H c . Landau further suggested that in
the normal layers B = H c , while B = in the superconducting ones.
Clearly the width of the laminae is strongly influenced by the mag
nitude of the interphase surface energy A. Indeed Landau finds that
for an infinite plate of thickness L oriented perpendicularly to the field
(£> = 1), the sum a of the thickness of the superconducting layer, a s ,
and that of the normal one, a„, is given by
LA
(VI.11)
60 Superconductivity
where W is a complicated function of the ratio of the external to the
critical field H e jH c . Numerical values for Y(HJH£ have been calcu
lated by Lifshitz and Sharvin (1951). A typical result is a value a « 1 4
mm for L = 1 cm and HJH C = 08. A similar equation has also been
derived by Kuper (1951), who predicts numerical values which are
smaller by a factor of two or three. Typical experimental results fall
in between these predictions.
These results have been obtained by a variety of methods, all
making use of the fact that in the intermediate state lines of flux pass
only through the normal laminae, and emerge from the specimen
wherever these laminae end on the surface. A number of authors
(Meshkovskii and Shalnikov, 1947; Shiffman, 1960, 1961) have
e2&
i\,\t\
&&
i\i\i\i\
Fig. 17
passed very fine bismuth wire probes across the surface of a specimen,
and observed the magnetoresistive fluctuations in the probe resistance
when passing from the end of a normal lamina to that of a super
conducting one. Others have spread on the surface of a flat specimen
fine powder, superconducting (Schawlow et al, 1954; Schawlow,
1956; Faber, 1958; Haenssler and Rinderer, 1960) or ferromagnetic
(Balashova and Sharvin, 1956; Sharvin, 1960). The former will shun
flux and cluster on the ends of the superconducting laminae, as shown
schematically in Figure 17; the latter will be attracted by flux and
move onto the ends of the normal laminae. The resulting powder
patterns can be easily seen and photographed. Another optical
method consists of placing a thin sheet of magnetooptic glass (for
example, cerium phosphate glass) on the specimen surface, and
observing the reflection of polarized light (P. B. Alers, 1957, 1959;
De Sorbo, 1960, 1961).
The surface energy 61
The frontispiece shows a series of photographs obtained by Faber
(1958) with superconducting tin powder on an aluminium plate,
taken with increasing external field oriented perpendicularly. The
dark areas are covered with powder and are therefore the ends of the
superconducting laminae. The gradual shrinking of these areas with
increasing field and the corresponding growth of the light, normal
regions is clearly visible. The domains show a peculiar type of
corrugation, not predicted by the Landau model, and adding to the
surface to volume ratio of the laminae.
6.3. Phase nucleation and propagation
H. London (1935) pointed out that the existence of a positive surface
energy at the interphase boundary must under suitable conditions
give rise to phenomena analogous to superheating and supercooling
in the more familiar phase transitions. In fact a stable nucleus for the
phase transition cannot exist at all if the surface energy is everywhere
positive. Indeed there are many experimental observations that when
a specimen is placed in a greater than critical magnetic field which is
then reduced, the normal phase persists in fields less than H c . This is
the superconducting equivalent of supercooling. A typical magnetiza
tion curve illustrating this is shown in Figure 18. The degree of this
'supercooling' is characterized by the parameter Si = H t IH c , or by
the parameter
<j>, m 1af  {HlHf)IHl (VI.12)
For tin, 5/ is commonly of the order of 09; in aluminium the degree
of supercooling is usually much larger, and values of S t as low as 002
have been observed.
Superheating is the name given to the persistence of the supercon
ducting phase at fields above H c . This is very rarely observed.
Garfunkel and Serin (1952) have shown that this is so because the
ends of any conventional specimen cannot resist the initiation of the
normal phase, probably because of large local field values resulting
from demagnetization effects. Centre portions of long tin rods could
be made to superheat to S t = 117.
Much information on the nucleation of the superconducting phase
and on its relation to the surface energy has been obtained by Faber
62 Superconductivity
(1952, 1955, 1957) in a series of measurements on supercooling in tin
and aluminium. His technique consisted of winding on a long cylin
drical specimen several small, spaced coils the field of which could be
made to add or to subtract from a field produced by a large solenoid
surrounding the entire sample. With the sample normal, the field of
the solenoid could be lowered to some value between H, and H c , and
the field could then be lowered locally by a suitably directed current
through one of the smaller coils. The superconducting phase then
nucleated in the portion of the sample under the coil, and spread
H, H c
Applied Field H e
Fig. 18
rapidly throughout the sample. In this fashion supercooling could be
studied at different portions of the sample. The transition was de
tected by pickup coils distributed along the specimen.
At a given temperature the degree of supercooling varied con
siderably from point to point in a given specimen but at a given point
frequently remained reproducible even when in between measure
ments the specimen was warmed to room temperature. This indicated
that nucleation must occur at particular spots, some of which pro
mote nucleation more effectively than others. As the surface energy
can be lowered and may even become negative due to strain, it is
reasonable to assume that the spots favouring nucleation are regions
The surface energy 63
of local strain, some of which exist in even the purest specimens. This
is supported by Faber's finding that any handling of the specimens
between measurements could change the location and effectiveness of
the nucleation centres. Strained regions probably contain a high
density of dislocations.
By correlating the size of the nucleating field H t with the time it
took to be effective, Faber could deduce the depth of the nucleating
flaw below the sample surface, and found this always to be between
10 4 and 10 3 cm. Etching down the surface to this depth would
uncover further flaws extending to a similar depth. It is thus reason
able to take 10 4 10~ 3 cm as being the approximate size of the
nucleating flaws. At temperatures well below T c , this length is con
siderably bigger than the width of the interphase boundary, and one
can therefore imagine such a flaw to consist of a region of negative
surface energy surrounded by a shell across which the surface energy
increases to the normal positive value of the bulk material. Faber
(1952) has shown that there is a potential barrier against the further
growth of this nucleus until one has reached a degree of supercooling
such that
, A
<f>, « +n,
(VI. 13)
where A is the surface energy parameter, r a length of the order of the
flaw size, and // a small constant determined by the flaw's shape and
demagnetization factor. The measurements in fact show that the tem
perature variation of </>, is very much like that of A, as determined
from other experiments.
Both Faber (see Faber and Pippard, 1955b) and Cochran et al.
(1958) found that supercooling was much enhanced after a specimen
had been placed temporarily in a field much higher than the bulk
critical value. This shows that certain superconducting nuclei can be
quenched only by such a high field and supports much other evidence
that in a nonideal specimen there can exist small regions of high
strain which remain superconducting in very high fields.
By means of a series of pickup coils along his specimens, Faber
(1954) was able to observe the propagation of the superconducting
phase once the transition had been initiated at some nucleus. From
CHRIST'S COLLEGE
I IDDADV
64 Superconductivity
his results he infers that the growth of the superconducting phase
occurs in a series of distinct stages. The nucleus, which is always near
the surface, first expands to form an annular sheath around the speci
men. This sheath then spreads along the entire length of the specimen
with a velocity of the order of 10 cm/sec, and finally the supercon
ducting phase spreads inwards to fill the entire sample.
The growth of a superconducting region is limited principally by
the interphase surface energy on the one hand, and by eddy current
damping on the other. If there were no surface energy, the super
conducting phase could propagate by means of very thin filaments
which displace no magnetic flux and therefore create no retarding
induced currents. For a sheath of finite thickness, on the other hand,
which propagates in the presence of an external field H c , eddy currents
are generated, and the magnetic energy gained in the phase transition
is balanced by the unfavourable surface energy as well as the eddy
current joule heating. Faber (1954, 1955) has shown that the resulting
velocity of propagation for very pure specimens under optimum con
ditions is given by
v = A(I/o)A 2 (lH c H e ]/H c y
(VI. 14)
where / is the electronic mean free path in the normal phase, a the
normal electrical conductivity, and A is a constant of the specimen.
By measuring the temperature variation of v, Faber has used this
equation to obtain the temperature variation of A for tin and for
aluminium.
The values of A(T) obtained in this way by Faber, as well as those
measured in different ways by Davies (1960), Sharvin (1960), and
Shiffman (1 960), can be fitted by a number of empirical functions of
temperature. According to the GL theory, the surface energy should
have the same temperature variation as A , at least very near T c , where
k is independent of temperature. Hence one would expect
A(t) = J(0)(l/ 4 ) ,/2 ,
which can also be written
j»«o«w
for/ « 1.
(VI.15)
(VI.150
The surface energy 65
The second of these functions appears to give a good fit to various
results for tin over a rather wide range of temperature, but Faber's
aluminium data can be represented only by the first of these. There
seems to be a definite difference in the temperature dependence of the
surface energy for these two metals which is at present not under
stood.
The uncertainty in the temperature dependence of A of course
introduces a degree of doubt about the extrapolated value at 0°K.
The table below lists the best available values of ^(0) for a number
of metals, from a comparison of all available experimental data. Also
listed are values of £ > th e range of coherence, as calculated from
equation IV. 17, as well as empirical values for A (0).
Element
Aluminium
Indium
Tin
10 5 <d(0)
(cm)
18
34
23
10 5 £
(cm)
16
44
23
10 5 Ao(0)
(cm)
050
064
051
6.4. Supercooling in ideal specimens
Near T c , A becomes large, and the flaws lose their effectiveness as
nucleation centres. Measuring H t in this region can, therefore, give
some information on supercooling in ideal, unflawed material. Faber
(1957) has found for aluminium, S, = 0036, for In 016, and for Sn
023 ; values of Cochran et al. (1958) for aluminium are in reasonable
agreement. These results can be compared with theoretical predic
tions arising from the GL model. Equation V.9' has an interesting
consequence with regard to the normal phase. One would expect that
with H e S* H c , the halfspace described by the equation would be
entirely normal, with </< = 0. This is indeed a solution, but the equation
is also satisfied by a second solution with ifj # 0. Assuming that for
this solution #/> <^ 1, so that H(z) M H e everywhere, and remem
bering that in the geometry chosen A(z) = H(z)z, the equation
becomes
7ES
66 Superconductivity
This has the form of a wave equation for a harmonic oscillator, which
is known to have periodic solutions tp which vanish for z= ±co
(which is the required boundary condition for the normal phase) if
«= V2^(/»+i),« = 0,1,2,...
tic
In other words, for any value of k, the normal phase of the super
conductor becomes unstable with regard to the formation of laminae
of superconducting material when
HJH C = K /(n + J) V2,
of which the highest value, with n = 0, is
H c2 IH c = V(2)k.
(VI. 17)
(VI. 18)
A distinction must now be made as to whether k < 1 / V2 or k> 1 /V2.
In the former case, which is that of most pure superconductors,
H c2 < B and the field H c2 is then the lowest field to which the
normal phase can persist in a metastable fashion. H c2 is thus the
lower limit to which an ideal superconductor can be supercooled,
and therefore in the region very near T c one would expect the experi
mental value of S t to equal \/(2) k (Ginzburg, 1956, 1958a; Gor'kov,
1959b, c).
The values of k calculated in this fashion from Faber's measure
ments of S t are: 01 64 for tin, 0112 for indium, and 0026 for alumi
nium. The first two of these agree very well with k values deduced
from experimental penetration depths. In aluminium the lack of
agreement is probably due to the appearance of nonlocal effects very
close to T c . Ginzburg (1958b) has noted that this is more likely to
invalidate calculations involving the penetration depth than those
regarding the surface energy and supercooling. Nonlocal effects
become important for the former when £ > A ; for the latter only
when
K
Thus Kvalues calculated from supercooling data are probably the
most reliable, except for the effect discussed in section 6.7.
The surface energy 67
For superconductors with a dimension small compared to X /k « £ ,
the order parameter is essentially constant throughout and one can
solve the GL equation with the simplifying assumption k a 0. The
critical fields of supercooling are then given by
H C 2 = V6/Z,
for a slab of thickness la,
H c2 = 2V5^H C
a
for a sphere of radius a, and
A
H c2 = V8jH c
for a cylinder of radius r (Ginzburg, 1958a). For all these geometries
H c2 decreases with increasing specimen size, approaching mono
tonically the value given by VI. 18, which depends only on the value
of k characteristic of the material.
The compatibility of the GL theory with the Pippard range of co
herence under those circumstances of temperature, size, or mean free
path which eliminate the need for a nonlocal electromagnetic formu
lation is brought out once again by the similarity of VI. 18 with the
corresponding expression derived by Pippard (1955). He finds, also
by minimizing the free energy, that
H  2 V 3A o„
n c2 — jn c 
(VI. 19)
This differs from VI. 18 only by a numerical factor of order unity
since «■« A /£.
6.5. Superconductors of the second kind
According to equation VI. 10, the surface energy becomes negative
when k > l/\/2. A similar conclusion follows from the Pippard non
local model when A > $ (equation VI. 4: seeDoidge, 1956). The exist
ence of a positive surface energy was shown to be necessary for much
68 Superconductivity
of the magnetic behaviour usually found in superconductors. It is,
therefore, not surprising that superconductors in which this energy is
negative display quite different characteristics. They are accordingly
called superconductors of the second kind.
For a bulk specimen of such a superconductor the volume free
energy in the superconducting phase remains lower than that of the
normal one in external fields up to the thermodynamic value H c
defined by equation II.4. The negative surface energy, however,
makes it energetically favourable for interphase boundaries to appear
at field lower than H c , and for superconducting regions to persist to
fields higher than H c . Goodman (1961) has shown that this can already
be deduced from the London model by the single addition of a
negative surface energy term.
The details of the behaviour of superconductors of the second
kind can be deduced from the GL theory, which is equally valid for
k > \/\/2 as for k < 1 /\/2. In particular, the analysis of the preceding
section still holds; that is, the normal phase has a stability limit at a
field H c2 given by equation VI. 18, which shows that for k > 1/V2,
H c2 > H c . Abrikosov (1957) has used the GL equations to analyze
in some detail the magnetic behaviour of superconductors of the
second kind, and finds the features indicated in the magnetization
curve shown in Figure 19 for a cylindrical sample parallel to an
external field H c . There is a complete Meissner effect only up to
H e = H ci < H c , at which point the magnetization changes with
infinite slope. For values of k not much larger than 1 /V2, Abrikosov
predicts in fact a discontinuity. At somewhat higher field, the magneti
zation approaches zero linearly, with a slope
4tt^= 118/(2^1),
and vanishes entirely at
BL = H c2 = V(2)kH c .
(VI.20)
(VI. 18)
The magnetization curve should be fully reversible. Abrikosov can
not solve the equations determining H cl for all values of k; for
k > 1 he obtains
V(2)kH c i/H c = In k+ 008 (VI.21)
The surface energy 69
In the limit k = 1/V2, H cl =H C = H c2 . Goodman (1962a) has inter
polated between the latter value and those given by VI.21 to get a
graphical representation of H cl /H c for all k . A numerical solution has
been obtained by Harden and Arp (1963).
The magnetization curve predicted by the GinzburgLandau
Abrikosov (GL A) model can be compared with experiment, as it is
possible to determine the value of k for a specimen by independent
measurement. Gor'kov (1959) has derived an expression for k valid
when the electronic mean free path is much smaller than the intrinsic
Hcl H c
Applied Field H e
Fig. 19
H c2
coherence length £ . This was shown by Goodman (1962a) to have
the convenient form
k = K +75xloy /2 p.
(VT.22)
k o is the parameter for the pure substance, y the Sommerfeld specific
heat constant, in erg cm 3 deg 2 , and p the residual resistivity in
ohmcm. For tin this predicts quite closely the resistivity at which the
surface energy becomes negative (Chambers, 1956).
Using this equation, Goodman (1962a) has shown that the GLA
model satisfactorily explains the magnetic behaviour of substances
70 Superconductivity
such as TaNb alloys investigated by Calverley and RoseInnes
(1960) and his own UMo alloys (Goodman et ai, 1960). Further
more, recent magnetization measurements on PbTl single crystals
(Bon Mardion et ai, 1962), indicate a considerable degree of rever
sibility. Detailed quantitative verification of the GLA magnetiza
tion curves, as is possible only near T c , was provided by Kinsel et al.
(1962), who used InBi specimens to compare values of k calculated
from equations VI. 18, VI.20, VI.22, and from Harden and Arp's
values of H ci /H c . The different values of k for a given specimen agree
to within a few per cent. Similar agreement can also be deduced from
the results of Stout and Guttman (1952) on InTl alloys. The GLA
model is thus well substantiated.
The negative surface energy need not be due to a short mean free
path. In principle, it is possible for the coherence length to be shorter
than the penetration depth, even in a pure superconductor; this is
most likely in superconductors with a high T c (cf. equationIV.17).
Indeed, Stromberg and Swenson (1962) have found that the magneti
zation curve of very highly purified niobium is that of a supercon
ductor of the second kind, with a value of H cl and H c2 corresponding
to k ~ 1 • 1 . This conclusion is consistent also with the results of Autler
(1962) as well as of Goedemoed et al. (1963).
Kinsel et al. (1963) have found with their InBi alloys that the
effective value of k as defined by equation VI. 18 rises gradually as
the temperature decreases below T c , increasing by about 25 per cent
as T approaches 0°K. This agrees with the calculations of Gor'kov
(1960). The experiments further show that at any temperature
k = 1/V2 continues to be the critical value for the onset of super
conductivity. Thus a specimen with k ~ 065 at its transition tem
perature is there a superconductor of the first kind, but becomes one
of the second kind at that temperature at which k reaches the critical
value.
The temperature dependent increase in k leads to a corresponding
decrease of the surface energy. Specimens for which k goes through
the value 1/V2 at some temperature are those for which at that tem
perature the surface energy changes from being positive to being
negative, as has been observed for indium alloys by Kinsel et al. ( 1 964).
There is reason to believe that neither a negative surface energy
The surface energy 71
nor the size effects to be discussed in the next chapter can increase the
critical field of a superconductor without limit. Both Clogston (1962)
and Chandrasekhar (1962) have pointed out independently that in
sufficiently high fields it is no longer correct to assume that the free
energy of the normal phase is independent of field. With a finite
paramagnetic susceptibility X p (which was ignored in deriving equa
tion II.4), this free energy is, in fact, lowered by an amount \X.H 2 .
Thus, in sufficiently high fields, this alone could already bring about
a transition from the superconducting to the normal phase. The limit
on the critical field imposed by this mechanism is estimated to be two
or three hundred K gauss, and this is consistent with the results of
Berlincourt and Hake (1962).
6.6. The mixed state or Shubnikov phase
The magnetization curve of type II superconductors clearly shows
that for H cX < H e < H cl , the material is neither in the usual super
conducting nor in the normal phase. Abrikosov (1957) has called this
region the mixed state, and De Gennes ([14]) has suggested naming it
the Shubnikov phase, honouring the scientist who first suggested the
fundamental nature of type II superconductivity (Shubnikov et al.,
1937).
It is evident from the importance of the negative surface energy that
in the mixed state the specimen must contain as large an area of
interphase surfaces as is compatible with a minimum of normal
volume. This could be brought about by a division of the material
into a large number of very thin normal and superconducting sheets
or laminae (Goodman, 1961, 1964; Gorter, 1964). According to the
GLA theory, however, the mixed state consists of a regular array
of normal filaments of negligible thickness which are arranged parallel
to the external field and are surrounded by superconducting material.
At the normal filaments the superconducting order parameter
vanishes, and then rises from these linearly with distance. It reaches
its maximum value as quickly as possible, that is over a distance of
the order of £. The magnetic field has a maximum value at the normal
filaments, and falls off over a distance of the order of A > £. This
means that the field decreases to zero only if the filaments are spaced
at distances at least of the order of A. This mixed state structure can
6
72 Superconductivity
be shown to have a lower energy than any laminar arrangement
([14], p. 111.81).
One can thus think of the mixed state as if the superconducting
material were pierced by a number of infinitesimally thin filamentary
holes, regularly spaced parallel to the external field and thus each
containing magnetic flux. From the discussion in Chapter II it there
fore follows that the total flux associated with each normal thread is
quantized in units of <f> . This flux does not penetrate far into the
superconducting material because of superconducting currents circu
lating in planes perpendicular to the filament. This creates a vortex
line of superconducting pairs along each normal thread, in striking
analogy to the vortices existing in liquid Helium II (Rayfield and
Rcif, 1964).
The flux and the currents associated with an isolated vortex line
extend over a distance of about A. The interaction between two
vortex lines can thus be appreciable only at distances less than A.
This means that when the formation of vortex lines becomes energeti
cally favourable at H = H ci , they can essentially immediately achieve
a density corresponding to a separation of about A without creating
much interaction energy ([14], pp. III. 74ff.). This causes the abrupt
decrease of the magnetization at H cl predicted by Abrikosov and
verified experimentally. It is not certain, however, whether the
magnetization actually decreases discontinuously at this field or
whether it merely drops with an infinite slope. The former would
correspond to a first order transition with a latent heat, the latter
only to an infinity in the specific heat.
With the external field increasing beyond H cX , more and more
vortex lines are formed until their spacing approaches
•Mh
e
as //nears H c2 (Abrikosov, 1 957). According to Abrikosov, the vortex
lines form a square array at all fields except very near H cU but
Kleiner et al. (1 964) and Matricon (1 964) have shown that a triangular
array has a somewhat lower energy throughout the mixed state. This
changes the coefficient in equation VI.20 from 118 to 116.
The surface energy 73
A fundamental feature of the vortex structure of the mixed state is
that the order parameter W is everywhere finite except along the
centre of the vortices, which are normal filaments of negligible
volume. Thus the material can still be considered as entirely super
conducting. Abrikosov (1957) showed in fact that in the mixed state
one can characterize the material by a mean square order parameter
y* 2 , and that near H c2 this varied linearly with the magnetization.
The correctness of this and therefore the validity of the vortex
structure has been substantiated by measurements of the specific heat
(Morin, et al., 1962; Goodman, 1962b; Hake, 1964; Hake and
Brammer, 1964) and of the thermal conductivity (Dubeck et al.,
1962, 1964).
