the science and technology of superconductivity: proceedings of a summer course held august...

The Science and Technology of Superconductivity Volume 1

The Science and Technology of Superconductivity Proceedings of a summer course held August 13-26, 1971, at Georgetown University, Washington, D. C.

Edited by

W. D. Gregory and

W. N. Mathews Jr. Department of Physics Georgetown University Washington, D. C.

and

E. A. Edelsack Office of Naval Research Washington, D. C.

Volume!

<:PPLENUM PRESS. NEW YORK-LONDON. 1973

ISBN-13: 978-1-4684-2999-2 001: 10.1007/978-1-4684-2997-8

e-ISBN-13: 978-1-4684-2997-8

Library of Congress Catalog Card Number 72-77226

© 1973 Plenum Press, New York Softcover reprint of the hardcover 1 sl ed ilion 1973

A Division of Plenum Publishing Corporation 227 West 17th Street, New York, N. Y. 10011

United Kingdom edition published by Plenum Press, London A Division of Plenum Publishing Company, Ltd. Davis House (4th Floor), 8 Scrubs Lane, Harlesden, London, NWIO 6SE, England

All rights received

No part of this publication may be reproduced in any form without written permission from the publisher

FOREWORD

Since the discovery of superconductivity in 1911 by H. Kamerlingh Onnes, of the order of half a billion dollars has been spent on research directed toward understanding and utilizing this phenomenon. This investment has gained us fundamental understanding in the form of a microscopic theory of superconductivity. Moreover, superconductivity has been transformed from a laboratory curiosity to the basis of some of the most sensitive and accurate measuring devices known, a whole host of other electronic devices, a soon-to-be new international standard for the volt, a prototype generation of superconducting motors and generators, and magnets producing the highest continuous magnetic fields yet produced by man. The promise of more efficient means of power transmission and mass transportation, a new generation of superconducting motors and generators, and computers and other electronic devices with superconducting circuit elements is all too clear. The realization of controlled thermonuclear fusion is perhaps totally dependent upon the creation of enormous magnetic fields over large volumes by some future generation of superconducting magnets. Nevertheless, whether or not the technological promise of superconductivity comes to full flower depends as much, and perhaps more, upon economic and political factors as it does upon new technological and scientific breakthroughs.

The basic science of superconductivity and its technological implications were the subject of a short course on "The Science and Technology of Superconductivity" held at Georgetown University, Washington, D. C., during 13-26 August 1971. This course brought together as speakers 25 experts in the field of superconductivity, including experimental and theoretical physicists, metallurgists, and electrical, mechanical, and cryogenic engineers, drawn from the academic, governmental, and industrial worlds. Together these speakers presented a comprehensive picture of the scientific and engineering fundamentals of superconductivity and its technological applications. The contents of this book are essentially the proceedings of that course.

v

vi FOREWORD

The book is divided into six parts. Part I consists of the introductory lectures by the editors, F. Bloch's presentation of the basic theory of the Josephson effect, D. J. Scalapino's discussion of the consequences of charge conservation for timedependent phenomena in superconductors, and Robert W. Stuart's review of the technology of refrigeration for superconducting devices. Part II is devoted to the consideration of superconducting materials. Part III deals with various technological applications of superconductivity. Part IV is a lightly edited transcript of the panel discussion on "The Scientific, Technological, and Economic Implications of Advances in Superconductivity" which was held during the course. Part V contains D. N. Langenberg's consideration of the technological implications of superconductivity for the next decade, and J. Bostock's brief summary of the course. Part VI contains four Appendices.

It is hoped that publication of this book will result in a greater awareness of the possibilities for exploiting the unique properties of superconductors, and that this increased awareness will in turn lead to an increase in the rate of advance of the science of superconductivity and its application.

The proceedings of the course are also available for distribution on video tape from Georgetown University.

We regret that H. T. Coffey's discussion of the application of superconducting technology for mass transportation and M. S. McAshan's review of the superconducting electron accelerator at Stanford were not available for publication •

. We are grateful to the PhYlIi..c.ai. Rev.(.ew for permission to reprint F. Bloch's paper on the theory of the Josephson effect.

We are grateful to the National Science Foundation and Georgetown University for sponsoring the course. We wish to thank Kenneth H. Fredgren and Constance Francis for their assistance in organizing and running the course, Ellen C. Cox, Madelyn Miller, Margaret Reinheimer, and Dawn E. Lechevet for their assistance in preparing the manuscript, and Mohamad Behravesh, Larry H. Capots, Nicholas C. Cirillo, Andrew J. Grekas, Richard Janik, Vishwanath Ka1vey, Rassmidara Navani, Morris A. Olson, and Thomas B. Thompson for their assistance in running the course and preparing the subject and author indexes. A special note of thanks is due Dr. Jon N. Lechevet for carefully reading the manuscript and putting the indexes in their final form.

William D. Gregory Wesley N. Mathews Jr. Edgar A. Ede1sack

CONTENTS OF VOLUME 1

Contents of Volume 2 • • • • • • • • • • • • • • • • • • • ix

PART I: INTRODUCTION AND BACKGROUND

Fundamentals of Superconductivity E. A. Edelsack

5

Phenomenological Theories of Superconductivity • • • • • • 25 W. D. Gregory

Elements of the Theory of Superconductivity W. N. Mathews Jr.

71

Josephson Effect in a Superconducting Ring • • • • • • •• 149 F. Bloch

Time-Dependent Superconductivity D. J. Scalapino

. . . . . . . . . . . . . Refrigeration for Superconducting Devices

Robert W. Stuart

PART II: SUPERCONDUCTING MATERIALS

Experimental Aspects of Superconductivity: Editors' Note ••••••

Superconductivity in Very Pure Metals W. D. Gregory

163

185

209

211

Tc's The High and Low of It . • • • • • • • . . . . .• 263 Bernd T. Matthias

vii

viii

The Metallurgy of Superconductors Robert M. Rose

Superconducting Intermetallic Compounds -The A15 Story •••• • • • •

Robert A. Rein


289

333

Theory of Superconducting Semiconductors • • • • • • • •• 373 C. S. Koonce

Enhancement Effects: Theory... • • • • • • • • • • • •• 389 C. S. Koonce

Enhancement Effects J. F. Schooley

Author Index •

Subject Index

405

xi

xxxiii


Contents of Volume 1 • • • • • • • • • • • • • • • • • •• vii

PART III: TECHNOLOGICAL APPLICATIONS

Superconducting Power Transmission • • • • • • • • • R. W. Meyerhoff

Application of Superconductivity in Thermonuclear Fusion Research • • • • • • • • •

A. P. Martinelli

Application of Superconductors to Motors and Generators • • • • • • •

Joseph L. Smith, Jr.

Superconducting Coils Z. J. J. Stekly

Physics of Superconducting Bascom S. Deaver, Jr.

Device s • . . • • • • • . • • •

Superconductivity in DC Voltage Metrology T. F. Finnegan

433

459

483

497

539

565

Electric and Magnetic Shielding with Superconductors • •• 587 TIlas Cabrera and W. O. Hamilton

Superconductive Computer Devices • 0

J. Matisoo

Superconductors in Thermometry J. F. Schooley

Millimeter and Submillimeter Detectors and Devices

Sidney Shapiro

ix

607

625

631

x CONTENTS OF VOLUME 2

Magnetometers and Interference Devices • • • • • • • • •• 653 Watt W. Webb

PART IV: PANEL DISCUSSION

The Scientific, Technological, and Economic Implications of Advances in Superconductivity

Edited by: W. N. Mathews Jr., W. D. Gregory, and E. A. Ede1sack

PART V: CONCLUSIONS AND SUMMARY

The Technological Implications of Superconductivity in the Next Decade • • • • • • • • • • •

D. N. Langenberg

A Summary of the Course J. Bostock

• • • 0 • • • • • • • • • • • • •

PART VI: APPENDICES

1. Program

2. Invited Speakers.

3. Participant List ••

4. Problems. • • • •

Author Index

Subject Index

681

719

735

757

765

767

771

779

801

VOLUME 1

Part I

Introduction and Background

FUNDAMENTALS OF SUPERCONDUCTIVITY

E. A. Edelsack

Office of Naval Research Washington, D. C.

This course on the science and technology of superconductivity COITles at a ITlost appropriate tiITle in the over-all developITlent of the field. The late 1950's and this past decade of the 1960's were in ITlany ways the golden age of superconductivity. During that period of SOITle fifteen years, superconductivity grew froITl little ITlore than a laboratory curiosity being studied at a few low teITlperature centers, to a full fledged science and technology, being actively pursued by scores of industrial and acadeITlic laboratories throughout the world. The years between 1959 and 1962 saw a reITlarkable growth in the superconducting literature. Figure I shows that in 1959 superconducting publications represented SOITle 6% of all low teITlperature publications. Three years later in 1962 it represented alITlost 17% of the total low teITlperature literature. This present decade is seeing a new and interesting developITlent -the transforITlation of superconductivity research out of the laboratory into the realITl of ITlilitary and industrial practicality. SOITle applications are ripe for iITlITlediate practical developITlent, others are still in the realITl of basic and applied research. In their totality, they represent an iITlpressive list of existing and potential applications. They include: ITlagnets, ITlotors, generators, ITlagnetoITleters, infrared and ITlilliITleter wave detectors, gravity ITleters, gyroscopes, boloITleters, nuclear particle detectors, voitITleters, aITlITleters, power transITlission lines, dc transforITlers, cOITlputer eleITlents, switches, passenger train suspension systeITls, acceleroITleters, antennas, aITlplifiers, ITlicrowave cavities, linear accelerators... This list is by no ITleans cOITlplete. It continues to grow with the passage of tiITle.

5

6

Z Qcn I-Z

I- Uo Z ::::>-

01-w z<C U a: o~ w U...J c... a:m

w::::> c...c... ::::> cn

20

18

16

14

12 GROWTH OF THE LITERATURE OF

SUPERCONDUCTIVITY 10

8

6~~ __ ~~~~ __ ~~~~ __ ~~~~~~ Before 1960 '60 '61 '62 '63 '64 '65 '66 '67' '68 1969,

PUBLICATION YEAR

Figure I

E. A. EDELSACK

This first lecture, presents a descriptive introduction to the fundamentals of superconductivity. It will be largely a panoramic survey of the major milestones in the development of the field. Figure II shows some of the major milestones we will be reviewing. This lecture is intended to provide a common stepping stone for appreciating the more advanced material of the subsequent speakers.

The story of superconductivity here on earth had its genesis in the stars, particularly in one star - our sun. Just a little over a hundred years ago in 1868, a British astronomer, using a newly invented instrument called a solar spectroscope observed unusual spectral lines from the hot incandescent gases of the chromosphere of the sun. These spectral lines could not be accounted for, as being produced by any known substance on earth, at that time. Thus the gas was given the name "helium" deriving from the word "helios, II the Greek word for sun. Twenty-seven years later in 1895, Sir William Ramsay, a British chemist, discovered helium on the earth and in 1908, the Dutch physicist, Kammerling Onnes, succeeded in liquefying helium gas at a temperature of 4 K. The subsequent discovery of the phenomenon of superconductivity was a direct consequence of this liquefaction of helium. A temperature interval down to about 1 K had now been opened for research.


MILESTONES IN SUPERCONDUCTIVITY

1911 Discovery, H. K. Dnnes. Persistent currents, critical fields.

1933 Flux exclusion, Meissner and Ochsenfeld.

1935 London Phenomenological theory and F. london's Proposal that superconductivity is a quantum phenomenon.

1950 Isotope effect, Tc a:: M-V" Reynolds et al.; Maxwell. Importance of electron-phonon interactions; Froholich.

1953 Experimental evidence for an energy gap. Goodman, others.

1957 Microscopic theory based on pairing. Bardeen, Cooper, and Schrieffer.

1960 Tunnel effect Giaever.

1961 Flux quantization. Deaver, and Fairbank; Doll and Nabauer.

1962 Supercurrent flow through tunnel barrier. Josephson.

Figure II

One of the first investigations which Onnes carried out in this newly available low temperature range was a study of the varia-

7

tion of the electrical resistance of metals as a function temperature. It had been known for many years that the resistance of metals fell when cooled below room temperature, but it was not known what limiting value the resistance would approach, if the temperature were reduced to very close to 0 K. Onnes, experimenting with platinum, found that, when cooled, its resistance fell to a low value which depended on the purity of the specimen. At that time the purest available metal was mercury and, in an attempt to discover the behavior of a very pure metal, Onnes measured the resistance of pure mercury. He found that at very low temperatures the resistance became immeasurably small, which was not surprising, but he soon discovered that the manner in which the resistance disappeared was completely unexpected. Instead of the resistance falling smoothly as the temperature was reduced towards 0 K, the resistance fell sharply at about 4 K, and below this temperature the mercury exhibited zero electrical resistance. (See Figure Ill) Furthermore, this sudden transition to a state of zero resistance was not confined to the pure metal but occurred even if the mercury was quite impure. Onnes recognized that below 4 K mercury passed into a new state with electrical properties quite unlike those previously known.

8

TranSition Pomt

Figure III

E. A. EDELSACK

Figure IV is a graph taken from annes' earliest paper which marks the discovery of superconductivity. In reading Onnes' first paper one can almost feel the sense of elation he must have had. He writes: "Mercury has passed into a new state, which on account of its extraordinary electrical properties may be called the superconductive state. There is left little doubt, that, if gold and platinum could be obtained absolutely pure, they would also pass into the superconductive state at helium temperatures. The behavior of metals in this state gives rise to new fundamental questions as to the mechanism of electrical conductivity."

Often the question is asked, "Is the resistance of a superconductor truly zero? A long series of measurements on superconducting rings and coils by annes and others were attempted to answer this question. In 1956 an experiment was performed by Professor Collins at Massachusetts Institute of Technology in which a superconducting ring carrying an induced current was maintained for over two years. The absence of any measurable decay of the persistent circulating current during this period allowed Professor Collins to place an upper limit of 10- 21 ohm-cm on the resistivity of the superconductor. This is to be compared to the value of 10-9 ohm-cm for the low temperature resistivity

Ol~

0.12~

iii 010 ~

~ ~ :! 007~ '" iii ... a:

002~

0.00 4"00

~n

1

a/ .~ • I

I

: HI I I

I I I

I I

I

I I I

I

4"30 4<>40

Resistance in ohms of a specimen of mercury vs. absolute temperature. This plot by Kamer1ingh Onnes marked the discovery of superconductivity in 1911.

Figure IV

of pure copper. More recently, Professor Little at Stanford University has calculated the upper limit of the lifetime of persistent currents in superconductors of finite dimensions by considering the effects of thermodynamic fluctuations in the superconducting specimen. These fluctuations provide a means

9

by which the persistent current can decay. Decay by these means should be observable in wires a few hundred angstroms in thickness. Figure V shows the results of these calculations. For all practical purposes the dc electrical resistance of a bulk superconductor is zero. I stress here dc because superconductors do have ac electrical resistance losses.

The temperature at which a material transforms from its normal or resistive state to the super conducting state is called the transition temperature, T c' The sharpness of the transition depends on the purity and state of the material. In some favorable cases it can occur within a temperature interval of less than 0.001 K.

10 E. A. EDELSACK

o Computed lifetimes for two Sn wires, olJe of 100 A

diameter and the other of 360 A

0 0

100 A wire 360 A wire Lifetime (OK) (OK)

_10-14 sec 3.7 3.7 -7 2.0 3.4 10_4 sec

10 sec 1.5 3.26 1 sec 1.25 3.10

10 days 1.0 3.06 106 years 0.75 2.90

Figure V

Figure VI shows the periodic table with those elements which now are known to be superconducting. Today over a thousand compounds and alloys are superconductors. Figure VII shows the distribution of superconductors as a function of transition temperature.

Following the discovery of superconductivity. Onnes soon found that a mercury wire loses its superconductivity and normal resistance reappears above a critical current. This was soon explained when Onnes discovered that a magnetic field in excess of a maximum value forces the superconductor into its normal resistive state. This value is called the critical magnetic field and is usually denoted by the symbol H. For all superconductors there exists a region of temperaturtfs and magnetic fields within which the material is superconducting. Outside this region the material is normal. This parabolic curve shown in Figure VIII can be approximated by

""-:",, [l-(~J 2]

FUNDAMENTALS OF SUPERCONDUCTIVITY 11

-r--

H He r0-

Ll ,Be, B C N 0 F Ne

No Mg ~ Si P S CI Ar

Se ~ w -"'4

Cr Mn Fe Co Ni Cu @.l ~ Ge As Se Br Kr K Co :~ V L_

~ l'N~ ~ ,""""""l

~ ~ .~

Rb Sr Y ITe : Rh Pd Ag In S~ Sb Te I Xe ':""v.w ::. •.. ,,'- .. , ~ • ...s ..... «

~R~:; ~ 0 ..

Cs Bo l'f-u ~Hf ~1° ; w Pt Au I!! H ' '.TI.1 'Pb Bi Po At Rn • Wo' ~ .9.: ,,-,

Fr Ro 1\\ ~ .

w'" ~1 L2! Ce P, Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb ''''' JI'' .' Ae ~!h:mpO~ u .!. Np Pu Am Cm Bk Cf Es Fm Md No

o Tc above 1.2 oK II II Tc below 1.2°K D Discovered at N.R.L.

Figure VI

where H is H at T= 0 K, T is the given temperature (sometimes called th~ tran1.ition temperature). T is the critical temperature (highest temperature at which a materi<al exists in the superconducting state in the absence of a magnetic field).

For specimens sufficiently thick that surface effects can be ignored, the critical current is that current which creates at the surface of the wire a magnetic field equal to the critical magnetic field. This criterion is known as Silsbee1s rule. The critical fieldtemperature curves for many of the early discovered superconductors are shown in Figure IX. These critical magnetic fields are all below 1000 gauss. These relatively low critical magnetic fields dashed all early hopes of using superconductors to build powerful magnets. Since one could obtain magnetic fields of between 15,000 to 20,000 gauss with conventional copper wound, iron core magnets, superconducting magnets were of no practical use at that time.

For many years after Onnes l discovery, which resulted in his receiving the Nobel Prize in 1913, it was believed that superconductors only differed from normal metals in having zero electrical

12

NUMBER OF KNOWN SUPERCONDUCTORS AS A FUNCTION OF T,

Figure VII

E. A. EDELSACK

resistance. It was thought that a superconductor was the limiting case of a normal conductor with the conductivity approaching infinity. This assumption was shattered in 1933 when Meissner and Ochsenfeld found that a solid cylinder of lead or tin situated in a uniform magnetic field expels the magnetic flux as it is cooled from the normal state below its critical temperature to the superconducting state. Figure X shows what occurs. The specimen becomes perfectly diamagnetic, cancelling all magnetic flux in its interior.

This discovery of Meissner and Ochsenfeld demonstrated for the first time, that superconductors were something more than materials which are perfectly conducting. They have an additional property that a merely zero resistance metal does not possess: namely, a bulk metal in the super conducting state never allows magnetic flux to exist in its interior. That is to say, inside a superconductor we always have B, the magnetic flux density, equal to zero. This perfect diamagnetic property of superconductors is perhaps the most fundamental macroscopic property of a superconductor. Today we call this property simply the "Meissner effect". In summary, the Meissner effect requires infinite conductivity, but infinite conductivity does not require the Meissner effect.


9 w it o

H

~ SuperconductiftCJ ~ stote

~ ~

TEMPERATURE

T

CRITICAL CURVE Hc(T) OF A METAL SEPARATING

ITS NORMAL AND SUPERCONDUCTING STATES.

Figure VIII

Historically, this early work of Meissner and Ochsenfeld opened the way to a correct thermodynamic description of a pure conductor. The state, with the flux excluded, was shown to be a thermodynamically stable state. From 1933 to 1950 the understanding of superconductors was based on thermodynamic and phenomenological theories.

