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Full text of "Superconductivity"

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, 0-79, and 
0-90. 

(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 

2-1 



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 1959-60. 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 Gorter-Casimir two-fluid 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 Non-local 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 non-local relations 

The Ginzburg-Landau 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 Ferrell-Glover sum rule 1 14 



,. 



XII 



XIII 






Contents 
Microscopic Theory of Superconductivity 

11.1 Introduction 

11.2 The electron-phonon 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 non-magnetic 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 co-operative 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 Gorter-Casimir. 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 IV-VII are discussed those aspects of the behaviour of 
superconductors which lead to the non-local 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 VIII-X can be read without a study of 
the preceding section (IV-VII) 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 
self-inductance 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 
ohm-cm on the resistivity of the superconductor.! This can be com- 
pared to the value of 10 -9 ohm-cm 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 ohm-cm 
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 co-workers, by 
Alekseevskii and co-workers, and by Zhdanov and Zhuravlev (see 





Table I 




Element 


T C (°K) 


H (gauss) 


Aluminium 


119 


99 


Cadmium 


0-56 


30 


Gallium 


109 


51 


Indium 


3-407 


283 


Iridium 


014 


~20 


Lanthanum-a 


~5 





Lanthanum-/* 


5-95 


1600 


Lead 


718 


803 


Mercury-a 


4153 


411 


Mercury-/? 


3-95 


340 


Molybdenum 


10 


— 


Niobium 


9-46 


1944 


Osmium 


0-7 


65-82 


Rhenium 


1-70 


201 


Ruthenium 


0-49 


66 


Tantalum 


4-482 


830 


Technetium 


11-2 


_ 


Thallium 


2-39 


171 


Thorium 


1-37 


162 


Tin 


3-722 


306 


Titanium 


0-40 


100 


Tungsten 


~001 





Uranium-a 


0-6 


~2000 


Uranium-y 


1-80 





Vanadium 


5-30 


1310 


Zinc 


0-92 


53 


Zirconium 


0-75 


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 two-fluid 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 semi-quantitative 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 non-local 



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 non-transition 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 quasi-infinite 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 superconducting-to-normal 
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 quasi-infinite 
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 

QABQ-Gjm = (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 right-hand 
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 non-transition 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/dyne-cm -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) 

9-75 9-62 

10-6 10-56 

41-5 41-6 



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 
co-ordinates 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 left-hand 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 Gorter-Casimir two-fluid model 

The so-called phenomenological two-fluid 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, two-fluid 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(l-iT) = 1-iT; 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 HllA-n 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)/(l-a)]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 Gorter-Casimir model can be used at best only semi-quantita- 
tively. Within this limitation, however, the concept of the two inter- 
penetrating 'fluids' of condensed and uncondensed electrons is very 
useful in obtaining a semi-quantitative 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 Gorter-Casimir model so as to yield 
more nearly the correct exponential variation of C es and the corre- 
sponding non-parabolic 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 two-fluid 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 semi-major and semi-minor 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, D-l/3. 
Combining III.l and III.2 yields : 



and 



M,= -H e /47r(l-D) 
H, = HJ{l-D). 



(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(l-D), 



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 -A-nM is plotted against H e . 

For all other ellipsoidal shapes, D ^ 0, and the non-uniformity 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, large-scale 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 small-scale 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 -A-nDMi = 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 quasi-infinite 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 simply-connected but non-homogeneous 
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 non-ideal 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 multiply-connected, 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 semi-infinite 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 semi-infinite 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 current-carrying 
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(R-r. 



- ^-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-^VamscR-rjx 

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 Bohr-Sommerfeld quantization rules 
and require that for the superconducting charge carriers 



(J> p-dl = nh, 



(IU.28) 



along any path enclosing the hole. According to 111.25 this then 

means that 

jMttA 2 m r he 

(b J-dl+&>A-dl = 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)A-rfl= f f H-c/S = 0. 

r 



* he 
J<fl+0 = n— • 

Q 



(111.30) 



(111.31) 



London called the left-hand 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 gauss-cm 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 Non-local 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 non-local 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 temperature-dependent factor 
in this defining equation, and in fact the Gorter-Casimir two-fluid 
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 non-local 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 
/ < 0-8, 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) = (1-f 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 


380-450* 


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 




470-600* 


Schawlow and Devlin, 1959 


Tl 


920 


Zavaritskii, 1952 
* Anisotropy. 



Pippard ( 1 950) investigated the change of the penetration of a small, 
r.f. field (9-4 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 non-linear 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 one-fourth of the total entropy 



The Pippard non-local 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 two-fluid 
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 non-local theory 41 

a broadened transition even for ideally homogeneous samples. 
Goodman (1962c) has discussed this in some detail. 