De Gennes and his collaborators (cf. [14], Vol. II) have studied the
properties of an isolated vortex line, as well as the interaction between
such lines. This leads to possible collective vibrational modes (De
Gennes and Matricon, 1 964), as well as to a surface barrier inhibiting
the motion of lines into or out of the superconducting material (Bean
and Livingston, 1963; [14], p. 111.85), De Gennes and Matricon
(1964) have also suggested the possibility of investigating the vortex
line structure of the mixed state by slow neutron diffraction. Prelimi
nary results have recently been reported (Cribier et al., 1 964).
In an ideal type II superconductor, homogeneous and devoid of
lattice imperfection, the vortex lines would be pushed out of the
material by the Lorentz force if the specimen carried any current
at right angles to the field (Gorter, 1962a, b). In any actual material,
the motion of the lines is inhibited by defects and inhomogeneities
which form potential barriers by which lines the are pinned. Anderson
(1963) has investigated the thermally activated 'creep' of lines at low
current densities, and has shown that on a local scale the density of
lines tends to remain uniform, so that bundles of lines move together.
This vortex or flux creep has been further discussed by Friedel et al.
(1963). With increasing current densities to creep changes into a
viscous flow of the lines, giving rise to resistive phenomena (Anderson
and Kim, 1964). Extensive experimental work on this has been done
by Kim era/. (1963, 1964).
Tinkham (1963, 1964) has shown that a quantized vortex structure
like that of the mixed state occurs even in pure films if they are very
74 Superconductivity
thin and placed in a perpendicular external field. This is in agreement
with magnetization measurements on such films by Chang et al. ( 1 963)
and penetration depth and critical field data of Mercereau and Crane
(1963). Guyon et al (1963) have investigated the dependence of the
critical field on thickness. For thin films so narrow as to contain only
a single row of vortices Parks and Mochel (1964) calculated that the
free energy should have a minimum at values of the perpendicular
external field at which the vortex diameter just equals the film width.
At T c this should result in a corresponding minimum of the film
resistance. They have observed such minima and take this as direct
evidence for the existence of quantized vortices.
6.7. Surface Superconductivity
As mentioned in Chapter V, the boundary condition applicable to
the GL order parameter W is that its derivative vanish. SaintJames
and De Gennes (1963) have shown that in an external field parallel to
the surface this leads to the persistence of an outer superconducting
sheath up to a field
H c3 = 1695 H c2
The thickness of this sheath is of the order of £. Its existence, explicitly
verified by many experiments (see, for example, Hempstead and Kim,
1963; Tomasch and Joseph, 1963), explains what had often been
a puzzling discrepancy between magnetic and resistive transitions.
The surface sheath exists also in Type I superconductors, but can be
detected only if H c2 > H c . As H c2 = \/2kH c , it follows that
H c3 = 240kH c , so that H c3 > H c for k > 042. Under this condition,
a measurement of H c3 /H c is in fact a way of obtaining k for Type I
materials (Strongin et al. 1964; Rosenblum and Cardona, 1964).
The existence of the surface sheath in Type I superconductors means
that if supercooling experiments are carried out on cylindrical
samples in a longitudinal field, as is usually the case, the ideal lower
limit for super cooling is H c2 rather than H c2 (cf. section 6.4). Thus the
experimental value of S, should be set equal to 1695V(2)k and the
values of k thus calculated are therefore correspondingly reduced.
CHAPTER VII
The Low Frequency Magnetic
Behaviour of Small Specimens
7.1. Increase in critical field
When one of the dimensions of a superconducting specimen becomes
comparable to the penetration depth, its critical magnetic field be
comes much higher than that of a bulk sample of the same material
at the same temperature. This follows already from the basic Gorter
Casimir thermodynamic description, according to which the free
energy difference per unit volume between the superconducting and
normal phases is
Hi
G n (0)GM = ■£•
(VII.l)
In an external field H e a superconductor acquires an effective mag
netization M(H e ) and becomes normal when
lie
M{H e )dH e = *■
077
(VII.2)
The integral is the area under the magnetization curve, and it was
pointed out in Chapter ni that for any ellipsoidal specimens VU.2
was satisfied when H e = H c . Actually this is true only when one
neglects the penetration of the external field into the sample, which
lowers the effective magnetization and the susceptibility of the
sample, as shown by equations IV.2 and IV.3. The susceptibility
determines the initial slope of the magnetization curve; a lower x
means that the curve has to go to a higher critical field to satisfy
equation VTI.2. Clearly, assuming this curve to remain linear with
slope x right up to a critical field H s \
"I
Hi
Xo
X
(VII.3)
75
76
and
Superconductivity
7T = v^3 — for a <^ Aq,
CVII.4)
(VII.5)
using the London equation to evaluate a = $ in IV.3 (Ginzburg, 1 945 ;
[11 p. 172). Similar expressions can be derived for spherical and
cylindrical samples. The resulting equations agree well with the fre
quently observed enhancement of the critical field in small specimens
when one uses for the penetration depth A the appropriate Pippard
value as calculated from IV.20. This is a good example of how ex
pressions derived from the London model can be used with the modi
fied value of A (Tinkham, 1958).
The field enhancement calculated from the GinzburgLandau
theory leads to nearly identical results. The essential difference is that
because of the additional terms V.6 in the GL free energy of the
superconductor, the penetration depth increases in the presence of an
external field (see equation V.16), so that the critical field for small
samples becomes even higher. For thin films of thickness 2a the
critical field is
si r la
(VII.6)
where /(*) is the same function of k which appears in equation V. 1 6,
and is very small for small values of k.
For very thin films, a < A , GL find that
which for very small k reduces to
H s A
ti c a
(VII.8)
This is the same expression which for thicker films gives the super
cooling field H c2 . Expressions similar to VII.6 and VII.8 have also
been derived for spheres and wires (Silin, 1951; Ginzburg, 1958a;
Low frequency magnetic behaviour of small specimens 11
Hauser and Helfand, 1962), and have been used by Lutes (1957) in the
interpretation of his measurements of the critical field enhancement
in tin whiskers.
It is possible to relate the thin film critical field to the basic super
conducting parameters £ anti K of the bulk material. The penetration
depth appearing in VII.8 should be given by 1V.20, in which £(/) is
determined by IV. 13 with the film thickness taken as the effective
mean free path (Tinkham, 1958). In the limit a < £ this yields
(Alloy*
(VII.8 ')
(Douglass and Blumberg, 1962). The use of the thin film susceptibility
as derived by Schrieffer (1957) with nonlocal electrodynamics leads
to 2040 per cent higher values of the numerical constant (Ferrell and
Glick, 1962;Toxen, 1962).
7.2. High field threads and superconducting magnets
The size effect on the critical field is particularly striking in experi
ments using extremely thin evaporated films. In their experiment on
the Knight shift in tin, Androes and Knight (1961) used films of thick
ness a 100 A and found H c (0) « 25 kgauss. Ginsberg and Tinkham
(1960) saw no effect on the superconducting properties of their
1020 A lead film in a field of 8 kgauss.
The equivalent of small superconducting specimens can exist also
in bulk material. In an inhomogeneous specimen there will be local
variation of the surface energy due to varying strain or to varying
electronic mean free path. If locally the surface energy is sufficiently
lower than the value elsewhere, it may be energetically favourable for
this region to remain superconducting in the presence of an external
field even when the surrounding material has become normal
(Pippard, 1 955). Under these conditions one can thus have a situation
quite analogous to that of small specimens: small superconducting
regions exist in a matrix of normal material (Gorter, 1935; Mendels
sohn, 1935; Shaw and Mapother, 1960). If their dimensions are
small compared to the penetration depth, the critical field of these
regions will be correspondingly raised, and it is known (Faber and
Pippard, 1955b; Cochran et al, 1958) that such regions can persist in
78 Superconductivity
high fields. In many instances these regions are threads which can
form continuous superconducting paths from one end of the speci
men to the other, resulting in a resistive transition much broader and
extending to much higher fields than the magnetic one (Doidge, 1 956).
The threads are, of course, likely to touch each other in many places,
resulting in what Mendelssohn (1935) called a superconducting
sponge. The multiple connectivity of such a structure generally leads
to highly irreversible magnetic transition with almost total flux trap
ping. Bean (1962) has used a simplified model with which to calculate
the magnetization curve of such a sponge. He has confirmed some
features of this model with an artificial filamentary superconductor
made by forcing mercury into the pores of Vycor glass (Bean et a/.,
1962). The possible relevance of this to superconducting magnet wire
will be discussed in Chapter XIII.
7.3. Variation of the order parameter and the energy gap with magnetic
field
From equations V.GL1 and V.GL2 one can also calculate the varia
tion of the order parameter *fi inside the thin films. For thicknesses 2a
very small compared to the width of the transition layer Xq/k, or in
the equivalent Pippard terms for 2a < g , if; can be considered con
stant, and one can take k K 0. This leads to (Ginzburg, 1958a)
X ) 3oUc7\V
H
"6 L \"c/ OA 6J/ L \ A o/ 30 \H
For very thin films VII.8 applies, so that
(VII.9)
<l<i
For such films, therefore, i/j(H s ) = 0, which means that the transition
into the normal state is of second order, without a latent heat and with
a discontinuity only in the specific heat, and not in the entropy. There
can be no supercooling, and therefore, no hysteresis. For thicker films
and bulk samples the transition in an external field, as discussed in
Chapter II, is always of first order. The critical thickness below which
there is a second order transition is
2a = V(5)A ,
Low frequency magnetic behaviour of small specimens 79
which has been verified by Zavaritskii (1951, 1952). Note that as the
penetration depth is inversely proportional to 0, A(//) for thin films
is much larger than A even in fairly small fields (Douglass, 1961c).
Douglass (1961a) has pointed out that because of the proportion
ality of the energy gap to «/r, as derived by Gorkov (1959, 1960),
equations VII.9 and VII. 10 represent the field dependence of the
energy gap in sufficiently thin films. Thus one can write
e\H s )
4
= for 2a < V(5)A .
(VII. 11)
For thicker films, VII.9 and VII. 10 do not apply, and GLl and
GL2 must be solved numerically. The resulting variation of the
energy gap at H e = H s asa function of film thickness has been calcu
lated by Douglass (1961a). It is displayed by the curve in Figure 20.
The points are gap values which Douglass (1961b) obtained from tun
nelling experiments (see Section 10.6). Similar results have been found
by Giaever and Megerle (1961), also by means of the tunnel effect, as
well as by Morris and Tinkham (1961) with thermal conductivity
measurements. With H e « H s , the empirical variation of the energy
gap with field closely agrees with the GinzburgLandauGorkov pre
dictions even at temperatures well below T c . In such high fields the
80 Superconductivity
order parameter is then small enough to make tenable the basic GL
assumptions as well as Gorkov's identification of the energy gap with
W, Tinkham (1962) has proposed ways in which the GL equations
can be extended to give agreement also with low field results over a
wide range of temperature. The limitations of these equations in this
region have been discussed by Meservey and Douglass (1964).
Bardeen (1962) has calculated the critical field and critical current
for thin films on the basis of the BCS theory. At higher temperatures
his results generally confirm the predictions of the GinzburgLandau
theory, including the vanishing of the energy gap and a resulting
secondorder transition at the critical field in sufficiently thin films.
At much lower temperatures, however, below about TJ3, Bardeen
finds that for any thickness the energy gap remains finite and the
transition a firstorder one. However, Maki (1963) as well as Nambu
and Tuan (1963) predict that the phase transition should be of the
second order at all temperatures. Merservey and Douglass (1964)
verify this down to t = 0 14.
CHAPTER VIII
The Isotope Effect
8.1. Discovery and theoretical considerations
The various phenomenological treatments based on the empirical
characteristics of a superconductor provide an astonishingly com
plete macroscopic description of the superconducting phase. How
ever, they do not give any clear indications as to the microscopic nature
of the phenomenon.
One of the first such clues arose through the simultaneous and inde
pendent discovery, in 1950, by Maxwell, and by Reynolds et al., that
the critical temperature of mercury isotopes depends on the isotopic
mass by the relation
T c M a = constant,
(vm.i)
r
4.00
Ave. mass no.: x
199.5 N
200.7 (nat)
202.0
203.3
J I I u
J ! I I
4.20
Fig. 21
81
82 Superconductivity
where Mis the isotopic mass and a m £. This is illustrated in Figure 21,
showing the variation of threshold field near T c for different isotopes.
The effect has since also been established in a number of other ele
ments. The following table contains the most reliable experimental
values of the exponent a, together with quoted probable errors.
Reference
Olsen, 1963
Reynolds et al., 1951
Matthias et al., 1963
Hein and Gibson, 1964
Shaw etal., 1961
Hake et al., 1958
Maxwell and Strongin, 1964
Gcballee/o/., 1961
Finnemoreand Mapother, 1962
Maxwell, 1952a
Serin et al., 1952
Lock et al., 1951
Maxwell, 1952b
Alekseevskii, 1953
Geballeand Matthias, 1964
In all the nontransition metals, with the exception of molybdenum,
the results are consistent with a = 1/2. However, small mass differ
ences and the possibility of impurity and strain effects limit the experi
mental reliability, as is made evident by the variations between differ
ent measurements on the same element. Thus one cannot rule out
deviations from the ideal value of a = £ which may be as high as 20
per cent in some cases. In view of recent theoretical work to be dis
cussed in Chapter XI, it is significant that the trend of the published
deviations from a = \ is toward lower values. The situation in the
transition metals ruthenium and osmium, however, appears to be
different. This will be further discussed in Section 1 1.5.
The inference to be drawn from the dependence of T c on the isotopic
mass is startling. A relation between the onset of superconductivity,
which is quite certainly an electronic process, and the isotopic mass,
which affects only the phonon spectrum of the lattice, must mean that
superconductivity is very largely due to a strong interaction between
the electrons and the lattice. Thus the discovery of the isotope effect
Element
a
Cd
051 ±010
Hg
0504
Mo
033
Os
021
Pb
0461 ±0.025
0501 ±001 3
Rh
04
Ru
<01
<005
Sn
0505 ±0019
046 ±002
0462 ±001 4
Tl
050 ±005
062 ±01
Zn
05
The isotope effect 83
clearly pointed out the direction in which a microscopic explanation
of the phenomenon had to be sought. In fact, Frohlich (1950) had
independently suggested just such a mechanism without knowing of
the experimental work. However, it took several more years until the
subtle nature of the pertinent electronlattice interaction was recog
nized and a valid microscopic theory began to be developed.
8.2. Precise threshold field measurements
The variation of critical temperature with isotopic mass was estab
lished by measuring the critical field H c as a function of temperature,
and then extrapolating this to zero field. Magnetic measurements of
course make use of the perfect diamagnetism of a superconductor, and
can be made in one of two ways : either the change in flux through the
sample at the transition induces an e.m.f. in a pickup coil which is
connected to a suitable galvanometer, or the changing susceptibility
of the sample is reflected in the change of the mutual inductance of
coaxial coils of which the sample forms part of the core. Either of
these methods can be applied with great accuracy in spite of simple
apparatus, and has the further advantage of measuring a bulk
property virtually unaffected by the possible presence of small regions
with different superconducting characteristics. By providing a
84 Superconductivity
misleading shortcircuiting path, such minor flaws can lead to very
erratic results when T c is measured by observing the variation of
electrical resistance.
The careful determination of critical field curves which arose as
almost a byproduct of the work on the isotope effect established a
4LEAD
Fig. 23
number of interesting characteristics. Figure 22 shows the variation
of the reduced critical field h m H c /H as function of t 2 m T 2 /T 2 C for a
number of tin isotopes measured by Serin et al. (1952). It is evident,
as was indicated earlier, that equation I.2a is only an approximation]
and that a better representation for h is a polynomial
N
W) = i  S of, (vni.2)
n=2
The isotope effect 85
A polynomial which fits the data for all tin isotopes as found by Lock
et al. (1951) to within onehalf of a per cent is
// = l10720/ 2 00944/ 4 + 03325/ 6 01660/ 8 . (VIII.3)
All measurements to date have indicated that to within the available
precision all isotopes of a given element follow the same critical field
polynomial. One also finds that H has the same mass dependence
as T c . This means that the superconducting condensation energy
Hq/Btt varies proportionally to the isotopic mass, and also that, as
shown by equations 11.15 or 11.16, the value of y is independent of
isotopic mass.
It has also been found that in going from one element to another,
the reduced threshold field curves show small but definite variations.
For all elements there are deviations from a strictly parabolic varia
tion, generally by a similar small amount in one direction, but in the
case of lead and mercury by an amount in the opposite direction.
Figure 23 shows these deviations as a function of reduced tempera
ture. It is important to emphasize the smallness of these deviations.
86 Superconductivity
so as not to allow them to obscure the basic similarity of the super
conducting behaviour of all elements in terms of reduced coordinates.
This not only sanctions the continuing discussion of superconduc
tivity in general terms with only occasional references to specific
elements, but also allows one to look for a microscopic explanation
of superconductivity, which in first approximation need not concern
itself with the distinctive characteristics of individual elements, but
only takes account of general and common features. The deviations
of the measured threshold fields from a simple parabolic variation
must be, according to the thermodynamic treatment developed in
Chapter n, correlated with the empirical deviations of the specific
heat from the corresponding change as the cube of the temperature
Serin (1955) showed this strikingly by plotting both these deviations
on the same graph, using the best available data for tin. This is shown
in Figure 24. Mapother (1959) has since established the correlation
between the experimental nonparabolic threshold fields and the
exponential variation of the specific heat.
CHAPTER IX
Thermal Conductivity
9.1. Low temperature thermal conductivity
In normal metals, heat is carried both by the conduction electrons
and by the quantized lattice vibrations, the phonons. The total
thermal conductivity consists of the sum of these two contributions :
Km — k en r k B
(IX.1)
where e and g denote the electrons and the lattice, respectively. The
electronic conductivity is limited by two scattering mechanisms : the
phonons and the lattice imperfections, and one can write at T< ©:
\\k cn = aT 2 + Po /LT.
(1X.2)
The first of the terms on the right gives the resistivity due to the
electron scattering by phonons, and predominates at higher tem
peratures; the second that due to scattering by imperfections, which
becomes important below the temperature at which k en has a
maximum:
TL* = Po/2aL.
(IX.3)
In these equations p is the residual electrical resistivity, L the Lorentz
number (244 xlO 8 wattohm/deg 2 ) and a is a constant of the
material which is inversely proportional to @ 2 . Note that for a given
material the addition of impurities increases p and thus raises T max .
In pure metals and dilute alloys, k en > k g „; it is only in metals con
taining as much as several per cent impurities that the two contri
butions are of the same order of magnitude.
The twofluid model allows one to predict qualitatively what hap
pens to the thermal conductivity of a metal when it becomes super
conducting (see Mendelssohn, 1955; Klemens, 1956). The condensed
'superconducting' electrons cannot carry thermal energy nor can
they scatter phonons. With decreasing temperature their number
increases, and that of the 'normal' electrons correspondingly
7 87
88 Superconductivity
decreases, which will result in a rapid decrease of the electronic heat
conduction. At the same time the conduction by phonons will be
enhanced, as these are no longer scattered as much by electrons.
In pure specimens, the decrease in k es will usually exceed any gain
in k gs , and the total conductivity in the superconducting phase will
then be much smaller than in the normal phase. This is illustrated,
for example, by the results of Hulm (1950) on pure Hg shown in
Fig. 25
Figure 25. There exist, however, pure materials in which the normal
conductivity is not very high but which are very free of grain boun
daries and other lattice defects. In the superconducting phase of such
substances at very low reduced temperatures the phonons are then
hardly scattered by anything except the specimen boundaries, result
ing in a large value of k gs . This has been observed, for example, by
Calverley et al. (1961) in tantalum and niobium.
Suppressing the electronic conduction in the normal phase by
adding impurities decreases the effect of condensing electrons out of
Thermal conductivity 89
the thermal circuit. For moderately impure specimens the super
conducting conductivity will then not be very different from the
corresponding normal one. This is shown, for example, by the results
of Hulm (1950) on a HgIn alloy, also displayed in Figure 25. The
results of Lindenfeld (1 96 1 ) on lead alloys shown in Figure 26 indicates
what happens with increasing inpurity content: as the phonon con
tribution to the normal conductivity becomes more appreciable, the
0.30
K (watt/cmdeg)
Pb + 6%Bi
•— Pb+3*h
Pb+6%ln
0.20
0.10
gain in k^ increasingly outweighs the decrease in k es , and the conduc
tivity in the superconducting phase becomes much larger than that
in the normal one.
9.2. Electronic conduction
If the effect on thermal conductivity by the superconducting transition
is indeed due to the disappearance of electrons from the conduction
process, then one should be able to write IX.2 for a superconductor as
!/*„ = x(ir)aT 2 +y(^)p Q ILT, (IX.4)
90 Superconductivity
where x(if) and y(ir) are functions only of the order parameter or
which indicates the fraction of condensed electrons. Equation 11.25
shows that IT is a function only of / = T/T c , so that one can write
instead
Uk„ = aT 2 /g(t) + Po LT/f(t). (IX.4)
The equation has been written in this form to agree with the nomen
clature introduced by Hulm (1950). He pointed out that if one chooses
a sample in which the electronic heat conduction is predominantly
limited by one or the other of the two scattering mechanisms, the
measured ratio k e Jk en then equals the appropriate ratio function g(t)
or /(r). To a first approximation, at least, these functions should be
universal functions for all superconductors and be related to the
microscopic nature of the phenomenon.
For a specimen for which T max < T c , as is the case for reasonably
pure Hg and Pb, and for extremely pure Sn and In, the heat conduc
tion just below T c is by electrons limited by phonon scattering. For
such samples
KJKn * sit).