13

I will briefly mention some of these theories. Subsequent lectures will discuss them in detail. A useful picture of the phase transition to the superconducting state was provided by a "twofluid" model theory developed by Gorter and Casimir in 1934. This theory assumed that below the transition temperature electrons in the superconductor were divided into two distinct groups. A fraction were assumed to remain "normal" while the remainder "condensed" into a superconducting aggregate. In this theory a superconductor is regarded as consisting of these two interpenetrating electronic fluids, the normal electrons and the superelectrons. The superelectrons are assumed to possess greater order than the normal electrons and the degree of overall order is related to the density of superconducting electrons. The predicted results of this theory are in good qualitative agreement with experiment. This theoretical model does provide a useful picture, which when combined with the London theory, yields very good results. Its limitations become apparent, principally, in situations in which size and surface effects are important.

14

.. .. ~ o

lloot--t--+----It---t----+--+---+--4

700~-~--~1,--~~--~-_+--+--~-~

~ ~~--~---+-\ u

x·

~~-~-_4--1~+_--H--'~-+--~---~~

100

Critical Field-Temperature Curves for a Number of Superconductors

Figure IX

E. A. EDELSACK

In 1935 two German physicists, Fritz and Heinz London. postulated an additional equation to Maxwell's equations as a basis for developing a phenomenological theory of superconductivity. Their theory was very successful in describing the macroscopic properties of superconductors. For example, it predicted that the magnetic field does not abruptly disappear at the surface of a superconductor, but falls off exponentially with distance into the metal with a characteristic length ~lled the penetration depth" • This distance is of the order of 10- cm. This predicted magnetic field penetration was confirmed experimentally in 1940. The London phenomenological theory also has its limitations. but it nevertheless provided a useful approximate description of the


MEISSNER EFFECT IN A SUPERCONDUCTING SPHERE COOLED IN A CONSTANT APPLIED MAGNETIC FIELD: ON PASSING BELOW THE TRANSITION TEMPERATURE THE LINES OF MAGNETIC INDUCTION ARE EJECTED FROM THE SPHERE.

Figure X

15

electromagnetic properties of a superconductor at low frequencies. Following the Londons I theoretical explanation of infinite conductivity and the Meissner effect, Fritz London wrote two magnificent volumes entitled, Superfluids. In these volumes he expanded on his earlier ideas of superconductivity as a quantum phenomenon. He suggested that superconductivity is " a quantum structure on a macroscopic scale" that requires "a kind of solidification or condensation of the average momentum distribution of the electrons." London made the rather startling prediction that the magnetic flux threading a superconducting ring is not continuous but quantized in units of hc/e. It was not until 1961 that this prediction was verified experimentally by Deaver and Fairbank at Stanford University, and independently by Doll and Nabauer in Germany. They found that the flux u~it is hc/2e rather than hc/e. It is of the order of 2xlO- 7 gauss/cm. The reason for this 2 will become evident in subsequent lectures on the BCS theory. It was Fritz London who first suggested that the super conducting state has the characteristics of a highly ordered single quantum state which extends over the whole volume of the superconductor. Thus the ideas of long range order and superconducting electrons acting as coherent waves were first introduced. Fritz and Heinz London obtained a crude, but important understanding of the wave

16 E. A. EDELSACK

functions of superconducting electrons and their energy spectrum. This early work of the Londons' was prophetic of the future development of the theory of superconductivity.

The period of the early 1950's culminated with the appearance of the Ginzburg-Landau theory. Two Russian theoreticians, Vitali Ginzburg and Lev Landau generalized the London theory by introducing ideas from quantum mechanics in the form of a complex order parameter with amplitude and phase to describe the properties of super conducting electrons. Details of this theory and the earlier London theory will be discussed in subsequent lectures.

During the period of the late 1940's and early 50's a great deal of research led to the discovery of many superconducting elements, alloys, and compounds. A series of experiments, which had significant impact on the development of the theory of superconductivity, involved measurements of the critical temperatures of various isotopic mixtures of mercury_ In 1950 researchers discovered that for a given element the critical temperature was inversely proportional to the square root of the isotopic mass

M is the average isotopic mass and a is a number with value of about 1/2

The implications of the dependence of the transition temperature on the isotopic mass are most Significant for subsequent theoretical developments. A relation between the onset of superconductivity, which is an electronic process, and the isotopic mass, which affects only the lattice of a material, seems to imply that superconductivity is due to a strong interaction between electrons and the lattice. However, it took a number of years until the subtle nature of the electron-lattice interaction was recognized and a valid microscopic theory developed. It is historically interesting to note that the possibility that the electronlattice interaction is at the heart of superconductivity was pointed out by Frohlich in 1950, just before the experimentally observed isotope effect was made known.

The next significant development in the field concerned the idea of an energy gap in the superconducting state. Significant experimental evidence began to accumulate in the early 1950's that superconductivity was characterized by a forbidden range of energies for the electrons (i. e. a gap in the energy states available to the conduction electrons in the superconductor). The distribution of energies of free electrons in a metal is such that at any instant of time each electron has a discrete and unique energy, commonly referred to as its energy state. The number of energy states available for occupation by free electrons of the


lattice is a function of their energy. Figure XI shows the density of energy states of a superconductor as a function of energy. This curve is usually called a density of states curve because its ordinate represent the number of energy states per unit energy interval. The energy gap is a maximum at absolute zero, and disappears at the critical temperature. The gap is symmetrical about a specific energy called the Fermi energy.

17

The Fermi energy is usually defined as that maximum energy level up to which the energy states are completely occupied at absolute zero temperature. The gap is usually about 10-3 to 10-4 electron volts in width at absolute zero. The half width of the energy gap is denoted by the symbol ~. This concept of the energy gap will be expanded and discussed in detail during subsequent lectures. It is a very important concept in the theory of superconductivity since it results in such phenomena as the exponential temperature variation of the superconducting electronic specific heat, the occurrence of a threshold for electron tunneling in superconductors, and perhaps most significant, the occurrence of persistent currents in superconductors. All elemental superconductors and most alloys exhibit a well-defined energy gap. There are a small class of superconductors which do not possess an energy gap. These so-called gapless superconductors are a very special non-typical case.

II en ~ ~ w ~ I ... a: I ;! w I en ...

z I ~

I c:» ~ a: c:» I I: w a: z w I II W Z

W I II ~ t: I I

Z I II

0 =» I II Z I I

a: I I, r I I

E,

ENERGY The density of energy states of a superconductor as a function of energy. The gap is a maximum at absolute zero and disappears at the critical temperature; its half-width is denoted ~.

Figure XI

18 E. A. EDELSACK

The next major advance in superconductivity occurred in 1957 when three physicists, John Bardeen, Leon Cooper and Robert Schrieffer developed a microscopic theory of superconductivity. This BCS theory has been most successful in providing theoretical results that are in remarkably good agreement with experimental data. The theory postulates that in the super conducting state, electrons of equal and opposite momenta and spins are weakly bound together. Since the electrons in a metal interact very strongly through their Coulomb repulsion, any binding or attractive force between them lTIUst take place indirectly, such as through the positive ions of the lattice of the material. For a material to be superconducting, this lattice-assisted attractive force must exceed the Coulomb repulsive force. This attractive force tends to couple together in "bound pairs" electrons of equal and opposite momenta and spins. This binding force is extremely weak, however, as revealed by the fact that it is disrupted and superconductivity destroyed by thermal vibrations at temperatures just a few degrees above absolute zero. This mutual coupling between electrons into pairs is known as a phonon interaction. The distance at which this electron-phonon interaction takes place depends on the regularity of the lattice structure. In the case of very regular structures, free of imperfections, attractive interactions can occur between electron pairs over distances as great as 10-4 cm. This distance is known as the coherence length and is given the symbol ~. This attractive interaction occurs among many electron pairs in the lattice. In order that the interaction between all these electron pairs should not interfere destructively, the velocity of the centers of mass of all the pairs must be the same. In the case of zero current, the centers of mass of the pairs will remain stationary. If a persistent current flows, the centers of mass will all have a common velocity. The involvement of the lattice in the attractive interaction explains the seemingly peculiar fact that superconductivity has never been observed in metals that are usually considered to be the best conductors, such as copper and silver, whereas it is a common phenomenon among the poorer conductors such as lead and tin.

The high conductivity of copper and silver is a consequence of the comparatively weak interactions of the electrons with the lattice of these metals. This reduces the scattering of single electrons that impairs conductivity in the normal or nonsuperconducting state but it also reduces the attractive interaction of electrons that leads to superconductivity. As you will see from subsequent lectures, the BCS theory predicts many of the properties of superconductors from first principles.

Up to now we have been reviewing the properties characteristic of all superconductors. In the case of the Meissner effect, there was complete cancellation within the superconductor of the


magnetic flux due to an externally applied magnetic field. If you recall the relation that

B = H+ 477 M where B is the magnetic flux in a material, M the magnetization, and H is the applied magnetic field.

For the cas e of the super conductor B, the magnetic flux equals zero, thus

The physical origin of the vanishing magnetic flux inside the superconductor is the presence of super conducting surface currents which create a magnetization M in the specimen. Figure XII shows the magnetization curve for an ideal superconductor. The relationship is linear until the applied magnetic field exceeds the critical field, then the surface supercurrents and associated magnetization vanish. For a number of years it had been observed that some types of superconductors, particularly alloys and impure metals, did not possess this ideal type of magnetization curve. This anomalous behavior was usually

-M

~ ________ ~oooo-oooo-__ H He

MAGNETIZATION CURVE OF A TYPE I SUPERCONDUCTOR

Figure XII

20 E. A. EDELSACK

ascribed to impurity effects. not considered to be of great importance. In 1957 the Russian theoretician. Abrikosov. showed that there was another class of superconductors with magnetic properties quite different from those previously measured. Today we know that these apparent anomalous magnetic properties of certain superconductors are inherent features of another class of superconductors now known as "Type Ill' to distinguish them from the earlier discovered elemental or Type I materials. This discovery of Type II superconductors. coupled with the development of the BCS theory. issued in the golden age of superconductivity. Most present technological exploitations of superconductivity depend on the properties of Type II superconductors. Figure XIII shows the magnetization curve of a Type II superconductor. Up to Hcl the magnetic field is completely excluded. but between Hcl and Hc2 the magnetic field penetrates the superconductor and increases until the material becomes completely normal at H '"). Materials which exhibit this kind of magnetization curvec~re called Type II superconductors. When a Type II superconductor is between H land H 2 it is said to be in the mixed state. Figure XIV shows cthe maggetic field penetration of a superconductor in the mixed state. Where the magnetic field penetrates the superconductor there exists islands or vortices of normal material surrounded by superconducting

-M

/' " I

/ : TYPE I I I

----------;---~----------~~--- H

MAGNETIZATION CURVE OF A TYPE n SUPERCONDUCTOR

Figure XIII


Superconductor mixed stote

Magnetic field penetration of a superconducting metal in the mixed state.

Figure XIV

material. These normal regions are sometimes called Abrikosov vortices. Since many of the Type II superconductors are mechanically hard, they are referred to as "hard" superconductors in contrast to Type I materials (such as lead and tin) which are often called "soft" superconductors.

21

In 1961 a group at Bell Telephone Laboratory discovered a Type II superconductor in the form of an intermetallic compound of niobium and tin. This material was capable of calrying a very high current density of almost 100,000 amperes/cm in magnetic fields up to 90 kilogauss. This important discovery marked the beginning of the development of superconducting magnets and other superconducting power devices. Figure XV shows the critical magnetic field-temperature curves for a number of Type II materials commonly in use today.

Up to now we have been discussing those properties of superconductors which are commonly referred to as macroscopic properties - such as zero resistance and the Meissner effect. Now I wish to mention those properties of superconductors which are often referred to as their quantum mechanical or microscopic properties. Theoretical and experimental research on these

22 E. A. EDELSACK

Figure XV

quantum properties have corne to the forefront in the last ten years, and today represent one of the most exciting areas of research and technology in the entire field of superconductivity. An example of these microscopic quantum properties is the phenomenon of electron tunneling in superconductors.

Tunneling is a quantum mechanical process arising from the wave nature of the electron. It is manifest by the transport of electrons through spaces that are forbidden by classical physics either because of a potential barrier or an energy gap. The tunneling of bound electron pairs between superconductors separated by a thin insulating barrier was first predicted in 1962 by a young British graduate student named Brian Josephson. (See Figure XVI)


This tunneling of electron pairs is now called Josephson tunneling or the dc Josephson effect. The type of sandwich structure of two superconductors separated by a thin insulating layer, 10 to 20 angstroms thick, usually the oxide of a material, is called a Josephson junction. As we shall see in subsequent lectures, these Josephson junctions can be built in many configurations and may be used as ultrasensitive detectors of minute magnetic fields as well as millimeter and infrared radiation. Josephson tunneling is but one of several types of tunneling that can occur. For example, single particle electron tunneling can occur between a normal metal and a superconductor. This type of tunneling was first investigated in 1960 by Giaever of the General Electric Research Laboratory. Also, single particle electron tunneling can occur between like or dissimilar superconductors.

The first experimental verification of the dc Josephson tunneling effect was reported in 1963 by Rowell of Bell Telephone Laboratory. The current that flows in a Josephson junction has a critical current density which is characteristic of the junction material and geometry. If the Josephson tunneling current density exceeds a critical value, an interesting and important phenomenon is observed. A voltage V appears across the junction and electromagnetic energy is emitted from the junction with a frequency V such that

hV = 2eV

SUPERCONDUCTOR

h = Planck's constant e = electronic charge

/ / SUPERCONDUCTOR

o.v.Q-----+-,-------- --,--->~ ELECTRON PAIR

.... "'OXIDE LAYER

Figure XVI

24 E. A. EDELSACK

(See Figure XVII). Since the voltage is usually in the range from a few microvolts to several millivolts, the emitted radiation is in the microwave and far infrared regions of the electromagnetic spectrum. In 1963, S. Shapiro experimentally confirmed these theoretical predictions of the ac Josephson effect.

From the discovery of superconductivity by Onnes in 1911 to the discovery of tunneling of electrons in superconductors by Josephson in 1962 - some fifty years of history of superconductivity condensed to about fifty minutes. In this process of condensation, of necessity I have taken the liberty of omitting a number of important developments which did not appear essential in the description of the field. Some of the developments discussed, reflect my bias toward practical applications. I hope that you may now have some idea of the interesting and diverse menu in superconductivity which lies ahead in the twenty sessions of this school over the next two weeks.

RADIATION DUE TO a.c. JOSEPHSON "-

CURRENT -""",

SUPER· CONDUCTOR

SUPER· CONDUCTOR

JOSEPHSON JUNCTION

BATTERY

VOLTAGE

d.c. JOSEPHSON

CURRENT

Under certain conditions (if the current source forces the current to exceed the maximum d.c. Josephson current) a voltage appears across the jilllction; at the same time, electromagnetic radiation is emitted from the insulating gap, demonstrating the presence in the gap of an alternating (a.c. Josephson) current.

Figure XVII

PHENOMENOLOGICAL THEORIES OF SUPERCONDUCTIVITY

W. D. Gregory

Department of Physics Georgetown University Washington, D.C. 20007

In the previous lecture you became acquainted with some o~ the ~acts and history o~ superconductivity. We will now begin a somewhat intensive study o~ these ~acts and the theories that have successfully explained them.

As you undoubtedly know from what has transpired so far, there is a microscopic theo~ of superconductivity, developed by Bardeen, Cooper and Schrieffer~l) (BCS), that explains essentially all of the phenomena. It would seem most logical to begin our study by learning about this theory, which is undoubtedly superior to anything else we might look at. Because of the complexity of this theory, it is clear that many of you would experience great difficulty if we were to proceed at such a rapid pace. And by avoiding discussions of some of the more elementary (but useful) earlier theories, you would miss the opportunity to develop some simple "back of the envelope" methods for making calculations useful to your future reading and research.

So, in this lecture we will attempt to describe superconductivity* at perhaps the most useful level - i.e., using all of the elementary physics and first order approximations that can be brought to bear on the subject. We will require nothing more complicated than basic physics and some calculus, but as you will soon see, a conglomerate theory, often good to 5%, will emerge. In addition, because of the variety of backgrounds of those present, the Directors of this course hope that the following material will also serve as a preparation for the remaining lectures for those who have been away from formal studies for some time.

* The basic definitions of terms associated with superconductivity that were developed in the previous lecture by E. A. Edelsack will ~ be repeated here. The reader may find it necessary to refer to the previous lecture occasionally.

2S

26 W. D. GREGORY

I. ELECTROMAGNETISM

A. Basic Facts

Since the most obvious changes that occur in the properties of a metal as it enters the superconducting state involve the electrical and magnetic properties, we will begin with a short review of electromagnetism. In particular, we will examine those aspects of electromagnetic theory that are treated differently when dealing with a superconductor.

Our starting point will be the Maxwell equations:

'V .f)= [41T ]p (la) -+

(lb) 'V B = 0 -+ _ [1.] aB (lc) 'V x E = c at -+ [41T] j + [1.] aD (ld)

'V x H = c c at

In equations (la) to (ld), the gaussian units form of the equations is obtained by keeping the expressions in brackets ([]) and the MKS units expressions are obtained by omitting the bracketed expressions. A good deal of the work on superconductivity has been expressed in the gaussian form, so we will use gaussian units in this lecture, unless stated otherwise. The development of the Maxwell equations, the treatment of the fields at boundaries between media, and other tools we will need in this lecture form the early part of most courses on E & M, so we will not discuss these things here but rather refer th9s~ who feel the need to some standard treatments of the subject. l2 )

B. Electromagnetic Properties of Superconductors

The Maxwell equations have general applicability; they do not apply just to normal metals. Superconductivity comes in when we specify what the B, li, 15, and E fields really are.(3) As with any other material, Maxwell's equations do not tell us·· this, but we must add additional specifications, usually called the constitutive equations. For example, for normal metals we find that the relation J = a~where a is the conductivity, reasonably well describes the connection between the current density ~ and electric field E, at least at low frequencies and high (non-cryogenic) temperatures. We adopt this constitutive relation for those cases where it describes the response of a normal metal to a field. Our criterion for accepting the relation is to demand that the consequences (in this case, Ohm's Law) are actually measured in experiments.

Similarly, we must search for constitutive relations for the

PHENOMENOLOGICAL THEORIES OF SUPERCONDUCTIVITY 27

superconductor that describe the experimental results. At this point, we have two such results from the previous lecture to compare to (1) the-zero r~sistance phenomenon discovered by Onnes, and (2) the lack of a B field in the bulk of a superconductor, discovered by ,Meissner and Ochsenfeld. These constitutive relations can be developed in a number of wafs. For example, for the normal metal, one can derive this relation from the Boltzman transport equation (somewhat from first principles) or a phenomenological approach can be used. In the spirit of this lecture, we will leave the "first principles" approach to the theorists (specificall¥ to Wes Mathews) and try a few simpler calculations for a superconductor.

C. Extension of the Normal Metal Constitutive Relations

A commonly used approach is to interpret the constitutive relations for a normal metal so that they fit a superconductor. For a normal material we have:

t = of .., .., D = £E

l!" = it + 4d~ .., .., .., .., M = XH or B = ~H

The first electric equation has beea discussed above. To make it fit a superconductor, we expect that J can remain finite..,even though no resistance is measured (0 .., ~). ~his happens if E, and hence the voltage drop in the metal (IE • dt) goes to zero. Although this fits some of the facts,we will see somewhat later that the correct ~urr~nt dens~ty constitutive relation is somewhat different (i.e., J ~ A where A is the vector potential).