4.4. The Pippard non-local 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)= - 



4-nXl 



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 semi-infinite 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 non-local 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 so-called anomalous 



44 Superconductivity 

skin effect were worked out by Reuter and Sondheimer (1948), who 
derived that 



3a CR(R-E)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 quasi-infinite 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(R-A)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 
non-local 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 non-local 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) = 3-52k B T c , IV.16 becomes 



g, = 0l8hv /k B T c . 



(IV.17) 



The Pippard non-local 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 2-1 x 10 -5 cm for tin, and 12-3 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 0-18, 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 non-local 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 size-limited 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 Ginzburg-Landau 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 non-local 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 Ginzburg-Landau 
Phenomenological Theory 

In 1950 Ginzburg and Landau (G-L) 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 non-local equation of the 
Pippard type. Furthermore, Gorkov (1959, 1960) has derived the G-L 
equations, under certain conditions, from his formulation of the BCS 
theory. 

G-L 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 Landau-Lifshitz 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 Ginzburg-Landau 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 Gorter-Casimir two-fluid model. 

The outstanding contribution of the G-L 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 gauge-invariance, G-L assume the extra energy 
term to be 

J-L/^-^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. 

G-L 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.G-L1) 



+ • 



Aire*"' 



mc 



I0I 2 A. 



(V.G-L2) 



In a very weak field, HxO, the function iff remains practically con- 
stant (that is, rigid), V0 = 0, 4> » 0o> and G-L2 reduces to 



V 2 A 



A-ne* 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). Non-local versions of the G-L 
treatment are obtained by substituting an integral expression for the 
second term in V.6. In its present local form the G-L 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 non-local 
electromagnetic character of superconductivity be ignored. 

The set G-Ll and G-L2 of coupled non-linear equations in ifj and 
A have been solved for essentially one-dimensional problems. Taking 
the z-axis to be normal to the infinite superconducting boundary, the 
field H along the>--axis, and the current J s and potential A along the 
x-axis, one obtains (using V.l) 









The Ginzburg-Landau 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 G-L 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 G-L equations G-Ll and G-L2 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 = 0-158, the second 0-149, 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 Ginzburg-Landau phenomenological theory 53 

skin-effect 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 non-local 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 2-3 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 G-L 
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 s-n 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. 125-130) 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 C-M 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 C-M 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 Ginzburg-Landau 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 half-space : 



00 



G s {H,z)- 



H(z)H c . H\ 



H: 



4tt 



+-P-G /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 = 1-89-° ■ 

K 



(VI.8) 



The thickness of the transition layer is thus, according to the G-L 
theory, of the order of A /k « 10A for most pure elements. The 
intimate relation between the G-L model and Pippard's range of 
coherence is shown by Gorkov's derivation of G-Ll and G-L2 from 
first principles. He finds an expression for the G-L 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 



0-96 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 G-L 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 2-5 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 tin-indium 
alloys just cited. Chambers (1956) found that the addition of 2-5 
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 « 0-6, which is close to the theoretical value of 0-707. 

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 = 0-8. 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 magneto-optic 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 1-af - {Hl-Hf)IHl (VI.12) 

For tin, 5/ is commonly of the order of 0-9; 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 = 1-17. 

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 pick-up 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 non-ideal specimen there can exist small regions of high 
strain which remain superconducting in very high fields. 

By means of a series of pick-up 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 G-L 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 
3-4 
2-3 



10 5 £ 
(cm) 

16 
4-4 
2-3 



10 5 Ao(0) 
(cm) 

0-50 
0-64 
0-51 



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 
0-23 ; values of Cochran et al. (1958) for aluminium are in reasonable 
agreement. These results can be compared with theoretical predic- 
tions arising from the G-L 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 half-space 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 non-local 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. Non-local effects 
become important for the former when £ > A ; for the latter only 
when 

K 

Thus K-values 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 G-L 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 G-L theory with the Pippard range of co- 
herence under those circumstances of temperature, size, or mean free 
path which eliminate the need for a non-local 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 G-L 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 G-L 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^= 1-18/(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+ 0-08 (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 Ginzburg-Landau- 
Abrikosov (G-L- 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 +7-5xloy /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 
ohm-cm. 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 G-L-A 
model satisfactorily explains the magnetic behaviour of substances 



70 Superconductivity 

such as Ta-Nb alloys investigated by Calverley and Rose-Innes 
(1960) and his own U-Mo alloys (Goodman et ai, 1960). Further- 
more, recent magnetization measurements on Pb-Tl single crystals 
(Bon Mardion et ai, 1962), indicate a considerable degree of rever- 
sibility. Detailed quantitative verification of the G-L-A magnetiza- 
tion curves, as is possible only near T c , was provided by Kinsel et al. 
(1962), who used In-Bi 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 In-Tl alloys. The G-L-A 
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 In-Bi 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 ~ 0-65 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 
G-L-A 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 G-L order parameter W is that its derivative vanish. Saint-James 
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 = 1-695 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 = 2-40kH c , so that H c3 > H c for k > 0-42. 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 1-695V(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 Ginzburg-Landau 
theory leads to nearly identical results. The essential difference is that 
because of the additional terms V.6 in the G-L 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 , G-L 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 non-local electrodynamics leads 
to 20-40 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 
10-20 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.G-L1 and V.G-L2 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 G-Ll and 
G-L2 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 Ginzburg-Landau-Gorkov 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 G-L 
assumptions as well as Gorkov's identification of the energy gap with 
W, Tinkham (1962) has proposed ways in which the G-L 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 Ginzburg-Landau 
theory, including the vanishing of the energy gap and a resulting 
second-order 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 first-order 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 non-transition 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 