(IX.5)
All pertinent measurements show the same qualitative features: g(t)
at / = 1 breaks away sharply from unity with a discontinuous slope,
and decreases as a power of / which is about 2 for Sn and In (Jones
and Toxen, 1960: Guenault, 1960), and 4 to 5 for Pb and Hg (Watson
and Graham, 1963; see also Klemens, 1956). Calculations by Kada
noff and Martin (1961), by Kresin (1959) and by Tewordt (1962,
1963a) appear to explain the experimental results for Sn and In,
but not for Hg and Pb.
For specimens for which T max ^ T c , the electronic conduction in
the superconducting phase is at all temperatures limited by impurity
scattering, so that for these
*«/*«, * /CO.
(IX.6)
Several investigations (see Klemens, 1956) have shown that at /= 1
/(0 approaches unity smoothly with a continuous slope, and that at
lower temperatures it decreases more slowly than git). The results
are in reasonable agreement with expressions for /(/) derived by
Bardeen etal. (BRT, 1959) and by Geilikman and Kresin (1959) on
Thermal conductivity 91
the basis of the BCS theory. The gradual change from a phonon
scattered to an impurityscattered electronic conduction in the same
material of increasing impurity is particularly well illustrated by the
recent results of Guenault (1960) on a series of monocrystalline tin
specimens.
When thermal conductivity measurements on superconductors are
extended to small values of t, as was first done by Heer and Daunt
(1949) and later by Goodman (1953), /(/) is found to decrease very
rapidly. Goodman pointed out that this could be represented by an
equation of the form
At) = aexp(*/0, (IX.7)
and suggested that this implied the existence of an energy gap between
the ground state and the lowest excited state available to the assembly
of superconducting electrons.
This conclusion can be inferred from thermal conductivity results
in the following manner. Simple transport theory shows that
k e = (1/3) lv C e , (IX.8)
where / is the mean free path, v tne average velocity, and C c the
specific heat of the electrons. Assuming that v , the Fermi velocity in
the normal metal, remains the same for the uncondensed 'normal'
electrons in the superconducting phase, and that in both phases the
mean free paths (which may differ in magnitude) vary only slowly
with temperature, then the temperature variation of k e Jk e „ must be
due entirely to that of the specific heats. In other words
f(t) * kjk„ M CJC e
(IX.9)
C en is known to vary linearly with temperature, so that IX.7 implies
that
C es = a'T c texpib/t). (IX.10)
That such a temperature variation of the specific heat corresponds to
an energy gap in the electronic spectrum can be shown as follows: If
a gap of width 2e lies below the lowest available excited state, the
number of thermally excited electrons will be proportional to
expi2e/2k B T), where k B is the Boltzmann constant, and the factor
2 arises because every excitation creates two independent particles,
92 Superconductivity
an electron and a hole. Thus the free energy of the superconducting
phase is equal to the condensation energy per particle multiplied by
the exponential factor, which remains unchanged, through two dif
ferentiations with respect to temperature, to appear in the specific
heat. The parameter b in IX. 10 is thus seen to equal 2e/2k B T c .
1.0 1 
.6
K
en
Aluminium
(.Zavaritskii)
.4
{Aluminium(Satterthwaite)
ITheory (BRT)
.4
t
Fig. 27
.6
.8
1.0
According to the microscopic theory to be discussed in Chapter XI,
the energy gap is a function of temperature. The parameter b can
therefore be written as
b =
<T)
*(0) x e(D
k B T c k B T c " € (0)
where € (0) is the gap value at 0°K. The detailed dependence of
kjk en on b has been calculated by BRT, and the function e(DMO),
calculated from the BCS theory, has been tabulated by Miihlschlegei
(1959). Measurements oikJk e „ can thus be used to infer the value of
<0)lk B T c .
The appropriate temperature dependence of kjk en has been
observed in a number of metals. The results for aluminium by
Satterthwaite (1960) and by Zavaritskii (1958a) are shown in
Thermal conductivity 93
Figure 27, together with a theoretical curve calculated with a gap
equal to 350 k B T c . The agreement is somewhat deceptive, since there
is good evidence that the gap width for aluminium is only 3 40 k B T c .
From an observed anisotropy in the temperature dependence of k„
at very low temperatures Zavaritskii (1959, 1960a, b) has been able
to infer a corresponding anisotropy in the width of the energy gap in
the spectrum of the superconducting electrons in the case of cadmium,
tin, gallium, and zinc. To the last he could apply theoretical expres
sions due to Khalatnikov (1959), from which he deduced a gap aniso
tropy of about 30 per cent. A similar result holds for cadmium. The
measurements of Zavaritskii also show that the gap anisotropy can
have different forms : in the case of gallium the value of the gap can
be approximated by an ellipsoid compressed along the axis of rota
tion; for zinc and cadmium this ellipsoid is stretched out along the
axis of rotation.
In cases where the energy gap is a function of the magnetic field,
measurements of k c Jk cn can be used to infer this field dependence.
This technique has been used by Morris and Tinkham (1961) for thin
films (see Section 7.3), and by Dubeck et al. (1962, 1964) for type II
superconductors in the mixed state (see Section 6.6).
9.3. Lattice conduction
Far below T c the fraction of 'normal' electrons becomes so small as
to make k cs <^ k gs . At the very lowest temperatures, the phonons are
primarily scattered by crystal boundaries in a manner which is the
same in the superconducting as in the normal phase. The charac
teristic T 3 dependence in this limit (Casimir, 1938) has been well
established experimentally (Mendelssohn and Renton, 1 955 ; Graham,
1958).
In the normal state there occurs at these temperatures still appre
ciable heat conduction by electrons, limited only by impurity scat
tering and varying linearly with temperature (see equation IX.2).
Thus in this range
kjk s = aT\ (IX.11)
where a is a constant of the material which can have values as high
as several hundred. For example, a suitable lead wire can have
94 Superconductivity
k n /k s « 10 5 at 01 °K. A number of authors (see Mendelssohn, 1955)
suggested using such wires in ultralow temperature experiments as
thermal switches which would be 'open', i.e. nonconducting, in the
superconducting phase, and 'closed' when the superconductivity is
quenched by means of a suitable magnetic field. Such heat switches
are now widely used (see, for instance, Reese and Steyert, 1962).
At somewhat higher temperatures, at which the phonons begin to
be scattered by the 'normal' electrons even in the superconducting
phase, there is necessarily a concurrent rise of the electronic con
duction. Experimentally it is very difficult to separate the conduction
mechanisms. Where this has been possible (Conolly and Mendels
sohn, 1962; Lindenfeld and Rohrer, 1963) the results have been
consistent with the pertinent calculations by BRT and by Geilikman
and Kresin (1958, 1959).
9.4. The thermal conductivity in the intermediate state
A number of experiments, in particular those of Mendelssohn and co
workers (Mendelssohn and Pontius, 1937; Mendelssohn and Olsen,
1950; Mendelssohn and Shiftman, 1 959), have shown that the thermal
conductivity of a superconductor in the intermediate state generally
does not change linearly from its value in the one phase to its value
in the other when at a given temperature the external field is varied.
Instead there appears an extra thermal resistance, which in some cases
can be very large, and which is attributed to the scattering of the
predominant heat carriers (electrons or phonons) at the boundaries
between the superconducting and normal laminae which make up the
intermediate state. For materials in which phonon conduction domi
nates this has been analyzed by Cornish and Olsen (1953) and by
Laredo and Pippard (1955). Strassler and Wyder (1963) have devel
oped a treatment for very pure specimens in which the conduction is
mostly by electrons. Experiments on the thermal conductivity in the
intermediate state thus yield strong confirmation that the laminar
structure, observed by various techniques at the surface of a specimen,
actually persists throughout a bulk sample.
CHAPTER X
The Energy Gap
10.1. Introduction
Ever since the initial discovery of superconductivity it had been known
but barely noted that the striking electromagnetic behaviour of a
superconductor at low frequencies is not accompanied by any corre
sponding changes in its optical properties: there is no visible change
at T c , although the reflectivity of a metal at any frequency is related
to its conductivity at that frequency. Thus at the very high optical
frequencies the resistance of a superconductor is a constant, inde
pendent of temperature, and equal to that of the normal metal. At
about the time of the discovery of the isotope effect steadily im
proving high frequency techniques had shown that atO°K the normal
resistance persisted down to frequencies of the order of 10 13 c/sec, but
that it remained zero up to frequencies of the order 10 10 c/sec. In 1952
already Shoenberg ([1], p. 202) concluded from this that at some fre
quency between these two limits '.. . quantum processes set in which
could raise electrons from the condensed to the uncondensed state
and thus cause energy absorption'.
As shown in the previous chapter, Goodman (1953) very shortly
after this inferred from his thermal conductivity results the existence
of an energy gap in the single electron energy spectrum. A similar
conclusion had been deduced a few years earlier by Daunt and
Mendelssohn (1946) from the absence of any Thomson heat in the
superconducting state. This indicated to them that the supercon
ducting electrons remain effectively at 0°K up to T= T c by being in
lowlying energy states separated from all excited states by an energy
gap of the order of k B T c .
In the years which followed, the existence of such a gap was firmly
established by a large number of experiments, and this, together with
the electronphonon interaction indicated by the isotope effect, pro
vided the keystones of a microscopic theory. This chapter will
95
96 Superconductivity
describe a few experiments which indicate the energy gap most clearly
and directly. The subject has been reviewed by Biondi et al. (1958)
and recently by Douglass and Falicov (1964).
10.2. The specific heat
After the resurgence of interest in specific heat measurements as a
result of the suggestive results of precise threshold field measurements,
of Goodman's thermal conductivity results, and of the first clear
experimental verification of a deviation from a T 3 law by Brown et al
(1953) on niobium, there have been in recent years a number of
measurements which clearly indicate the exponential variation of C„
corresponding to an energy gap. The first of these were the results of
Corak et al. (1954) on vanadium and by Corak and Satterthwaite
(1 954) on tin ; and since then the exponential variation of C es has been
established in a number of elements. The appropriate column in
Table III lists the energy gap values of these elements deduced from
the specific heat measurements. Note that in units of k B T c these gaps
are of very similar size for widely varying superconductors. This again
bears out the basic similarity of all superconductors in terms of re
duced coordinates.
It is perhaps useful to consider briefly the difficulty of obtaining
good values for C es . What is measured, of course, in both the super
conducting and in the normal phase, is the total specific heat. It is then
necessary to separate the electronic from the lattice contribution in
the normal phase in order to be able to subtract the latter from the
total specific heat in the superconducting phase. Unfortunately, even
at low temperatures, C ga is small compared to C„ only for metals
with large Debye temperatures. These are just the hard, highmelting
point metals which are difficult to obtain with high purity, without
which superconducting measurements are misleading. The softer and
lower melting point metals, on the other hand, have a very unfavour
able ratio of electronic to lattice specific heat.
Measurements by Goodman (1957, 1958), Zavaritskii (1958b), and
Phillips (1959) on aluminium have shown at very low temperatures
(/ < 02) a deviation of C es from a simple exponential law (Boorse,
1959). Cooper (1959) has pointed out that this can be a consequence
of anisotropy in the energy gap. At the lowest temperatures most
The energy gap 97
electron excitations would be expected to occur across the narrower
portions of the gap, and this would be reflected in an upward curva
ture of C es when plotted semilogarithmically against 1/r. Figure 28
compares a number of measurements which show this curvature with
BCS .
3CWT) 3
Fig. 28
the exponential law expected from the BCS theory. According to a
theory of Anderson (1 959) (see Section 12.2) the gap anisotropy of an
element diminishes with the addition of impurities. Indeed Geiser and
Goodman (1963) have found in aluminium specimens of different
purity that the deviation of C cs from an exponential form decreases
with increasing impurity.
98 Superconductivity
10.3. Electromagnetic absorption in the far infrared
The magnitude of the energy gap 2e(0) can be characterized by a fre
quency i/ ? such that hv ? = 2e(0). It is at this frequency that one would
expect the change from the characteristically superconducting re
sponse to low frequency radiation, to the normal resistance main
The energy gap 99
The measurements of the transmission of such radiation through
superconducting films will be discussed in a later section. Richards
and Tinkham (1960), Richards (1 961), and Ginsberg and Leslie (1962)
have observed directly the absorption edge at the gap frequency in
bulk superconductors. Radiation from a quartz mercury arc infrared
JVL
10 15 20 25 30 35 40 45 50
FREQUENCY (cm 1 )
Fig. 29
tained at high frequencies. Unfortunately, the frequencies corre
sponding to gap widths inferred from the specific heat measurements
are 10 u 10 12 csec~ \ which is an experimentally awkward range at
the upper limit of klystronexcited frequencies, yet very low for
mercury arc ones. Only recently have Tinkham and collaborators
developed the techniques needed to detect the very low radiation
intensities available in this far infrared region.
Table III
Element
Energy gap
(2<Q)lk B T c )
A
B
C
D
E
F
Aluminium
316
29
337 b , 343 c
33 d
35
Cadmium
...
...
33
Gallium
...
...
...
35
Indium
41
39
39
363 a , 345 b
35
Lanthanum
285
...
37
Lead
414
40
...
433 a , 426 b
418 d
39
Mercury
46
...
...
...
37
Niobium
28
...
44
384 e ,36*,
359 8
37
Rhenium
...
33
Ruthenium
31
Tantalum
< 30
36
360°, 35',
365 u
36
Thallium
32
28
Thorium
...
35
Tin
36
33
35
36
346 a , 347 b
365 d
36
Vanadium
34
...
36
34'
36
Zinc
...
...
...
25
...
34
A — from infrared absorption (lead: Ginsberg and Leslie, 1962;
lanthanum: Leslie et al., 1964; all others Richards and Tink
ham, 1960).
B — from infrared transmission (Ginsberg and Tinkham, 1960).
C — from microwave absorption (aluminium: Biondi and
Garfunkel, 1959; tin: Biondi et ai., 1957).
D — by fitting specific heat data to exponential (Goodman, 1 959).
E — from tunneling ("Giaevcr and Megerle, 1961; b Zavaritskii,
1961; c Douglass, 1962; d Douglass and Merservey, 1964;
Townsend and Sutton. 1962; 'Giaever, 1962; "Sherrill and
Edwards, 1962; "Dietrich, 1962).
F— calculated from XI.32 (Goodman, 1959).
100 Superconductivity
monochromator was fed by means of a light pipe into a cavity made
of the superconducting material under investigation. The cavity con
tained a carbon resistance bolometer, and was shaped so that the
incident radiation would make many reflections before striking this
detector. For frequencies lower than v g , the superconducting walls of
the cavity do not absorb, and much radiation reaches the bolometer.
At v g , absorption by the walls sets in, and the signal from the bolo
meter decreases sharply. Figure 29 shows normalized curves of the
fractional change in the power absorbed by the bolometer, in arbi
trary units, plotted against frequency for all the metals investigated
by Tinkham and Richards. The gap values obtained are listed in
Table III. The absorption edges for Pb and Hg show a certain struc
ture, which has also been found in the same elements in infrared
transmission measurements (Ginsberg et al., 1959). Ginsberg and
Leslie (1962) have shown that this structure persists even in a lead
alloy containing 10 atomic per cent of thallium, so that it is probably
not due to gap anisotropy. The effect may be due to states of collective
excitations lying in the gap (Tsuneto, 1960) which have not been taken
into account in the BCS theory. However, calculations of Maki and
Tsuneto (1962) lead one to expect that the energy of collective excita
tions should be drastically shifted by impurity scattering.
Richards (1961) has reported measurements on single crystals of
pure tin and of tin containing 01 atomic per cent indium. His results
show that the position of the absorption edge varies with crystal
orientation, which clearly indicates the anisotropy of the gap. Further
more this anisotropy decreases with increasing impurity, which
strongly supports Anderson's suggestion (1959) that the anisotropy
becomes smoothed out in impure samples. The absorption edges
observed by Richards have a structure which, unlike that seen in Pb
and in Hg, occurs for frequencies greater than v g . These postcursor
peaks do not seem to change with impurity, and have not yet found
an explanation.
10.4. Microwave absorption
Although the resistivity of a superconductor vanishes at 0°K for fre
quencies up to v g , there is a finite resistance at higher temperatures
even at lower frequencies (H. London, 1940). One can understand
The energy gap 101
this from a simple twofluid picture, according to which at any finite
temperature a fraction of the electrons remains ' normal'. H. London
pointed out that in the presence of an alternating electric field these
electrons absorb energy as they would in a normal metal, and that
such a field is needed to sustain an alternating current even in a super
conductor because of the inertia of the superconducting electrons.
Into a normal metal an alternating field penetrates to a skin depth
8, which leads to anomalous results if the mean free path l> 8, as is
the case at high frequencies and low temperatures (see p. 42). In the
superconducting phase, the theory of the anomalous skin effect still
applies in principle, but has to be modified both because for high fre
quencies the superconducting penetration depth A is much smaller
than the skin depth 8 (except very near T c ) and decreases very rapidly
with decreasing temperature, and because the number of 'normal'
electrons also drops sharply below T c . Both of these lead to a reduc
tion of the resistance in the superconducting phase as compared to
that in the normal one : the ratio of the resistances decreases rapidly
below T c , changes more gradually at lower temperatures where both
A and the order parameter are fairly constant, and finally vanishes at
0°K where there are no more 'normal' electrons.
Unpublished calculations of the variation of RJR„ with tempera
ture and with frequency have been made by Serber and by Holstein
on the basis of the ReuterSondheimer equations, the London theory,
and the twofluid model. Typical results are the solid curve labelled
065Ar B r c and the dashed one labelled 237 k B T c in Figure 30. With
frequencies up to 8 x 10 10 c/sec there is general experimental agree
ment with these calculations, as shown, for example, by the recent
results of Khaikin (1958) on cadmium and of Kaplan et al. (1959) on
tin. Their temperature dependence for a given frequency can be
represented by an empirical function, suggested by Pippard (1948):
#0 = / 4 (l/ 2 )0/ 4 r 2 . (X.1)
The frequency dependence is as v 4/3 at low frequencies, tending to
ward a constant value at higher frequencies.
However, surface impedance measurements at frequencies con
siderably higher than 8 x 10 10 c/sec show appreciable deviations from
the predictions of the simple twofluid model. Figure 30 shows the
Superconductivity
102
ratio RJR n as a function of reduced temperature for aluminium as
measured at three microwave frequencies by Biondie/A/ (1957) The
frequences are given in units of k B TJh. For 065, the results agree
well with the temperature variation calculated without regard to an
energy gap. For 237, however, such calculations would give the
dashed curve, and it is evident that for / > 07, the measured ratio
considerably exceeds the predicted one. The same is true for
hv  304k B T c , except that in this case the deviation already begins at
O.^r^k ' ' ' J ' 1 t I
0.6 07 0.8 0.9 1.0 U
Fio. 30
Clearly an additional absorption mechanism occurs for these fre
quencies, and of course this is due to the boosting of condensed elec
trons across the energy gap. If this gap had a constant width at all
t< 1 the appearance of this extra absorption would depend only on
the frequency. Its temperature dependence, however, clearly shows
that the energy gap varies with temperature, tending toward zero as
/M. As a result, photons of energy 237k B T c , for example, are not
sufficient to bridge the gap at / = 0, but become effective at that tem
perature at which the gap has shrunk to a width of 237 k B T A series
of measurements of the resistance ratio as a function both of frequency
and of temperature thus serves to map out the temperature variation
no™? u aP ° f 3ny g,Ven ^Perconductor. Biondi and Garfunkel
(1959) have obtained values of the resistance ratio by measuring
The energy gap 103
calorimetrically the amount of energy absorbed by an aluminium
wave guide, over a range of frequencies ranging from 065k B T c
(15 x 10 10 c/sec) to 391^7^ (10 x 10 10 c/sec) at temperatures down
1.0 1.5 2.0 2.5 3.0 3.5
Energy (in units of kT c )
Fig. 31
4.0
to 035°K. The accuracy of the measurements was such that the ab
sorption of 10 " 9 watt could be detected. Their results give a tempera
ture variation of the gap which is in close agreement with the predic
tions of the BCS theory.
Mattis and Bardeen (1958) and Abrikosov et ul. (1958) have de
veloped a theory of the anomalous skin effect in superconductors on
the basis of the BCS theory. Miller (1960) used the work of the former
to calculate the surface impedance for many different frequencies and
temperatures. The close agreement between his results and the
8
104 Superconductivity
measurements of Biondi and Garfunkel is shown in Figure 31, in
which points calculated by Miller are superimposed on smooth
curves representing the empirical values. The theoretical treatments
are equally successful in the lower frequency range in which there are
no gap effects.
10.5. Nuclear spin relaxation
When the nuclear spins of a substance are aligned by the application
of an external field, they again relax to their equilibrium distribution
predominantly by interaction with the conduction electrons. In this
interaction, a nucleus flips its spin one way as the electron spin flips
the other way so as to conserve the total spin. The electron can do this
only if there is available an empty final state of correct energy and
spin direction, and the nuclear relaxation rate in a normal metal
depends therefore both on the number of conduction electrons (itself
proportional to the product of the density of states and the energy
derivative of the Fermi function) and on the density of states in the
vicinity of the Fermi surface.
To predict the temperature variation of this relaxation process in
the superconducting phase one is tempted to use again a simple two
fluid model, according to which the number of 'normal' electrons
available decreases rapidly below T c . To this should, therefore, corre
spond a decrease in the relaxation rate as compared to that in the
normal phase. But the energy gap severely modifies the density of
states available to the interacting electrons. In the gap there are, by
definition, no available states at all, and the missing states are 'piled
up ' on either side. The presence of the energy derivative of the Fermi
function in the relaxation rate expression makes this rate essentially
proportional to the square of the density of states evaluated over a
range k B T on either side of the Fermi energy. At temperatures near
7;., the gap is still very narrow and < k B T, so that the pileup of
states on either side results in an appreciable increase of the rate over
that in the normal phase. At lower temperatures, the gap becomes
wider than k B T, and the relaxation rate rapidly diminishes, approach
ing zero as T*0.