.., The magnetic equations express the difference between the B

and it fields resulting from the "magnetizeability" it of the material. In normal materials this can be thought of as a magnetic dipole moment per volume due to (i) circulating currents in the material and (ii) alignment of intrinsic magnetic moments associated with elementary particles in the atomic structure (including, of course, the intrinsic moments of the conduction electrons also). In this picture, to obtain the Meissner effect, we must assume

X = - ~ for the material at all times while in the superconducting

state (perfect diamagnetism). We shall call this treatment of the B and K fields the magnetization picture.

Actually, the largest contribution to it in many superconduc-tors (at least at low fields and on a macroscopic or non-atomic scale) is the circulating current portion. Consequently, one can conceive of a second waf of calculating the magnetic fields at low frequencies in those superconductors where the complete metal is superconducting. That is, attribute the field totally to the currents

28 w. D. GREGORY

in+the M~ell equation (ld). Then, one makes no distinction between a B and H field, and in fact, defines only one magnetic field. ( 3) This treatment of the magnetic fields in a superconductor we call the current picture.

Both methods of treating the magnetic field have valid uses in describing superconductors. We will consider the better known examples of each.

D. The Magnetization Picture

Most of the data on the behavior of superconductors in magnetic fields is presented using this picture. An example of this for a very long cylinder of a type I superconductor is shown in Figure 1.

We note a number of facts one can obtain from the magnetization measurements. First, the Meissner effect (B=O in the superconducting state) obviously results if we calculate B = H + 4wM for H ~ Hc. Secondly, we note that a field greater than Hc destroys the "perfect diamagnetism" and we find essentially ~ magnetic moment, typical of the normal state.

Finally, and most important, if the field is reduced below Hc again, one finds the magnetization follows the same path as in an increasing field (with some reservations necessary regarding a slight hysteresis seen in very ideal single crystal samples.to be discussed later). In no event does the magnetization follow the dashed curve in Figure 1 for the simple case we are considering. which is what one would expect if a superconductor was merely a perfect conductor. To see this, we observe that a perfect conductor would develop essentially zero e.m.f. due to a time changing field, i.e., e: -+ O. But by rewriting equation (lc) we find e: = _ .2. (fs . dS)'

at so this implies that the magnetic flux remains constant in time, and to do this one m~ readily calculate that the magnetization must follow the dashed line.

Magnetization data provide us with a surprising amount of information about Type I superconductors. To summarize, magnetization curves show that the magnetic flux is alwayS expelled from these superconductors (B( t )=0, Meissner effect); that. this effect is not connected to the "perfect conductivity" phenomenon but is a newand separate property; and finally we observe the existence of a critical field, Hc ' capable of destroying superconductivity and returning the metal to the normal state.

All of this was obtained assuming "ideal" samples and experimental conditions. The magnetization picture can be pushed a bit


-41iM

R

-H c

29

H

Fig. 1. Top-magnetization vs ~ield ~or a long cylinder o~ Type I superconductor (solid line). The magnetization o~ a "per~ect II conductor is the same in increasing field but ~ollows the dashed line in decreasing ~ield. Bottom - The change in electrical resistance with magnetic field is shown for comparison.

30 W. D. GREGORY

further to explain some "non-ideal" experiments.

For example, let us suppose the sample is not a "long" cylinder, but instead that the cylinder is "short". "'"T"Short" and "long" here will be taken to mean with respect to the diameter o~ the cylinder). It is easy to see, using the Maxwell equations (la) to (ld), that a buildup o~ magnetization on the ends o~ the specimen will contribute to the local magnetic ~ield in the interior o~ the sample, producing a di~~erent magnetization curve ~rom that shown in Figure 1. We assume that we can treat the contribution o~ the sample to the total field with magnetization only, and that there is no current injected into the sample externally, so that equation (ld) becomes

V x 1 = 411 "J = 0 (3) c

Since the curl o~ the gradient of any scalar function is zero, equation ( 3) tells us we can represent H as

-+ -+ H = - Vrp + K (4)

m

where ¢ is a scalar function, K is a constant vector and we choose the min~s sign on V¢m as our convention.

The implication of equation (4) is that we can obtain H from a scalar potential, just as in electrostatics, provided no currents are introduced into the problem. The solution for rpm is obtained by putting equation (4) into equation (lb).

so

or

-+ -+-+ V • B = V • (H + 411M) = 0

-+ -V • Vrp = -411V·M m

V2¢ = 411V'M m

We note that equation (5) is just Poisson's equation for a scalar potential. We obtain-+the same equations for the electrostatic problem (i.e., when aB = 0 in equation (lc». In the elec

at trostatic case. V'N -+-p sO-V'M has the effect of volUI:le "charge" density.

We can now obtain an explicit expression for rpm without expending any effort, since we have reduced the problem to one that is well known in electrostatics. A quick glance at the standard E & M texts referenced earlier(2) will show that a unique solution for ¢m is given bJY

V'· M(~') d3Jt' rpm = - -'-----:=-'--.=......:.::~ ( 6 ) I~ - ~'I


where x is the point at which the potential is to be measured and t' the point at which the contribution to the field H = - V~m from M is to be calculated. (Our notation here will be that V' operates

+ + ) only on x', V only on x.

To proceed further~ we must make some preliminary assumption about the variation of M throughout the sample vol~e~ If we assume the sample is uniformly magnetized, then V I • M(x I) = 0 ~ cept at the sample surfaces. If the magnetization is in the same direction as the externally applied field (non-tensor susceptibility), then a little algebra will show that equation (6) reduces to an integral over the s~rtace of t~e specimen. That is, V'·M(~') has the property that V I ·M(x' )=0 if x' is not at a surface and if one integrates through a surface, one obtains

J V' ·M(:t' )d3:t' JM dS' = - n n It - t'l .:. + Ix - x' I

where dS is an area element perpendicular to the surface. equationn(6) becomes J r:i(~') • dS'

~ = + m I~ _ i'l

Sam~le surI"ace

Now

(8 )

Since M' is constant in magnitude up to the surface, equation (8) can be rewritten J Cos a' dS'

~ = M' m l;t - ;t, I

SamI11e surTace

where a' is the angle between dS' and M' at each point on the sample surface. Finally, we can obtain the contribution to the field due to the surface magnetization

H = - V ~ = + M' J (;t - t') Cos a I dS I m l;t _ t'I 3 (10)

SamI1le surI'ace

We will put aside more complicated cases for the moment and consider simpler sample shapes, such as a "not-too-long-cylinder". We see that equation (10) may be written as

H '" -M411n (11)

where we express the result of the integration in equation (10) as -411n. Here, n depends only on the geometry. The number n is often called a demagnetizing coefficient. This treatment of the magnetic fields interior to a magnetizable material is not unique to superconductors. You will find further interesting discussions of de-

32 W. D. GREGORY

magnetizing effects tn works on ferromagnetism (see for example, the text by Bozorth( ». The demagnetizing tens9S)has been calculated for general ellipsoidal samples by Stoner. {

We now must observe field as defined above. "aligned" by an external field is then

that ~ is but one contribution to the H Clearly, the magnetization is being field, which we shall call He' The net H

~ ~ ~

H = H + H (12) e m The last result may be a bit ~urpr1s1ng to those who ~e ac

customed to thinking of the field H as the external fie~d He only. One might be tempted to lump ~ with the~agnetization M since it is a contribution to the "total" field, B, resulting from the some effect due to the sample. However, a careful re-reading of our deve10p~ent ~11 show that the field we ca1cula~ed is truly an "if" fie1d.l3lThe "B" field is related to the total,. "H: fie1~ and the magnetization Mby~our original ~efinition, B = H + 4nM, so we can, if we wish, write B in terms of H and He only:

~ ~

B = -(l-n) H + He n

(13)

Using equation (13), we can now see how a short (non-ideal) sample will have a magnetizati~n curve different from Figure 1. In the superconducting state, B = 0 so from equation (13) we ob

~

tain H = He 1-n (14)

This tells us that the total H field will exceed the external value (plotted on the abcissa on Figure 1) by a factor of l/(l-n). Since for the ideal (long) sample (where n=O) we find superconductivity

..... .....1 destroyed whenIHel~IHq, we might naturally ask what happens in a shorter sample when it I = IHcl (or /Hel = (l-n) IHcl>?

On a phenomenological level, we can convince ourselves that the sample cannot go into the normal state in any simple fashion at this point. The argument for this is based on the fact that if we imagine any simple normal-superconductor (N-S) boundary within the sample, applications of the boundary condit.!ons ..... would show that there are regions of the s~er~onductor where IHI>IHcl and regions or the normal material where !H I < IHc I.Tbis argument is developed further, with illustrations, by Sb9~nberg.(6) An alternate possibility was proposed by MendelssOhn,lrJi.e., that the N-S transition takes place through an intermediate state having a complex distribution of normal and superconductiI)g regions (often called a "Mendelssohn sponge"). Pierles and F. Londonl~) further proposed a model for the behavior of the magnetization in the intermediate regi~n. Simply stated, we expect the intermediate region to start whenIHI=IHql= (l-n)~el and certainly to stop (all material is normal) whenIHel=IHcl. In be-


-4f1M Intermediate State t--- -----t

33

H

Fig. 2. Magnetization of a short cylinder of T,ype I superconductor with demagnetizing coefficient n, showing the intermediate region.

-+ B

Fig. 3. Sketch o! the region near a metallic surface with a tangential a.c. B field showing the relationship between the mean-free path, i, and the skin depth, ~, for use in the "ineffectiveness concept" calculation of the anomalous skin depth.

34 W. D. GREGORY

tween, a simple linear mixture by volume of N and S regions i~ assumed, so that the magnetization goes to zero linearly as IHel goes from (l-n)lttcl to I~cl (see Figure 2).

In a later lecture ("Superconductivity in Very Pure Metals"), we shall see that the Pierles-London hypothesis works rather well. Data obtained on 99.9999% pure Ga single-crystal cylinders exhibit magnetization curves very much like Figure 2. This is a rather surprising achievement for a phenomenological theory, for two reasons. First, there is no reason to believe that the fields interior to a superconductor would be calculable with the demagnetizing arguments. Secondly, the assumption of a sponge-like mixture of N and S material has not been established in detail here at all. We shall see later, in discussing the Ginzberg-Landau ordering theory, that the requirement for a distribution of N and S material can be established with energy arguments. In fact, a calculation of the actual distribution can be made for some simpler geometries. These more detailed arguments will only serve to justifY the simpler assumptions above.

E. Current Density Picture; The Londons' Equations

A little while ago we made the point that, in some cases, we could describe superconductors equally well by lumping the effects generating magnetic fields within the sample into a contribution to the current density j in equation (ld), as compared to the use of a magnetization M in the section above. We can now see just when such an approach would be valid and useful. If we consider a sample of geometry such that the surfaces perpendicular to Mare negligible and/or quite far away from the region under considert-) tion, then H = He only, and we can safely drop the distinction, 3 between H and B by setting M = 0 and assuming all contributions to the field are lumped into j. This is a particularly useful picture, then, for a long cylinder.

In this picture, we can do a phenomenological calculation of just what j will be for a superconductor, and with one simplifYing assumption thrown in for good measure, we can get the Meissner effect out of our calculation. We start by assuming we have a long cylindrical superconductor in a 10ngitudi~I)B (or H) field (does not matter what we call it now - remember?l3). In the phenomenological spirit, we will assume that the current density is that for a perfect (resistanceless) conductor and hope that we can fix it up a bit to account for the distinction between perfect conductivity and the Meissner effect. Under these assumptions, the electrons in the superconductor are constantly accelerated, with the force F given by


* dv = F = m dt qE

3S

In equation (15) we are assuming Newton's Laws of motion for the electrons with velocity v and charge q. As any good "solidstater" knows, we can use such a model only if we treat the mass of the electrons as an effective mass m* that takes into account the effects on the acceleration due to the fields produced within the sample, as well-+as th,;. "external" field, t. The current density, j, is given by .I = nqv, where n is the number of electrons per unit volume. U~ine equation (15), we calculate

or

-+ doT dv .I = - = nq-dt dt

.+ nq2 E .I = m~

(16)

This can be used to calculate the field by taking the time deriva-tive of equati~n (ld) ~ 2 -+

V x it = 41T .I = 4nnq E c m*c

If we now take the curl of the last expression we obtain

V x (V x R) = V (V • ff) _ '12 ~

= 4nnq2 V x E cm*

Using equation (lb)

and thus,

. -+ -+ V • H = 0 so V' H = 0

2-+ 4 2+ V H = ~

c2m*

This is the point wh~re we must add some a~sumption, since what we desire is fi, not H. The Londons',( 9) who are responsible for this development (or at least its end result), made the s~mplest assumption possible, i.e., that if one time integrates H,

-+ where C is

J~ dt = R(t) + ~(r) constant in time. Then equation (17) becomes

where 2 2 A = m*c

4nnq2

(18)

This equation for the behavior of the magnetic field in a su-

36 W. D. GREGORY

perconductor is known as the Londons' equation. The quantity A is, by dimensions, a lenflh. The significance of A, and, indeed, of the entire equation 18), can be determined by finding a solution for a simple case. Let us suppose our superconducting cylinder has a diameter D large compared to A. As we have now set up the problem (D»A) the "cylinder" is the equivalent of a plane surface. The symme~ry along the plane allows us to speculate that the variation of H, ~2H, takes place only normal to the plane, and we will call that direction the x direction. We now see that a solution to the London's equation (18) is

H(x) = H(o) e-X/ A (19) when H(O) is the applied field just outside and parallel to the surface.

The exponentially decaying solution for the field interior to a superconductor we have just obtained can be developed in a more general way and can be established as the unique solution. We see that the field well into the interior of a bulk sample is indeed zero at all times, so on a macroscopic scale the 1-1eissner effect is predicted by equation (19). However, we also see that we must alter our detailed thoughts about the Meissner effect to include a finite field within a distance A, called the penetration depth, from the sample surface.

F. Implications of the London's Theory: The Constitutive Relation for Superconductors and Flux Quantization

The London's equation derived above can be presented in a different fashion. If we take the curl of both sides of equation (16), we obtain: .

-+ 'V x J ... (20)

or -+

('V x J) B = 0

This is equivalent to equation (17) if we assumed the result of time integration of both sides gives:

-+ 411>.2 -+ ('V x J)(--) + B = ° (21) c

Since 'V'B=O, we can define ~ as B='V~where ~ is a vector potential and we obtain

2 'V x (j 411A + A) = 0

c

So, up to the gradient of a scalar -+ A (22)


Equation (22) can be viewed as the constitutive relation for a superconductor. Note that this relation is between ~ and 1 and not between j and E, as for a normal metal. A more elegant way of arriving at the Londons' equation and the Meissner effect is to assume this constitutive relation to begin with, accepting it as fact when the experimental data (Meissner effect with a penetration depth) are predicted. In so doing, we would avoid the questionable procedure of establishing the Meissner effect, which distinguishes a superconductor from a perfect conductor, by starting with the equations for a perfect conductor. This would have been more pleasing to theorists but perhaps not sufficiently intuitive for our purposes. We should also point out that the derivation we have given, assuming the force on the "zero resistance" electrons is qE, ignores a ~ontribution to this force from the magnetic field, .l -+ -+ llO)

v x B. c There is a more important reason for advocating the postulate

of equation (22) rather than our cruder derivation. Other properties of superconductivity can be derived from this assumption, besides the Meissner effect, thereby making it a more fundamental postulate, as the true constitutive relation should be.

If we rewrite equation (22) using j = ne~ and the definition of A from equation (18), we obtain

-+ n -+ V = - ~ A m*c (23)

This tells us two things. First, the canonical momentum in a magnetic field (the momentum used in the DeBroglie relation) is given by

-+ -+ n ~ p = m*v +.:.. A

c

so apparently in a superconductor, p=O.

(24)

Second, if we think of a su~erconducting ring with a normal metal interior, we expect that J p' dl. = fvxp· d1l ::f 0 in the interior (normal) portion since p::fO there. So, from equation (24) we can obtain for a path integration around the loop

S P . d! = S m*~ d ! + % Sit. d 1 Now the path integral of the canonical old quantum theory. That is,

fp'dl=Nh

momentum is quantized in the

where N = integer. Using Stokes theorem

38 W. D. GREGORY

JI·~=Svx!'4=Ji'4=. == magnetic flux

For a path deep in the superconductor (i.e. the ring is thicker than A everywhere~J~'dl = 0 because of the penetration depth (Meissner) effect. Thus

• = N h c = N • (26) q 0

where .0 is called the flux quantum.

Equation (26) tells us that flux should be quantized in multiply connected superconductors, if the superconductor is thicker than the penetration depth, A, everywhere. This prediction was ( verified almost at the same time ~~wo groups, Doll and NRbauer 11) in Germany and Deaver and Fairbank in the U. S. They found that the flux quantum $ is -2xlO-"( G-cm2 • This results only if you assume that q = 2e; £hat is, the effective charge is twice the electron charge, implying that the "super" electrons are actually electron pairs. This !act, coupled with the observation that the canoni-cal momentum p is zero for a superconductor, will be verified by the microscopic theory.

In subsequent lectures Professors Deaver and Webb will discuss more of the implications of flux quantization, particularly the devices that are possible using this phenomenon. Prof. Hamilton will discuss attempts to use the zero flux quantum (N=O in equation 26) to produce magnetic shielding. So, rather than extend these arguments at this time, we will move on to other topics. Before we do, we should note that when a great deal of flux is trapped in a ring, (N large), the ring behaves much like a perfect conductor as far as the net magnetization of the ring is concerned. (See Fig. 1). That is, the quantum steps are blurred in this case and the ring tries to keep the flux trapped inside at the last value it had when the material went superconducting. Flux trapping is well known to experimentalists, particularly those who study magnetization curves of alloys, where normal regions might occur at points of non-stoichiometry or damage in their samples. If you wish to understand the physics of tqe superconducting ring, ~~u can read an account in Shoenberg's textJ6) Also, Chandrasekhar I3 Jpoints out the problem of interpreting the persistent current in a superconducting ring as purely a "zero resistance" phenomenon, since it would be a required condition for quantized flux anyway. Professor Bloch will present some very interesting thoughts on this subject in a later lecture.

G. Non-Local Constitutive Relations and A.C. Effects

Up to this point we have treated a superconductor, and as a matter of fact, a normal metal, as if the constitutive relation is

... ... one existing between the current density J at a point x in the ma-terial and the field f (normal metal) or the vector potential t (superconductor) at the same point, t. Such relations are called local connections between currents and fields. If we took the approach discussed early in this lecture and attempted a microscopic derivation of these relations, in the case of'both normal and superconducting metals we would find many situations where the fields or potentials at a point were related to the currents at many other points in the material, a non-local situation.

In a normal metal, we can see how this might happen if we think about the significance of the definition of conductivity in equation (2). The conductivity is a measure of the drift velocity that might be imparted to the conducting charges by the electric field. By not assuming constant acceleration by the field, as in the "derivation" of the Londons' equation, we are assuming that something else limits the velocity these particles can attain. This "something else" is the collisions the particles undergo. Obviously, if these charges undergo many collisions per unit of time in the metal, they will do so over small distances, so that the limi t on the drift velocity (v = ~) will require only an averaging of this process over "small" dis~inces. Hence, the limiting parameter, a, will relate the driving force E and the drift velocity: (or current density j) locally.

To illustrate what happens when the collisions do not occur over "small" distances, and to specify better what we mean by "small" here, we will consider a simple example. This is the problem of calculating the depth of penetration of a.c. and r.f. magnetic fields into a metal. Assuming a local consitutive relation, one finds that the fields decay exponentially from the value at the surface of the metal with a characteristic distance

2 1 <5 = ( c ) -2 ( 21 )

21T\lwa

For this case a can be obtained from simple kinetic theory arguments and is given by

a = nq2t (28) m*

where t is the time between collisions.