0-51 ±010 


Hg 


0-504 


Mo 


0-33 


Os 


0-21 


Pb 


0-461 ±0.025 




0-501 ±001 3 


Rh 


0-4 


Ru 


<01 




<005 


Sn 


0-505 ±0019 




0-46 ±002 




0-462 ±001 4 


Tl 


0-50 ±005 




0-62 ±01 


Zn 


0-5 



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 electron-lattice 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 pick-up 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 short-circuiting 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 by-product of the work on the isotope effect established a 



4-LEAD 







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 one-half of a per cent is 

// = l-10720/ 2 -0-0944/ 4 + 0-3325/ 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 co-ordinates. 
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 non-parabolic 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 (2-44 xlO -8 watt-ohm/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 two-fluid 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 Hg-In 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/cm-deg) 



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 impurity-scattered 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 texpi-b/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 
expi-2e/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 3-50 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 0-1 °K. A number of authors (see Mendelssohn, 1955) 
suggested using such wires in ultra-low temperature experiments as 
thermal switches which would be 'open', i.e. non-conducting, 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 
low-lying 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 electron-phonon 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 co-ordinates. 

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, high-melting 
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 
(/ < 0-2) 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 semi-logarithmically 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 




J-VL 

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 c-sec~ \ which is an experimentally awkward range at 
the upper limit of klystron-excited 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 


2-9 


3-37 b , 3-43 c 
3-3 d 


3-5 


Cadmium 








... 


... 


3-3 


Gallium 


... 






... 


... 


3-5 


Indium 


4-1 


3-9 




3-9 


3-63 a , 3-45 b 


3-5 


Lanthanum 


2-85 






... 




3-7 


Lead 


414 


4-0 


... 




4-33 a , 4-26 b 
418 d 


3-9 


Mercury 


4-6 




... 


... 


... 


3-7 


Niobium 


2-8 




... 


4-4 


3-84 e ,3-6*, 
3-59 8 


3-7 


Rhenium 






... 






3-3 


Ruthenium 












31 


Tantalum 


< 30 






3-6 


3-60°, 3-5', 
3-65 u 


3-6 


Thallium 








3-2 




2-8 


Thorium 






... 






3-5 


Tin 


3-6 


3-3 


3-5 


3-6 


3-46 a , 3-47 b 
3-65 d 


3-6 


Vanadium 


3-4 


... 




3-6 


3-4' 


3-6 


Zinc 


... 


... 


... 


2-5 


... 


3-4 






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 two-fluid 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 Reuter-Sondheimer equations, the London theory, 
and the two-fluid model. Typical results are the solid curve labelled 
0-65Ar B r c and the dashed one labelled 2-37 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 two-fluid 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 0-65, the results agree 
well with the temperature variation calculated without regard to an 
energy gap. For 2-37, however, such calculations would give the 
dashed curve, and it is evident that for / > 0-7, 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 2-37k 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 2-37 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 0-65k B T c 
(1-5 x 10 10 c/sec) to 3-91^7^ (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 0-35°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 pile-up 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 piled-up 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 
superconducting-normal 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 = 1-2°K) 
and Pb (T c = 7-2°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 s-n 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 current-voltage characteristics in this 
case; the limits of the negative resistance region are very sharp. Thus 
the current-potential 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 far-infrared range of wavelengths between 0-1 
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. 168-176). 

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^Hi-rt^) +[( 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 = 0-27. 
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 Kramers-Kronig (K-K) 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 delta-function 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 infra-red 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 K-K 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 Ferrell-Glover 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 London-type 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 K-K 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 K-K relations to show that 
associated with such a delta function 

a,(oi) = A8(oj-0) (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 Ferrel-Glover sum rule, 
must lead to the appearance of a delta function X.12. In turn this 
leads to a London-type 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 0-23 for lead, 0-26 
for tin, and 0- 19 for thallium. These, as well as Glover and Tinkham's 
value of 0-27 for both tin and lead, are in remarkable agreement both 
with the Faber-Pippard data (0-15 for tin and indium) and with the 
BCS prediction for all metals (0-18). 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 Ferrell-Glover sum-rule 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 Ginzburg-Landau 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 10-20 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 self-energy 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 self-energy 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 electron-electron 
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 all-important 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#int|-k!»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 co-ordinates. 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^l-MMl-^}" 2 . (XI.5) 

k kk" 

The summation is over all those k-values 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 (l-h k )} 1 ' 2 = v w 



l-2h. 