The measurements of Hebel and Slichter (1959), of Redfield (1959)..
and of Masuda and Redfield (1960a) fully confirm this consequence
The energy gap 105
of the energy gap. In particular, a detailed analysis by Hebel (1959)
has shown that the empirical results are compatible with the manner
of piling up predicted by the microscopic theory. Hebel's results and
some empirical values are given in Figure 32, in which the ratio of the
relaxation rate in the superconducting phase to that in the normal one
is plotted against temperature. The temperature variation of what can
be considered as the attenuation of the nuclear alignment is markedly
different from the corresponding change in the attenuation of an
'0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
t
Fig. 32
ultrasonic elastic wave in a superconductor. As this difference is one
of the most striking consequences of the BCS theory, its discussion
and the general description of ultrasonic attenuation in supercon
ductors is postponed until a later chapter.
In his calculations, Hebel avoids singularities on either side of the
gap by introducing a parameter r which represents a smearing of the
density of states over an energy interval small compared to the width
of the gap. It is possible to interpret this in terms of an anisotropy of
the gap, since the relaxation process samples the gap over all direc
tions simultaneously. With this interpretation, the data on aluminium
106 Superconductivity
of Masuda and Redfield (1960a, 1962) indicate an anisotropy of the
order of 1/10 of the gap width, and recent measurement by the same
authors (Masuda and Redfield, 1960b; Masuda, 1962b) indicate that
this anisotropy decreases in impure aluminium. Anisotropy of mag
nitude similar to that in aluminium has been found by Masuda
(1962a) in cadmium.
10.6. The tunnel effect
The most recent and the most direct measurement of the energy gap
has been provided by the work of Giaever (1960a), who essentially
superconductor
normal metal
insulator
Fig. 33
measured the width of the gap with a voltmeter. He accomplished this
by observing the tunneling of electrons between a superconducting
film and a normal one across a thin insulating barrier. Quantum
mechanically, an electron on one side of such a barrier has a finite
probability of tunneling through it if there is an allowed state of equal
or smaller energy available for it on the other side. Figure 33 shows
the density of states function in energy space for a sandwich consist
ing, from left to right, of a superconductor, an insulator, and a normal
The energy gap 107
metal, all at 0°K. In the last of these, electrons fill all available states
up to the Fermi level E F ; in the superconductor, there is a gap of half
width e(0), and states up to E F — e(0) are filled. With such conditions
there can be no tunneling either way, as on neither side of the barrier
are there any available states.
A potential difference applied between the two metals will shift the
energy levels of one with respect to the other. It is evident from
Figure 33 that tunneling will abruptly become possible when the
applied voltage equals e(0). The subsequent variation of tunneling
current with applied voltage of course depends on the details of the
density of states curve of the superconductor on either side of the gap.
At first, there is a very rapid rise of current with voltage due to the
€(0)
Voltage
Fig. 34
large density of piledup stages; for voltages much exceeding e(0), the
tunnelling samples the density of states well beyond the gap, and the
variation of I vs. V approaches the purely ohmic character of a junc
tion of two normal metals. This is summarized in Figure 34, which
gives with the solid line the current voltage characteristic of the
superconductingnormal junction at 0°K. The dotted line indicates
the behaviour at < T< T c , the modification being due to the fact
that at finite temperatures on both sides of the junction some electrons
are excited across the gap or the Fermi level, respectively. The dashed
line shows the behaviour at T > T c , i.e. for a junction of normal metals.
Nicol et al. (1960) and Giaever (1960b) have extended such experi
ments to cases where both metals of the junction are superconductors,
but with very different critical temperatures, such as Al (T c = 12°K)
and Pb (T c = 72°K). The gaps of the two will be correspondingly
'08 Superconductivity
different, and for such a junction the density of states function at 0°K
is shown in Figure 35. A tunneling current will begin to flow when
the potential difference between the two metals is €(0) Pb +e(0) AI . In
this case, however, the modification due to finite temperature is more
significant than with an sn junction. Imagining the density of states
curve of Figure 35 with a few excited electrons beyond both gaps, and
a few available states remaining below both, one recognizes that now
superconductor 1
superconductor
insulator
Fig. 35
the current / at first increases with increasing potential V, then de
creases for e(0) Pb e(0) A1 < K<€(0) Pb + € (0) A1 , and then increases
again. Figure 36 shows the currentvoltage characteristics in this
case; the limits of the negative resistance region are very sharp. Thus
the currentpotential characteristics yield the energy gap values at a
given temperature for both metals.
The energy gap values obtained by this method for several super
conductors are listed in Table III, and can probably be considered as
the most reliable of all experimental determinations. Measurements
as a function of temperature closely support the thermal variation
of the energy gap predicted by the BSC theory. The films used are thin
The energy gap 109
compared to the penetration depths, and because of their size their
critical fields are very high. This and its use in investigating the varia
tion of the energy gap with magnetic field was discussed in Chapter
VI I. Recent tunneling studies have verified other aspects of the energy
gap, in particular its relationship to the phonon spectrum of the
superconducting lattice. This will be summarized in Chapter XI.
Simultaneous tunneling of two electrons has been observed by
Taylor and Burstein (1962), in agreement with the calculations of
6,e 2 e,+e 2
Voltage
Fig. 36
Schrieffer and Wilkins (1962). This is not to be confused with the
tunneling of Cooper pairs, as predicted by Josephson (1962), which
will be discussed in Section 1 1.7. The results of Taylor and Burstein
also indicate the possibility of tunneling assisted by the simultaneous
absorption of a phonon. Theoretical aspects of this have been dis
cussed by Kleinman (1963) and Fibich (1964).
10.7. Far infrared transmission through thin films
In a series of experiments, Tinkham, Glover, and Ginsberg (Glover
and Tinkham, 1957; Ginsberg and Tinkham, 1960) have measured
110 Superconductivity
the transmission through thin superconducting films of electromag
netic radiation in the farinfrared range of wavelengths between 01
and 6 mm. Their results lend themselves to an ingenious analysis
leading to a number of very fundamental conclusions about the inter
relation of the energy gap, the response to high frequency radiation,
Fig. 37
and the existence of perfect conductivity and of the Meissner effect in
the limit of zero frequency (see Tinkham [10], pp. 168176).
In Figure 37, the curve labelled TJT n is one which can be drawn
through the empirical values of the ratio of the transmissivity in the
superconducting phase, T s , to the normal value, T„, all suitably
normalized for film resistance and substrate refraction, and plotted
against frequency. The transmissivity of a substance is related to its
conductivity. One can approximate the conductivity of the film in the
normal state by a real number, a n , which to a good approximation is
The energy gap 1 1 1
independent of frequency in the range under investigation. The super
conducting conductivity can be written as the complex quantity
It then follows from general electromagnetic theory that
(X.2)
£ = ( T^Hirt^) +[( I ^> ,/2 ^]~}~ • (X 3 >
Microwave work on bulk superconductors, such as the measurements
of Biondi and Garfunkel (1959), have shown that at T< T c and
ho < k B T c , the surface resistance vanishes. It follows from this that
the real, lossy part of the conductivity must also vanish in this range,
or a, « 0, so that the low frequency measurements of T s /T„ can be
used to evaluate the corresponding values of a 2 /a„. For a number of
samples of tin and lead with widely varying normal conductivity, all
the data of Glover and Tinkham fit a universal curve represented by
°il° n = (\la)(k B TJtiw), a = 027.
As a n is independent of frequency, X.4 implies that
a 2 cc 1/cu.
(X.4)
(X.5)
This is just the frequency dependence which follows from the London
equation
curlJ + _ 2 H = 0,
(X.6)
since this with Maxwell's equation
curlE = H
c
leads to
c 2 1
a 2 =
4ttA 2 ,
(X.7)
An imaginary conductivity which is inversely proportional to the
frequency thus corresponds to the consequences of X.6 : the Meissner
CHRIST'S COLLEGE
I mm nw
112 Superconductivity
effect and a finite penetration depth A. However, the magnitude of A
calculated from the experimental transmission results with the aid
of X.7 exceeds the London value \ L = mc 2 jATT 2 ne 2 by at least a factor
often. Furthermore, there is nothing in the London theory to explain
why a 2 /cr„ for different superconductors should satisfy a universal
equation like X.4. On the other hand the Pippard treatment predicts
for these films, in which £ as l< A, that (see equation IV. 18a) :
where (equation IV.9) :
Hence
A 2 = (lo/OAl,
£ = afiv /k B T c .
CT 2
On
ne 2 I 1 1
X— X— X —
m £ to o„
For a normal metal
mvQ
and hence the Pippard theory leads to
Z 2
O n
i k B T t
a htxi
for all superconductors. This is strikingly verified by the results of
Glover and Tinkham.
The real and imaginary parts of any linear response function, such
as the electrical conductivity, are related by a pair of integral trans
forms known as the KramersKronig (KK) relations. In terms of the
conductivity these take the form:
i<«)
+ 00
co x a 2 (co x )dco\
t0 2 — (0 2
+ 00
o 2 (to)
■.
co 2 — to 2
(X.8)
The energy gap 1 1 3
Substituting X.7 into the first of these two relations shows that the
imaginary conductivity o 2 must be accompanied by a real conduc
tivity which takes the form of a deltafunction at the origin:
a, (to) = (c 2 /8A 2 ) 8(co  0). (X.9)
Similarly, in terms of the empirical value X.4 for o 2 /o„ one would
have
o x _ TT\ k B T r
o~ 2a h
S(o)O).
(X.10)
Such an infinite real conductivity at zero frequency of course does not
introduce losses.
6 8 10 12
h<y/k B T c
Fig. 38
Turning now to the high frequency far infrared transmissivity data,
the peak and subsequent decrease of T s /T„ indicates that at a fre
quency roughly corresponding to the peak, a real, lossy component
ct, of the superconducting conductivity must appear. In the absence
of such a component T s /T„ would continue to rise. The appearance of
a real component of conductivity at or near some critical frequency is,
of course, highly suggestive of an energy gap. Taken by themselves,
the data of Tinkham et al. do not determine the gap quite unam
biguously (see Forrester, 1958). However, accepting the existence of
a gap from other experiments allows a fully consistent interpretation
of the transmission results from which the magnitude of the gap as
well as other interesting quantities can be derived.
1 14 Superconductivity
The calculations of Miller (1960) of the variation of a x \a n are
shown in Figure 37, the ordinate being scaled in units of
2e(0)
where 2c(0) is the width of the gap at 0°K. An energy gap implies
that, as for a normal metal, the imaginary part of the conductivity
vanishes for frequencies beyond the gap. Using X.3 one can then
calculate cr,/<x n from the measured values of TJT„ to a first approxi
mation, and then apply an iterative procedure using the KK relations
as well as the sum rule about to be mentioned to obtain final values
of ajcr,,. Figure 38 gives the result thus obtained by Ginsberg and
Tinkham for lead, showing the precursor peak also found for mer
cury. One ignores this in deriving energy gap values from the limit
<y i/ CT /. > 0 The resulting gap widths are listed in Table III.
10.8. The FerrellGlover sum rule
The intimate connection between the experimentally verified decrease
of aj/a„ near co g , corresponding to the existence of a gap, and the low
frequency Londontype imaginary conductivity a \ <x 1/eo, corre
sponding to infinite conductivity and the Meissner effect at zero
frequency, was first pointed out by Ferrell and Glover (1958) and
further elaborated by Tinkham and Ferrell (1959). The first of these
papers pointed out that at extremely high frequencies, such that hut
far exceeds any of the binding energies of an electron in the metal,
the real part of the conductivity vanishes. The appropriate KK rela
tion for the imaginary conductivity then becomes, since a t is an even
function,
a 2 (o>) « —
TTOi
I
9f(a>i)dh>.
CX.11)
At these very high frequencies all electrons are free in both the normal
and the superconducting phases, and one would thus expect cr 2 (co)
and, therefore, the integral in X.l 1 to have the same value in both
phases. In other words, there exists the sum rule that this integral
remains unchanged under the superconducting transition. From this
The energy gap 1 1 5
it follows that any area A removed from under the o,(o>) curve by
the energy gap must reappear somewhere else, and it can do so only
at the origin in the form of a delta function of strength A. This being
the case, one can then again apply the KK relations to show that
associated with such a delta function
a,(oi) = A8(oj0) (X.12)
must be a contribution to the imaginary conductivity of magnitude
aiiai) = IA/tho. (X.13)
The argument has now come through a full circle. An energy gap
corresponds to a disappearance of a^w) in the superconducting
phase over some frequency range in which this conductivity is finite
in the normal metal. This, according to the FerrelGlover sum rule,
must lead to the appearance of a delta function X.12. In turn this
leads to a Londontype imaginary conductivity ff 2 ccl/£o, which was
seen to correspond to the Meissner effect and infinite conductivity.
One sees further that in terms of the parameter a of X.4, one can
write
A\o n = (7Tl2)(k B T c lh)(Ma). (X.14)
Determining A/a„ from their transmission data and using this rela
tion, Ginsberg and Tinkham obtain values for a of 023 for lead, 026
for tin, and 0 19 for thallium. These, as well as Glover and Tinkham's
value of 027 for both tin and lead, are in remarkable agreement both
with the FaberPippard data (015 for tin and indium) and with the
BCS prediction for all metals (018). The agreement is particularly
convincing if one considers the simplifications of the theory on the
one hand, and the wide variety and considerable difficulty of the
experiments on the other.
From X.13 and X.7 it is evident that
A 2 = c 2 I8A.
(X.15)
Thus the FerrellGlover sumrule leads to an inverse proportionality
between the square of the penetration depth and the energy gap. Such
a relation is implicit in the Pippard model and the Bardeen theory,
and appears explicitly in the GinzburgLandau treatment as extended
by Gor'kov.
CHAPTER XI
Microscopic Theory of Superconductivity
11.1. Introduction
In reviewing the contents of the preceding chapters, which give an
empirical description of superconductivity, perhaps the most striking
feature to be noticed is how much quantitative information can be
given about superconductivity in general without speaking about the
specific properties of any one of the many superconducting elements.
The astonishing degree of similarity in the superconducting behaviour
of metals with widely varying crystallographic and atomic properties
indicates that the explanation for superconductivity should be in
herent in a general, idealized model of a metal which ignores the com
plicated features characterizing any individual metallic element. It
should, therefore, be possible to find in the simple model of the ideal
metal the possibility of an interaction mechanism leading to the super
conducting state, and to derive from this at least qualitatively the
properties of an ideal superconductor.
One would judge from this that an explanation for superconduc
tivity should be fairly easy, until he realizes the extreme smallness of
the energy involved. A superconductor can be made normal by the
application of a magnetic field H c which at absolute zero is of the
order of a few hundred gauss. The energy difference between the
superconducting and the normal phase at absolute zero, which is
given by Hq/Stt, thus is of the order of 10~ 8 e.v. per atom. How
very small this is can best be judged by remembering that the Fermi
energy of the conduction electrons in a normal metal is of the order
of 1020 e.v. The simple model of Bloch and Sommerfeld gives a
reasonably accurate description of the basic characteristics of a metal
although it completely ignores, among other things, the correlation
energy of the conduction electrons due to their Coulomb interaction.
This energy is of the order of 1 e.v. !
As a further difficulty in arriving at a microscopic theory of super
conductivity one must add the extreme sharpness of the phase
116
Microscopic theory of superconductivity 117
transition under suitable conditions. The absence of statistical fluctu
ations shows that the superconducting state is a highly correlated one
involving a very large number of electrons. Thus it is necessary to find
inherent in the basic properties common to all metals an interaction
correlating a large number of electrons in such a way that the energy
of the system relative to the normal metal is lowered by a very small
amount. The discovery of the isotope effect in a number of super
conducting elements clearly indicated that in these the interaction in
question must be one between the electrons and the vibrating crystal
lattice, and indeed Frohlich (1950) had suggested such a mechanism
independently of the simultaneous experimental results.
11.2. The electron phonon interaction
Frohlich and, a little later, Bardeen (1 950) pointed out that an electron
moving through a crystal lattice has a self energy by being 'clothed'
with virtual phonons. What this means is that an electron moving
through the lattice distorts the lattice, and the lattice in turn acts on
the electron by virtue of the electrostatic forces between them. The
oscillatory distortion of the lattice is quantized in terms of phonons,
and so one can think of the interaction between lattice and electron
as the constant emission and reabsorption of phonons by the latter.
These are called 'virtual' phonons because as a consequence of the
uncertainty principle their very short lifetime renders it unnecessary
to conserve energy in the process. Thus one can think of the electron
moving through the lattice as being accompanied or 'clothed', even
at 0°K, by a cloud of virtual phonons. This contributes to the electron
an amount of selfenergy which, as was pointed out by Frohlich and
by Bardeen, is proportional to the square of the average phonon
energy. In turn this is inversely proportional to the lattice mass, so
that a condensation energy equal to this selfenergy would have the
correct mass dependence indicated by the isotope effect. Unfortu
nately, however, the size turns out to be three to four orders of
magnitude too large.
It was only seven years later that Bardeen, Cooper, and Schrieffer
(BCS, 1957) succeeded in showing that the basic interaction respon
sible for superconductivity appears to be that of a pair of electrons
by means of an interchange of virtual phonons. In the simple terms
118 Superconductivity
used above this means that the lattice is distorted by a moving elec
tron, this distortion giving rise to a phonon. A second electron some
distance away is in turn affected when it is reached by the propagating
fluctuation in the lattice charge distribution. In other words, as shown
in Figure 39, an electron of wave vector k emits a virtual phonon q
which is absorbed by an electron k'. This scatters k into k — q and
k' into k' + q. The process being a virtual one, energy need not be
conserved, and in fact the nature of the resulting electronelectron
interaction depends on the relative magnitudes of the electronic
energy change and the phonon energy fico q . If this latter exceeds the
Fig. 39
former, the interaction is attractive — the charge fluctuation of the
lattice is then such as to surround one of the electrons by a positive
screening charge greater than the electronic one, so that the second
electron sees and is attracted by a net positive charge.
The fundamental postulate of the BCS theory is that supercon
ductivity occurs when such an attractive interaction between two
electrons by means of phonon exchange dominates the usual repulsive
screened Coulomb interaction.
11.3. The Cooper pairs
Shortly before the formulation of the BCS theory, Cooper (1956) had
been able to show that if there is a net attraction, however weak,
Microscopic theory of superconductivity 1 19
between a pair of electrons just above the Fermi surface, these elec
trons can form a bound state. The electrons for which this can occur
as a result of the phonon interaction lie in a thin shell of width ^ hu) q ,
where hco q is of the order of the average phonon energy of the metal.
If one looks at the matrix elements for all possible interactions which
take a pair of electrons from any two k values in this shell to any two
others, he finds that because of the Fermi statistics of the electron
these matrix elements alternate in sign and, being all of roughly equal
magnitude, give a negligible total interaction energy, that is, a
vanishingly small total lowering of the energy relative to the normal
situation of unpaired electrons. One can, however, restrict oneself to
matrix elements of a single sign by associating all possible k values in
pairs, kj and k 2 , and requiring that either both or neither member of
a pair be occupied. As the lowest energy is obtained by having the
largest number of possible transitions, each represented by a matrix
element all of the same sign, one wants to choose these pairs in such
a way that from any one set of values (kj, k^, transitions are possible
into all other pairs (k", k£. As momentum must be conserved, this
means that one must require that
k 1 + k 2 = k,' + k^ = K
(XI.1)
that is, that all bound pairs should have the same total momentum K.
(See, for example, Cooper, 1960.)
To find the possible value of kj and k 2 which satisfy XI. 1 and at
the same time lie in a narrow shell straddling the Fermi surface k F
one can construct the d iagram shown in Figure 40, d rawing concentric
circles of radii k F  8 and k F + 8 from two points separated by K. It is
clear that all possible values of k, and k 2 satisfying XI. 1 are restricted
to the two shaded regions. This shows that the volume of phase space
available for what has become known as Cooper pairs has a very
sharp maximum for K = 0. Thus the largest number of possible
transitions yielding the most appreciable lowering of energy is
obtained by pairing all possible states such that their total momentum
vanishes. It is also possible to show that exchange terms tend to reduce
the interaction energy for pairs of parallel spin, so that it is ener
getically most favourable to restrict the pairs to those of opposite
spin. One can, therefore, summarize the basic hypothesis of the BCS
9
120 Superconductivity
theory as follows: At 0°K the superconducting ground state is a highly
correlated one in which in momentum space the normal electron states
in a thin shell near the Fermi surface are to the fullest extent possible
occupied by pairs of opposite spin and momentum. The most direct
verification of the existence of these pairs arises from the flux quanti
zation measurements mentioned in Chapter III.
The energy of this state is lower than that of the normal metal by a
finite amount which is the condensation energy of the superconducting
state and which at 0°K must equal Hi/Sir per unit volume. Further
more, this state has the allimportant property that it takes a finite
quantity of energy to excite even a single ' normal ', unpaired electron.
For not only does this require the very small amount of energy needed
Fig. 40
to break up a bound pair, but more importantly the occupation of a
single k state by an unpaired electron removes from the system a large
number of pairs which could have interacted so as to occupy k and
— k. Hence the total energy difference between having all paired
electrons and having a single excited electron is finite and equal to a
large multiple of the single pair correlation energy. In terms of the
single electron spectrum, therefore, theBCS theory correctly yields an
energy gap. It has already been shown that such an energy gap not
only leads to the observed variation of the specific heat, the thermal
conductivity, and the absorption of high frequency electromagnetic
radiation, but also that it is correlated with the existence of perfect
diamagnetism and perfect conductivity in the low frequency limit.