Let us now consider the case where the t is large and the mean collision distance l is therefore also large. We will use Figure 3 for this(~\scussion. The argument follows one originally given by Pippard.l+~e asked what would change if t is large. Well, from Figure 3 we can see that certainly the number of carriers/volume (n) that would do their scattering within a distance <5 from the surface, where the fields are at all finite, would be quite small. An estimate of this would be that those carriers that travel at an

40 W. D. GREGORY

angle 6 greater than 6 = Tan-l (o/l) with respect to the surface would have a high probability of scattering outside the skin depth and would no longer by influenced by the field (for o/l«l, 6=0/l). That is, at distances from the surface much greater than 0 the fields are essentially zero. These carriers would then be ineffective for contributing to currents and one should alter the value of n in equation (28) to take this into account. If we assume the carriers have an isotropic velocity distribution without the applied field, then this means n will be cut down linearly by the factor 6'=0/l for small values of 6' and the new expression for o is 2 --

02 = c

2 7f llwao/lK

where K is a factor that describes the scale of this effect that would be worked out in a better derivation. Factoring out 0 from the right side of equation (29) we obtain

2 I o = (Vac ) 3" anomalous 27fllWK (30)

We have labeled 0 as "anomalous" in equation (30) to distinguish it from the "normal" value in equation (27). Note that the anomalous skin depth is independent of the conductivity, since by the simple kinetic theory model, aal and all is constant. (This is approximately true even when the simple kinetic theory arguments do not strictly apply.) Also, the depth of penetration is proportional to w-I / 3 instead of w-I / 2 as in the "normal" case. Remember that these results apply when oil is small, so that we expect this to occur at high frequencies (0 small) and low temperatures (not as much scattering, so l is large). Experiments on many metals at cryogenic temperatures indicate that the anomalous expression is correct at frequencies where the mean free path is indeed large compared to the skin depth.

The "Pippard ineffectiveness model" calculation we have just done illustrates the meaning of the terms "large" and "small" in the earlier discussions of local and non-local constitutive relations. l-le expect the local ("normal") relations to hold where the distance over which the fields vary (0) is larger than the distance over which the interactions (in this case, collisions) are effective (this distance is i). Conversely, we expect that we must somehow average the interactions when the fields vary over distances that are small compared to the interaction distance (o«l), since it is these interactions that are supposed to generate the variation of the fields in the first place. Again, it is easier to see how this occurs for a normal metal. so let us first extend the anomalous skin effect model for a normal metal before we look at the case of a superconductor.


To develop a general expression for the conductivity or, equivalently, the relation between j and E for the normal material, we would have to examine more closely the method for obtaining equation (28) for o. This is the result of a straigh~f~rward calCUlation where we obtain the distribution function f(r,v,t)for the carriers as a function of position, velocity, and time from the Boltzmann transport equation:

-+-(ll. + :t . tJ-+-f + ~ , tJ f) ( af) at r m :t = at collision (31)

You will recall that f has the significance that f(r,v,t)d3:t d3; is the number of carriers within a small velocit~ volume d3; of veloci~ ~ and a small real space volume d3; of r at time t. The force F in this case is, of course, given by the applied fields. To solve equation (31) in the normal skin depth limit, we approximate the collision term, (}.f) coll' which describes the effect of

the interactions, by ~f/T where T is the typical time it takes to alter f by collisions and ~f is the small resulting alteration. This set of approximations produces the conductivity of equation (28). If we must now do this calculation more rigorously, we are forced to abandon the approximation (~) ~~f/T and we must

at coll af find a solution using the full expression for (at) 11 in terms of the scattering cross section for collisions, 1.e:~

(~!) 11" = (dO. ~3v2od2)I;1-;21 (f2fi - f2fl) co 1S1on ) ' J ~ -+- -+- -+-, -+-, , , ' ( where vl ' f l , v2 ' f2 and vl ' v2 ' f l , f2 are these quantit1es be-

fore, after) collisions, respectively, for binary 9Q~tisions involving two particles 1,2. Reuter and Sondheimer ~.L;;'> have found a method of solving this integro-differential equation that, si~ly state~. s8f~ one must average fields over the collision distance so that J and E are related by

(32)

Here:; is the position at which j is evaluated and ;, is the position where E is evaluated. We will not attempt to prove this solution here, but we note that it does result in a non-local relation between j and E, as expected.

At about this point, we should get back to superconductivity, We can now use the results just obtained to "guess" a non-local constitutive relation for the superconductor, as well as where this relation might be necessary. Let's handle the latter point first.

42 W. D. GREGORY

In analogy to the normal metal, we should need a non-local relation where the variation of the fields (or vector potential, 1) takes place over distances large compared to the typical distance over which the interaction causing the 1, A relation varies. The first distance is just the penetration depth A. The second distance has not been defined or discussed up until now, so let us choose the name 1';. We then carry the analogy further and postulate the use of an average just like the expression tor a normal metal. i.e ••

C;--;') (-;--;') . t.. exp Cl~-~' I) I ~ ~ 14 t r-r'

~ ~ f3~ J(r) ex: d r'

Pippard(14) was the first to carry through the arguments leading to equation (33) and in subsequent microwave experiments with superconductors established the validity of this result for certain met~s.(15)The number I'; is found to be of the order of 10-5 to 10- cm. Those materials that require the use of the non-local relation are often called Pippard superconductors and the local type materials are called London superconductors, for obvious reasons.

Later in this lecture, while discussing the Ginzburg-Landau (GL) theory, we will be able to attach a bit more significance to the "interaction distance", 1';, which is often called a coherence length. Also, in later lectures we will see that the coherence length can be related to other properties calculated in the microscopic theory (BCS). The complete anomalous skin effect calculation for superconductors has been worked out by Mattis and Bardeen.(l6) Any further discussion of effects in time varying fields will be reserved for later discussion of these other theories. We should indicate here, however, that when a.c. and r.f. fields are applied to superconductors, the superconductor is no longer "resistanceless". This occurs partly because of a quantum absorption effect, partly because of a "shaking" of vortices (flux quanta) formed in certain types of superconductors (understood in the GL theory) and partly because of material preparation effects. Complete understanding of the a.c. loss effects is still in a state of flux (pun definitely intended).

II. THERMODYNAMICS

A. Basic Facts

We have seen so far that the superconducting state is a special condition of some metals that appears suddenly below well defined temperatures and magnetic fields. We also saw that this change of phase occurs reversibly for long, thin bulk samples with


the long axis parallel to an applied magnetic field. At least under those conditions, then, we might expect to apply reversible thermodynamics and treat the superconducting transition as a change of phase of a metal. We will try this prescription, after first collecting together some of the facts about thermodynamics that we will need.

We will presume that you are familiar Vi tl1 the usual way of stating the three "Laws" of thermodynamicsll7). The three laws specify two thermodynamic functions, the internal energy, U, and entropy, S, and place limits on them. These functions are so defined that they are the same every time all of the thermodynamic variables are all the same. As a result, every time we come back to the same state for a system, (same collection of variables T, P,V,M,H,E, etc.) we get the same value of U or S, so we don't have to measure all of these variables, but only U or S, to know where we are. Such variables are called state functions. (It is easy to see that same of the other thermodynamic variables are not necessarily state variables. For example, processes that involve both injection of heat and the performance of work by a system can pro-duce different changes of these quantities between the beginning and end of the process, depending on how you add the heat and extract the work. Because total energy is conserved, only the total (internal) energy is unique~ the state of the system for su~ process.)

Since state variables are so useful for specifying the condition of the system, several others have been defined and are in common use. They are combinations of S, U, and the other thermodynamic variables. In general, these other state variables are designed to measure a needed quantity or are arranged so that they are at extreme values (maximum or minimum) in equilibrium, (i.e., when we stop changing things). To save time, we will tabulate below many of these variables in Table I for a system where the important variables are ft ,ft, and T.

The final step necessary to make productive use of these quantities is to interconnect them. Nature provides us with one immediate connection of these varibles, the equation of state (PV= NkT for an ideal gas, M=NDH/T for a simple paramagnetic system, etc.). Other useful relations can be obtained by connecting the partial derivatives of the functions in Table I. This is a straightforward task, but we will save time by not derivin~ the results. The derivations can be found in standard texts. (17) Instead, let us talk about a simple diagram for remembering the results. It is called a Maxwell diagram (Figure 4). We will illustrate use of the diagram for a magnetic system where the required variables are M,R, and T. The following operations are possible: (1) For each thermodynamic function F, U, G, and H, that forms the

W. D. GREGORY

TABLE I

State functions and potentials for magnetic systems.

(M, H, T variables, differential of ";lork = -HdM)

Function Definition

U, internal. energy dU == dQ, - d\-l

S, entropy <is = ~ T

F(or A) free energy U - TS

G, Gibb's potential. F - HM

){, entbal.py U - HH

T

G

H

Fig. 4. Maxwell diagram connecting tbe tbermodynamic potentials for a magnetic system.

middle element of one side of the diagram, the thermodynamic variables that appear in the function form the two ~ of that side. We can obtain the derivatives of these functions with respect to these variables Qy sketching the following diagram (illustrated for (aG)

aT H

/<~ / I

/ @ P

S H

so

That is, from the function G move to the variable that is to be changed, T, and the result appears at the end of the arrow. The other variable (in this case, H) is obviously held constant. If you must move in the wrong direction along the arrow, then take the negative of the result (in this case, -8). Derivations using only the corners are also possible. For example, if you wish (aM) you can obtain this diagram:

aT H

so aM 3S) (aT') H = (aR' or

The key is to perform an identical operation on the opposite side of the diagram. Again, the sign of the result depends on the symmetry of the arrows. Obviously, here they are symmetric, but if we wanted (aH) we would find:

aT M

46 w. D. GREGORY

You can invent other rules for yourself. It is hard to find one that does not produce a legitimate relationship that can be established with exact calculations.

B. Phase Transitions: The Superconducting Phase Transition

We now have the tools required to study superconducting phase transitions. They key is to recognize that the function G is a minimum for a system in equilibrium, so if we move from the normal to superconducting state, it must be that the Gibbs potential is lower below the transition for the superconductor and lower above the transition for the normal metal, and the two values are equal at the transition. Thus we expect

Let us now assume that the total contribution to G by the magnetic field is due to the energy lost by the expulsion of magnetic field from the sample due to the Me1ssner effect. That is

GS(Hc,T) = GS(O,T) - SH.d M H 2 _c_ v 4n

For the last form of equation (35) we used the magnetization picture and assumed X = -1 throughout the sample volume, V.

4n

We m~ now use Figure 4 and equation (35) to obtain the difference in some of the thermodynamic functions at the transition. For example, from the figure we can deduce (lG) =-8

aT H

and l18 - -a

aT

(36)

H aH = - 4~ (aT c) V

If the transition is reversible, l1Q=Tl1S where 6Q = the jump in

47

heat required to raise the temperature b,y AT and thus C.~ is given by AT

2 H 2 dQ dT = l1C TV.JL (i- )

dT2 n d2H VT c

4n dT2

We have available measurements of Hc(T) for superconductors, which we can use to evaluate equations (36) and (37). To within about 5%, we find

H = H (1 _ (TIT )2) c 0 c (38)

for most elemental superconductors. dHc

Using equation (38), we note first that dT ~ 0 when T < Tc so A~O, implying that the normal state entropy is greater than the superconducting state entropy, 8N~SS' In turn, using the order-disorder interpretation for the entropy, this says the superconducting state is one of greater order than the normal state. Similarly, we see that since dHc < 0 as ~Tc' l1C is ~ zero,

dT i.e. , there is always a ~ in heat capacity at the superconducting transition. Note that at T='r c' Hc=O, so /'i3=0 at this point. At any other field (Hc>O, T<Tc) l1SiO and so there is an extra term in the heat capacity, i.e. extra heat is required, proportional to He' This extra heat required is called a latent heat of transition (similar to a heat of vaporization or melting in a solid-liquid-gas system). When one finds that the first discontinuity that occurs in the derivative of the Gibbs potential occurs at the mth derivative, this transition is labeled a~ an "mth order" transition in a scheme first developed by Ehrenfest. UB) We note that at T=Tc ' Hc =0, the supercon-

48 w. D. GREGORY

ducting transition is a 2nd order transition with a continuous entropy and no latent heat required to make the transitl.on. Wnen Hc>O. T <l.'c' the transition is 1st order. And remember, the superconducting state is one of greater order in all cases.

C. The Critical Field Curve

With a bit of hand waving it is possible to "derive" the form of the critical field curve, equation (38), from a crude gu~ss at the form of the heat capacity as a function of temperature.~19' Using this form for ~C(T), we simply integrate equation (37) twice, using some general arguments to evaluate the needed constants, to obtain llJT).

In the normal state, we can measure CN(T) and argue rather easily to the form we find:

The linear term is the form for the heat capacity of a system of Fermions at low temperature. These would be the electrons. The T3 term is the form for a three-dimensional Bose system. The lattice vibrations or phonons would satisfy for this part.

In the superconducting state, early measurements of CS(T) were interpreted as essentially cubic, i.e.,

C (T) ::: bT3 (40) s We now know this is a deviation from the truth, but let us proceed with this assumption and discuss the problems later. The net form for AC=CN-CS is then

~C = A T + (B_b)T3 (41)

From equation (37) this equals

TV i (H 2) = (AT + (B_b)T3) 8n dT2 c

Integrating equation (42) V d(H 2) -8 ~ = AT + (B-b)

n dT

where D is the constant of

once, we obtain

T3 + D 3

integration. As T-+O, cally that dHc

dT -+0 so D=O. Integrating again, we

VH 2 AT2 (B-b)

T4 E = --+ -+ C 2 12 &;T

(42)

we know empiri--get

(44)


2 V Again as T~O, Hc~o so E=Ho an and equation (44) reduces to

VH 2 c

'8ir

We must still obtain A and (B-b) to complete our derivation. Let us go back to the heat capacity expression, integrate this once to obtain the entropy and then use our knowledge of the entropy at T=T~ to evaluate one of these constant. Integrating equation (37) combined vith equation (41) we get

T ~T (~S) = AT + (B_b)T3

T3 = AT + (B-b) 3"

(remember ~S~O as T~O by the Third Law of Thermodynamics). the condition at T=Tc ' ~S=O, we then get

T 2 A = _ (B-b) _c_

3

(46)

Using

49

A final fact we equation (45) we get

have available is that H ~ as T~T , so using c c

or

AT 2 O=_c __

2

A = - 4 T 2

c

(A.l ) T 2

c

Putting equations (47) and (48) into equation (45) we get

or

H 2V V H2=_0_ 8n c 8n

H 2 c

T2 [- 2-

T 2 c

4 + T + 1] ~ c

(48)

50 W. D. GREGORY

So finally we obtain the critical field curve by taking the square root of equation (49).

Hc = Ho [1 - (T/Tc)2] (50)

This derivation of Hc(T) illustrates how one may apply thermodynamics to a superconductor. The form for H (T) agrees with experiment to about 5%. Note that one consequegce of this form is that there should be a "law of corresponding states" for superconductors. This means that one would expect the normalized critical field, h = Hc/Hq, to vary with the square of the normalized temperature, t = TfT~ for all superconductors, i.e.,

2 h = (l-t ) (51)

That most elemental superconductors do obey such a universal law to within about 5% is indeed remarkable, since we know that some of our assumptions, particularly those regarding CS(T), were inaccurate. To go further and correct those assumptions within a macroscopic model is not worthwhile. We shall see later in the lectures on BCS theory that the deviation from the law of corresponding states yields information about the details of the strength of the microscopic interaction causing superconductivity. Our derivation of this accurate a result must be regarded as a fortuitous cancellation of inaccuracies.

D. Non-Reversible Effects: The Surface Energy Problem

Now that we have seen what can be done with reve:r'sible thermodynamics, let us consider a simple example of an irreversible situation. The phenomenon we wish to consider is the effect of surface energies on phase transitions. These effects occur in many phase transitions, with very nearly the same general features. We will use the somewhat simpler liquid-gas transition to dev~lop the problem. The purpose of this example is to illustrate that the superconducting transition, which we have treated up until now as a reversible process, can in some instances exhibit irreversibility due to surface energy effects, particularly in ideal samples.

Let us consider a small drop of the liquid in equilibrium with its vapor while undergoing the liquid to gas phase transition. As with our earlier discussion of the bulk superconducting transition, the change from liquid to gas, and vice versa, takes place when the Gibbs potential of the two phases is equal. In general. the Gibbs potential must be minimum. For the small drop, we can readily see that this requires taking a surface energy into account, as well as


the Gibbs energy associated with the bulk of the material. This is the result of surface tension on the liquid.

In a liquid, the surface tension is proportional to the length along the surface, 6t, and perpendicular to it,

F = y6!

This tension is what keeps a needle floating on the liquid, in spite of the greater density of the needle. The surface tension coefficient y is defined by equation (52), i.e., the force along the surface perpendicular to the wire necessary to pull the wire of length At. For a spherical drop, t!is tension is parallel to the surface and if the drop increases or decreases in radius, there is a corresponding dimension change As along the tension force, which produces an energy change,

W = Fils = y6l11s (53)

= "ylla where lla is just the increment in surface area of the drop. This produces a contribution to the mechanical work due to the increase of surface area of the drop, so the net work done is given by

dW = PdV - yda

Putting this into the first law, we obtain the internal energy change as

dU = dQ dW

= dQ PdV + yda

Now that we know the internal energy, we can define the free energy and the Gibbs potential in analogy to the magnetic system where M, n, and T were the variables of interest. Here we use V for M, and P for n, noting that the work done by ~he~(p,V,T) system (for an infinite drop) is + PdV compared to -H'dM for the (M,n,T) system, so the sign of t~e work term for the (P,V,T) system is opposite that for the (~,H,T) system. This will merely change the sign of the arrow moving from P to V in the Maxwell diagram (Figure 4). where P replaces H and V replaces M.

With these considerations, we can see that the internal energy can be written as

4 3 2 U = -If r u + 4lf -yr • 3 co

The remainder of the problem proceeds quite analogous to the derivation of the critical field curve. The full details are

52 w. D. GREGORY

available in the text by HUang(20). With the approximations that the equation of state of the gas is the ideal gas law and that the density of the liquid is much greater than that of the gas, one obtains for the equilibrium P(T) relation:

P (T) = P (T) exp ( ~T 1) r 00 P .(.-- r

We can see from Figure ( 5 ) that the pressure at equilibrium deviates strongly from that for an infinite drop, especially for small drop sizes. In general, we have a higher vapor pressure at a given temperature, indicating that condensation is inhibited. Such an effect is called supercooling. If we had performed the calculation of surface energy effects on the gas, we would find an opposite phenomenon called superheating.

Using Figure (5·), we can see that the supercooling and superheating at phase transitions are not terribly stable processes. Consider a drop of size r to be the average size of a liquid drop in the system. Also ass~e that very few drops are bigger or smaller than this. Thus the actual pressure in the system is close to Pro(T). Now ask what happens to one of the few drops with size r>r. The pressure it must have to be in equilibrium is less than Pr fT), so if this drop attempts to control its own destiny and c~e into equilibrium, it will condense vapor on itself. Of course this is a self defeating process, so the drop just gets bigger and bigger. The opposite occurs with a drop of size r<r. This drop requires a pressure greater than Pro(T) to be in equ~librium, so it evaporates liquid in a self defeating fashion, until it vanishes. The moral of this story is that big drops get bigger and small drops get smaller, (that is, "the rich get richer, and the poor get poorer") and unless one is very careful not to do anything that would start a few drops condensing, a catastrophic condensation to liquid can be expected. This equilibrium condition would be correctly termed metastable. We would expect to see such supercooling only in very pure systems with uniform temperatures and pressures, i.e., when no center of nucleation of the liquid phase is possible.