2e k 



By defining 
equation XI.6 simplifies to 



e(0)= KStMl-Ml 
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 non-linear 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 0-3, 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 electron-phonon 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 0-5 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 cut-off may be determined by the lifetime of the 
'normal' electrons which can be excited across the gap. These elec- 
trons are not the bare, non-interacting electrons of the simple Bloch- 
Sommerfeld model. Instead they are so-called quasi-particles 
'clothed' by their interactions with each other and with the lattice 
(see \1], pp. 184-95). 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 quasi-particles 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 quasi-particles 
of larger energy are so strongly damped as not to be available for pair 
formation. It is thus necessary to modify the BCS cut-off 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 quasi-particles in a superconductor. According to his pre- 
liminary result this upper bound is 2-2 x 10~ 7 sec, which is only about 
five times as large as the average time calculated by Schrieffer and 
Ginsberg (1962) for quasi-particle 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 
electron-electron 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 10-30 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 d-band tend not to follow the motion of the s-electrons. This 
results in 'anti-shielding' 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: {0-MM ,/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 quasi-continuous 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& 3-4, 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 quasi-particle 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 two-fluid 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 + V-2f 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(l-2/*)(l-2/*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 = Wm-TS = [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 T-Z {A.lnA. + (l-/*)m(l-A)}. (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 



Ml-hM«* S[Mi-M]" 2 (i-2A0 



l-2h k 
This time one defines 



= V 



2e 4 



<T)= F£[M1-/'a<)] ,/2 (1-2/,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 non-linear 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) = 3-52 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 4-0 k B T c even in the non-physical 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 multi-phonon 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 on-line 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 quasi-particle 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 quasi-particle 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 quasi-particles. 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 self-limitation of the Josephson current by 
the magnetic field it generates itself, and De Gennes (1963) has derived 
an expression for the current from the Ginzburg-Landau-Gor'kov 
equations. 

A more realistic cut-off 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: 



— « 8-5 exp(-l -44TJT), 

yT c 



2-5 < 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 

3-65 
318 
307 
2-60 
2-60 
2-58 
2-57 
2-25 
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 2-43, 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 quasi-particle 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. 212-24) 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 two-fluid 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] 3-2 ±01 

parallel to [1 10] 4-3 ±0-2 

perpendicular to [001] and 18° from [100] 3-5 ± 01 



Microscopic theory of superconductivity 139 

11.10. Electromagnetic properties 

To describe the many-particle 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 Ginzburg-Landau 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 
G-L order parameter, so that the dependence of the latter on tem- 
perature, magnetic field, and co-ordinates, 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. 261-263; 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 d-electron 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 18-2°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 non-magnetic 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 co-ordinate £ //, 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 solvent-solute 
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 co-ordinate, 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 time-reversed wave-functions. 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 'well-behaved' 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 In-Sn 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 
Ti-Fe and Ti-Co 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 electron-electron 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 non-magnetic 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 Ti-Cr, Ti-Fe, and Ti-Co 
also carried localized magnetic moments. In addition there is calori- 
metric evidence (Cape and Hake, 1963) that in Ti-Fe samples only a 
small fraction of the volume is superconducting. These results throw 
considerable doubt on Matthias' speculation that iron-group 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 Nb-Mo 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 s-f magnetic 
interaction is rather long range, while the d-d one is a short-range 
inter-action 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 so-called 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 non-mag- 
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 lanthanum-rare 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 

electron-spin 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 spin-flipping 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 Ferrell-Glover 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, o-j 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 non-superconducting 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 Ginzburg-Landau 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 
zero-momentum 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 Rose-lnnes 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 co-workers (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 one-half 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 lead-bismuth 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 /3-wolfram 
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 cold-worked 
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. Rose-Innes and Heaton 
(1963) have used Ta-Nb 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 cross-sectional 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 4-3 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 large-scale 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 4-2°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 flip-flop 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 read-out cryotron, the position 
of the bi-stable 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 non-desired branches. Under 
steady-state 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 two-state 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 
greater-than-unity 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 cross-bar 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 cross-bar, 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 cross-bar 
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 cross-bar is very thin and 
narrow and therefore has a low critical current. When the induced 
current exceeds this critical value, the cross-bar 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 cross-bar is 
restored, and now the flux threading the hole is trapped, as long as the 
cross-bar 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 cross-bar current / c . Pulse 1 is 
too small to induce a critical value of I c . Pulse 2 results in I c > I cril ; 
the cross-bar 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 read-out pulse 5 succeeds in driving the 
cross-bar well beyond the critical value. Note that this is a destructive 
read-out. 