11.4. The ground state energy
The recognition of the basic electron interaction mechanism respon
sible for superconductivity does not remove the major difficulty
Microscopic theory of superconductivity 121
mentioned earlier, namely that the correlation energy in question is so
very much smaller than almost any other contribution to the total
electronic energy. BCS therefore take the bold step of assuming that
all interactions except the crucial one are the same for the supercon
ducting as for the normal ground state at 0°K. Taking as the zero of
energy the normal ground state energy and including in this all
normal state correlations and even the self energy of the electrons due
to virtual phonon emission and reabsorption, BCS proceed to calcu
late the superconducting ground state energy as being due uniquely
to the correlation between Cooper pairs of electrons of opposite spin
and momentum by phonon and screened Coulomb interaction.
The interaction leading to the transition of a pair of electrons from
the state (k t , k  ) to (k' t , k' I ) is characterized by a matrix
element,
]^ fc/ = 2(k'i,k'tl#intk!»M), (XI.2)
where i/ int is the truncated Hamiltonian from which all terms com
mon to the normal and superconducting phases have been removed.
V kk  is the difference between one term describing the interaction
between the two electrons by means of a phonon, and a second one
giving their screened Coulomb interaction. The basic similarity of the
superconducting characteristics of widely different metals implies that
the responsible interaction cannot crucially depend on details charac
teristic of individual substances. BCS therefore make the further
simplifying assumption that V kk  is isotropic and constant for all
electrons in a narrow shell, straddling the Fermi surface, of thickness
(in units of energy) less than the average energy of the lattice, and
that V kk  vanishes elsewhere. Measuring electron energy from the
Fermi surface, and calling e k the energy of an electron in state k, one
can state this formally by the equations:
and
V kk .= V forhfcl.M «&»„
V kk  = elsewhere.
(XI.4)
The basic BCS criterion for superconductivity is equivalent to the
condition
V< 0.
122 Superconductivity
It is well to note clearly at this point that this simplification of the
interaction parameter F necessarily leads to what can be called a law
of corresponding states for all superconductors, that is, virtually
identical predictions for the magnitudes of all characteristic quantities
in terms of reduced coordinates. Any empirical deviation from such
complete similarity is, therefore, no invalidation of the basic premise
of the BCS theory, but merely an indication of the oversimplification
inherent in XI.4. (See footnote, page 1 30.)
Let h k be the probability that states k and — k are occupied by a
pair of electrons, and (\—h k ) the corresponding probability that the
states are empty. W(Q), the ground state energy of the superconducting
state at 0°K as compared to the energy of the normal metal, is then
given by
^(0)= S^ArSW^lMMl^}" 2 . (XI.5)
k kk"
The summation is over all those kvalues for which V kk  9* 0» so that
using XI.4 one can simplify to
wm  S 2e k h k  v 2 {h k (i MMi h)) m  (Xi.50
k kK
The first term gives the difference of kinetic energy between the super
conducting and normal phases at 0°K. The factor 2 arises because for
every electron in state k of energy e k there is with an isotropic Fermi
surface another electron of the same energy in — k. This first term can
be either positive or negative, and is smaller than the second term
which gives the correlation energy for all possible transitions from a
pair state (k, — k) to another (k', — k'). For such a transition to be
possible, k must initially be occupied and k' empty. The simultaneous
probability of this is given by h k {\ — h k >). The final state must have k
empty and k' occupied, and this has probability h k {\ — h k ). The
square root of the product of these probabilities multiplied by the
matrix element for the transition and summed over all possible values
of k and k' gives the total correlation energy.
W(0) must of course be negative for the superconducting phase to
Microscopic theory of superconductivity 123
exist, and to see whether this is possible XI.5' can be minimized with
respect to h k . This leads to
[h k (lh k )} 1 ' 2 = v w
l2h.
2e k
By defining
equation XI.6 simplifies to
e(0)= KStMlMl
kf
1/2
=*!)•
h,=
where
E k = [4+e 2 (0)]
1/2
(XI.6)
(XI.7)
(XI.8)
(XI.9)
Substituting XI.8 back into XI.7 one obtains a nonlinear relation
for€(0):
Ky g(0)
e{0) "2Z[ £ I + eW 2 '
(XI. 10)
This can be treated most readily by changing the summation to an
integration and transforming the variable of integration from k to e.
Assuming symmetry of states on either side of the Fermi surface
( c = 0), and introducing the density of single electron states of one
spin in the normal state at e = 0: M0), XI. 10 becomes
JlCOq
M0)
[e 2 +€ 2 (0)] ,/2
The limit of integration is the phonon energy above which, according
to XI.4, V=0.
The solution of XI. 11 is
e(0) = /mysinMl/MO) V\. (XI.12)
Putting this back into XI.9 and XI.7 and finally into XI.5', one finds
that the ground state energy of the superconducting state is given by
W(0) = 
2MQ)(W 2
exp[2/M0)F]l
(XI. 13)
124 Superconductivity
The numerator of this quantity follows from dimensional reasoning
from any theory which postulates an interaction between electrons
and phonons and allows this interaction to be cut off at some average
phonon energy hw q « k B 6, beyond which the interaction becomes
repulsive. A term like this had been contained in the earlier attempts
of Frohlich and of Bardeen, and, as mentioned before, is much too
large. The success of the BCS theory lies in the appearance of the
exponential denominator which reduces W(0) by many orders of
magnitude. Although a precise calculation of the average interaction
parameter V for a specific metal continues to be among the most
important questions still to be solved, various estimates (Pines, 1958 ;
Morel, 1959; Morel and Anderson, 1962) indicate that the values
of N(0) Vx 03, derived from a knowledge of H Q , are reasonable.
Thus the denominator has a value of about e 7 .
The isotope effect follows from the numerator of XI. 1 3, as it would
from any theory involving electronphonon interaction with a cut
off frequency related to the Debye 6 and hence to the isotopic mass.
Equation XI. 13 shows that
H 2
~ = ^(0) cc {hu> q ) 2
(k D Q) 2 cc Mfol
(XI. 14)
For a group of isotopes, one finds H cc T c , so that
T c « Mr* 12 . (XI.15)
Any appreciable deviation of the isotope effect exponent from the
value 05 could indicate that the simplifying BCS assumption of a cut
off for both Coulomb and phonon interaction at hw q has to be modi
fied (Tolmachev, 1958; Swihart, 1959, 1962). Bardeen (1959) has
pointed out that the cutoff may be determined by the lifetime of the
'normal' electrons which can be excited across the gap. These elec
trons are not the bare, noninteracting electrons of the simple Bloch
Sommerfeld model. Instead they are socalled quasiparticles
'clothed' by their interactions with each other and with the lattice
(see \1], pp. 18495). As a result the wave functions describing them
are not eigenfunctions of the system, so that the particles have a finite
lifetime. The effect of this on the pair interaction has been further
Microscopic theory of superconductivity 125
discussed by Ehashberg(1961),Bardeen[9],Schrieffer(1961),Betbeder
Matibet and Nozieres (1961), and Bardasis and Schrieffer (1961).
The damping of the quasiparticles is found to be very small even up
to energies well beyond the Fermi energy. This is in contradiction to
the BCS assumption embodied in XI .4, as the justification of the cut
off of the Coulomb interaction at hu) q is essentially that quasiparticles
of larger energy are so strongly damped as not to be available for pair
formation. It is thus necessary to modify the BCS cutoff by taking into
account the existence of the repulsive interaction for e k > hw q . This
does not appreciably affect the gap at the Fermi surface (e k = 0), but
will result in its variation with e k , as will be further discussed in
Section 11.7.
With a compound tunnelling arrangement in which electrons are
injected into a layer of superconducting lead and then have the possi
bility of tunnelling through a second junction into normal metal,
Ginsberg (1962) was recently able to place an upper limit on the life
time of the quasiparticles in a superconductor. According to his pre
liminary result this upper bound is 22 x 10~ 7 sec, which is only about
five times as large as the average time calculated by Schrieffer and
Ginsberg (1962) for quasiparticle recombination into pairs by means
of phonon emission. This has also been calculated by Rothwarf and
Cohen (1963).
Swihart (1962) as well as Morel and Anderson (1962) have studied
the isotope effect for different forms of the energy dependence of the
electronelectron interaction. They find that the exponent of the iso
topic mass in equation XI. 1 5 is less than the ideal value of one half by
amounts of 1030 per cent which increase with decreasing TJ6.
However, the isotope effects in ruthenium (Geballe et ol., 1961,
Finnemore and Mapother, 1962), osmium (Hein and Gibson, 1964)
and perhaps also in molybdenum (Matthias et a I., 1963) appear to be
too small to be explained by these calculations.
This raises questions about the origin of the attractive interaction
responsible for the formation of Cooper pairs in these as well as
perhaps in other metals. Matthias (see for example, 1960) has repeat
edly suggested that in all transition metals there exists an attractive
magnetic interaction responsible for superconductivity. However,
both Kondo (1963) and Garland (1963a) have tried to explain the
126 Superconductivity
apparently anomalous superconducting behaviour of the transition
metals as a consequence of the overlap at the Fermi energy of the .v
and d bands of the electronic spectrum, and not because of a magnetic
interaction. Rondo assumes a larger interband interaction; Garland,
on the other hand, believes that the electrons of high effective mass
in the dband tend not to follow the motion of the selectrons. This
results in 'antishielding' the interactions between ^electrons, leading
to an attractive screened Coulomb interaction between them being
added to the usual attractive interaction by exchange of virtual
phonons.
Garland (1962b) calculated the magnitude of the isotope effect for
all superconducting elements and obtains results which agree closely
with all available experimental results, including in particular the
reduced effect in transition metals. This also results, at least quali
tatively, from Rondo's calculations. Garland was also able to explain
the anomalous pressure effect in transition metals (Bucher and Olsen,
1964).
11.5. The energy gap at 0°K
From XI.5' one can see that the contribution of a single pair state
(k, k) to this total condensation energy is
W k = 2* k h k 2Vj: {0MM ,/2 . (XI.16)
The first term represents the kinetic energy of both electrons in the
pair state k, and the second term the total interaction energy due to all
possible transitions into or out of the state.
At 0°R the lowest excited state of the superconductor must corre
spond to breaking up a single pair by transferring an electron from a
state k to another, leaving an unpaired electron in  k. The condensa
tion energy is then reduced by W k . The first term of this can be made
arbitrarily small, and is analogous to the excitation energy in a normal
metal, for which there is a quasicontinuous energy spectrum above
the ground state. The second term of W k , however, is finite for all
values of k, which is why in the superconducting phase the lowest
excited state is separated from the ground state by an energy gap.
Microscopic theory of superconductivity 127
Comparing XI.16 with XI.7 one sees that this energy gap has the
value 2e(0), which according to XI. 12 equals
2c(0) = 2^^/sinh [1 /N(0) V\. (XI. 1 7)
As 1/N(0) V& 34, this can be approximated by
2 € (0) = 4Aco 9 exp [ 1 /tf(0) V\. (XI. 1 8)
11.6. The superconductor at finite temperatures
As the temperature of the superconductor is raised above 0°R, an
increasing number of electrons find themselves thermally excited into
single quasiparticle states. These excitations behave like those of a
normal metal; they are readily scattered and can gain or lose further
energy in arbitrarily small quantities. In what follows they are simply
called normal electrons. At the same time there continues to exist the
configuration of all electrons still correlated into Cooper pairs, and
displaying superconducting properties, being very difficult to scatter
or to excite. One is thus led again to a twofluid point of view.
As at 0°R, one can write down an analytic expression for the ground
state energy W{J) containing a kinetic energy term and an interaction
term. In both, the presence of the normal electrons must be accounted
for, which is done by introducing a suitable probability factor f k .
Letting f k = probability of occupation of k or of k by a single
normal electron, then
1 2f k = probability that neither k nor k is occupied by a
normal electron.
This leads to a kinetic energy term
Wn] K . E , = 2£ M[f k + V2f k )h k ],
k
(XI. 19)
where the summation is over the same range as at 0°R, and h k retains
the same definition, though no longer the same value. The second
term in the brackets clearly gives the probability that the pair state
(k, — k) not be occupied by normal electrons but by a correlated pair.
The correlation energy at a finite temperature is
kk'
x(l2/*)(l2/*0. (XI.20)
128 Superconductivity
The last two terms ensure that the correlated pair states not be occu
pied by normal electrons. It is obvious that the presence of these
terms decreases the pairing energy.
The thermal properties of the superconductors can now be found
quite readily by writing down the free energy of the system and
requiring this to be at a minimum. The free energy is
G = WmTS = [W(T)\ K . B MW(Tj\ C0 „TS, (XI.21)
where J is the temperature and S the entropy. This last is due entirely
to the normal electrons; the electrons which are still paired are in a
state of highest possible order and do not contribute at all. Thus the
entropy is given by the usual expression for particles obeying Fermi
Dirac statistics :
TS = 2k B TZ {A.lnA. + (l/*)m(lA)}. (XI.22)
k
Substituting XI. 19, XI.20 and XI.22 into XI.21, and minim ising this
free energy with respect to h k , one now obtains
MlhM«* S[MiM]" 2 (i2A0
l2h k
This time one defines
= V
2e 4
<T)= F£[M1/'a<)] ,/2 (12/,0,
and again obtains
hi
Hi
(XI.23)
(XI.24)
(XI.25)
where E k is now defined as E k m [e k + e 2 (!T)] ,/2 .
One sees that, as at 0°K, 2e(T) represents the contribution of a
single pair state to the total correlation energy, and that to break up
one such pair at any finite temperature removes from the supercon
ducting energy at least this amount. In other words, the supercon
ducting state continues to contain an energy gap 2e(T) separating the
lowest energy configuration at any given temperature from that with
one less correlated pair.
Microscopic theory of superconductivity 129
To evaluate the magnitude of the energy gap one must first find an
expression for f k , which one obtains by minimizing the free energy
with respect to f k . This yields
f k = [ C xp(E k /k B T)+]) 1 . (XI.26)
XI.26, XI. 18, and XI.24 yield for e(T) sl nonlinear relation which,
changing as before from a summation over k to an integration over
e, becomes
Jiuia
[e2+ 7^^(^H' <XL27)
o
Wv = J
The critical temperature T c is reached when all pair states are broken
up so that e(T c ) = 0. Hence
1 f de . €
W = J 7 tan W c
o
(XI.28)
As long as k B T c < hw q , the solution of this can be written as
k B T c = M4/K^exp[ 1/W(0) V\. (XI.29)
The exponential dependence of the transition temperature has been
verified by Olsen et at, (1964) by means of measurements of its varia
tion with pressure in aluminium.
11.7. Experimental verification of predicted thermal properties
Combining equations XL 18 and XI.29 yields for the width of the
energy gap at 0°K
2e(0) = 352 k B T c . (XI.30)
This is in remarkable quantitative agreement with empirical values
obtained from the wide variety of measurements mentioned in Chap
ter X. Table HI shows that for the most widely different elements
the energy gap does not appear to deviate from this idealized value
by more than about 20 per cent. The theoretical temperature variation
of the gap width is displayed in Figure41 ; this has also been well con
firmed by a number of experiments.
130 Superconductivity
Muhlschlegel (1959) has tabulated values of the energy gap, the
entropy, the critical magnetic field, the penetration depth, and the
specific heat, all in reduced coordinates, as functions of the reduced
temperature. All these are in close agreement with experimental
results.
These agreements clearly vindicate the basic BCS approach , accord
ing to which the similarities between superconductors outweigh their
differences, so that an approximate law of corresponding states should
1.0
OB
em o.6
02
0.2
0.4 0.6
t
Fig. 41
0.8
t.O
hold.f This similarity principle had of course emerged from much pre
vious experimental evidence. However, differences between metals
and anisotropics in a given metal do exist, and the experimental evi
dence for gap variations from one metal to another, as well as for gap
anisotropics, clearly indicates the need to refine the details of the BCS
calculations. For one thing it is of course desirable to take into account
t Deviations from such a law can occur even with the BCS assumption
of constant Kif in solving equations XI.27 and XI.28 one takes into account
higher order terms in k B T e lhw q (Muhlschlegel, 1959). The resulting correc
tion factors appearing in equations XI. 30, XI.35, and XI.36 are, however,
too small to explain the empirical deviations from similarity discussed in
this section. Thouless (I960) has shown that in the BCS formulation the
energy gap at 0°K is only 40 k B T c even in the nonphysical limit
Microscopic theory of superconductivity 131
the dependence of the interaction parameter Fon k and k', so as to
be able to calculate directional effects. Even more challenging are the
previously mentioned attempts to relax, even in an isotropic model,
the assumption XT.4 that Fis strictly constant for e k < hoj q and is then
cut off abruptly. A better knowledge of the variation of Fwith e k in
turn would allow the more precise calculation of the corresponding
dependence of the energy gap on e k . The actual form of this variation
undoubtedly more nearly resembles the solid line in Figure 42 rather
than the dotted line which corresponds to the simple BCS assumption.
Usually one is interested in excitation energies of the order of k B T c
and the BCS assumption is then fully applicable as long as k B T c < hu) q ,
which is called the weak coupling limit. As hw q &k B ®, where 6 is
the Debye temperature, this requires that
T c < e.
For a number of superconducting elements, in particular for Pb and
Hg, this condition does not hold.
Swihart (1962, 1963) as well as Morel and Anderson (1962) have
investigated the consequences of an energy dependence of the inter
action Kmore realistic than that assumed by BCS. In particular they
take into account that, as was mentioned earlier, lifetime effects are
too small to justify cutting off the Coulomb repulsion at hm q . There
fore these authors include in the interaction a repulsive part (V> 0)
for energies e k > hw q . The resulting variation of the energy gap at
0°K as a function of e k has been shown by Morel and Anderson to
have the form represented schematically in Figure 42. Swihart found
132 Superconductivity
that a rise of this gap function on moving from the Fermi surface
leads to the correct specific heat jump for lead at T c . The relation
between calorimetric and magnetic properties indicates that such a
gap variation is also consistent with the observed critical field curve
for lead and probably also with that for mercury.
For even higher quasi particle energies the energy gap continues
to change sign periodically at multiples offiw q . This is consistent with
the observations of Rowell et al. (1962), who found maxima in the
tunneling conductance with that periodicity. Excitation of high
energy quasi particles involve multiphonon interactions.
A precise calculation of the energy gap variation with e k cannot
content itself with assigning to the phonons an average energy hcu q .
Instead it must take into account the details of the phonon spectrum,
as determined, for example, by neutron diffraction. In particular it
is necessary to recognize the different frequency distributions for the
longitudinal and transverse phonons. Such a calculation was carried
out for lead by Culler et al. (1962), using an online computer facility.
The relation between the phonon spectrum and the tunneling
characteristics has been fully discussed by Scalapino and Anderson
(1964).
In considering an energy gap which changes sign as a function of
e k , it must be remembered that in an experiment involving thermal
or electromagnetic absorption across the gap, the quantity actually
observed is the energy E k , denned by XI. 9. This involves only the
square of the gap, and is therefore always positive. However, the
details of the variation of the variation of the gap with e k can be
verified by tunneling experiments, in which the conductance dl/dV is
directly proportional to the density of states in the superconductor
(Bardeen, 1961a, 1 962a; Cohen et al. 1962).Schrieffer<?/a/. (1963) have
however pointed out that for tunneling one cannot use the standard
expression for the quasiparticle density of states. This is because
when an electron tunnels from one side of the barrier to the other,
the initial and final states are not quasiparticle eigenstates of the
individual metals making up the tunnel. Instead the appropriate
density of states to use is
E k
N ™ = m) Hvv^}
Microscopic theory of superconductivity 133
in which the energy gap e(0) is taken to vary with e k . Rowell et al.
(1963) have closely verified the expected structure by tunneling experi
ments with lead, tin, and aluminium. Indications of this structure
had been seen earlier by Giaever et al. (1962) in lead and by Adler
and Rogers (1963) in indium.
The tunneling discussed thus far in this section and in Section 10.6
involves the passage of one or more quasiparticles. Josephson
(1962) has predicted an additional tunneling current when both sides
of the tunnel are superconducting. This current can be considered
10
r4
4.0
_
Pb
■ /sla
3.6
SaTh
V S + n V^g
26(0)
Al
kT c
3.2
1
Re
Ru
In
IS,

f Tl
Zr
2.4
1
i
10' 3 10' 2
T c /€)
Fig. 43
10"
as being due to the direct passage of coherent Cooper pairs from one
side of the insulating barrier to the other. As has been elaborated by
Anderson (1963) and by Josephson (1964), the relative phase of the
superconducting wave functions on either side of the barrier has
physical meaning because it is a quantity conjugate to the number of
electrons on each of the two sides and because this number is not
constant, that is, not fully determined. As a result the energy of the
system depends on this phase difference, and in turn this gives rise
to a flow of pairs across the barrier in the absence of an applied
potential difference.
The Josephson current is very difficult to detect because it is quen
ched by a magnetic field of a few tenths of a gauss. It was first observed
1 34 Superconductivity
by Anderson and Rowell (1963) and by Shapiro (1963). The tempera
ture dependence has been studied by Fiske (1964), following calcu
lations by Ambegeokar and BaratofF (1963). Ferrell and Prange
(1963) have discussed the selflimitation of the Josephson current by
the magnetic field it generates itself, and De Gennes (1963) has derived
an expression for the current from the GinzburgLandauGor'kov
equations.
A more realistic cutoff can also yield theoretical justification for
the apparent correlation of e(0) with TJ@. Such a correlation was
suggested by Goodman (1958), whose plot of energy gap values
against TJ0 for 17 different superconductors is shown in Figure 43.
Goodman used gap values deduced from empirical values of y, H ,
and T c by combining XI. 1 3, XT. 1 8, and XI.20, and remembering that
y = ffl^jAftO). This yields
k B T c V3
Appropriate values of 2e(0)/(£ B r c ) are listed in Table III.