Of course, this is not the only example of a metastable supercooling or superheating phenomenon. Those of you who have worked with freezing liquids may be aware of the fact that a very pure liquid. will remain liquid well below its "advertised" melting point, if left undisturbed. A striking example of this is found in pure Ga (99.9999% pure, to be exact~ It can be kept in a refrigerator freezer in liquid form for months without freezing. Of course, if you poke the liquid with a stick or a single crystal of gallium, you will nucleate the growth of the solid phase.


p

(T constant)

I I I

poo - ---t--- -- - - --I I

~--------~------------~~r ro Fig. 5. Equilibrium Pressure, Pr , vs drop radius, r, for a small drop of liquid in equilibrium with its vap9r. P was calculated taking surface energy effects into account. r The figure is taken from Huang, Ref. 20.

t -41TM

Fig. 6. Magnetization curve for a Type II superconductor. The "Type III", "surface superconductivity" behavior, producing a long tail on the magnetization curve that disappears at Hc ' is shown on an exaggerated scale to make it visible. 3

53

S4 W. D. GREGORY

The point of all this, of course, is that there is a direct analogy to a superconductor. In a superconductor, the formation of a boundary between the normal and the superconducting phase as the phase change occurs will also involve a surface energy. As a result, we can expect to see supercooling and superheating of the superconducting transition. This will depend on which phase dominates to begin with, and-therefore will be an irreversible effect. These effects are also seen onlY in pure, undamaged samples, since inhomogeneities are possible nucleation sites, just as in other phase transitions. In a later lecture we will show some data taken on pure single crystal specimens which exhibit these phenomena. The role of surface energies in superconductors is treated more exactlY in the Ginzberg-Laudau theory, which we will discuss in a short while.

III. Some Classical Models for Superconductivity

The previous portions of this lecture have laid the groundwork for discussion of some of the classical models for superconductivity that we have not yet discussed. In this segment we will take up two such types of theories - two fluid theories and ordering theories. Both of these types of theories use the fact that the superconducting state was found to have a lower entropy and hence greater order than the normal state.

A. Two Fluid Models

Many of the observations we have made so far indicate that in the superconducting state there are electrons that are extremelY well ordered. On the other hand, penetration depth data and thermodynamic (heat capacity) data indicate that perfect order is not achieved instantaneously at the superconducting transition, but simPlY starts to appear at Tc and continues until absolute zero is achieved. Observations such as these gave rise to the development of two fluid theories for superconductors, in which a normal, disordered phase was postulated to decrease starting at T=Tc ' with the simultaneous appearance of an ordered superconducting phase. The transfer from "normal" to "superconducting" electrons was complete onlY at T=O. If one does the bookkeeping in this fashion, it is possible to estimate the fraction of the conducting fluid that is "normal" or "superconducting" at every temperature by fitting calculated free energies to the thermodynamic data. In turn, the fraction of superconducting fluid vs normal fluid obtained from this postulate can be used to predict rather accuratelY the temperature dependence of some quantities, such as the penetration depth, which we have ignored so far.

We will consider here the method used by Gorter and Casimir(2l) to fit such a two fluid hypothesis to superconducting data. They


assumed that the Gibbs potential would be the sum of a "normal" and "superconducting" part, as follows

55

In equation (58). W is the fraction of electrbns that are superconducting. so that (l-W) is the traction that are normal. ~(T) are gs(T) are the specitic Gibbs potentials tor the normal ana superconducting parts of the superfluid at a temperature T. The Gibbs potential G can be obtained from heat capacity data. using manipulations discussed in the segment dealing with thermodynamics. The "normal" and "superconducting" parts ot G can be deduced from data obtained with and without an applied magnetic tield that exceeds the critical value.

The functions a(l-W) and b(W) then represent the fraction of the "normal" and "superconducting" fluid that contribute to the Gibbs potential. At first glance, one would expect that

and a(l-W) = 1 - W

b(W) = W

are the correct forms for a and b. However, one quickly finds that the thermodynamic data can be fit by this prescription at only one temperature using this definition of a and b. If one allows a more general form for a and b such as

a(l-W) = (1_W)1/2

b{W) = W

then the oata can be fit over a wide temperature range. In this case one finds that

W{T) = 1 _ (TIT )4 (60) c

The fact that the "normal" and "superconducting't po11tions ot the conducting fluid cannot be thought of as each contributing linearlY to the total Gibbs potential is a disturbing result. if we put too much faith in the model. A better wq to view this result is to calculate some other properties using the result. and accept or reject it entirely on the basis of its usefulness in predicting superconducting parameters.

The first comparison that we will make is to the Londons' result. specifically assuming d.c. fields. In this case, we found that the magnetic field drops off exponentially from its value at the surface of the sample with a characteristic penetration depth, A, given by

56 W. D. GREGORY

(61)

where n is the density of carriers per volume shielding the field from the interior. Since one would not expect the "normal" carriers to be moving for applied d.c. fields, we would interpret n to be ~S. the density of superconduct1ng carriers, given by the tvo fluid nypothes1s as

ns(T) = noW(T)

4 = no(l-t ) where t = T/Tc (62)

Here n(O) is the density of total carriers per volume. Putting this into equation (61) we find the temperature dependent penetration depth predicted by the two-fluid model would be

X(T) = X(O) 1/2 (63)

(l-t 4) where X(O) is the non-temperature dependent value of X that we originally derived (equatioj (61». Measurements of such as those of Schawlow and Devlin(22 show this temperature dependence is aCGurate to about 5%. A more precise temperature dependence can be obtained using the microscopic theory.

It is possible to extend this discussion of the temperature dependent penetration depth to non-zero frequencies. The assumptions are that the "superconducting" electrons obey the Londons' equations (and Maxwell's equations) and the "normal" electrons obey the Maxwell's equations. The proper current for Maxwell equation (ld) is assumed to be the sum of a normal and superconducting part

j = js + jN (64) ":t' .. ..

where JS comes from the Londons' treatment and IN=aE. The solution for the fields and currents for an infinite plane geometry of the type treated earlier for the d.c •• non-temperature dependent Londons' equation derivation is:

H = HOe -Kx . E = E e -Kx ; J Joe

-Ke = (65) , 0

where ! 1 1 [(m+l)2+ "2

K = i (m - 1) ] X 2

x4 1 m = (1 + 4 4) "2

0


and A is given by equation (61), 0 by equation (27). The distance x is the distance from the boundary and H , EO' and J 0 are the fields and currents evaluated at the b9~~. These relations have been used successfully by Pippard~]W to discuss the microwave behavior of superconductors and by Werthamer( 23) to calculate the behavior of Josephson Junction sources of high frequency radiation. The results in equation (65) also compare f~yorablY to the more precise calculation of Hattis and BardeenO: } from the BeS theory. As can be seen from equation (65), the general behavior of an r.f. field in a superconductor is that it will propagate very slowlY as a damped wave. Some interesting experiments attempting to utilize this fact are discussed in ref. (24).

The two-fluid hypothesis is quite successful at explaining the temperature dependence of many superconducting properties (recall that the temperature dependence of the thermodynamic parameters will be reproduced by the two-fluid model, since the function WeT) was forced to fit these datal) More sophisticated versions of the two-fluid model have been developed by Ginzburg \25) and H.W. Lewis(~that take in account the existence of a gap in the energy spectrum of superconductors. L. TizJ27)has also developed a twofluid model for superconductors analogous to the form applied to liquid helium.

B. Ordering Theories

Further use can be made of the fact that the entropy in the superconducting state is lower than in the normal state. Many other phase transitions from a disordered to ordered state occur in naturE. These have been successfully described using a parameter to measure the order, ~~ which in the case of a superconductor, using the considerations from the two-fluid theories, is obviously a measure of the fraction of "superconducting" fluid, 1. e. ,

(66) It is possible to show in a general theory of phase transitions that the Gibbs potential depends only on the ordering parameter, so expanding in a series should be possible when this parameter is small (near Tc) in the following fashion:

(67)

58 W. D. GREGORY

This type of theory was first developed by Ginzberg and Landau. The relative value of the constants a and e can be obtained by minimizing the Gibbs potential at each temperature. With such a treatment, one can predict the behavior of the critical field near T : c

H 2 = 4'JT C

(T _T)2 a 2 + (a~) T=T c

(68)

We see that this checks with the empirical expression for H near Tc (equation (38». c

An even more striking result can be obtained by expanding G in a series for H near H =0. I.f we consider the variable of the expansion to be the spa'£ial variation for the order parameter, we obtain the following differential equations for this variation for the one-dimensional problem of an N-S boundary, where the z direction is perpendicular to the boundary and the fields, currents and vector potential are along the y, x, and x directions respectively:

2m* h

(69)

Even in this simple case the equations are rather complex, so we will merely state the pertinent results. Using the order parameter, one finds that the formation of the N-S boundary results in a positive contribution to the Gibbls potential when a dimensionless parameter called the Ginzburg-Landau Kappa, K,

2 2H 2 ,,2 = ~_c_ ). 4

~c2 0 (70)

= ).2/[.2

1 1 is such that K~ 'f2"' When K> i2' the boundary surface energy is

negative. Thus, formation of the boundaries is discouraged for low


Ie materials and encouraged (Gibbs potential is lower) for high K materials. This means that the transfonnationlfrom the N to S state takes place with a minimum of boundary for KS 2. This is the case

for the materials we have discussed so far. We call these Type I materials. The case where Ie: ! defines a class of materials, called

2 Type II, for which fonnation of an N-S boundary is energeticallY favorable above a field value of Hcl . From this field value up to a value of

Hc2= i2 ICH cl (71)

the ratio of normal to superconducting material increases~ until finallY at Hc2 all of superconductivity is destroyed. This gives rise to the type of magnetization curve with a partial Meissner effect between HCI and Hc2 QS shown in Figure (6). The regions of nonnal material are found to be long, thin filaments at the center of a whirlpool (or vortex) of supercurrent. In higher and higher fields, more and more vortices are formed, until finallY the whole material has become normal at Hc2 • In analogy to the intermediate state in Type I materials, the region from H~to Hc2 of combined N and S material is called the mixed state. In an alternating field, energy can be lost in the nonnal coresof the mixed state by "shaking" the vortices. One can also pin these vortices at dislocations, causing the energy dissipation to be a field and temperature dependent phenomenon.

In a Type I material, the field "Hc2" is lower than H and represents the field that one would supercool to in decre~ing field. That is, since the surface energy associated with the N-S boundary is positive, the system attempts to avoid formation of these boundaries unless some other effect, such as impurities or lattice defects, nucleates the process.

These conclusions were drawn from calculations assuming the distribution of N and S mate~i~s was within the bulk of the sample. St. James and deGennes. 29 -have shown that a similar surface energy problem can be treated at the sample boundary and they find that a superconducting surface sheath will remain up to a field Hc3 which is larger than Hc2 • Since this sheath is very thin (of the order of A or ~ in thickness) it was observed onlY recently in rather sensitive measurements as a very small extra tail on the magnetization curve, which we show exaggerated in Figure ( 6). The parameter ~, which obviousl.v measures the spatial variation of the superconducting phase, can be interpreted as the coherence length for the superconducting interaction. Note carefully that this w~ of defining ~ is different from the Pippard definition (equation (33».

Effects associated with time and spatial dependence of the

60 w. D. GREGORY

superconducting phase are some of the most intriguing problems still remaining to be solved in superconductivity. The GinzburgLandau theory in a time dependent form is a very useful tool for treating these problems and you m~ wish to look into this further in your reading. As you can see, many of the results have been anticipated by some of our earlier intuitive considerations.

IV. A Simple Quantum Mechanical Model for SuperCl.onductiyity

Although our considerations up to this point have been classical, many of the phenomena we have discussed show evidence of the quantum mechanical behavior of superconductors. Before we move on to a discussion of the correct microscopic theory, which takes the quantum mechanics into account rigorously, we will collect these observations and discuss a simple quantum mechanical approach to superconductivity.

If we consider the charge carriers in a superconductor to be a collection of particles that must be treated by quantum statistics, you will recall that there are two kinds of rules possible for developing these statistics. In all cases, a correct quantum treatment requires that we think not of particles but of the wave function ~(;,t,etc.) that describe the particle and has the significance that 1~12 is the probability of finding the particles at a position;: at a time t with a whole host of other properties (such as momentum, energy, intrinsic angular momentum) described by the "etc.". Since these probability "waves" are not sharply defined spatially but can, in fact, overlap, we must realize that the particles, on a microscopic scale, are indistinguishable, and we must determine the proper rule for treating these particles when their wave functions overlap. The two simplest possibilities are: the overlap is possible, or it is not possible. These are the rules, then, that set apart two types of quantum statistics. If the first rule is correct, we call such particles Bose Einstein particles and we do our numbers assuming that such particles re~ have all of the same physical properties at the same time (i.e., have the same "quantum numbers"). This is simply are-statement of the rule that their wave functions may overlap. The resulting distribution of particles as a function of energy, called the distribution function, f BE , is then given by

fBE = (E-IJ) /kT e -1

1


where ~ = chemical potential.

On the other hand, if we assume that no two particles can have all the same quantum numbers, then we obtain the Fermi Dirac distribution, f FD :

1 fFD = -r:( E~--~") /~k=T--

e + 1

If we further calculate the heat capacity for a simple noninteracting systems of particles obeying each of these types of statistics, we find at low temperatures

CBE = AT3 (74)

CFD = BT

It has been found that particles with half integer values of intrinsic angular momentum,such as electrons, will obey the Fermi Dirac statistics and those with the whole integer values of intrinsic angular momentum will obey Bose-Einstein statistics. Examples of the latter particles would be quanta of light (photons) or of vibrations of a crystalline lattice (phonons), or any collection of particles with a net integer spin.

We alluded to these results previously in discussing the critical field curve, where we required expressions for the heat capacity of the normal and superconducting phases. You will recall that we found both a Bose and Fermi Dirac contribution to the normal state heat capacity, as would be expected for the lattice vibrations and electrons, respectively. In the superconducting phase, the contribution was thought to be proportional to T3, implying the electrons in the superconducting state also had "Bose" like properties. This would be reinforced by two other observations: (1) We found evidence in the flux quantization experiments that pairs of electrons do the quantizing, so a net integer spin is possible if these pairs are thought of as the "superconducting" charge carriers. (2) We also found that the pairs had a net zero canonical momentum, p=O. At low temperatures, this would be a result expected of Bose particles, since the distribution function, equation (72), would tend to infinity when E=~. To avoid this nonphysical result, Bose systems often ~dergo a condensation to a state where all of the particles have zero momentum.

Although these arguments may sound reasonable for establishing that the superconducting phase is one of the Bose condensed pairs of electrons, we know from the microscopic theory that this result is over simplified. For one thing, the heat capacity in the superconducting phase is not strictly proportional to T3. Also, we find from the correct microscopic approach that the proper statis-

62 W. D. GREGORY

tics are a mixture of the two simple cases. Nevertheless, there are treatments of the superconducting phase that have succeeded in using the condensed Bose fluid hypothesis, to a degree.(30)

B. ic henomenon

If we treat the superconducting pairs as all having "condensed" to the same state, then the wave function for each pair would be the same,

1/Ii = Ri e i '1 = Rei, (75)

Further, if we consider the pairs to be very weakly interacting (which is not really true), then a standard result of quantum theory tells us that the wave function for the whole system of N particles is the product of the individual wave functions, which will look like this:

N 1/1 - II total -

(76)

i=l Equation (76) says that the entire system can be thought of as having one wave function with one magnitude, RN, and one phase, ,N. Thus we should be able to think of the superconducting state, in this model, as a giant wave, with all of the properties we have associated in our previous physical experience with other wave phenomena, such as light and sound. These effects would include constructive and destructive interference, diffraction, and so on. If we assume this model is true, we could derive from it the Londons' equations and the principle of flux quantization as direct results. So, in a way, the assumption of a "quantum wave" might be thought of as the most basic property, at least for discus-sions of the electrodynamics of a superconductor. Of course, we should consider in more detail why one might expect to get away with the assumption of non-interacting pairs, when in fact the pair wave functions overlap and interact rather strongly. We will do this in the following lectures dealing with the microscopic theory. Here, we will discuss this most important use of this model so far presented - prediction of the a.c. and d.c. Josephson effects.

C. The Josephson Effect

About a decade ago, Brian J osephsorl31>Suggested experiments in which the superconducting wave exhibited some of the characteristic wave properties. Classically, these properties are the result of adding waves of different phases. In a superconductor, this can be achieved by adding the waves from two disconnected pieces of super-


conductor which, hopefully, would have a different phase associated with each piece. Furthermore, if electric and magnetic fields are introduced, they will also alter the phase of the superfluids, allowing one to construct analogs to optical and acoustical systems, almost at will.

To see how these results occur, we will consider the simple system considered by Josephson, now called a Josephson junction, illustrated in Figure ( 7 ). This is nothing more than a connection between two pieces of superconductor with a barrier in between. You will note that this arrangement should afford sufficient disconnection between the pieces of superconductor so that they will have different phases to their wave functiQns, but sufficient connection so that the wave functions from each piece can overlap, and hence the supercurrent can pass from one piece to another, a process called tunneling.

If we can treat the electrons on each side of the b.arrier as if they have one wave function, we can describe the time development of the tunneling process with the time dependent Schr8edinger equation

i~ 2..t = E1/I (77) at

As written, equation (77) will hold at points well within each superconductor. Near the barrier, we expect some of the wave function from the left, 1/11' to leak into ~he right side and, vice versa, some of the wave function on the right, 1/12' to leak over to the left. We can describe this process by postulating that the time development of 1/11 and 1/12 are perturbed by this process, to first order, by the addition of small and equal amounts of the other wave functions on the right side of equation (77). If we also assume that the battery driving the current produces a voltage drop across the barrier, V, then we can place the zero of energy to be such that the electrons on the left have energy, E, given by +qV/2 and those on the right by E = -qV/2. With these assumptions, the Schr8edinger equations for the two systems, at points near each side of the barrier,are

a 1/11 ill at = ~ V1/Il + K1/I2

where K is a small number. with energy units, representing the strength of the mixing of wave functions from one side to the other. .

(78)

The equations (78) and (79) represent two coupled equations

64 W. D. GREGORY

for ~l' and ~2. These can be uncoupled if we postulate

~ 1 = P 1 exp a 61)

~2 = P 2 exp (i 6 ) 2 (80)

We then finJ 32 ~hat . 2

K Jp l P2 sin 6 (81A) Pl = ii

P2 = -~ K V Pl P2 sin 6 (81B)

• =!S..R cos (_ 9.. y) (81C) 1 l'i - 2 l'i

Pl

62 !S.~ cos (+ 9.. y) (81D) = l'i P2

2 l'i

Since the current density passing between the superconducting pieces is proportional to the rate of change of the magni~ude of the wave functions, Ja:p, we find from equations @lA) and @lB ) that p 1 = P 2 and so

J = ~ J p 1 P2 sin 8(t) (82 )

From equation (81C) and (81D ) we find, after performing a time integration,

8 ( t) = 6 ( 0) + ~ ~ V ( t ) dt (83)

We may now determine what we can expect for various types of voltages applied to the junction. For example, if V=O (no voltage) then 6(t) = 6(0) and if 8(0) is not an integer value of n, then J~O. This means that at zero voltage. a current can flow! This remarkable result was in fact ObS~~~d very soon after Josephsons' prediction by Rowell and Anderson • This effect is called the d.c. Josephson Effect. (See Figure 7(B»

If we ask what happens with a constant, non-zero, applied vol-


s s

1 Barrier

I

(A)

t v---4----

Sn- Sn

(B)

Fig. 7. (A) Idealized sketch of a Josephson Junction. (B) T,ypica1 I-V curve obtained for a Josephson Junction. Data are taken from James E. Hercereau in "Superconducti vi ty" t R.D. Parks ed •• Harce1 Dekker, New York (1969).