The memory is sensed by means of a wire below the cross-bar, 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 cross-bar changes, that is, whenever the cross-bar 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 cross-bar 
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 two-dimensional matrix of 
memory elements with the drive wire forming part both of an x- and 
a y-circuit, 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. 
Rose-Innes (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 

Bardeen-Cooper-Schrieffer (BCS) 
theory, 12, 117-40, 150 
(See also energy gap, interac- 
tion parameter V, quasi 
particles) 
and G-L theory, 52, 139 

Pippard non-local relations, 
44 
anomalous skin effect, 103 
basic hypothesis, 120 
coherence effects, 136-8 
collective excitations, 100 
critical current and field in thin 

films, 80 
critical field, 130 
critical temperature, 129, 143 
electromagnetic properties, 103, 

114, 139-40 
electron -phonon interaction, 

118-20, 125 
ground state energy, 120-6 



183 



Bardeen-Cooper-Schrieffer (BCS) 
theory - cont. 
high frequency conductivity, 103, 

114 
Knight shift, 139^10 
nuclear relaxation rate, 104-6, 137 
penetration depth, 37, 52, 130 
range of coherence, 48, 115 
similarity principle, 122, 130 
specific heat, 97, 130, 134-6 
thermal conductivity, 91, 92, 136 
thermal properties, 127-36 
weak coupling limit, 131 

Bulk modulus, 16 

Coefficient of thermal expansion, 17 

Coherence : see range of coherence 

Coherence effects, 136-8 

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, 118-21, 
125, 140, 144, 151 

Critical current, 4, 80, 155, 161 

Critical magnetic field, 4 
in BCS theory, 130 
in G-L theory, 51 
of lead and mercury, 132 
of small specimens, 46, 58, 75-8, 

80, 109, 155 
of superconducting elements 

(table), 5 
of thin films, 46, 76-7, 80, 109 



184 



Index 



Critical magnetic field - cont. 
precise measurements, 83-5 
pressure effects, 16—17 
relation of thin film value to A 6 

and lo, 77 
relation to thermal properties, 

13-19,21,86, 132, 135 
similarity of reduced field curves, 

4, 85, 135 
temperature dependence, 4, 18, 

84-6, 130, 135 
very high values, 27, 70-1, 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, 

145-50 
effect of non-magnetic impurities, 

141-4 
in BCS theory, 129-43 
Matthias' rules, 6, 141, 148 
of superconducting elements 

(table), 5 
Crowe cell, 161-3 
Cryotron, 157-63 
Cylindrical specimens, 13, 16, 24 
{See also thin wires) 

Debye temperature, 10, 87, 124-5, 

131, 134, 143 
Demagnetization coefficient, 23 
Density of electron states, 104-8, 

123, 127, 129, 134, 143 
Diffusion, 152 
Dilute alloys 

critical temperature, 141-4 

magnetization curve, 144-5 



Dilute alloys - cont. 
specific heat, 143 
thermal conductivity, 87-8, 90 
variation of surface energy, 59, 70 

Effective charge, 49, 52 
Elastic properties, 9 
Electron irradiation, 143 
Electron-electron interaction, 12, 

82-3, 95, 117-25 
Energy Gap 

{See also anisotropy of energy 
gap) 
correlation with TJ9, 134 
deduction from 
infrared absorption, 98-100 
infrared transmission, 113-14 
microwave absorption, 100-4 
nuclear spin relaxation, 104-6 
specific heat, 11,91,96-7 
thermal conductivity, 92, 95 
tunnelling, 79, 106-9 
ultrasonic attenuation, 137-8 
dependence on 
field, 79-80, 92, 109, 139 
phonon spectrum, 132 
position, 150-2 
quasi-particle energy, 131-3 
size, 79, 139 

temperature, 90, 103, 108, 129, 
139 
in BCS theory, 120, 126-34, 139 
in thin films, 79-80 
of superconducting elements 

(table), 99 
relation to 
G-L order parameter, 52, 79, 

139 
Meissner effect and perfect 
conductivity, 110, 114-15, 
120 
penetration depth, 1 1 5 
range of coherence, 44 
Thomson heat, 95 
Entropy, 14, 18,38,78,128 






Index 



185 



Ferrell-Glover sum rule, 114-15, 

150 
Ferromagnetism, 145-8 
Flux creep, 73 

Flux quantization, 32-3, 72, 120 
Free energy, 13-4, 19-20, 47-8, 

57, 75, 93, 128, 148-9 



Gapless superconductivity, 149 
Gauge invariance 
in BCS theory, 139 
in G-L theory, 49 
Geometry, influence of, 23-6 
Ginzburg-Landau (G-L) theory, 12, 
48-54, 150 
basic equations, 50, 139 
critical field of small specimens, 

75-7 
extension to lower temperatures, 

49,80 
free energy, 48-9, 57, 75 
limitations, 49, 52-3, 66, 80 
non-local modifications, 48, 50 
range of coherence, 58 
relation to 
BCS theory, 48, 52, 139 
London equations, 50, 54 
small specimens, 76-80 
superconductors of second kind. 