(XI.32)
11.8. The specific heat
One can obtain the electronic specific heat in the superconducting
phase by twice differentiating with respect to temperature the free
energy expression XI.21. At sufficiently low reduced temperatures,
for which 2e(7") > k D T c , this yields
yTc
where K { and Kj, are first and third order modified Bessel functions of
the second kind. This simplifies in the temperature regions indicated
to the following exponential expressions:
— « 85 exp(l 44TJT),
yT c
25 < TJT < 6,
26 exp ( 1 62TJT), 7 < TJT < 1 1 .
(XT.34)
Experimental data at this time exist only in the first of these two
regions where they are in good agreement with the BCS values,
Microscopic theory of superconductivity 135
except for the upward deviation at the lowest temperature which was
mentioned earlier (see Figure 28).
Further numerical predictions of the BCS theory include
yT c
= 2 43.
(XI.35)
The following table is taken from [7] (p. 212) and shows how closely
Element
Lead
Mercury
Niobium
Tin
Aluminium
Tantalum
Vanadium
Zinc
Thallium
CJTJ
yT c
365
318
307
260
260
258
257
225
215
most experimental values agree with this. The theory also yields that
(XI.36)
^ = 0170,
Hi
from which one can calculate (see equation 11.15) that the predicted
coefficient of t 2 in the polynomial expansion of the threshold field is
a 2 m 107.
(XI.37)
This agrees exactly with the experimental value for tin (VIII.3) and
closely with that for several other elements. In terms of the deviation
of the threshold field curve from a strictly parabolic variation as dis
played in Figure 23, any value of a 2 greater than unity corresponds
to a curve below the abscissa; only mercury and lead are seen to have
deviations corresponding to values of a 2 smaller than unity.
The BCS calculations are based on an isotropic model, in which the
interaction parameter Kdoes not depend on the direction of A: and k ' .
Pokrovskii (1961) and Pokrovskii and Ryvkin (1962) have investi
gated the effects of anisotropy on thermal and magnetic properties.
10
136 Superconductivity
They find that in anisotropic superconductors the specific heat ratio
in Xl.35 should be smaller than 243, the quantity in XI. 36 larger than
01 70, and therefore the coefficient a 2 larger than 107. In the second
of the papers cited these results are compared with extensive experi
mental data.
The thermal conductivity in superconductors has been calculated
on the basis of the BCS theory for several of the pertinent mechanisms.
Bardeen et al. (BRT, 1959) and Geilikman (1958) have derived the
ratio of the electronic conductivity in the superconducting phase to
that in the normal one when this is primarily limited by impurity
scattering (equation IX.6). Their results have been well confirmed
experimentally, as was discussed in Chapter IX. The derivation of
BRTforthecaseofelectronicconductionlimitedbyphononscattering
(equation IX.5) does not, however, lead to the empirical behaviour.
Calculations by Kadanoff and Martin (1961) and by Kresin (1959)
are in better agreement, but further theoretical work is needed for this
conduction mechanism, in which quasiparticle life times may again
be important (see [7], pp. 272 ff.). According to calculations of
Tewordt (1962, 1963), however, these appear to have little effect on
this conduction mechanism.
BRT as well as Geilikman and Kresin (1958, 1959) have derived
the lattice conductivity limited by electron scattering. Experimentally
it is very difficult to separate out this part of the heat transport.
Where this has been possible (Connolly and Mendelssohn, 1962;
Lindenfeld and Rohrer, 1963) the results have been in general
agreement with the theoretical predictions.
11.9. Coherence properties and ultrasonic attenuation
One of the most striking predictions of the BCS theory arises as a
direct consequence of the pairing concept, and experimental verifica
tion of this point is thus of particular importance. In a normal metal
the scattering of an electron from state k t to state k' t is entirely
independent from the scattering of an electron from  k  to k' ]•
or of any other transition. The coherence of the paired electrons in
the kf and k j states in the superconducting phase, however,
makes these two transitions interdependent. The details of the theory
(see [7], pp. 21224) show that the contribution of the two possible
Microscopic theory of superconductivity 137
transitions interfere either constructively or destructively depending
on the type of scattering phenomenon involved. There is constructive
interference in the case of electromagnetic interaction, such as the
absorption of electromagnetic radiation, and the hyperfine inter
action which determines the nuclear relaxation rate. The experi
mental results expected in these two cases are therefore qualitatively
those which follow from a twofluid model consideration of the total
number of electrons available as well as from the density of available
states. It has already been mentioned how this explains the observed
rise in the nuclear relaxation rate just below the critical temperature
(Figure 32).
On the other hand, the contributions of the two transitions inter
fere destructively in the case of the absorption of phonons, such as
occurs in the attenuation of ultrasonic waves. This destructive inter
ference so decreases the probability of absorption that the effect of
the increase in density of states on either side of the gap is completely
wiped out, and the absorption just below T c drops very sharply. For
low frequency phonons, ha> <^ 2e(0), the ratio of attenuation coeffi
cient in the superconducting and normal phase o.J<x„ drops below T c
with an infinite slope, and is given by
? = 2/{l+exp[2e(T)lk B T]}.
a„
(XI.38)
This function is shown in Figure 44, which includes experimental
points on both tin and indium by Morse and Bohm (1957). It should
be contrasted with the theoretical prediction for nuclear relaxation
rate, shown in Figure 32.
Measurements of the ultrasonic attenuation in single crystals of tin
in different crystal directions has yielded very convincing demonstra
tion of the anisotropy of the energy gap. When an electron absorbs a
phonon, energy and momentum can both be conserved only if the
component of the electronic velocity parallel to the direction of sound
propagation is equal to the phonon velocity, which is the velocity of
sound S. Since, however, the Fermi velocity of the electrons, v , is
several orders of magnitude larger than S, this is possible only for
electrons which move almost at right angles to the direction of sound
138 Superconductivity
propagation. Thus a measurement of the attenuation of sound propa
gated in a particular crystalline direction involves only those electrons
whose velocity directions lie in a thin disk at right angles to this direc
tion. The value of the energy gap appearing in equation XI. 38 i > thus
one averaged over this particular disk. Such measurements have been
1.0 r
0.8
0.6
0C n
0.4
02
• TIN
x INDIUM
performed on variously oriented tin single crystals both by Morse et al.
(1959) and by Bezuglyi et al. (1959). Their results are in good agree
ment and are summarized in the following table:
Wave vector q 2e(0)/k B T c
parallel to [001] 32 ±01
parallel to [1 10] 43 ±02
perpendicular to [001] and 18° from [100] 35 ± 01
Microscopic theory of superconductivity 139
11.10. Electromagnetic properties
To describe the manyparticle wave function of the superconducting
state in the presence of an external field, BCS treat the electromagnetic
interaction as a perturbation, and obtain an expansion in terms of the
spectrum of excited states in the absence of the field. This wave func
tion is then substituted into an equation of the form III. 20 to calculate
the current density. Mattis and Bardeen (1958) and also Abrikosov
et al. (1958) have expanded this to treat fields of arbitrary frequency.
The result of the former has been used by Miller (1960) to calculate
values of cri/o n and of a 2 la n over a wide range of temperatures and
frequencies. His calculations are in excellent agreement with all the
experimental results using weak fields at high frequencies, described
in Chapter X, if the energy gap is taken as a parameter to be adjusted
to its empirical value.
The treatment of a magnetic field as a perturbation in the BCS
formulation makes it very difficult to extend it to high field values
(H m H c ). This can be done more readily from a representation of the
BCS ideas in terms of Green's functions which has been developed
by Gor'kov (1958). A simplified version of this method has been pre
sented by Anderson (1960). The electromagnetic equations occurring
in this formulation were shown by Gorkov (1959, 1960) to be equiva
lent to the GinzburgLandau expressions in the region near T c and
under circumstances where A > £. As was pointed out in Chapters V
and VII, Gor'kov showed that the energy gap is proportional to the
GL order parameter, so that the dependence of the latter on tem
perature, magnetic field, and coordinates, also applies to the former.
The successful application of these ideas to a number of experimental
results has been mentioned in Chapter VII.
An apparent shortcoming of the original BCS treatment is its lack
of gauge invariance. It was suggested by Bardeen (1957) and worked
out by various authors that this can be remedied by taking into
account the existence of collective excitations. A discussion of this
with full references is given in [7] (pp. 252 ff.).
There exists as yet no fully satisfactory explanation that the Knight
shift in superconductors does not vanish in any of the elements in
which is has thus far been studied : mercury (Reif, 1 957), tin (Androes
and Knight, 1961), vanadium (Noer and Knight, 1964) and aluminium
'40 Superconductivity
(Hammond and Kelly, 1964). The Knight shift is defined as the frac
tional difference in the magnetic resonance frequency of a nucleus
in a free ion and the same nucleus in a metallic medium. It is due to
the field at the nucleus created by the free electrons, and is usually
taken to be proportional to the electronic spin susceptibility. A literal
interpretation of the Cooper pairs of opposite spin would lead one
to expect that in a superconductor this susceptibility and hence the
Knight shift should vanish at 0°K. A number of authors (see [7],
pp. 261263; Anderson, 1960; Suhl, 1962; Cooper, 1962) have
suggested why this may actually not be the case, and although none
of these explanations appears fully adequate, they have shown that
the Knight shift offers no fundamental disagreement with the idea of
the BCS theory.
It is, furthermore, possible that the Knight shift in some of these
elements is not primarily due to spin paramagnetism. Clogston et al.
(1962, 1964) deduce from the temperature variation of the Knight
shift in vanadium that in the superconducting state the dominant
contribution due to the delectron spin does vanish, as the simple
theory would predict. This, however, leaves a finite Knight shift
due to orbital paramagnetism which involves electrons too far from
the Fermi surface to be involved in pairing. Thus this contribution
to the Knight shift in vanadium is not affected by the superconducting
transition, and perhaps the orbital part is the dominant one in tin
and mercury.
CHAPTER XII
Superconducting Alloys and Compounds
12.1. Introduction
Ever since the discovery of superconductivity there have been many
searches for new superconducting materials. Roberts (1961) has
recently listed more than 450 alloys and compounds with critical
temperatures ranging from 016° up to 182°K. In the appearance of
superconductivity among these substances there exist certain regu
larities which were discovered by Matthias (1957) and to which
reference was made in Chapter I. One might consider as an ultimate
goal of any complete microscopic theory the ability to derive these
Matthias rules from first principles. This would be equivalent to being
able to calculate with some precision the actual critical temperature
of any superconductor. At the moment our understanding of super
conducting and of normal metals is still very far from such
achievements.
One of the many ways of increasing this understanding is a sys
tematic study of superconducting alloy systems in which solvent or
solute are used as controlled parameters. This has been done in a
number of experiments.
12.2. Dilute solid solutions with nonmagnetic impurities
Serin, Lynton, and collaborators (Lynton et al., 1957; Chanin et al.,
1959) have investigated the superconducting properties of dilute
alloys of various solutes into tin, indium, and aluminium, up to the
limit of solid solubility. For low impurity concentrations, of the order
of a few tenths of an atomic per cent, T c decreases linearly with the
reciprocal electronic mean free path, independently of the nature of
the solute. When plotted against the reduced coordinate £ //, where
 is the coherence length of the pure solvent, the fractional change
in T c is the same for elements as different as Sn and Al (Serin, 1960).
This is shown in the initial portions of both curves in Figure 45. The
existence and the magnitude of this seemingly general effect lend
141
N
142 Superconductivity
strong support to Anderson's model of impure superconductors
(Anderson, 1959). He suggested that the energy gap anisotropy is
smoothed out by impurity scattering and disappears when the elec
tronic mean free path is comparable to
fiv
MO)
~ p
 so
+.02
+.01
.01
02
%
04
05
ELECTRONEGATIVE
5 6 7 Aoll
Fig. 45
This should then result in a lowering of T c by an amount approxi
mately equal to the square of the fractional anisotropy. Nuclear
resonance in aluminium (Masuda and Redfield, 1960a, 1962) and ultra
sonic and infrared absorption in tin (Morse et al., 1959; Bezuglyi,e/ al.,
1959; Richards, 1961) have shown that the gap in these elements
varies by about 10 per cent from its average value, so that T c should
be lowered by about 1 per cent when / « P . The measurements of T c
confirm this very well. Recently Caroli et al. (1962), Markowitz and
Kadanoff (1963), and Tsuneto (1962) have shown in terms of the
microscopic theory that Anderson's idea of the smoothing of an
Superconducting alloys and compounds 143
anisotropic energy gap indeed leads to a lowering of T c of the observed
magnitude. Hohenberg (1963) has calculated the dependence of T c ,
the energy gap, and the density of states on the concentration of
impurities.
This general mean free path effect on T c has also been found in
tantalum by Budnick (1960). It has been verified by using a number
of different ways of scattering the electrons : by mechanical deforma
tion and cold work in aluminium (Joiner and Serin, 1961), by size
effects in indium (Lynton and McLachlan, 1962), by quenching
(De Sorbo, 1959), electron irradiation (Compton, 1959), neutron
bombardment (Blanc et al., 1960), and by using isoelectronic ternary
compounds (Wipf and Coles, 1959) in tin.
Figure 45 shows that for P /l > 1 , the effect on T c deviates from the
initial linear decrease in a way which depends on whether the solute
is electropositive (valence smaller than that of solvent) or electro
negative (valence larger). Chiou et al. (1961) have extended such
measurements to higher concentrations. They found that for both
types of impurities T c ultimately rises to values above that of the
solvent, and were able to repissent the variation of T c with impurity
concentration in all cases by an empirical relation containing two
parameters adjusted according to the particular solventsolute
combination.
According to the BCS theory (equation XI.29), T c depends on three
parameters : an average phonon frequency m q (which is proportional
to the Debye temperature ©), the density of normal electron states at
the Fermi surface, N(0) (which is proportional to the Sommerfeld y),
and the BCS interaction parameter V. Specific heat measurements on
tin alloys have recently enabled Gayley et al. (1962) to find the effects
of the addition of indium, bismuth, and indium antimonide on the
values of y and of © for tin. One can use equation XT.29 to calculate
the corresponding change in T c . This seems to account for most of
the difference in the behaviour of electropositive and electronegative
solutes, at least in the case of indium and bismuth, but not for the
increase in T c at high solute concentrations. One concludes that this
increase is mainly due to effects of alloying on the interaction
energy V.
Any attempt to calculate Kin the presence of impurities has to take
144 Superconductivity
into account that with scattering the wave vectors k are no longer
good quantum numbers. Hence the question arises of the criterion
for pairing of the electrons. Abrahams and Weiss (1959) and
Anderson (1959) have pointed out that in impure superconductors
Cooper pairs are formed of two electrons the wave functions of which
are identical except for the reversal of the time coordinate, and which
have the same energy. The former authors have used this to deduce
Applied Field He
Fig. 46
several impurity effects, the magnitude of which is difficult to esti
mate. Anderson (1959, 1960) has discussed the general implications
of the use of timereversed wavefunctions. Detailed microscopic
calculations of the impurity effects on the superconducting parameters
have been attempted by Caroli et al. (1962) and by Markowitz and
Kadanoff(1963).
It is interesting to note that the work on carefully homogenized and
annealed solid solutions has shown these to be 'wellbehaved' super
conductors according to several criteria. Transitions occur within a
few millidegrees, and very little flux remains in suitably oriented
cylindrical samples after an external field has been removed (Budnick
Superconducting alloys and compounds 145
et al., 1956). Also the absorption edge of infrared radiation at the
gap frequency can be very sharp (Ginsberg and Leslie, 1962).
Detailed magnetization curves (Lynton and Serin, 1958) however,
show that the transitions for such alloys are nevertheless not fully
reversible, as shown in Figure 46 for 311 per cent InSn cylinders
transverse to an external field. In decreasing field the magnetic
moment does not attain its full diamagnetic value until H vanishes.
This indicates that flux is initially trapped, but then leaks out as
suggested by Faber and Pippard (1955b).
12.3. Compounds with magnetic impurities
Matthias and collaborators have traced the occurrence of super
conductivity in a large number of compounds containing para
magnetic and ferromagnetic impurities (Matthias, 1960). Their
results can be summarized as follows:
Ferromagnetic transition elements with 3d electrons (Cr, Mn, Fe,
Co, and Ni) put into fourth column superconductors (Ti, Zr) raise
T c more than do corresponding amounts of transition elements with
Ad electrons (Re, Rh, Ru, etc.) (Matthias and Corenzwit, 1955;
Matthias et al., 1959b). At the same time magnetic measurements on
TiFe and TiCo alloys indicated the absence of localized moments.
The effect of the 4d electrons can be attributed to the increase in
the number of valence electrons per atom toward five, a number
particularly favourable for superconductivity. The extra rise with
3d electrons is attributed to a magnetic electronelectron inter
action favouring superconductivity. For the same reason adding Fe
(3d electrons) to a Ti . 6 V . 4 compound lowers T c less than does an
equal amount of Ru (4d electrons): in both cases T c is decreased
because the number of valence electrons per atom rises beyond five,
but with Fe the apparent magnetic interaction counteracts this in
part. It must be pointed out, however, that ferromagnetic transition
elements with 3d electrons put into fifth column superconductors (Nb,
V) lower T c in approximate agreement with the expected effect due to
the change in valence electrons per atom (MUller, 1959). There does
not appear to be any added effect due to the magnetic nature of the
impurities. Why such effects should appear with fourth column
metals but not with fifth column ones is far from clear, as in neither
146 Superconductivity
case are there any localized magnetic moments associated with the 3d
solute atom.
Quite recently Cape (1963) has measured the electrical and mag
netic properties of very carefully prepared alloys of Ti containing
0.2 to 4 at % Mn. Depending on the method of preparation these
specimens are either in a single hexagonal close packed (hep) phase,
or contain an admixture of a second, body centred cubic (bec) phase.
Localized moments exist only in the hep phase, which however is not
superconducting. This is consistent with the usual suppression of
superconductivity by impurities retaining localized moments (see
below). The nonmagnetic bec phase, on the other hand, has a tran
sition temperature which is raised above that for pure Ti by an amount
commensurate with the increase in the number of valence electrons.
Hake et al. (1962) had earlier deduced from their measurements of
transport properties that the hep phase of TiCr, TiFe, and TiCo
also carried localized magnetic moments. In addition there is calori
metric evidence (Cape and Hake, 1963) that in TiFe samples only a
small fraction of the volume is superconducting. These results throw
considerable doubt on Matthias' speculation that irongroup im
purities which do not carry a localized magnetic moment enhance
superconductivity by means of a magnetic interaction between
electrons.
While 3d impurities in fifth column metals (for example Nb) do not
show any evidence for a localized moment, they do when put into
sixth column metals (for example, Mo), and in fact Matthias et al.
(1960) found that the change in behaviour occurs in NbMo solutions
at a concentration of about 60 per cent Mo. One would therefore
expect some special effects on T c to appear in 3d compounds with
sixth column metals. Until the recent discovery of the superconduc
tivity of Mo, no such metal was known to be superconducting. For
that reason, this effect was studied on superconducting Mo . 8 Re . 2 ,
and indeed small amounts of 3d impurities lower T c far more than one
would expect from valence effects. It is in fact this which made the dis
covery of the superconductivity of Mo so difficult: a few parts per
million of iron are enough to depress T c below the measurable range
(Geballe et al, 1962). A less abrupt decrease in T c is obtained when
rare earth elements with 4/electrons are put into lanthanum (Matthias
Superconducting alloys and compounds 147
et al., 1 958b, 1959a). The magnitude of this decrease, for each per cent
of rare earth impurity, is correlated with the spin rather than with the
effective magnetic moment of the solute. This is shown in Figure 47
in which  AT C for each per cent, the spin, and the effective moment
/x eff are plotted for the different rare earths. A higher effective moment,
in fact, appears to tend to raise T c , perhaps for the same reason as in
the case of the 3d impurities in fourth column metals : erbium, with
spin 3/2 and large moment lowers T e less than does an equal per
centage of neodynium, which has the same spin but a smaller moment.
Fig. 47
All these compounds containing 4/electrons show ferromagnetic
behaviour at somewhat higher concentrations of the rare earth
solutes, with the Curie temperature rising with increasing number of
4/ electrons. Such dilute ferromagnetism has not been observed for
compounds with 3d electrons which indicates that the sf magnetic
interaction is rather long range, while the dd one is a shortrange
interaction effective only through nearest neighbours, which is
impossible in dilute solutions (Matthias, 1960).
Interesting analogies in the variation of the Curie temperature and
the superconducting critical temperature are found by investigating
the magnetic characteristics of socalled Laves compounds AB 2 ,
where B is germanium or a noble metal (Ru, Os, Ir, Pt) and A is either
148 Superconductivity
a rare earth with 4/ electrons (A') or one of the group Y, Sc, Lu, or
La {A"), none of which contain 4/ electrons (Suhl et al., 1959). A'B 2
is always ferromagnetic, A"B 2 always superconducting. Comparing
the Curie temperatures of the former with the critical temperatures of
the latter one finds a similar dependence on spin and on the number
of valence electrons per atom. This is but one of a number of interest
ing correspondences which Matthias has found between supercon
ductivity and ferromagnetism. There are, for example, several groups
of isomorphous compounds which are either superconducting or
ferromagnetic (see, for example, Matthias et al., 1 958a ; Compton and
Matthias, 1959; Matthias et al., 1962). Also, the appearance of locali
zed moments when a ferromagnetic impurity is put into a nonmag
netic transition element seems to depend on the number of valence
electrons in a manner similar to the criterion for the appearance of
superconductivity (Matthias, 1962). Matthias hasfrequentlysuggested
that an electron configuration favourable to superconductivity may
also be favourable to ferromagnetism.
The possible coexistence of superconductivity and ferromagnetism
in the same substance has been investigated in lanthanumrare earth
binary compounds (Matthias et al., 1958b) and in Laves compound
mixtures (A'i^ x A%)B 2 (Matthias et al., 1958c; Suhl et a/.,1959).