6S

66 W. D. GREGORY

tage, we find that the phase is given, from equation (83), by

15 (t) = 15 (0) + it Vt ( 84)

Thus, the phase oscillates given by

between 0 and 2n with a frequency, v,

~ _ 2eV ",= 11- 11

As a result, the current also oscillates at this frequency. If the effective capacitance and inductance of the junction have values typical of "thin film" type junctions, the time average current for Vd >0 will be zero. This is called the a.c. Josephson effect. The typg'of IV curve one would expect is shown in Figure (8 ).* It is clear that this oscillating current will radiate at the oscillation frequency. This radiation can be observed, as will be discussed in great detail in later lectures. The frequency is such that q must be assumed to be twice the electronic charge. This may be thought of as another confirmation of the pairing hypothesis but Professor Bloch will present a slightly different view on this and the similar result for flux quantization is his lecture.

If we were to develop the Josephson effect more carefully, including m~netic fields, we would find that the magnetic vector potential A, or to be more precise, SA 'di around the barrier separating the two superconductors. can also change the phase. Using Stokes' theorem, we can show that this is equivalent to a dependence of 15 on the magnetic flux, in the Junction, fB.dS=~. Thus we would expect to find that the current varies periodically with magnetic field applied to the junction. Several of the later lectures will present data that demonstrate this effect and show how one can use it to measure very small magnetic fields.

Finally, we should mention that the classical interference and diffraction experiments from optics have been repeated using the superconducting "wave" where the Josephson junction replaces a slit, the current is proportional to the amplitude of the wave, the phase to V and A, etc. A typical two slit experiment (two Josephson junctions in parallel) is shown in Figure ( 9 ).

* This was first reported by Shapiro(33) •


z, r-,----,----.----,----.----,----~

n = 4

20

4 .5 ! !E ~ 10 II:: a

5

10 . 20 30 40 0

0 60

VOL TAGE (fL V)

Fig. 8. I-V curve for a Josephson junction exhibiting the a.c. Josephson effect. At each of the voltages marked by the inte-

67

ger n, the Josephson radiation of frequency 2e V resonates v = n h

in the junction structure, producing a vertical current step. (Data are taken from D.N. Langenberg, D.J. Scalapino, B.N. Taylor and R.E. Eck, Phys. Rev. Letters 15, 294 (1965)).

A

-~ -400 - lOO - 200 ~ o 100 200 300 400 ~

MAGNETIC FIELD (MILLIGAUSS I

Fig. 9. Behavior of Josephson junctions in a magnetic field, exhibiting behavior analogous to diffraction and interference effects with light. The maximum Josephson current at zero voltage may be thought of as the amplitude of a quantum wave while the magnetic field controls the phase, producing a combination of interference and diffraction effects for two junctions in parallel. Traces (A) and (B) are for two different sets of junctions. (Data taken from James E. Mercereau, in "Superconductivity", R.D. Parks, ed., Marcel Dekker. New York, (1969)).

68 W. D. GREGORY

V. Conclusions

The phenomenological theories of superconductivity are by no means limited to those discussed in this lecture. We have selected the topics in the lecture mainly to introduce the Physics of superconductivity and to assist further reading. Same subjects"have not been treated at all, for a number of reasons. For example, theories involving the enerSl ~ in the electronic energy spectrum of superconductors and various experiments (such as Giaver tunneling experiments) that demonstrate that this "gap" exists, have not been discussed. The presence of an energy gap in the excitation spectrum is a natural result of the microscopic theory and is best treated in that context. In most treatments of superconductivity on a phenomenological level, much more discussion of the voluminous experimental results reported in the literature would be included. However, the remainder of this course is devoted to detailed discussion of some of these results. Hopefully, the separation of the simpler theories from extensive discussion of data at this early stage will assist you in understanding the remaining lectures.

There is only so much one can do with phenomology and we have about reached the practical limit at this point. It is now time to turn our attention to a better approach - the microscopic theory.

References

1. J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108, 1175 (1957).

2. An excellent reference text for Electricity and Magnetism is "Classical Electrodynamics", J.n. Jackson, John '''iley and Sons (New York) 1962. Other possible references are listed at the" end of each chapter in Jackson.

3. Some confusion has devel~ped in ~he literature on superconductivity over the use of "H" and "B" to designate the magnetic !ield in a sunerconductor. If one defines+a m~gneti~ation, M, then the B and H fields are related by B = H + 4wM ~ they must be considered to be volume averages of the field. If all of the "sources" of magr.etic fields are considered :1;0 be lumped in the current, :t, then the field defined is a "B" field and it may be treated as a microscopic quantity, defined at a point in space. In the derivation of the Londons' equation, many authors call the field "R" when it should be treated as a "B" field by the criteria just stated. +Since this has ~ecome the accepted practice, we will use "H" rather than "B", when discussing the Londons' equation. Although not strictly correct, no harm is done in this case. (For further discussion of this problem, see Ref. 3, p. 150 ff).


4. R.M. Bozorth, "Ferromagnetism", Van Nostrand, New York, (1951). 5. E.C. Stoner, Phil. Mag. 36,803 (1945). 6. D. ShBenberg, "Superconductivity," 2nd. ed., Cambridge, New

York (1952). 7. K. Mendelssohn, Proc. Roy. Soc. A 152,34 (1935). 8. R. Peierls, Proc. Roy. Soc. A 155,16:13 (1936); F. London,

Physica 3,450 (1936). ---9. F. London and H. London, Proc. Roy. Soc. A 149,71 (1935);

Physica 2, 341 (1935). There is a second Londons' equation describing the behavior of the electric field in the superconductor that is analogous to equation 18. 'I'le will use both of these in describing the behavior of a superconductor in an r.f. field, equation (65).

10. If the Lorentz force term, ~ ~ x B, is included, we will obtain the same result providea we also treat the acceleration, ~ + +

dt- , as a total time derivative, ~~ = ~~ + [~'17)~ 11. R. Doll and t-1. Nlibauer, Phys. Rev. Letters 7, 51 (1961). 12. B. S. Deaver and W.M. Fairbank, Phys. Rev. Letters 1, 43 (1961). 13. B. S. Chandrasekhar in "Superconductivity" R.D. Parks, Ed.,

Marcel Dekker, New York, (1969). 14. A.B. Pippard, Proc. Roy. Soc. A 203, 98 (1950); 195 (1950),

A 216, 547 (1953). 15. G.E.H. Reuter and E.H. Sondheimer, Proc. Roy. Soc. A 195, 336

(1948). -16. D.C. Mattis and J. Bardeen, Phys. Rev. 111,412 (1958). 17. A particularly useful treatment of thermodynamics for super

conductivity, because of the attention given to magnetic systems, is cont'ained in "Thermal Physics," P.M. Morse, H.A. Benjamin, New York, (1969) (2nd ed.).

18. See, for example, P. and T. Ehrenfest, "The Conceptual Foundations of the Statistical Approach in Mechanics," H.J. Moravcsik, Trans., Cornell Univ. Press (Ithaca, N.Y.)(1959): also, Commun. Phys. Lab. Univ. Leiden, Suppl. No. 75b.

19. The derivation of Hc(T) given here follows that of Schoenberg. Ref. 6.

20. K. H'lang, "Statistical Mechanics," Wiley (New York) 1966. 21. C.J. Gorter and H.B.G. Casimir, Phys. Z. 35,963 (1934); Z

Techn. Phys. 15, 539 (1934). -22. A. L. SchawloW-and G.E. Devlin, Phys. Rev. 113,120 (1959). 23. N.R. Werthamer, Phys. Rev. 147,255 (1966).---24. W.D. Gregory, L. Leopold, J. Bostock, R.F. Averill and D.

Repici, Proceedings of the 12th Int. Low Temp. Conf., Kyoto (1971), p. 445.

25. V.L.Ginzburg, Zh. Eksperim. i. Teor. Fiz. 14, 134 (1946); Fortschr. Phys. 1, 101 (1953).

26. H.W. Lewis, Phys-:- Rev. 102, 1508 (1956). 27. L. Tiza, Phys. Rev. 80,~3 (1950); 84,163 (1951).

70 W. D. GREGORY

28. V.L. Ginzburg and L.D. Landau, 7.h. Eksperim. i. Ther. Fig. 20, 1064 (1950).

29. D. Saint-James and P.C. de Gennes, Phys. Letters (Holland) 7, 306 (1963).

30. J.r.1. Blatt, "Theory of Superconductivity," Academic Press, New York, (1964).

31. B.D. Josephson, Phys. Letters (Holland) 1, 251 (1962). 32. R. P. Feynman, R. B. TJeighton, and M. Sands, "The Feynman Lec

tures on Physics, Vol. III," Addison-Wesley (Reading, Mass) 1965.

33. P.W. Anderson and J.M. Rowell, Ph,vs. Rev. Letters 10, 340 ( 1963).

311. S. Shapiro, Phys. Rev. Letters 11, 80 (1963).

ELEMENTS OF THE THEORY OF SUPERCONDUCTIVITY

W. N. Mathews Jr.

Department of Physics Georgetown University Washington, D. C. 20007

INTRODUCTION

The purpose of this series of lectures is to present an introduction to the modern theory of supercondu1tivity in as straightforward and physical a fashion as possible.

The mathematical clothing in which the modern theory of superconductivity is customaril~ clad is drastically different from that of the original BCS paper. The underlying physical concepts are, however, essentially the same. We thus begin our study of the microscopic theory with a comparatively qualitative discussion of the problem which confronted BCS and the basic physical ideas which went into their solution of this problem. 3 Then, with the physical problem well in mind, we turn to the formalism which is used in most of the current theoretical research on supercon1uc,fivity, the powerful and elegant formalism of Green's functions. ' -7 Finally, we apply the Green's function method to the problem of superconductivity. We primarily concern ourselves with the equilibrium properties and the response to a weak, static magnetic field of a weak-coupling superconductor, the subjects of the original BCS paper.

PHYSICAL BASIS OF THE MICROSCOPIC THEORY OF SUPERCONDUCTIVITY

Pre-BCS Status of the Theory of Superconductivity

Midway through the "decade of the fifties the list of the main facts to be explained by a theory of superconductivity stood as

71

72 w. N. MATHEWS Jr.

follows:

(1) In the absence of an external Inagnetic field the transition froIn the norInal to the super conducting phase is of the second order. Specifically, the free energy and its first derivatives are continuous at the transition, and the specific heat shows a discontinuity given by

(1)

(See Fig. 1.) Here T is the critical teInperature, the teInperature at which the phase trcrnsition occurs, and ')I is related to the norInal state electronic specific heat, C en, by the equation

C = ')IT, en

(2)

where T is the absolute teInperature.

(2) The quasiparticle excitation spectruIn has an energy gap of width 2.6.. With increasing teInperature this gap decreases Inonotonically froIn a InaxiInuIn value of 2t:. (O)ex kB T at T = 0, where kB is the BoltzInann constant, to zero at T • (§ee Fig. 2.)

c

(3) A sufficiently weak applied Inagnetic field is excluded froIn the interior of a bulk superconductor. This is the MeissnerOchsenfeld effect. The depth of penetration of the Inagnetic field at the surface, A , has a teInperature dependence approxiInately given by

(3) where

t = TIT c (4)

is the reduced teInperature. (See Fig. 3.)

(4) When the applied Inagnetic field reaches a critical value, H , a first order (except for t = 1) transition froIn the superconductfng to the norInal phase occurs. The teInperature dependence of the critical field is given approxiInately by

where Ho = Hc(O).

(See Fig. 4.)

(Sa)

(Sb)


Free Energy

T o Tc

Figure i-a. Qualitative sketch of the free energies of the normal (FN ) and super conducting (FS) phases versus temperature. The two curves cross with the same slope at T=T •

c

o

Electronic Specific Heat

,/ ,/

,/ ,/

,/

,/

O'y Tc

.,.,/

---L

T Tc

Figure i-b. Qualitative sketch of the electronic specific heat versus temperature in the absence of a magnetic field. There is a finite discontinuity 0: Y T c at T c.

73

74 W. N. MATHEWS Jr.

Energy

Excited states

T 2.c:,.

1 Fermi sea

Figure 2-a.Qualitative representation of the quasiparticle excitation spectrum of a superconductor. There is an energy

1

o

gap of width 2.c:,. centered at the Fermi energy. The states below the gap make up the Fermi sea and are completely occupied at T=O, while the excited states lie above the gap •

.c:,.(T) 1.c:,.(0)

t

1 Figure 2-b. The reduced energy gap, l:,.(T)/.c:,. (0), vs. reduced

temperature, TIT , according to BCS. The slope at t= 1 is infinite. c


A(t)/;>"(O)

2.0

1.5

1.0+----------

o Figure 3. Approximate temperature dependence of the

penetration depth.

75

t

(5) For a pure superconductor, the relation between the electric current density, J, and the magnetic vector potential, A, is given in the transverse~gauge (\7 .~ = 0) by ~

,:! (~) = Jd3x , ~(~_~I).~(~I). (6a) ~

The kernel, K, is approximately given by

~

K (~) _ _ 3 R R e -R/ ~ 0

4'ITCA~o ~ ~ (6b)

where

(7)

a ~. 2, v F is the average Fermi speed, and the coefficient A is related to the penetration depth, A' by the equation

(8a)


1

O~---------------------------------------rl----~t

Figure 4. Approximate temperature dependence of the critical field. The slope at t= 0 is zero in accord with the third law of thermodynamics, while the slope at t= 1 is finite.

and is given at absolute zero by

A(O) =t(Z/3)eZ [N(O)/VO]vF2 } -1, (8b)

with e the magnitude of the charge of the electron N(O) the normal density of states at the Fermi surface for electrons of a given spin. and V 0 the volume of the system.

(6) For different isotopes of the same material, the critical temperature, T • and the isotopic mass, M, are related

d . c accor lng to

O!~ T M - constant,

c

with Q' ~ t in a number of cases.

(9)

It was clear at this time that the transition is primarily electronic in character, in the sens e that it is the propertie s of the electrons which differ in the normal and superconducting phases. It was also clear that an independent-particle model of the conduction electrons could not be expected to yield an

ELEMENTS OF THE THEORY OF SUPERCONDUCTIVITY 77

understanding of superconductivity. Thus the simplest model upon which a theory of superconductivity could be based was seen to be that of a collection of interacting electrons. The difficulty, then, was in finding the correct interaction. This difficulty becomes apparent when the energy associated with the various interactions is compared with the energy difference between the normal and superconducting phases.

The ener~ difference between the two phases is of the order of N(O)(kB T ) , which can be interpreted as meaning that electrons wnhin a band of energy of width approximately kB T centered on the Fermi surface have their energy changed 15y c roughly ~T ; i.e., about 0.10/0 of the conduction electrons experience ab energy change of the order of 10-3 eV.

The first interaction which comes to mind is the Coulomb interaction. Unfortunately this interaction produces an energy change of about 1 eV per §olJduction electron. Moreover, the work of Bohm and Pines ' has shown that the long range portion of the Coulomb interaction gives rise to collective excitations of the conduction electron fluid, plasmons, which have an excitation energy comparable with the Fermi energy and so play no primar'¥) role in the transition. (See, however, Rothwarf's recent work. 1 ) The residual, screened, repulsive Coulomb interaction is not considered to be a likely candidate for the role of an interaction leading to superconductivity.

The second possibility for the interaction responsible for superconductivity is the electron-phonon interaction. In fact, the isotope effect indicates that this interaction does indeed play an important role. The question is how. Eventually the isotope effect proved to be the key clue in putting together the pieces of the puzzle of superconductivity, but following the path indicated by that clue took over five years.

At the same lft~ that the isotope effect was discovered experimentally, ' Frohlich PI§posed a theory of superconduc-tivity based on this interaction. Shortly afterward Bardeen al~~ attempted an approach based on the electron-phonon interaction. Both of these theories yielded an Ci = t isotope effect, but failed in other respects. They did not yield a phase with superconducting properties, and, moreover, the energy difference between what were supposed to be the normal and superconducting phases was far too large, about 10-4 eV per conduction electron. The difficulty with both of these theories is that they are based on the diagonal or self-energy part of the electron-phonon interaction. It is now believed that this part of the energy is substantially the same in both the normal and superconducting phases.


Thus it is seen that the Coulomb interaction and the diagonal part of the electron-phonon interaction do not lead to superconductivity. The effects of both interactions are essentially the same in the normal and superconducting phases, The modern point of view is, in fact, that these two interactions are substantially taken into account by appropriately defining the excited states of the normal phase. The consequences of this viewpoint are four-fold. First, as we mentioned above, the long-range part of the Coulomb interaction gives rise to plasmons, density fluctuations of the conduction electron fluid, which have high excitation energy, and so play no primary role in the phase transition. Second, the low-lying single-particle excitations are particle-like, though not identical with particles; they are referred to as quasiparticles. Third, because of the residual interactions, a quasiparticle can decay into a number of other quasiparticles. Hence quasiparticles have a finite lifetime and a state with one quasiparticle, unless it has zero excitation energy, is not an exact eigenstate of the system. Nevertheless, for most purposes it is permissible to treat those quasiparticle states with lifetime » 'til e, where e is the excitation energy of the state, as eigenstates. Fourth, the energy necessary to add or remove a quasiparticle from the system, the energy of the quasiparticle, depends on the numbers of other quasiparticles present. This is a new feature which does not appear in an independent particle model. This scheme for describing a collection of interacting fermiong is known as the Landau theory of the normal Fermi liquid. 15, 1

The problem confronting BCS, then, was to find an effective electron-electron interaction which could be used to explain the main facts of superconductivity. The Coulomb interaction and the diagonal part of the electron-phonon interaction had been shown to be inadequate for this purpose.

The Electron-Phonon Interaction

. The isotope effect, on the one hand, implies that the electronphonon interaction plays a primary role in superconductivity. On the other hand, however, the effects of the diagonal part of the electron-phonon interaction are basically the same in the normal and superconducting phases. Thus it must be the remaining or off-diagonal part of the electron-phonon interaction which is of importance for superconductivity. To understand how this can be so and what it means, we must consider the electron-phonon interaction in more detail.

The basic processes in which the electron-phonon interaction is manifested are the absorption and emission of a phonon by an electron; these processes are depicted in Fig. 5. The next


Figure Sa. Absorption of a phonon (_) by an electron (04-).

Figure Sb. Emission of a phonon by an electron.

simplest process, depicted in Fig. 6, is the emission and subsequent absorption of a phonon by an electron; this is what we mean by the diagonal part of the electron-phonon interaction. Clearly then, the process by which the electron-phonon interaction leads to an effective electron-electron interaction is that shown in Fig. 7, the exchange of a phonon between two electrons.

We hasten to point out that the phonons involved in the process of Fig. 7 are "virtual" phonons. The word virtual has three aspects to its meaning in this context. First, we mean that the phonons involved are not statistical phonons, those phonons ordinarily present in an equilibrium system at a given temperature and described by the usual Bose-Einstein distribution function,

f (w)= l/(e{3W_ l ), BE

(10)

where w is the phonon energy and {3 = l/kBT. Obviously, if the effective interaction of Fig. 7 is to be responsible for superconductivity, the phonons involved cannot be statistical phonons; as the temperature decreases superconductivity becomes stronger, e. g., the energy gap increases, while the number of statistical phonons decreases very rapidly. Second, in using the term virtual we mean that the phonons involved are not constrained by a strict energy conservation. This is because the process of Fig. 7 is a quantum process and thus the uncertainty principle

80

Figure 6.

w. N. MATHEWS Jr.

Emission and subsequent absorption of a phonon by an electron.

applies. Hence such a phonon can be present for a time T without obeying energy conservation provided

(11)

Third, the word virtual may also be interpreted to mean intermediate, since the state with the phonon present is an intermediate state, in the sense of perturbation theory.

An effective interaction such as that of Fig. 7 occurs in other branches of physics. The ordinary Coulomb interaction is of this character, except that it is virtual photons rather than phonons which are interchanged. The strong interaction of nuclear physics can be thought of as arising from the exchange of virtual mesons by nucleons.