67-72 
supercooling, 65-7 
surface energy, 57-9 
G-L order parameter, 48 
and free energy, 48-50, 57 
effect of magnetic field, 53, 78, 

139 
gradual spatial variation, 49-50, 

57-8, 73 
proportionality to energy gap, 

52, 79, 139 
relation to penetration depth, 49, 
53^, 78 
G-L parameter «-, 5 1 -3, 58, 66, 68-70 
and range of coherence, 58 



G-L parameter k - cont. 
critical value for negative surface 

energy, 59, 66, 67, 70 
deduction from 
penetration depth, 51-3 
supercooling, 51-3, 66 
in thin films, 54, 78 
relation to 
normal conductivity and 

specific heat constant, 69 
surface energy, 58 
temperature dependence, 70 
Gorter-Casimir thermodynamic 

treatment, 11, 13-19 
Gorter-Casimir two-fluid model, 
11, 19-21 
{See also two-fluid model; 
two-fluid order parameter) 
application to G-L theory, 49 
relation to penetration depth, 36 
Gyromagnetic ratio, 22 

Impurity effects : see mean free path 

effects 
Infrared absorption, 98-100 
Infrared transmission, 45, 99, 109- 

114, 115 
Interaction parameter V, 121 
anisotropy, 131 
BCS cut-off, 121, 124-5, 131 
effect of 

non- magnetic impurities, 1 43-4 
magnetic impurities, 149 
influence on isotope effect, 

124-6 
quasi-particle lifetime effects, 

124-5, 131 
variation with quasi-particle 
energy, 131 
Intermediate state, 14, 23-6, 29, 

59-61,94 
Isotope effect, 12, 81-3, 95, 117, 
124-6 
absence in transition metals, 12, 
82, 125 



186 



Index 



Isotope effect - cont. 
effect of quasi-particle life time, 

124-6 
in the BCS theory, 124 
table of values, 82 

Josephson effect, 109, 133-4 

Knight shift, 77, 139^K) 
Kramers-Kronig relations, 112, 114 

Latent heat, 1 5, 78 
Lattice parameters, 10 
Laves compounds, 147-8 
Lifetime effects, 124-6, 131 
Localized magnetic moment, 145-6 
London theory, 1 1 , 28-32, 36, 41-2 
basic equations, 29, 42, 44,46, 1 1 1 
incorrect values of penetration 

depth, 29, 37-8, 43 
microscopic implications, 11, 31, 

43 
non-linear extension, 38 
prediction of penetration depth, 
29,36 
Low frequency behaviour 
diamagnetic description, 22-6 
influence of geometry, 23-6 
relation to high frequency re- 
sponse and energy gap, 1 14-15 
small specimens, 75-80 

Magnetic field distribution, 7-8, 

22-4 
Magnetic field dependence of 

energy gap, 79-80 

entropy, 37 

free energy, 13-14, 49 

G-L order parameter, 53, 78 

penetration depth, 35, 38-9, 53, 76 
Magnetic field penetration: see 

penetration depth 
Magnetic susceptibility, 14, 23, 

34-5, 75, 77, 83 



Magnetic impurities, 145-50 
Magnetization 
area under magnetization curve, 

16, 26, 75 
dilute alloys, 144 
filamentary superconductors, 

78 
ideal superconductors, 13-14, 

24-6,68 
small specimens, 75 
superconductors of second kind, 
68-9, 156 
Magnetostriction, 16 
Matthias' rules, 6, 141, 148 
Mean free path effects on 
anisotropy of energy gap, 97, 100 

106, 142 
critical temperature, 141-4 
G-L parameter k, 69 
infrared absorption, 100 
nuclear relaxation rate, 106 
penetration depth, 35, 38, 41-3, 

45-6,58, 112 
range of coherence, 42-3, 45-7, 

152 
surface energy, 58-9, 70, 77 
Mechanical effects, 16-17, 143 
Mendelssohn 'sponge', 78, 155 
Microwave absorption, 100-4 
Mixed state, 71-74 

Neutron bombardment, 143 
Nuclear spin relaxation, 104-6 
Nuclcation of superconducting 
phase, 61-3 

Order in the superconducting 
phase, 14, 19 

Order parameter, see G-L order 
parameter; two-fluid 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, 38-9, 53, 76 
mean free path, 35, 38, 41-3, 

45-6,58,112 
range of coherence, 43, 46, 112 
size, 38, 46, 79 
temperature, 35-8, 51-3, 130 
in BCS theory, 37, 52, 130 
in Pippard theory, 45-6, 112 
incorrectness of London values, 