Both magnetic (Bozorth et al., 1960) and calorimetric measurements
(Phillips and Matthias, 1960) have shown that ferromagnetism and
superconductivity occur in the same sample, but the evidence is not
entirely conclusive in ruling out the possibility that these two phe
nomena merely exist side by side in different portions of the specimen.
Anderson and Suhl (1959) have shown that the actual coexistence of
ferromagnetism and superconductivity on a microscopic scale is
energetically possible if the ferromagnetic alignment occurs in the
form of extremely small domains probably of the order of 50 A. They
call this 'cryptoferromagnetic' alignment.
Suhl and Matthias (1959) have treated the general problem of the
lowering of T c due to the presence of magnetic impurities by extending
an argument of Herring (1958), according to which the polarization
due to the coupling of the conduction electrons with the spins of the
paramagnetic impurity ions lowers the free energy in both the normal
and in the superconducting phases. The free energy is lowered by each
Superconducting alloys and compounds 149
electronspin scattering interaction by an amount proportional to the
reciprocal energy difference between the initial and final electron
state. In the normal state this difference can be arbitrarily small, in
the superconducting case this difference cannot be smaller than the
energy gap. As a result the free energy of the normal phase is lowered
more than that of the superconducting one, and the onset of super
conductivity therefore occurs at a lower temperature. Suhl and
Matthias ignore the small changes in the interaction matrix element
V, and as a result their prediction (dTJdc * 0) for very small magnetic
impurity concentrations is probably wrong. Abrikosov and Gor'kov
(1960) show that magnetic impurity effects on V initially lowers T c
linearly with impurity concentrations. Atmuch higher concentrations
Suhl and Matthias find that S7y3c>co, which is supported by
experiment (Hein et al., 1959). Similar calculations have been carried
out by Baltensperger (1959). Suhl (1962) has recently reviewed this
and similar work.
Abrikosov and Gor'kov (1960) as well as De Gennes and Sarma
(1963) show that magnetic impurities will lower the energy gap more
rapidly than the transition temperature. There should thus be a
range of concentration for which the alloy has a finite critical tem
perature at which its DC resistance disappears without the existence
of an energy gap. Indeed Reif and Woolf ( 1 962) have found this para
doxical behaviour to exist. They measured the electrical resistance as
well as the tunneling characteristics of a number of lead and indium
film containing magnetic impurities. The gap decreased twice as
rapidly as the transition temperature, and an indium film containing
1 at % Fe, for example, had no resistance below 3°K but a perfectly
ohmic tunneling conductance.
Phillips (1963) has pointed out that such gapless superconductivity
does not violate any fundamental principle. The spinflipping scatter
ing of the conduction electrons by the magnetic ions gives the former
a very short lifetime. This broadens the electron states, particularly
those nearest the gap, so much as to spread the states into the gap.
At a certain impurity concentration states will have spread throughout
the width of the gap, making it disappear. The density of states,
however, will still have maxima at what used to be the edge of the gap,
and as long as this exists the material will have zero DC resistance.
1 50 Superconductivity
The details of this have been worked out by Skalski et al. (1963). In
terms of the FerrellGlover rule and the frequency dependence of the
conductivities shown in Figure 37, the situation can be described by
saying that at some concentration there is a finite real conductivity a i
at all frequencies. For some further range of impurity concentration,
however, oj will still be less than a N for cu/w g < 1 , resulting in a reduced
delta function at the origin. At even higher concentrations, ct, x a N
for all frequencies. The sum rule is now satisfied without an infinite
DC conductivity, so that the metal is normal in every respect.
12.4. Superimposed metals
A recent series of experiments by Meissner (see Meissner, 1960 for full
references) has revived interest in the question whether thin layers of
superconducting material deposited on a normal metal would them
selves become normal, and whether conversely sufficiently thin layers
of normally nonsuperconducting metal would become superconduct
ing when in contact with a superconductor. Such superimposed metals
differ from the sandwiches used in the tunnelling experiments by the
absence of an insulating layer.
Parmenter (1960a) has constructed a theory for such direct metallic
contacts in which he attempts to introduce directly into the BCS
formulation a dependence of the energy gap on position by postulating
a spatial variation of the parameter h k appearing in equation XI.5.
This adds to the kinetic energy portion of the ground state energy (the
first term in equation XI.5) terms involving the square of the gradient
of h k 12 and of (1 h k ) 112 , broadly analogous to the extra energy term
V.6 in the GinzburgLandau theory. Near the boundary of a super
conductor this leads to a significant variation of the energy gap over
distances which are of the order of 10 6 cm, that is, two orders of
magnitude smaller than the coherence length £ . The configurational
surface energy resulting from this gap variation turns out in this
theory to be about 10 5 cm (Parmenter, 1960b) which is an order of
magnitude smaller than the generally accepted value.
To investigate the behaviour of superimposed metallic layers under
this theory, Parmenter postulates a set of plausible boundary con
ditions involving the continuity of h k and of the normal component
Superconducting alloys and compounds 1 5 1
of the gradient of h k , which, however, have yet to be justified from
more fundamental considerations. From these he concludes that a
normal layer sandwiched between two superconductors will itself be
supreconducting if it is no thicker than about 10 5 cm. A supercon
ducting layer between two normal metals will remain normal up to
some similar critical thickness.
Cooper (196 1 ) has given a more intuitive argument for the possibil
ity that the superconducting properties of thin metallic films may be
strongly affected by direct contact with other metals. He emphasized
that in the BSC theory one must clearly distinguish between the range
of the attractive interaction between electrons, and the distance over
which as a result of this interaction the electrons are correlated into
Cooper pairs. The range of the interaction is very short (10~ 8 cm) ; the
'size' of the wave packets of the pairs, on the other hand, is of the
order of the coherence length, that is, 10 4 cm. This, as Cooper points
out, is analogous to the difference between the range of the nuclear
interaction and the much larger size of the resulting deuteron wave
packet. Because of this long coherence length the Cooper pairs can
extend a considerable distance into a region in which the interaction
between electrons is not attractive. Thus when a thin layer of super
conducting material is in contact with a layer of normal metal, the
zeromomentum pairs formed because of the attractive interaction in
the superconductor extend into both layers. As a result the ground
state energy of this thin bimetallic layer is characterized by some aver
age over both metals of the parameter N(0) V, which in turn deter
mines the energy gap of the layer and its transition temperature,
according to equations XI. 1 8 and XI.29. The form of this average of
course depends on the nature of the boundary between the two metals ;
the better the contact, the more effective is a superimposed layer in
changing the properties of the substrate. Regardless of how one
accounts for this, one would expect the average to depend also in some
manner on the relative thickness of the two layers. The thicker the
normal layer, the smaller the average interaction, and the more the
energy gap width and the transition temperature are decreased from
the values they would have if only the superconductor were present.
Similarly a combination of two superconductors would be expected
to have a T e somewhere between the T c values of the two materials,
11
1 52 Superconductivity
varying from one extreme to another as the relative thickness of the
two layers is varied.
These qualitative conclusions presuppose that both layers of the
bimetallic film are sufficiently thin so that the coherent electron pairs
extend over the entire volume. One expects the critical thickness for
this to be of the order of the coherence length, although it is not clear
whether this should be the ideal value £ , or the mean free path limited
value £(f). If one of the two superimposed metals is much thicker than
whatever critical length is appropriate, then presumably the average
interaction is determined by the ratio of the smaller thickness to the
critical length.
The experiments of Smith et al. (1 96 1 ) with lead films of about 500 A
on or between silver films varying from 100 to 7000 A indicate a de
crease of the transition temperature of the lead with increasing silver
thickness, supporting earlier work of Misener and Wilhelm ( 1 935) and
of Meissner (1960). Similar results have been obtained by Hilsch and
Hilsch (1961 ) with combinations of copper and lead films. These agree
with the calculations of De Gennes and Guyon (1962) and the
more extensive treatment of Werthamer (1964). However, the work
of Roselnnes and Serin (1961) has shown that results can be strongly
influenced by varying evaporation procedures, even under condi
tions which quite preclude ordinary bulk intermetallic diffusion.
Interpenetration of metals seems to occur quite readily with super
imposed layers, possibly by the mechanisms of surface and defect
diffusion. Because of this the experimental situation is at this time far
from clear.
CHAPTER XIII
Superconducting Devices
13.1. Research devices
The characteristics of superconductors have for a long time already
been put to use in many low temperature experiments. It is very com
mon to use niobium wires, for example, in electrical connections to
samples which one wishes to isolate thermally as well as possible. Such
wires are superconducting with a high critical field throughout the
entire liquid helium temperature range, and combine low thermal
transport with perfect electrical conductivity. The use of lead wires as
heat switches at temperatures below 0T°K has been mentioned in
Chapter DC.
More specialized research devices using superconducting com
ponents of varying complexity have been suggested or used frequently,
and it is possible in this cursory survey to mention only a few of these.
A number of such devices have been developed to detect very small
potential differences, as occur, for instance, in studies of thermo
electric powers. Pippard and Pullan (1952) improved earlier designs
by Grayson Smith and coworkers (Grayson Smith and Tarr, 1935;
Grayson Smith et al., 1936) by using a single turn of superconducting
wire to construct a galvanometer capable of detecting e.m.f.s of
10" ,2 V. With a resistance as low as 10~ 7 ohm this required a current
sensitivity of only 10~ 5 amp ; the time constant L/R was kept short by
the single turn design which reduced the effective inductance. A super
conducting magnetic shield made possible controlling fields as low as
001 gauss.
A different approach to the measurement of very small potentials
was suggested by Templeton (1955b) and by De Vroomen (De
Vroomen, 1955; De Vroomen and Van Baarle, 1957). These authors
designed 'chopper' amplifiers in which the small d.c. signal is con
verted into an alternating one by passing through a superconducting
wire which is modulated into and out of the normal state by being
153
154 Superconductivity
placed in an alternating magnetic field. The resulting oscillating
potential across the wire is then amplified in a conventional manner.
These devices can operate stably with a noise level at about 10~ ' ' V.
Templeton (1955a) has also designed a superconducting reversing
switch to suppress undesirable thermal voltages in measurements of
potential differences of the order of about 10~ 6 V.
Many low temperature experiments as well as superconducting
magnets require rather high direct currents at very low voltages.
To avoid the use of thick electrical leads which would bring too
much heat into the helium dewar, Olsen (1958) has designed a
superconducting rectifier and amplifier which, together with a low
temperature transformer, allows one to feed in a low alternating
current through thin leads. The rectification occurs as the current
flows through a superconducting wire placed in an external, nearly
critical field, such that the field due to the current in one direction
is sufficient to make the wire normal during about onehalf of each
cycle.
D. H. Andrews et al. (1946) made use of the change in resistivity
at the superconducting transition in designing a bolometer. A different
superconducting radiation detector has been suggested by Burstein
et al. (1961) who pointed out that a tunnelling device (Chapter X)
suitably biased would respond to absorption of electromagnetic
radiation in the microwave and submillimetre range. RF detection
with a tunneling device has been achieved by Shapiro and Janus
(1964). For work at high frequencies superconducting metals may
also be used to construct resonant cavities of extremely high Q. This
has been discussed by Maxwell (1960), and preliminary experiments
have been reported by Fairbank et al. (1964) as well as by Ruefenacht
and Rinderer (1964). Thought is also being given to the use of
superconducting cavities in high energy proton linear accelerators
(Parkinson, 1962; Fairbank et al., 1964). Many of the devices listed
in this section as well as others have recently been discussed by
Parkinson (1964).
13.2. Superconducting magnets
As early as 1931, De Haas and Voogd found critical fields as high as
15 kgauss in some leadbismuth alloy wire. Other instances of rela
Supei conducting devices 155
tively large values of the critical field have been observed for many
alloys and for strained or impure samples of the superconducting
elements. For niobium published values of the critical field at 0°K
vary from about 1950 to 8200 gauss. Quite recently Kunzler et al.
(1 961 b) discovered Nb s Sn to have a critical field of about 200 kgauss,
and similar critical fields have since been found in other substances.
These seem to be either intermetallic compounds of the /3wolfram
structure, or body centered cubic alloys. When suitably prepared
these materials remain superconducting while carrying current den
sities as high as 5 x 10 4 amp/cm 2 in fields almost up to the critical
value.
The critical fields of these materials are much too high to be the
thermodynamic critical fields H c as defined by equation II.4. How
ever, earlier chapters have discussed two reasons why superconduc
tivity can persist in a given specimen to fields higher than H c . One
possibility is that the material is sufficiently inhomogeneous, so as to
display the characteristics of a 'Mendelssohn sponge', as discussed
in Section 7.2. In such a specimen superconductivity persists in a
filamentary structure, the dimensions of which are much smaller
than the penetration depth. As a result, the filaments remain super
conducting to a field H s > H c , as given by equation VII.8. On the
other hand, a quite different mechanism for high field superconduc
tivity was discussed in Chapter VI, where it was shown that super
conductors of the second kind remain in a superconducting mixed
state up to U c2 > H c (Abrikosov, 1957). Superconductors of the
second kind are materials which may be quite homogeneous and
which have a negative surface energy, generally because of their very
short electronic mean free path. Goodman (1961) was the first to
suggest the possible relevance of this mechanism to explain the high
critical field, found by Kunzler and others, and there is convincing
evidence that this is indeed the case (see for example, Berlincourt and
Hake, 1963). The specific heat results of Morin et al. (1962) on V 3 G a
are consistent with the behaviour expected for superconductors of
the second kind (Goodman, 1963b), and so are the critical fields
observed by Berlincourt and Hake (1962, 1963) in the low current
limit for a number of high field alloys and compounds. Hauser
(1962) as well as Swartz (1962) have further shown that the
156 Superconductivity
magnetization curves of suitably prepared specimens of various
compounds are consistent with the identification of these materials
as superconductors of the second kind. A systematic study of the
role of defects on the magnetization curve has been carried out by
Livingston (1963, 1964).
However, Gorter (1962a, b) has pointed out that a homogeneous
superconductor with uniform negative surface energy cannot in the
presence of a transverse magnetic field carry the high current densities
which are actually observed in most of the compounds and alloys
under discussion. This can best be understood in terms of the vortex
structure of the mixed state which is created by the external field (see
Section 6.6). When a current passes through the specimen at right
angles to the vortices, it interacts with the latter so as to push them
out of the specimen. This, as was mentioned in Section 6.6, can be
prevented only if the vortices are pinned down by local variation of
the surface energy, as would be present if the specimen were inhomo
geneous. Indeed, there is much evidence that the high current carrying
capacity is associated with the presence of dislocation in coldworked
specimens (Hauser and Buehler, 1962). Annealed samples may still
have a very high critical field while carrying a low current density,
but turn normal when the latter is increased. RoseInnes and Heaton
(1963) have used TaNb wire to show very strikingly how sample
treatment can change the current carrying capacity without changing
the critical field.
Thus the present picture of high field superconductors is that
basically they are materials characterized by a negative surface
energy. They are further able to carry high current densities in high
fields if through cold work they are made to contain a high density
of dislocations which pin down the current carrying regions. A nearly
uniform distribution of these dislocations explains why the critical
current increases as the crosssectional area of the specimen (Lock,
1961a; Hauser and Buehler, 1962).
The ability of some superconductors to carry high current densities
in high fields, of course, suggests their use in the winding of magnets.
Yntema (1955) described a superconducting solenoid wound with
niobium wire and producing up to 7 kgauss, but this received little
attention. In 1960 Autler wound a niobium solenoid creating a field
Superconducting devices 157
of 43 kgauss, and since then the interest in the subject has grown
explosively, with much scientific and technical activity in a large
number of laboratories. Kunzler?/ al. (196 la) and others used Mo 3 Re
to wind solenoids producing up to 1 5 kgauss; much higher fields were
achieved soon thereafter as a result of work with Nb 3 Sn (Kunzler
et al. 1961b), Nb 2 Zr (Kunzler, 1961; Berlincourt et al., 1961) and
NbTi (Coffey et. al., 1964). Solenoids wound of these materials
have produced fields up to 100 kgauss, and both suitable
superconducting wire as well as entire solenoid assemblies have
become commercially available. At the moment, the size of these
is still measured in inches, but largescale superconducting coils
producing fields well in excess of 100 kgauss seem quite feasible.
Kropschot and Arp(1961) have recently reviewed the subject of super
conducting magnets, and have discussed the considerable technical
and economic advantages of such devices. Much information can also
be found in [11] as well as in Berlincourt (1963).
13.3. Superconducting computer elements
Much research and development work is currently being devoted to
attempts to use superconductors both as switching devices and as
memory storage elements in electronic computers. The basic idea for
a superconducting switching element originated with Buck (1 956) who
invented the cryotron. This consists of a layer of thin (0003 in.)
niobium wire wound on to a thicker (0009 in.) tantalum wire. A
sufficiently large current through the former, called the control
winding, can quench the superconductivity of the latter, called the
gate. The two materials are chosen because the convenient operating
temperature of 42°K is only a little below the critical temperature
of Ta, but much lower than that of Nb, so that a control current
sufficient to 'open the gate' is still much less than the critical cur
rent of the control. The diameter of the gate is furthermore kept
large so as to maximize the amount of gate current, I g , which can be
controlled by the control current, I c . Calling H c the critical field of
the tantalum gate at the operating temperature, and D its diameter,
then
(/,)m« = H c ttD,
(xni.i)
158
and
Superconductivity
*c t
n
(XIII.2)
where n = number of turns/unit length of control winding. Thus
(XIII.3)
This is the 'gain' of the cryotron, which must be kept at a value
greater than unity in order that the gate current of one cryotron can
be used to control another.
.+
;!
Fig. 48
A great variety of logical circuits can be built up by making use of
this reciprocal control of a number of cryotrons. Most of these cir
cuits contain the basic flipflop or bistable element, shown in Figure
48. Current through this element can flow in either one or the other
branch and, once established in one, will flow in it indefinitely since
it makes the other one resistive. The choice of branch can be dictated
by placing a further cryotron gate in series with each branch, and con
trolling this by an outside signal, which can 'open the gate', making
the corresponding branch resistive and forcing the current into the
other path. This is shown in Figure 49, which also indicates that if
each branch also controls the gate of a readout cryotron, the position
of the bistable element can be read. Figure 50 shows other basic
logical circuits using cryotrons; the current through the heavy line
Superconducting devices 1 59
flows only if: (a) cryotron A or B is open, (b) cryotrons A and B are
open, (c) neither A nor B are open. More complicated logical circuits
are discussed by Buck (1956) as well as in review articles by Young
(1959), by Haynes (1960), and by Lock (1961b).
Basically all these cryotron circuits consist of a number of parallel
superconducting paths between which the current can be switched by
the insertion of a resistance into the nondesired branches. Under
steadystate conditions the power dissipation is zero as long as there
is always at least one path which remains superconducting. The speed
READ
"ZERO"
INPUT
"ZERO"
37
INPUT
1 "ONE"
§ *
V
±
Fig. 49
READ
"ONE"
with which the resistance can be inserted, that is, the speed with which
a given gate can be made normal, depends on the basic phase transi
tion time and is small enough ( as 10~ 10 sec) not to be a limiting factor
at this time (see, for instance, Nethercot, 1961 ; Feucht and Woodford,
1961). On the other hand, the switching time from one current path
to another is determined by the ratio L/R, where L is the inductance
of the superconducting loop made up of the current paths, and R the
resistance introduced by an opened gate. The usefulness of wire
wound cryotrons is severely limited by the fact that this time is no less
than 10~ 5 sec, even if the gate consists of a tantalum film evaporated
on to an insulating cylinder. Because of this all current research and
development effort is directed toward making thin film cryotrons
160 Superconductivity
consisting of crossed or parallel gate and control films separated by
insulating layers, and placed between additional superconducting
shielding films called ground planes. The resistance of the thin film
gates is comparable to that of a wire gate, but the ground planes con
fine magnetic flux to a very small region and thus result in L/R values
of the order of 10" 8 10~ 10 sec. Cryogenic loops with a time constant
of 2x 10 9 sec have been operated (Ittner, 1960b). An account of
many of the design considerations governing such thin film cryotrons
can be found in several papers in [9].
INPUTS
a\or
IB _<ztXp
xr5=fo
(neither B T T A
c\ a — cnto QXlo —
t nor B ^^Zpr*^
Fig. 50
Suggestions for superconducting memory devices were advanced
simultaneously by Buckingham (1958), Crittenden (1958), and Crowe
(1958). Their devices are basically quite similar and make use of the
fact that a current induced in a superconducting ring will persist in
definitely. Since the current can circulate either way one has the
possibility of a twostate memory storing one bit of information with
no dissipation of power other than that required to maintain the low
temperature. Of the three suggestions it is that of Crowe on which in
recent years most attention has been concentrated and which will be
briefly described here. Before doing so it might be noted that per
sistent current memory devices have in common with switching
cryotrons that a current in one superconducting circuit quenches the
Superconducting devices 161
superconductivity in another. There is, however, no need for a
greaterthanunity gain, as the controlled current is not in turn used
to drive another unit. One therefore often calls the memory elements
low gain cryotrons.
The Crowe cell basically consists of a thin film of superconducting
material (for example, lead) with a small hole, a few millimetres in
diameter, which has a narrow crossbar running across it. This is
shown schematically in Figure 51. A drive 'wire' in the form of a
second narrow strip lies just above the crossbar, separated only by
a thin insulating layer. As long as the entire configuration remains
superconducting, the magnetic flux threading the hole must retain its
SENSE
WIRE.