The interaction between electrons via an exchange of virtual phonons can also be viewed clas sica11y. An electron moves through a lattice of positive ions, interacts with it, largely via the Coulomb interaction, and so distorts or polarizes the lattice. A second electron experiences an interaction not with the unperturbed lattice, but rather with the distorted lattice. Thus, one electron affects another with the lattice as intermediary. Since lattice vibrations can be expressed in terms of phonons, i. e., the phonon is merely a quantized unit of lattice vibration, we refer to the resultant interaction as the effective phonon-mediated electronelectron interaction.

From this classical picture it is apparent that the phononmediated interaction is not instantaneous. The lattice vibration, or equivalently the intermediate phonon, propagates with a finite speed characteristic of the lattice. Thus the interaction is retarded.

If this phonon-mediated interaction is to lead to superconductivity it must be strong enough to dominate the residual screened Coulomb interaction. Thus we would expect that the stronger the


Figure 7. The exchange of a phonon between two electrons.

electron-phonon interaction, the more likely is superconductivity. But, in the normal phase the stronger the electron-phonon interaction, the more difficult it is for an electron to move through the material, i. e., the poorer is the conductivity. So it would appear qualitatively that superconductivity is more likely in the poorer conductors. This is in fact borne out by experiment. For example, lead and tin are poor conductors and fine superconductors, whereas copper and silver are excellent conductors but do not become superconducting. Aluminum is an intermediate case, being both a fair conductor and a fair superconductor.

At this point we can qualitatively understand much of the recent discussion of other mechanisms for superconductivity. A whole class of possibilities arises from the replacement of the phonon in Fig. 7 by other "intermediate bosons". The excitation exchanged by the electrons must be a boson because the kinematics of the exchange requires it to carry zero or unit spin. The list of excitations which might be exchanged by electrons to prody.se an effective elfctgon-electron interaction includes plasmons and excitons. I ,I (One could also consider an interaction mediated by spin waves or magnons. However, these excitations involve parallel spin correlations which tend to suppress superconductivity.) These mechanisms have in fact been considered, but to date no decisive re suIts have been forthcoming.

Since the superconducting phase has lower energy than the normal phase, it seems reasonable that an effective electronelectron interaction can lead to superconductivity only if it is attractive. (We will discuss this point further in the next section.) We must thus inquire whether the sum of the screened Coulomb interaction and the phonon-mediated electron-electron interaction can be attractive.


We shall give a comparatively simple treatment, 19 which is, however, qualitatively correct, in which the metal is regarded as a plasma of quasiparticles and ions. This is the so-called "jellium" model.

Consider the response of our plasma to an external test charge density, Pext. We begin with the two Maxwell equations,

'i/O E = 4'/T (6 P + P ) '" ext'

(12a)

and

'i/o D = 4".p '" ext, (12b)

for E and D, the usual electric field and displacement vectors, respectively. Here 6 p is the charge density induced in the plasma by the external charge density. If we Fourier transform in space and time in the usual way, we obtain

i k ·E '" '"

= 4". (6 p + p ) ext' (l3a)

and i k • D = 4". P

'" '" ext (13b)

for the various Fourier components. But now, ~ and 12 are purely longitudinal fields. Thus we can define the scalar, wave vector- and frequency-dependent, longitudinal dielectric coefficient, f (.!s, w), by the relation

D (k, w) = dk, w) E (k, w). -." ,...,., -." -." f"tJ

(14)

Equation (l3b) can then be written as

i k • dk,w)E(k,w) = 4".p (~,w). -." -." -." . ......, ext .-

(13c)

Moreover, since we are dealing with longitudinal fields, we can take

(15)

Consequently, equation (13c) can be written as

k 2 E" (~w) cp (~w) = 4".p (~w), ext

(16)

whence the actual electric potential in the plasma is given by

(17)


where

2 (,Dext(~' w) = 41T p ext(~ w) Ik

is the external potential.

83

(18)

We see that ( (k,w) is a direct measure of the effectiveness of the electron-ion system in screening an external test charge. Clearly the electron-electron interaction is screened in the same way; i. e., the effective electron-electron interaction is given by

V eff = V ('!::)h (~,w), (19)

where V (k) is the Fourier transform of the bare or unscreened electron-electron interaction. In equation (19) h k and ware the momentum and energy transfer, respectively. Thus the effective interaction can be represented as in Fig. 8 with

w = ( - (

£+~ £: If the bare interaction is a Coulomb interaction,

then

2 V (~) = e II ~ I,

2 2 V (k) = 47T e Ik V

'" 0'

where V 0 is the volume of the system.

(20)

(21a)

(21b)

The immediate problem is to calculate dk,w). The necessary prescription follows from equations (13a) andl'13c):

1 I dk, w) = 1 + 6 p (k, w) I p (k, w). ~ ~ ext ~

So we must calculate the induced charge density,

6 p = 6 p e + 6 p i,

which is the sum of the electronic and ionic contributions.

We f:i:rst note that from equations (13a) and (15) we obtain

k 2,/'I (k,w) = 47T [6P(k,w) + P t(l),w) J "f' ,.,. ~ ex •

(22)

(23)

(24)

We next note that if the external charge density perturbs the system only slightly, then we need keep only that part of 0 P

which is linear in P t' in which case E depends only upon the ex

I I I I I

Figure 8. Representation of the effective interaction,

equilibrium properties of the system. In this approximation, the equation of motion of the ionic electric current density is

2 M ol"i/;,t = - nZe \l cp ,

where M is here the ionic mas s, n is the electron number density, and Ze is the ionic charge. In writing down this equation, we hq,ve required overall charge neutrality. The ionic electric current density is also related to the induced ionic electric charge density by the equation of continuity,

;, (<5Pi)/ ot + \l'li = O.

Upon combining these two equations we obtain

2 2 2 2 M;, (<5Pi)/;,t = nZe \l cp.

After Fourier transforming and using equation (24), we find

w 2 <5 P i (~, w) = Wo 2 [ <5 P (ls, w)+ P ext (~w) ] '

with

(25)

Wo= 'h,(411'nZe 2 /M)1/2.

If we now restrict our considerations to values W «€F' the Fermi energy, and w« W , where

(26)

of w such that

p 2 1/2 1/2

wp = 'h, (411'ne 1m) = (M/mZ) Wo


is the plasma frequency and m is the electronic quasiparticle mass, and to values of ls such that !lsl «the equilibrium Fermi wave number, then we can use the Fermi-Thomas approximation. In this approximation the electronic quasiparticle fluid is as sumed to be in local thermodynamic equilibrium, so that there is a local Fermi wave number given by

2 (fl k F ) 12m = E.F + ecp.

The local electron number density is given by

3 2 n = kF 131T ,

or

where

3/2 n = no(l+ ecpl ( )

F

2 3/2 2 (2m(/h) /31T

F

is the equilibrium electron number density. Thus if ecp« E. F , in accord with our linear response approximation,

~ / 2 OPe - -(3 2)(no e / (F) cp • (27)

Equations (24) and (27) together yield

OPe r;;;: -(ks 2/k2) (Op + P ext)' (28)

where

(29)

Then equations (25) and (28) can be combined to give

o P 2 2 2 2

= (w / W - k / k) (Op + p ) o s ext· (30)

If we now use equation (30) for op in equation (22), we find

( (ls, w) = [w2 (k2+ ks 2) - Wo 2k2 ] / w2k 2 • (31a)

The spontaneous modes of vibration, i. e. the phonons, of the electron-ion plasma are determined by the requirements

t= 0, cp ~O, which, as can be seen from equation (16), implies p( e(g, w) = o. For each.!: we thus find one frequency, wk ' given by

wls 2 = Wo 2 k 2 I (k2+ ks 2). (32)

In terms of wk ' E is given by ,..,

2 2

~+ wls ) 1 k -k2+k 2 2 2 E" (~, w) s W -wk

(31b)

So we see that in this approximation the effective interaction is

(33)

The first term is the Fourier transform of a screened Coulomb inte raction,

v c

2 ~) = e

and the second term is an interaction due to the exchange of virtual phonons between two quasiparticles.

(34)

The most interesting feature of the effective interaction given by equation (33) is that it can be negative; i. e. the phonon-mediated interaction can overcome the screened Coulomb interaction to result in a net attractive interaction. The effective interaction (33) is attractive as long as

I E" p+ k - ( I < wk • ~ ...... g

This condition is easily satisfied in a superconductor since the important quasiparticle excitation energies are of the order of kB T c or les s, with T c typically a few Kelvins,

-4 ~ < k T '" 10 e V, .. ..J?...... B c

whereas typical phonon energies range up to roughly kB T D' where T D , the Debye temperature, is typically a few hundreaKelvins,

wk.:s kB T D '" 10 - 2 e V •

The interaction of equation (33) is greatly oversimplified for application to a real material, as is evident from its derivation. In fact, this interaction, if taken seriously in a quantitative sense, would predict that all metals become superconducting at sufficiently low temperatures. The point of this treatment has been


merely to establish that there can be a net effective quasiparticle interaction which is attractive. This we have done. One might object on the grounds that we have in fact dealt with a highly idealized system in a very approximate way, so that the applicability to a real material of even the qualitative results of this treatment should be questioned. One answer to such an objection is that screening tends to diminish the difference between an electron-ion plasma and a real metal; for example, the screened lattice potential has a much weaker spatial variation than the bare lattice potential, and is thus closer than the bare lattice potential to what one would expect in a plasma.

There remains one final piece to the puzzle. Granted the existence of an effective attractive interaction among the quasiparticles, how does such an interaction lead to superconductivity?

Cooper's Problem 20

The remaining piece of the puzzle was supplied by Cooper who showed that if there is a net attractive interaction among the quasiparticles, no matter how weak, then the normal state is unstable against the formation of bound pairs of quasiparticles.

Specifically, Cooper considered the problem of two quasiparticles interacting only with one another above a quiescent filled Fermi sea. The quasiparticles in the Fermi sea enter the problem only via the Pauli exclusion principle, so that the two interacting quasiparticles can scatter one another only to states above the Fermi sea.

If we as sume no spin-dependent interactions, then the total spin of the pair is a constant of the motion and the wave function for the two interacting quasiparticles may be written as the product of a spatial wave function and a spin wave function. The spatial wave function can, with complete generality, be written as

(35)

The prime on the sums over single -particle states is to remind us that the sums are restricted to states above the Fermi sea.

The c.on's are simply normal state, spatial, single-quasiparticle wave functions, and thus obey the single-quasiparticle Schrodinger equation,

(36)

88

where the single -quasiparticle Hamiltonian is

2 Ke(l) =.£1 /2m+ U ~l) -£Fo

w. N. MATHEWS Jr.

(37)

The interaction U includes the potentials associated with the crystal lattice, non-magnetic impurities, boundaries, any external scalar fields, etc. In subtracting the Fermi energy, IE. F' we are defining the zero for the excitation energy, E ,to be at tile Fermi surface. Thus the prime on the sums in equatfon (35) impliesE >O,e._>O.

n n

The full Schrodinger equation obeyed by 'lr is

(H-E) 1li = 0, (38) with

H (1,2) = K e (1) + Ke(2) + V eff ~1,~2), (39)

where V eff is the effective interaction between the two quasiparticles.

With the use of equations (35) - (39) we obtain

a(n, Ii) (£n + .fri.-E) + r;' a(nl, iiI) < n, IiI V eff I nl, iiI >= 0, (40) nl, iiI

where

(41)

For a general potential, V ff' equation (40) is a very complicated integral equation for whi~1i general solutions are prohibitively difficult to obtain. In the spirit of Cooper's original paper and the BCS theory, we now make two assumptions. First, we regard ii as a function of n; that is, We assume that we have some prescription for determining the state ii to go with a given state n in equation (35). Second, we assume a simplified interaction such that

-, I I -, < n, n Veff n , n > = ° otherwise

Equation (40) then simplifies to

a(n, Ii) = V

~ + c_-E n n L;'a(n', iiI), n l

(42)

and

1 = V~I n

1

89

(43)

where the sum in equation (43) is over quasiparticle states n such thatO<£ 6",,<W. n, II c

If the effective interaction involves the exchange of phonons, then a cutoff as in equation (42), with Wc the maximum phonon energy, is a reasonable first approximation.

For the ground state of our system of two quasiparticles interacting above a quiescent Fermi sea, ther~ will be no electric current. This is the Ctfe if and only if the state Ii is the time reverse of the state n. Then ~= En and we obtain the equation

1 = V 'E n

1

2E - E n

(44)

determining the energy eigenvalues E. We readily see that this equation has a series of positive energy solutions and, for V>O, a single negative energy solution. We can easily obtain an analytic result for the negative energy solution if we assume that the density of states, N (£). is slowly varying in the interval ° <E.< W •

Then equation (44) further simplifies to c W

1 = N (0 ) V J d ~ -,::-=:1----:::::---° 2E -E (45)

from which it follows that

E = - 2wc / [exp [ 2/N(O)V] - 1 } • (46)

From this result we note two important facts. First, the energy is a very sensitive function of the interaction strength. Second, and most important, there is a negative energy solution as long as there is a net attractive interaction.

We have as yet said nothing about the spins of the two quasiparticles in the bound pair. There are just two possibilities: the two quasiparticles are either in a singet state with opposite spins or in a triplet state with parallel spins. Which of these two possibilities occurs depends on the details of the interaction, but all the evidence presently available points to singlet pairing.

For a pure superconductor the quasiparticle states may be characterized by wave vector, k, and the pairing is between states with opposite wave vectors. With the simplified interaction of equation (42), a(k, -k) depends only on the magnitude of k, and thus

~ ~ ~


'II ~l ~2) is symmetric. Since the total wave function must be antis'ymmetric, it follows that the spin part of the wave function must be antisymmetric and the pairing occurs in the singlet state. Thus for the interaction of equation (42) the pairing in a pure superconductor is the usual Cooper pairing; i. e., a quasiparticle with (kt) is paired with one with ( -!:n.

We thus conclude that as long as the net quasiparticle interaction is attractive near the Fermi surface, the Fermi sea is unstable against the formation of bound pairs.

Physical Basis of Pairing

Our considerations have led us to an eff ective Hamiltonian of the form

H == f JCe (i) + i ~ V (i, j) i~ j eff

(47)

where JC is the quasiparticle Hamiltonian, with eigenvalues E. , and Vef/is an effective interaction which can be attractive undRr reasonable circumstances. Of course we cannot solve the eigenvalue problem for this Hamiltonian exactly, we must somehow approximate it. However, since the physical effect we are seeking is a very subtle and delicate effect, any approximation must be made with extreme care so as not to wash out the soughtafter effect. It was at just this point that the work of BCS was so very clever. The most convincing arguments for the approximation to be made are still the original ones of BCS, and we can do nothing more effective than to restate those arguments.

The reasoning of BCS may be described as follows. The problem under consideration is that of a collection of interacting fermions described by a Hamiltonian H== H + U. We denote the eigenvalues and eigenstate vectors of H b§> E. and W., respeCtively, . H E ,T, 0 1 1 1. e. --oW i == i :r i' and the matrix elements of U between the eigenstate vectors of Ho by Uij • Suppose that there is a subset of the eigenstate vectors of Ho in which the matrix elements of U are negative. Then as a tentative choice for the ground state vector of H we take

I

W == ~ a. \¥. , ill

where the prime indicates that only members of the subset are included in the sum, and where all the non-zero a.'s have the same sign. The resultant ground state energy is 1

2 , E == L a. E.+ ~ aia.U ..

ill .. J lJ 1, J


If the matrix elements Uij are negative, the interaction term in the energy is negative, and a coherent low energy state results. The ener gy maybe minimized by a variational choice of the a.' s.

1

The superconducting ground state vector is most naturally taken to be a superposition of normal state vectors in which quasiparticles are excited to low energies above the Fermi sea. All of the normal phase correlation energies and self-energies are included in the normal state vectors. The remaining interaction is the sum of the screened Coulomb and phonon-mediated interactions, The prescription for the approximation to be used in solving the Schrodinger equation for the superconductor consists of a statement of which nornal state vectors are to be superposed to yield the state vector for the superconductor ana how the accompanying coefficients are to be chosen. The prescription used by BCS and in subsequent work may be arrived at on the basis of the following considerations:

(1) The effective quasiparticle interaction connects a large number of nearly degenerate normal states with one another via non-zero matrix elements.

(2) The super conducting ground state vector is to be a linear superposition of low-lying normal state vectors.

(3) The normal state vectors all obey Fermi-Dirac statistics;i. e., they are antisymmetric under interchange of two quasiparticles. Thus the matrix elements of the effective interaction between a random selection of normal state vectors will alternate randomly in sign. It follows that a state vector formed as a linear superposition of randomly selected normal state vectors will yield little interaction energy; i. e., the matrix elements of the interaction will add incoherently.

(4) What we need to do is select a subset of normal states between which the matrix elements are predominantly negative for an attractive interaction. The obvious way to do this is to select normal states in which the quasiparticle states are occupied in pairs. That is, we associate the quasiparticle states in pairs, e. g. (n, a) and (ii, a), and we consider only those normal states in which (n,O') and (ii, 17) are both occupied or both unoccupied. Of course, the effective interaction then becomes a pair interaction.

(5) To obtain the maximum number of matrix elements, and thus the lowest energy, the pairs should be chosen in such a way that any pair can be scattered into any other pair by the interaction. Thus, for example, if linear or angular momentum is conserved by the effective interaction, then each pair should have the same linear or angular momentum. For a pure superconductor the wave vector is a good quantum number. The pairing in this case


can be taken as (ls, a ; E, if) with]s+ is= 9 the same for each pair.

(6) The super conducting ground state is a non-current carrying state. Thus, in the absence of spin-dependent interactions, the proper pairing is (n, a ; ii, a ), where nand ii refer to spatial quasiparticle states which are the time reverse of one another. For the pure superconductor this reduces to (ls, (J ; -ls, ij ).

(7) Under most circumstances the exchange or Fock term in the interaction energy has a sign opposite to that of the direct or Hartree term. But the Fock term is non-zero only if the two interacting quasiparticles have parallel spin, and can thus be eliminated by taking the two members of a pair to have opposite spin. Thus in the absence of spin-dependent interactions the pairing is (nt, iid where nand ii are the time reverse of one another. For a pure superconductor this reduces to (kt, -kl ). Parallel spin or triplet pairing may conceivably, however,lead to a lower energy state if the angular dependence of the effective interaction is such that the exchange and direct terms have the same sign.

Thus we arrive at a coherent superconducting ground state which is a superposition of normal states in which the quasiparticle states are occupied in pairs which are the time reverse of one another. Of course, to obtain a lower energy, the superconducting ground state must be predominantly made up of normal states such that the matrix elements of the net electron-electron interaction between these states are negative.

For a pure superconductor carrying an electric current the pai.ring is between quasiparticles with wave vectors 1s and,E, where k+ k=,9 is the same for all pairs. For an excited superconducting state there will be unpaired quasiparticles present as well.

The pairing discussed above is not the sort with which we are familiar from the problem of two particles interacting via an attractive potential; i. e., we are not discussing a pair of particles bound in a highly localized fashion. In fact, the spatial extent of a Cooper pair turns out to be of the order of 1O-4cm • This conclusion can be reached by the following well known argument. For a quasiparticle near the Fermi surface in a pure material the wave number difference 6. k corresponding to an energy difference kBT c is given by

6. k '::: (k /2 E.F)k T • F B c


Thus the spatial extent of the corresponding wave function is

6. r .:: 1/ 6. k ':::'(2 EF/k T ) (11k ). B c F

4 8 -1 -4 Since £ Ik T ~ 10 and k ~ 10 cm , we find 6. r ~ 10 cm.