29,37-8,43, 112 
methods of measurement, 35-6 
relation to 
energy gap, 115 
entropy, 38 

frequency variation of con- 
ductivity, 111-2 
G-L order parameter, 51, 78 
surface energy, 55-7 
susceptibility, 34-5 
thin film critical field, 77 
values in superconducting ele- 
ments (tables), 38, 65 
Perfect conductivity of supercon- 
ductors, 4, 29-30, 114-15 
Perfect conductor, 4, 6-8, 27-8 
Persistent current, 3, 6, 26, 158-60 
Phase propagation, 63-5 
Phonon spectrum, 132 
Pippard non-local theory, 12, 41-6 
(See also range of coherence) 
basic equations, 42, 44, 45 
critical field in thin films, 46 
field penetration through thin 

films, 45 
penetration depth, 42-3, 45-6 
reduction to local form (London 

limit) 12, 45-6 
relation to energy gap and BCS 

theory, 44-5 
susceptibility of thin films, 77 
Pressure effects, 12, 16-17 



Quantized flux, 32-3, 72, 120 
Quasi-particles, 124-5, 132 
Quenching, 143 

Range of coherence, 1 1 
and superimposed metals, 150-1 
dependence on mean free path, 

42, 45-7, 152 
in BCS theory, 44 
in G-L 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, 44-6 
sharpness of transition 40-1 
surface energy, 57 
uncertainty principle 40, 45 
values for Al, In, Sn (table), 65 
Relation between magnetic and 
thermal properties, 13-21, 86, 
96, 132, 135 
Rutgers' relation, 15, 17 

Semiconductors, superconducting, 

6 
Silsbee's rule, 5 

Similarity, 85-6, 96, 116, 121-2, 130 
Size effect on 

critical field, 46, 76-7, 155 

critical supercooling field, 67 

critical temperature, 143 

energy gap, 79-80 

magnetic susceptibility, 34 

penetration depth, 38, 46, 79 

range of coherence, 46 
Skin depth, 35-6, 43 
Small specimens 

critical field, 46, 75-8 

in G-L theory, 46, 54, 67, 75-80 

in Pippard theory, 46 

low frequency behaviour, 75-80 



188 



Index 



Small specimens - cont. 
penetration depth, 35-8, 43, 

45-6, 79 
range of coherence, 45-6 
Sommerfeld specific heat constant, 9 
in dilute alloys, 143 
independence of isotopicmass,85 
relation to 
critical field, 19-21,135 
G-L parameter *, 69 
Specific heat of the electrons 
comparison of magnetic and 
calorimetric data, 17, 19-21, 
86, 134, 135 
dependence on temperature, 9, 
11,18,20-1, 86,91,96-7,134 
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, 96-7, 132 
thermal conductivity, 91, 95 
Rutgers' relation, 15, 17 
Specific heat of the lattice, 9-10, 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, 26-7, 62, 77, 155 
Superconducting alloys and com- 
pounds, 5-6 
dilute alloys, 58-9, 87-90, 141-5 
ferromagnetism, 145-8 
high critical fields, 154-6 
Laves compounds, 147-8 
magnetic impurities, 145-50 
Matthias' rules, 6, 141 
non-magnetic impurities, 141-5 
rare earth and transition metal 

solutes, 145-9 
thermal conductivity, 88-9 



Superconducting devices 

cavities, 154 

computer elements, 157-64 

d.c. amplifiers, 153-4 

galvanometers, 153 

heat switches, 94, 1 53 

leads, 153 

magnets, 78, 154-7 

memory devices, 1 60-4 

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, 
6-8 

discontinuity of 
specific heat, 9, 15, 17, 78, 

134 
entropy, 14 

free energy, 13-14,48 

in dilute alloys, 144 

in thin films, 78-81 

length and volume changes, 
16-17 

order, 72, 78-81 

reversibility, 7, 13, 17, 144 

speed, 159 
Superconductors of second kind, 

67-73, 155-6 
Supercooling, 52-3, 61-3, 65-7, 

74 
Superheating, 61 
Superimposed metals, 150-2 
Surface currents, 22 
Surface energy, 55-74, 150, 156 

dependence on temperature, 64-5 

effect of strain, 62, 77, 156 

in G-L theory, 57-8 

in inhomogeneous specimens, 77, 
156 



Index 



189 



Surface energy - cont. 
in Pippard theory, 56-7 
mean free path effect, 58-9, 67, 

77 
negative values, 58-9, 62, 67, 69, 

70, 157 
relation to 
intermediate state, 59 
phase nucleation and propa- 
gation, 61-6 
range of coherence, 56-8 
values for AI, In, Sn (table), 65 
Surface impedance, 35-6, 52-3, 95, 
101-4, 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, 84-6, 130, 

135 
energy gap, 90, 103, 108, 130 
G-L parameter k, 70 
penetration depth, 35-8, 51-2, 

130 
specific heat, 9, 11, 18, 20-1, 86, 

91,96-7, 130, 134 
surface energy, 64-5, 70 
surface impedance, 101-3, 139 
thermal conductivity, 87-94, 136 
two-fluid order parameter, 19-21, 