Fio. 51
original value, which we shall take to be zero. Therefore if a current
is passed through the drive wire, it will induce currents in the crossbar
and the remainder of the film. The direction of this induced circu
lating current will be such as to keep the flux from penetrating, and
results in a flux distribution indicated in Figure 52a, which shows a
schematic cross section of the cell. The crossbar is very thin and
narrow and therefore has a low critical current. When the induced
current exceeds this critical value, the crossbar becomes normal. The
flux now changes to the configuration shown in Figure 52b, as the
remainder of the film remains superconducting. If finally the drive
current is again removed, the superconductivity of the crossbar is
restored, and now the flux threading the hole is trapped, as long as the
crossbar remains superconducting, by a persistent current which is
in the opposite direction of the originally induced flow. Even when
162 Superconductivity
the drive wire current is now removed, the flux distribution remains
that of Figure 52c.
The idealized operation of a Crowe cell (Garwin, 1957) is indicated
in Figure 53, which shows on equal time scales, but arbitrary vertical
scales, the drive current I d , and the crossbar current / c . Pulse 1 is
too small to induce a critical value of I c . Pulse 2 results in I c > I cril ;
the crossbar becomes momentarily normal, and after the drive pulse
is removed a persistent current l v is stored. Pulse 3 is now a 'read
out' pulse which has no effect since it induces a current in a direction
opposite to that of the persistent current. With pulse 4, however, the
persistent current is reversed, storing the other possi bility of the two
state memory, and now readout pulse 5 succeeds in driving the
crossbar well beyond the critical value. Note that this is a destructive
readout.
The memory is sensed by means of a wire below the crossbar, also
very close to it but electrically insulated. A current pulse will be in
duced in the sense wire because of its proximity whenever the flux
linking the crossbar changes, that is, whenever the crossbar becomes
normal. Thus we note on Figure 53 that the sense wire response I s to
pulse 3 is nothing, which can be taken as ' Read 0', while its response
to 5 is a pulse which can be taken as 'Read 1 '.
Superconducting devices 163
The operation of the Crowe cell is rendered more complicated than
is indicated in the preceding simplified account because the crossbar
heats up through joule heat when it becomes normal, and the thermal
recovery time may be appreciable. Crowe (1957), Rhoderick (1959),
Von Ballmoos (1961), and several papers in [9] discuss the resulting
complications.
Id _
STORE READ STORE READ
"0" "1"
STORED STORED
"0" "0 
V
v
STORED
"I"
k READY
K READO" IV
Fig. 53
Crowe cells can be arranged into a twodimensional matrix of
memory elements with the drive wire forming part both of an x and
a ycircuit, as indicated in Figure 51 . Driving pulses I x , I y are then so
chosen that either alone is not sufficient to activate the device, but
that both together do. The reader is again referred to [9] for a number
of papers on superconducting memories built up of such matrices.
RoseInnes (1959) has estimated the consumption of liquid helium
required to keep a memory like that cold, and finds this to be of the
order of two litres per hour for an array of one million cells. This is
well within the capacity of closed cycle helium refrigerators such as
the one described by McMahon and Gifford (1960).
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Index
Adiabatic Magnetization, 15
Anisotropy of energy gap
and BCS theory, 131
decrease with impurity, 97, 100,
106, 142
deduction from
infrared absorption, 100
nuclear spin relaxation, 106
specific heat, 96
thermal conductivity, 92
ultrasonic attenuation (table),
138
Anomalous skin effect, 44, 101 , 103
Anomaly in lattice specific heat, 9
Atomic mass, effect on T c , 6
{See also isotope effect)
Atomic volume, effect on T c , 6
BardeenCooperSchrieffer (BCS)
theory, 12, 11740, 150
(See also energy gap, interac
tion parameter V, quasi
particles)
and GL theory, 52, 139
Pippard nonlocal relations,
44
anomalous skin effect, 103
basic hypothesis, 120
coherence effects, 1368
collective excitations, 100
critical current and field in thin
films, 80
critical field, 130
critical temperature, 129, 143
electromagnetic properties, 103,
114, 13940
electron phonon interaction,
11820, 125
ground state energy, 1206
183
BardeenCooperSchrieffer (BCS)
theory  cont.
high frequency conductivity, 103,
114
Knight shift, 139^10
nuclear relaxation rate, 1046, 137
penetration depth, 37, 52, 130
range of coherence, 48, 115
similarity principle, 122, 130
specific heat, 97, 130, 1346
thermal conductivity, 91, 92, 136
thermal properties, 12736
weak coupling limit, 131
Bulk modulus, 16
Coefficient of thermal expansion, 17
Coherence : see range of coherence
Coherence effects, 1368
Cold work, 143
Collective excitations, 100
Colloidal particles, 29, 35, 46
Compressibility, 17
Condensation energy, 19, 85, 91,
117, 126
Condensation of electrons in mo
mentum space, 12, 14, 19, 21, 31,
41
Cooper pairs, 12, 32, 52, 11821,
125, 140, 144, 151
Critical current, 4, 80, 155, 161
Critical magnetic field, 4
in BCS theory, 130
in GL theory, 51
of lead and mercury, 132
of small specimens, 46, 58, 758,
80, 109, 155
of superconducting elements
(table), 5
of thin films, 46, 767, 80, 109
184
Index
Critical magnetic field  cont.
precise measurements, 835
pressure effects, 16—17
relation of thin film value to A 6
and lo, 77
relation to thermal properties,
1319,21,86, 132, 135
similarity of reduced field curves,
4, 85, 135
temperature dependence, 4, 18,
846, 130, 135
very high values, 27, 701, 109,
155
Critical field for supercooling, 66,
74,76
for surface superconductivity, 74
Critical temperature, 3
{See also isotope effect)
dependence on
atomic mass, 6
atomic volume, 6
discontinuity of specific heat, 9,
15, 17, 132, 135
effect of magnetic impurities,
14550
effect of nonmagnetic impurities,
1414
in BCS theory, 12943
Matthias' rules, 6, 141, 148
of superconducting elements
(table), 5
Crowe cell, 1613
Cryotron, 15763
Cylindrical specimens, 13, 16, 24
{See also thin wires)
Debye temperature, 10, 87, 1245,
131, 134, 143
Demagnetization coefficient, 23
Density of electron states, 1048,
123, 127, 129, 134, 143
Diffusion, 152
Dilute alloys
critical temperature, 1414
magnetization curve, 1445
Dilute alloys  cont.
specific heat, 143
thermal conductivity, 878, 90
variation of surface energy, 59, 70
Effective charge, 49, 52
Elastic properties, 9
Electron irradiation, 143
Electronelectron interaction, 12,
823, 95, 11725
Energy Gap
{See also anisotropy of energy
gap)
correlation with TJ9, 134
deduction from
infrared absorption, 98100
infrared transmission, 11314
microwave absorption, 1004
nuclear spin relaxation, 1046
specific heat, 11,91,967
thermal conductivity, 92, 95
tunnelling, 79, 1069
ultrasonic attenuation, 1378
dependence on
field, 7980, 92, 109, 139
phonon spectrum, 132
position, 1502
quasiparticle energy, 1313
size, 79, 139
temperature, 90, 103, 108, 129,
139
in BCS theory, 120, 12634, 139
in thin films, 7980
of superconducting elements
(table), 99
relation to
GL order parameter, 52, 79,
139
Meissner effect and perfect
conductivity, 110, 11415,
120
penetration depth, 1 1 5
range of coherence, 44
Thomson heat, 95
Entropy, 14, 18,38,78,128
Index
185
FerrellGlover sum rule, 11415,
150
Ferromagnetism, 1458
Flux creep, 73
Flux quantization, 323, 72, 120
Free energy, 134, 1920, 478,
57, 75, 93, 128, 1489
Gapless superconductivity, 149
Gauge invariance
in BCS theory, 139
in GL theory, 49
Geometry, influence of, 236
GinzburgLandau (GL) theory, 12,
4854, 150
basic equations, 50, 139
critical field of small specimens,
757
extension to lower temperatures,
49,80
free energy, 489, 57, 75
limitations, 49, 523, 66, 80
nonlocal modifications, 48, 50
range of coherence, 58
relation to
BCS theory, 48, 52, 139
London equations, 50, 54
small specimens, 7680
superconductors of second kind.
6772
supercooling, 657
surface energy, 579
GL order parameter, 48
and free energy, 4850, 57
effect of magnetic field, 53, 78,
139
gradual spatial variation, 4950,
578, 73
proportionality to energy gap,
52, 79, 139
relation to penetration depth, 49,
53^, 78
GL parameter «, 5 1 3, 58, 66, 6870
and range of coherence, 58
GL parameter k  cont.
critical value for negative surface
energy, 59, 66, 67, 70
deduction from
penetration depth, 513
supercooling, 513, 66
in thin films, 54, 78
relation to
normal conductivity and
specific heat constant, 69
surface energy, 58
temperature dependence, 70
GorterCasimir thermodynamic
treatment, 11, 1319
GorterCasimir twofluid model,
11, 1921
{See also twofluid model;
twofluid order parameter)
application to GL theory, 49
relation to penetration depth, 36
Gyromagnetic ratio, 22
Impurity effects : see mean free path
effects
Infrared absorption, 98100
Infrared transmission, 45, 99, 109
114, 115
Interaction parameter V, 121
anisotropy, 131
BCS cutoff, 121, 1245, 131
effect of
non magnetic impurities, 1 434
magnetic impurities, 149
influence on isotope effect,
1246
quasiparticle lifetime effects,
1245, 131
variation with quasiparticle
energy, 131
Intermediate state, 14, 236, 29,
5961,94
Isotope effect, 12, 813, 95, 117,
1246
absence in transition metals, 12,
82, 125
186
Index
Isotope effect  cont.
effect of quasiparticle life time,
1246
in the BCS theory, 124
table of values, 82
Josephson effect, 109, 1334
Knight shift, 77, 139^K)
KramersKronig relations, 112, 114
Latent heat, 1 5, 78
Lattice parameters, 10
Laves compounds, 1478
Lifetime effects, 1246, 131
Localized magnetic moment, 1456
London theory, 1 1 , 2832, 36, 412
basic equations, 29, 42, 44,46, 1 1 1
incorrect values of penetration
depth, 29, 378, 43
microscopic implications, 11, 31,
43
nonlinear extension, 38
prediction of penetration depth,
29,36
Low frequency behaviour
diamagnetic description, 226
influence of geometry, 236
relation to high frequency re
sponse and energy gap, 1 1415
small specimens, 7580
Magnetic field distribution, 78,
224
Magnetic field dependence of
energy gap, 7980
entropy, 37
free energy, 1314, 49
GL order parameter, 53, 78
penetration depth, 35, 389, 53, 76
Magnetic field penetration: see
penetration depth
Magnetic susceptibility, 14, 23,
345, 75, 77, 83
Magnetic impurities, 14550
Magnetization
area under magnetization curve,
16, 26, 75
dilute alloys, 144
filamentary superconductors,
78
ideal superconductors, 1314,
246,68
small specimens, 75
superconductors of second kind,
689, 156
Magnetostriction, 16
Matthias' rules, 6, 141, 148
Mean free path effects on
anisotropy of energy gap, 97, 100
106, 142
critical temperature, 1414
GL parameter k, 69
infrared absorption, 100
nuclear relaxation rate, 106
penetration depth, 35, 38, 413,
456,58, 112
range of coherence, 423, 457,
152
surface energy, 589, 70, 77
Mechanical effects, 1617, 143
Mendelssohn 'sponge', 78, 155
Microwave absorption, 1004
Mixed state, 7174
Neutron bombardment, 143
Nuclear spin relaxation, 1046
Nuclcation of superconducting
phase, 613
Order in the superconducting
phase, 14, 19
Order parameter, see GL order
parameter; twofluid order para
meter
Penetration depth, 29, 58
defining equations, 28, 34, 36
Index
187
Penetration depth  cont.
dependence on
field direction, 39, 53
frequency, 39
magnetic field, 35, 389, 53, 76
mean free path, 35, 38, 413,
456,58,112
range of coherence, 43, 46, 112
size, 38, 46, 79
temperature, 358, 513, 130
in BCS theory, 37, 52, 130
in Pippard theory, 456, 112
incorrectness of London values,
29,378,43, 112
methods of measurement, 356
relation to
energy gap, 115
entropy, 38
frequency variation of con
ductivity, 1112
GL order parameter, 51, 78
surface energy, 557
susceptibility, 345
thin film critical field, 77
values in superconducting ele
ments (tables), 38, 65
Perfect conductivity of supercon
ductors, 4, 2930, 11415
Perfect conductor, 4, 68, 278
Persistent current, 3, 6, 26, 15860
Phase propagation, 635
Phonon spectrum, 132
Pippard nonlocal theory, 12, 416
(See also range of coherence)
basic equations, 42, 44, 45
critical field in thin films, 46
field penetration through thin
films, 45
penetration depth, 423, 456
reduction to local form (London
limit) 12, 456
relation to energy gap and BCS
theory, 445
susceptibility of thin films, 77
Pressure effects, 12, 1617
Quantized flux, 323, 72, 120
Quasiparticles, 1245, 132
Quenching, 143
Range of coherence, 1 1
and superimposed metals, 1501
dependence on mean free path,
42, 457, 152
in BCS theory, 44
in GL theory, 49, 58
relation to
energy gap, 44
field dependence of penetra
tion depth, 40
mean free path effect on T c ,
142
penetration depth, 40, 446
sharpness of transition 401
surface energy, 57
uncertainty principle 40, 45
values for Al, In, Sn (table), 65
Relation between magnetic and
thermal properties, 1321, 86,
96, 132, 135
Rutgers' relation, 15, 17
Semiconductors, superconducting,
6
Silsbee's rule, 5
Similarity, 856, 96, 116, 1212, 130
Size effect on
critical field, 46, 767, 155
critical supercooling field, 67
critical temperature, 143
energy gap, 7980
magnetic susceptibility, 34
penetration depth, 38, 46, 79
range of coherence, 46
Skin depth, 356, 43
Small specimens
critical field, 46, 758
in GL theory, 46, 54, 67, 7580
in Pippard theory, 46
low frequency behaviour, 7580
188
Index
Small specimens  cont.
penetration depth, 358, 43,
456, 79
range of coherence, 456
Sommerfeld specific heat constant, 9
in dilute alloys, 143
independence of isotopicmass,85
relation to
critical field, 1921,135
GL parameter *, 69
Specific heat of the electrons
comparison of magnetic and
calorimetric data, 17, 1921,
86, 134, 135
dependence on temperature, 9,
11,18,201, 86,91,967,134
discontinuity at T c , 9, 15, 17, 78,
132, 135
in BCS theory, 130, 135
relation to
critical field, 15, 17, 21,86, 96
energy gap, 11, 90, 967, 132
thermal conductivity, 91, 95
Rutgers' relation, 15, 17
Specific heat of the lattice, 910, 18
Spherical specimens
critical field of small spheres, 76
magnetization, 8, 26
penetration depth of small
spheres, 29, 35, 46
supercooling in small spheres, 67
Spin, effect on T c , 147
Strain, 267, 62, 77, 155
Superconducting alloys and com
pounds, 56
dilute alloys, 589, 8790, 1415
ferromagnetism, 1458
high critical fields, 1546
Laves compounds, 1478
magnetic impurities, 14550
Matthias' rules, 6, 141
nonmagnetic impurities, 1415
rare earth and transition metal
solutes, 1459
thermal conductivity, 889
Superconducting devices
cavities, 154
computer elements, 15764
d.c. amplifiers, 1534
galvanometers, 153
heat switches, 94, 1 53
leads, 153
magnets, 78, 1547
memory devices, 1 604
radiation detectors, 154
rectifiers, 154
reversing switches, 154
Superconducting elements (table),
5
Superconducting filaments, 78, 155
Superconducting ring, 3, 8, 26, 30,
32
Superconducting transition
contrast with perfect conductor,
68
discontinuity of
specific heat, 9, 15, 17, 78,
134
entropy, 14
free energy, 1314,48
in dilute alloys, 144
in thin films, 7881
length and volume changes,
1617
order, 72, 7881
reversibility, 7, 13, 17, 144
speed, 159
Superconductors of second kind,
6773, 1556
Supercooling, 523, 613, 657,
74
Superheating, 61
Superimposed metals, 1502
Surface currents, 22
Surface energy, 5574, 150, 156
dependence on temperature, 645
effect of strain, 62, 77, 156
in GL theory, 578
in inhomogeneous specimens, 77,
156
Index
189
Surface energy  cont.
in Pippard theory, 567
mean free path effect, 589, 67,
77
negative values, 589, 62, 67, 69,
70, 157
relation to
intermediate state, 59
phase nucleation and propa
gation, 616
range of coherence, 568
values for AI, In, Sn (table), 65
Surface impedance, 356, 523, 95,
1014, 111, 139
Table of
critical fields and temperatures,
5
energy gap values, 99
energy gap anisotropy, 138
isotope effect exponents, 82
penetration depth values, 38, 65
ranges of coherence, 65
specific heat discontinuities, 17,
135
superconducting elements, 5
surface energies, 65
Temperature dependence of
critical field, 4, 18, 846, 130,
135
energy gap, 90, 103, 108, 130
GL parameter k, 70
penetration depth, 358, 512,
130
specific heat, 9, 11, 18, 201, 86,
91,967, 130, 134
surface energy, 645, 70
surface impedance, 1013, 139
thermal conductivity, 8794, 136
twofluid order parameter, 1921,
36
Thermal conductivity, 8794
in BCS theory, 92, 136
of thin films, 79, 93
Thermal conductivity  cont.
relation to
energy gap, 913, 95
gap anisotropy, 92
specific heat, 91
Thermal expansion coefficient, 17
Thermodynamics of superconduc
tors
BCS theory, 128
GL theory, 489
GorterCasimir treatment, 11,
1319
relation between magnetic and
thermal properties, 1321, 86,
96, 132, 135
Thin films
(See also superimposed metals)
critical current, 80
critical field, 47, 7580
critical thickness for second order
transition, 78
cryotrons, 15763
energy gap, 79, 93
infrared transmission, 46, 10915
in GL theory, 54, 7580
in perpendicular field, 73
magnetic behaviour, 33, 734,
7580
penetration depth, 29, 356, 46,
79
relation of critical field to X b and
&»77
second order transition, 78
supercooling, 67
susceptibility, 345, 76, 77
thermal conductivity, 79, 93
total field penetration, 36, 45
variation of GL order para
meter, 78
Thin wires, 29, 35, 67, 76
Thomson heat, 95
Threshold magnetic field: see criti
cal magnetic field
Timereversed wave functions,
144
190
Index
Transition metals
absence of isotope effect, 12, 82,
1256
effect on T c , 14550
Trapped flux, 3, 267, 32, 120, 144,
161
Tunnelling, 1069, 1324
Twofluid model
(See also GorterCasimir two
fluid model)
and BCS theory, 127
extension of GL theory, 49
relation to
nuclear spin relaxation, 104
penetration depth, 36
thermal conductivity, 87
Twofluid order parameter, 1921,
36
gradual spatial variation, 40,
567
relation to
penetration depth, 36
surface energy, 567
thermal conductivity, 89
rigidity in London theory, 39
Ultrasonic attenuation, 105, 1368
Uncertainty principle, 31, 40, 45
Valence elections, effect on T c .
143, 145
Vortex lines, 724
6,
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IONIZATION AND BREAKDOWN IN GASBS
F. Llewellyn Jones
LOW TEMPERATURE PHYSICS L. C. JaCKSOl
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MASERS AND LASERS G. J. F. TrOUp
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MOLECULAR BEAMS K. F. Smith
NUCLEAR RADIATION DBTBCTORS J. Sharp
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THE THBORY OF GAMBS AND LINEAR
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o
ALTERNATING CURRENT MEASUREMENTS David Owen
APPLICATIONS OF INTERFEROMETRY W. Ewait Williams CQ
APPLICATIONS OF THERMOELECTRICITY H. J. Goldsmid ^J
atmospheric electricity B. F. J. Schonland
ATMOSPHERIC TURBULENCE O. G. Sutton ^3
atomic spectra R. C. Johnson ^J
cartesian tensors George Temple
classical mechanics J. W. Leech
THB CONDUCTION OF ELBCTRICITY THROUGH GASES K. G.
Emeteus
the cosmic radiation J. E. Hooper and M. Scharff
dielectric aerials D. G. Kiely
dipole moments R. J. W. le Fevre
the earth's magnetism Sydney Chapman ,—4
ELASTICITY, FRACTURE AND FLOW J. C. Jaegar
THB electric arc J. M. Somerville C^
elements OF pulse circuits F. J. M. Farley ^3
BLBMENTS OF TENSOR CALCULUS H. Lichnerowicz ^_{
fluid dynamics G. H. A. Cole ^t
frequbncy modulation L. B. Arguimbau and R. D. Stuart
FRICTION AND lubrication F. P. Bowden and D. Tabor
FUNDAMENTAL OF DISCHARGE TUBE CIRCUITS V. J. Francis
gbnbral circuit theory Gordon Newstead
THE GENERAL PRINCIPLES OF QUANTUM THEORY G. Temple
glass G. O. Jones
heaviside's electric circuit theory H. J. Josephs
high energy nuclear physics W. Owen Lock
HIGH FREQUENCY TRANSMISSION LINES Willis Jackson
INTEGRAL TRANSFORMS IN MATHEMATICAL PHYSICS C. J.
Tranter
AN INTRODUCTION TO ELECTRONIC ANALOGUE COMPUTERS
M. G. Hartley
AN INTRODUCTION TO ELECTRON OPTICS L. Jacob
AN INTRODUCTION TO FOURIER ANALYSIS R. D. Stuart
AN INTRODUCTION TO THE LAPLACE TRANSFORMATION
J. C. Jaegar
AN INTRODUCTION TO SERVOMECHANISMS A. Porter
AN INTRODUCTION TO VECTOR ANALYSIS B. Hague
AN INTRODUCTION TO TENSOR CALCULUS AND RBLATIVITY
Derek F. Lawden
AN INTRODUCTION TO PHASEINTEGRAL METHODS
J. Heading
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