36 
Thermal conductivity, 87-94 
in BCS theory, 92, 136 
of thin films, 79, 93 



Thermal conductivity - cont. 
relation to 

energy gap, 91-3, 95 

gap anisotropy, 92 

specific heat, 91 
Thermal expansion coefficient, 17 
Thermodynamics of superconduc- 
tors 
BCS theory, 128 
G-L theory, 48-9 
Gorter-Casimir treatment, 11, 

13-19 
relation between magnetic and 

thermal properties, 13-21, 86, 

96, 132, 135 
Thin films 

(See also superimposed metals) 
critical current, 80 
critical field, 47, 75-80 
critical thickness for second order 

transition, 78 
cryotrons, 157-63 
energy gap, 79, 93 
infrared transmission, 46, 109-15 
in G-L theory, 54, 75-80 
in perpendicular field, 73 
magnetic behaviour, 33, 73-4, 

75-80 
penetration depth, 29, 35-6, 46, 

79 
relation of critical field to X b and 

&»77 
second order transition, 78 
supercooling, 67 
susceptibility, 34-5, 76, 77 
thermal conductivity, 79, 93 
total field penetration, 36, 45 
variation of G-L order para- 
meter, 78 
Thin wires, 29, 35, 67, 76 
Thomson heat, 95 
Threshold magnetic field: see criti- 
cal magnetic field 
Time-reversed wave functions, 
144 



190 



Index 



Transition metals 
absence of isotope effect, 12, 82, 

125-6 
effect on T c , 145-50 
Trapped flux, 3, 26-7, 32, 120, 144, 

161 
Tunnelling, 106-9, 132-4 
Two-fluid model 

(See also Gorter-Casimir two- 
fluid model) 
and BCS theory, 127 
extension of G-L theory, 49 
relation to 
nuclear spin relaxation, 104 
penetration depth, 36 
thermal conductivity, 87 



Two-fluid order parameter, 19-21, 
36 
gradual spatial variation, 40, 

56-7 
relation to 
penetration depth, 36 
surface energy, 56-7 
thermal conductivity, 89 
rigidity in London theory, 39 
Ultrasonic attenuation, 105, 136-8 
Uncertainty principle, 31, 40, 45 



Valence elections, effect on T c . 

143, 145 
Vortex lines, 72-4 



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Monographs on Physical Subjects — continued 

IONIZATION AND BREAKDOWN IN GASBS 
F. Llewellyn Jones 

LOW TEMPERATURE PHYSICS L. C. JaCKSOl 

magnetic amplifiers George M. EtringC 
magnetic materials F. Brailsford 

MASERS AND LASERS G. J. F. TrOUp 

THE MEASUREMENT OF RADIO ISOTOPBS 
Taylor 

MBCHANICAL AND ELECTRICAL VIBRATION 
J. R. Barker 

mercury arcs F. J. Teago 

THE METHOD OF DIMENSIONS Alfred W. P 
MICROWAVE LBNSBS J. Brown 

microwavb spectroscopy M. W. P. Stran 

MOLECULAR BEAMS K. F. Smith 
NUCLEAR RADIATION DBTBCTORS J. Sharp 

the nuclear reactor Alan Salmon 
optical masers O. S. Heavens 

ORDER-DISORDER PHENOMENA E. W. ElCQ 
PHOTONS AND ELECTRONS K. H. Spring 
PHYSICAL CONSTANTS W. H. J. Childs 

physical formulae T. S. E. Thomas 

THE PHYSICAL PRINCIPLES OF WIRBLBSS 

J. A. Ratcliffe 

PRINCIPLES OF APPLIED GEOPHYSICS 
D. Parasnis 

relativity physics W. H. McCrea 
sbismology K. E. Bullen 
SBMI-CONDUCTORS D.A.Wright 
SHOCK TUBES J. K. Wright 
THB SPECIAL THBORY OF RELATIVITY M< 
Dingle 

superconductivity Ernest A. Lyman 

THE THBORY OF GAMBS AND LINEAR 

gramming S. Vajda 

THERMIONIC VACUUM TUBBS W. H. Aldoi 

Sir Edward Appleton 
thermodynamics Alfred W. Porter 
wave filtbrs L. C. Jackson 
wave guides H. R. L. Lamont 
WAVB MBCHANICS H. T. Flint 
x-ray crystallography R. W. James 
x-ray optics A. J. C. Wilson 



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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 

HI-GH FREQUENCY TRANSMISSION LINES Willis Jackson 

INTEGRAL TRANSFORMS IN MATHEMATICAL PHYSICS C. J. 
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AN INTRODUCTION TO ELECTRONIC ANALOGUE COMPUTERS 
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AN INTRODUCTION TO ELECTRON OPTICS L. Jacob 

AN INTRODUCTION TO FOURIER ANALYSIS R. D. Stuart 

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