F B c F

But now, the t8umber of pairs per cm 3 near absolute zero is typically ~ 10 • Thus in a cube 10-4 cm on a side there are about a million pairs. Obviously there is considerable spatial overlap between the pairs and a picture of isolated pairs is inapplicable. The pairing actually occurs in "quantum number space" - e. g., in ~-space for a pure material.

GREEN'S FUNCTIONS

Second Quantization

In dealing with a collection of interacting particles we are faced with the task of carrying out some sort of perturbation treatment with symmet2~zed or antisymmetrized wave functions describing perhaps 10 particles. In many cases perturbation theory through some finite order will not suffice; rather some infinite subset of terms in some perturbation series must be kept in order to obtain meaningful results. In such cases the formalism of second quantization is very convenient - it permits the calculations to be done more easily and understandably. Here we confine our considerations to a collection of interacting fermions, and we give only an outline of second qua~tizationi fuller treatments are readily found in the literature. 4,6,7,22- 4,*

We begin by introducing operators c (t) and cat (t) which destroy and create, respectively, a partftle with quantum numbers a and wave function cp ( Here a stands for all the necessary quantum numbers, inclffding spin.) at time t. These operators obey the following equal time anticommutation rules:

[ca , cal }= [cJ, ca't} = 0 , [c"" c~,t } = 6, (48) u. u. ()I, a

Here

[ A,B} ;: AB+BA

is the anticommutator of A and B. For a= a' these rules are consequences of the Pauli exclusion principle - no two particles can have the same quantum numbers. For a t- a ' these rules stem from the requirement that the many-particle wave function be antisymmetric under the interchange of particles.

* See also Appendix A of Ref. 1.

A one-particle operator, Ea·, appears in second quantized notation as i 1

t r ai = L. < a I a I a I> C C I . I 1 ,.",." 1 a,a <.. symbolizes the matrix element of a between the

states a and a I,

(49b)

Similarly, a two-particle operator, l; a .. , may be written as ¥ j 1J

L e .. = ~ <otl.,('leu.lcl'~ot'II>Ccl.T'k'''t C.,("~III ) 1. t:J.A.J 01. p.~cJ!IId.11I

where

( 0(. J..' Ie 101. '1 cI.;" ........ -) 12. I/,-

(50a)

Thus, for example, if we use the notation of equation (36) and consider a pure material, the Hamiltonian of equation (47) may be written as

As we discussed in the previous section, the important terms in the interaction are those corresponding to interactions between Cooper pairs, i. e., those with pI = -p and t1 1= - t1. If only these terms are kept, if the diagonal part of the interaction is dropped, and if the energy of the filled zero temperature Fermi sea is subtracted off, then we obtain the reduced Hamiltonian of BCS:

_ ,t Hred - ~ E:p CP<r CPO"" - 2. L eb +

l,(f'''''''' ~ ..e ~ (52)

~t>F'

+ hr.' "eu q~/,.e) C1't C~/. C-.e+C,e+ • We can further understand the meaning of the creation and

destruction operators by noting that the one-particle wave function

can be expressed as

cp =ctq, a a 0

(53 )

where cp is the vacuum state containing no particles at all. a

The considerations of the previous section indicate the following choice for the superconducting ground state vector:

Ws= n (un+vncn~c.J~) q,o n

(54)

Here ii is the time reverse of n. The coefficients u and v are to be determined by minimizing the expectation valu~ of H n •

red

We could now proceed to work out the basic properties of superconductors following the path laid out by BCS. We choose instead to use the method now in current use, the method of thermodynamic Green's functions. In so doing we shall need a second set of operators, the wave field operators,

'Ys (~,t)= ~ ~ (~,S)~(t) ) ",t(!.Jt) =! (P/~)Sr Ci(t) (55)

which destroy and create, respectively, a particle at the spacetime point (~, t). From equations (48) we find that ¢ and ¢ tobey the following anticommutation rules:

{'I${~Jf) 1ISA~/Jf)} ={'lGi{~J:t)) "stCt/i)} =Dj (56)

In terms of these operators the Hamiltonian of equation (47) may be written as

(57)

+ ~ tSI 5d3 x5d3)(' 'V:(~ ,!) ""s~(:6~:I:) ~:F~~J!) lIf/l!~-t) <Its (~J t). There is one further operator which we shall find of immediate

use, the number operator, given by

N(i;) = ~ CI{t)~{t) = f Ji~x tJ;{(l)"lJ ~ ('!/t). (58)

The eigenvalues of this operator are simply the total numbers of particles in the system. This is readily apparent because, for example, c t c 'It = 0 or 1 depending on whether the state Q'is empty or ogcupied in the state vector 'It. With the use of the number operator,

n = c t c (59) pO" pO" pO" ,.... ,....

we can rewrite equation (52) as

H ... ed = L e" l\~<t' + L (-€,,' (1- n "0") + h.,.. .... IV ~ .(1" I'V #'oJ £. N"

J:»PF P~"s:- (52')

+ ~pl Vef§(.e'>.e' C!'t C~,~ C-t + e,t ... ) ~ IV

which is readily seen to be equivalent to equation (3.1) of BCS.

Grand Canonical Ensemble

Our model of a superconductor is a collection of Fermi quasiparticles interacting with one another via an attractive potential. For such a system in equilibrium at a temperature T, the statistical average or expectation value of any operator may be computed as a grand canonical ensemble average:

-tHE -~N) ~ <C,NI91{ ,N> e CN

<9 > = I: ,N

(60a)

r: tHE -~ N)

e I:N

C ,N

Here N denotes the number of quasiparticles in the system, and I: stands for all the remaining quantum numbers necessary to describe the system. Thus the sum over C and N is a sum over all the possible quantum mechanical states of the system with all possible numbers of particles. Of course < r, NI a I ~,N> stands for the expectation value of the operator a when the system is in the quantum state with quantum numbers, and Nand E N is the corresponding energy of the system. The thermodyna.hAc state of the system is now defined by the quantities {j = l/kB T and IJ.. the chemical potential. For given values of Nand T, we may think of ~ as the energy necessary to add one more quasiparticle to the system, i.e. ~ = ( 0 E/o N)T' where E = < H:>is the energy of the system.

We shall use the more compact notation,

< e> [ - 8(H- fLN) ]

= Tr_ e e Tr [e -8(H- fL N) ]

(60b)

or

< e> = Tr (De). (60c)

As usual the trace, denoted by Tr, is merely the sum of diagonal matrix elements. The operator

80. -8(H- fL N) D = e e ( 61)

is the grand canonical density operator, and the grand potential, 0., is related to the grand partition function, Z. by

Z = r: - 8 (E - fL N) e ~N

1:' N[ _ 8(H-fL N)] = Tr e

-(Jo. (62) = e

When external time-dependent fields are present, matters are not quite so simple. However, we need only interpret the angular brackets as standing for a grand canonical average with an appropriately chosen equilibrium ensemble.

Time -Ordering

The Greenls functions which we shall use in our analysis of superconductivity are grand canonical ensemble averages of timeordered products of '" I S and ~ tiS. T he time -ordering is indicated by the time -ordering symbol, ']I'. When applied to a product of several ~ IS and i> t IS the time-ordering operation orders them from right to left in ascending time order and multiplies the result by (-1) P, where p is the number of interchanges (of fermion operators) from the original order. Thus. for example,

{ '" (~,:/;) ~:l~~tl)

for

-lfS!{!')t')'Is{~Jt)

t > tl

t < tl or more succintly

t t = ,', (x,t·u• I (xl,t)a(t-t l)-." (XI,tl)J. (x,t)9 (tl-t).

rs,-- "',s I"'W 'rs'''' 'rs'"

Here

11 t> 0

9(t) = for o t < 0

is the standard step function.

One- and Two-Particle Greenls Functions

We define the one- and two-particle Greenls functions (or propagators) according to the equations*

(63)

(63 1 )

1 t G (ls,1'sl) =r < 'll' ~s(l) fsl (1') > (64)

and

(We henceforth use the abbreviated notation 1 to mean ~1' t l ), l' to mean ~l .. t1')' etc.)

The one-particle Greenls function represents the propagation of a particle added to the system at l' with spin s I and removed at 1 with spin s. We represent this pictorially as follows:

G(ls,1'sl) = Is ( IISI.

* We shall for the most part employ the notation and techiliques of Ref. S. See also, however, Ambegaokarls article, Chapter 5, in Ref. 1 and Ref. 25.

Similarly,

G 2 (ls, 2A-;1's' ,2 '4J) = 2..4..(

99

l's' > .. , describes the propagation of two particles added to the system at l' with spin s' and 2' with spinN and removed at 1 with spin sand 2 at spinAl. The propagation of the two particles described by G 2 is correlated by their interaction with one another and the remainder of the system.

There are several reasons for carrying out our calculations with the language of Green's functions. First, the single-particle Green's function contains the observable properties of greatest interest: the energy, the excitation spectrum, and all the thermodynamic properties of the system. Second, the statistical mechanical expectation value of any single-particle operator of the system can be expressed in terms of G. Third, the equation of motion and diagrammatic methods of calculating G are very powerful, often permitting one, for example, to account for an infinite subset of perturbation theory terms; in many cases this is necessary in order to obtain physically meaningful results. Fourth, the method of Green's functions is readily extended to nonhomogeneous systems and non-equilibrium situations. Full dis i 4-7 cussions of some of these points may be found in the literature. '

The Boundary Condition

In addition to the one-particle Green's function we define two correlation functions:

1 t G>(ls, 1's')=[ < ~s(l)¢s' (1') >

< 1 t G (ls, l's')= -:- < ", (1')", (1»

1 is' 'Y s

We note that <

G (ls,l's')

G(ls,l's')= for >

G (ls,l's')

t <t ' 1 1

(66)·

(67)

If we use the usual Heisenberg picture of quantum mechanics to describe the time development of the operators of our system, then for a time-independent situation the wave field operators evolve in time according to

a (t)=eiHt/t/ a (O)e- iHt/tl (68a)

As a lDatter of convenience we shall use a lDodified Heisenberg picture in which, instead of equation (68a), we have

i(H- /-LN)t/ f! -i(H-/-L N)t/ f! e (t)= e e(O)e • (68b)

-i(H-/-L N)t/ f! The tilDe developlDent operator of this picture, e , is identical to the weighting operator which enters into the grand canonical average, e - 8(H- j.L N), if we set t= -if! 8 0 We now use this fact to derive a fundalDental relation between G< and G> •

We note that

=Tn. ['IS ()(1JO) e-f3 (H-JtNJI/s;(1I)]/.i Z ""

[ _~(H-fiN) tl,f{ ') tv ( . .J /. z = Tit e lS' J IS ~ i») .J..

= 1 ~~ i1'J 'f;;{~,o~ • Consequently

> < G (ls,l's')1 =-G (ls,l's')1

t l = -if! (3 t 1=0

(69a)

If we take all tilDes to lie on a straight line joining the tilDes 0 and -if! f3 ' and interpret ']I' to lDean tilDe ordering according to the value of J t I dt', then equation (69a) can be written as

o

G(ls,l's'l = -G(ls, l's'),

t = -if! 8 1

since 0 and -ift8are then the "earliest" and "latest" tilDes respectively.

(69b)

At this point we seelD on the verge of errant nonsense. (IlDaginary tilDes indeed~) Nothing could be further frolD the truth. The use of ilDaginary tilDes greatly facilitates our analysis - as will becolDe readily apparent. The key point is that equation (69a)

is a relation between the correlation functions G< and G>, whereas equation (69b) is actually a boundary condition on G.

One can in fact go freely between real and complex times. Equation (69b) results if instead of a straight line one uses any curve between 0 and -ifi {3 which falls between the lines lmt= 0 and lmt= - fi {3. Of course one must prove that G is well defined in this region. It is then obviously trivial to shift the end points of the curve to to and to -ifi (3, with to real. The final step in regaining the real time function is to take the limit as to _ -00 • *

T he boundary condition (69b) will prove to be of key importance in our subsequent analysis.

Real and Complex Time Fourier Transforms

For the purposes of this subsection we consider a timeindependent system. Then G(ls, l' s ') depends upon tl and tl' only via their difference, tl-t l ,. Since here we concern ourselves only with the time dependence of G, we suppress the position and spin variables.

The boundary condition in the form (69a) implies the relation

< -{3w > G (w) = e G (w),

where the real time Fourier transforms of G are defined as

< < G> (w)= -+ 2. {dte iwtl fiG> (t).

fi -00

The inverse transforms are

:; -1 . I < G (t)= + i r ~;. e -lWt fi G> (w).

-00

(70)

(71a)

(71b)

The boundary condition in the form given by equation (69b) may be guaranteed by expressing G as a Fourier series,

1 G(t -t ')=-

1 1 -i ~

where

z = (TT/-i~) (21) + 1), I)

* See Chapter 8 of Ref. 5.

(72a)

(73)

and the sum runs over all integral values of v' Since tl and tll here lie within the strip Q 2! Imt 2! -tz ~, we refer to this as the complex time Fourier transformation. The inverse transformation is

(72b)

where the integral is along that curve, C, within the strip, on which tl and tll are taken to lie, as discussed in the previous subsectlon.

Spatial Fourier Transforms

For the case of a spatially homogeneous system,G(ls, llSI) depends on2}l and,2511 only via their difference, ,25t~l'. It is then convenient to Fourier transform in space accordlng to

and

whereit. may be G < G> G , ,or • ~

Interpretation of G:5 (p, w)

(74a)

(74b)

Consider a system which is translationally invariant in both space and time. If we write

(75)

then we readily find that

G< (h W\ = 1 Sdt elwt/Ji.~ct (0) Ch(f' s s' t' J '/ - I' Sit' $ ) tv :Ii: -00 N IV

(76a)

and

(76b)

We also note that in the absence of spin-dependent interactions, G:> I with s 101 s vanishes. We can thus make the interpretations ss

< G ss(,E,w) =

G> (p,w) = ss ~

the number of particles per unit energy interval with momentum fI p spin s, and energy w ,.....,

(77) the number of available one-particle states per unit energy interval with momentum fiE' spin s, and energy w

It is now apparent that the boundary condition in the form (70) is a statement of detailed balance:

- ~ w (density of particles in p, s, w) = e (density of available

states in p, s,w). (78)

The Spectral Function

We define the spectral function, A, by

t A(ls,l's')= <[~s(l),~s' (l')} >,

or equivalently

A(ls,l's')=i [G>(lS,lISI)_G«lS,lISI)] •

(79)

(80)

From equation (79) we see that a knowledge of the non-equal time anticommutation relations gives A. From equation (80) we see that the spectral function contains information on the excitation spectrum of the system.

From equation (79) we note that

(81)

For a time -independent system this equation implies a sum rule on A:

(82)

Here we have Fourier transformed A according to

1 0:> • t/ A(w) = -;; f dte lW tI A( t), (83)

-0:>

where we have suppressed position and spin variables. For a time-independent system, equation (80) implies

A(w) = G> (w)+ G< (w). (84)

This result and the detailed balancing condition, (70), lead to expressions for G::>= in terms of A:

G < (w) = f(w)A(w) , G> (w) = [ l-f(w)J A(w).

Here

(Jw f(w)=l/(e -1)

(85)

(86)

is the usual grand canonical ensemble occupation probability for a fermion mode of energy w. Considering equations (77), (85), and (86), we interpret A(w) as a weight function, governed by the sum rule (82), which tells us what happens to a particle of energy wadded to the sys tern. We shall return to this point.

With the use of equations (71b) and (72b), the detailed balancing condition (70), and equation (84) for A, we readily obtain

G(zv)= r dw A(w) -00 211" z V -w

This equation defines G(zv) only on the infinite set of discrete points, zv, given by equation (73). The unique analytic continuation to all non-real z is just the function

00

G(z) = f dw A(w) -00 2TT z-w

(87)

Using the Dirac trick,

1 =p_l __ + i 17 0 (w-w l ),

w-wl±i T/ w-w l

where '1 is a positive infinitesimal and P stands for the principle value, we find that A (w) is given by the discontinuity of G{z) acros s the real axis:

A{w) = i [G{w+ iT7) - G{w-iT7 ) ] • (88)

The results (85) and (88) are central to the method of calculation which we shall employ. We will calculate G<'e., z) directly. Equation (88) will then enable us to calculate A~ ,w), and subsequently equation (85) will enable us to obtain G:::;; (r"w).

Eq uations of Motion

We consider a system of fermions described by a Hamiltonian of the form given by equation (57), with the one-electron Hamiltonian, :ICe' given by

:IC (i) =..!:. [-!L \7 1 + ~ ~(l) ] Z - ecp (l)+ U (l) -I-" • e Zm 1 c 0

(89)

Here c is the speed of light in vacuum, /:!; and cp are the selfconsistent electromagnetic vector and scalar potentials, respectively, and the single-particle potential Uo takes into account all other effects - e. g., those of the crystal potential, surfaces and boundarie s, impuritie s, etc.

We assume, for the moment, that the effective interaction is instantaneous in time. We write

and we use the notation

3 f dZ= f d X z fdtZ•

(90)

In including the factor of 1J. in equation (89) we have chosen our zero of energy at the electronic chemical potential or Fermi energy. This is equivalent to replacing the Hamiltonian, H, by H-I-" N, where N is the electronic number operator. The usual Heisenberg picture is then, as we discussed, modified accordingly. The equations of motion of the wave field operators are then


We first work out the equation of motion of ~ • We immediately obtain s

and

rps(l),Vl = ~ fd2 V (1,2) ,II t (2),h (2)Jo (1), l' J Sl "'Sl rsl rs

where we have assumed that

V (1, 2) = "V (2,1).

Thus

[it! :\ -JCe(l~ ~s(1)=~.fd2V(1,2) fs; (2)~S,(2) I/Js(l). (92a)

t The equation of motion of~s is somewhat trickier. We find

= -Jd3x2 ~{~-~~){:M [(f VL)~- ~~ (V~.~{2)+ + ~l").V~)+({ ~(2»)2} e £Pit) + Uo(2)_)d lJr!cZ»)

where we have integrated by parts and assumed that the integrated parts vanish. Thus

where

[~st(l)'TJ = -JCe*(l)~st(l), 2

JC *(1) = _1 [~V'l - ~ A(l)] -ecp (l)+U (l)-~ e 2m 1 c ~ 0

(89*)

Also

[~s t(l). ~ =

Thus

- [ifz _0_ + JC *(1) j ", t(l)= ~ IdZV (1,2),'" t (1)", t (2)", I (Z). (92b) otl e 't's s' 't's 't'S' 't's

We can now regain the generality lost in requiring the effective interaction to be local in time. This requirement limits our considerati~n~50Z%*ak-coupling superconductors. As is well known, , , to properly describe intermediate- and strongcoupling superconductors we must work with the full interacting electron-phonon system. Using the cOITlplete Hamiltonian for the interacting electron-phonon system, we can readily work out the coupled equations of motion of the electron and phonon wave field operators. We take the electron-phonon interaction, as renormalized by the Coulomb interaction among the electrons, into account only to the extent that it gives rise to an effective electronelectron interaction. This is equivalent to taking all correlations except those associated with the residual screened Coulomb interaction and those specific to the superconducting phase into account only via renormalization. We can then eliminate the phonon operators from the electronic equation of motion. In this way we obtain equations (9Z) with

'V=V+Y, (93) c p

where Vc is the residual screened Coulomb interaction and V is the retarded, phonon-induced, electron-electron interaction. p

Now we use the equations of motion of I/J and Ik, t to work out eq uations of motion for the one -particle Green I s function. From equation (9Za) we readily obtain

+ = -i~ IdZ V (1, Z)GZ(ls, Zs ll;l l s ' , Z S"), s"

where the Z+ take1 account of the fact that in the equation of motion of W (1), ¢ II(Z) stands to the left of ¢ II(Z). But equation (133'1) ena~les us to write s

r'lr ", (1),/1 t (l')J = 'lr[~'" (1)", t(l')l + (j .<:(1-1'). L't's lS' otl't's 't'S' J S,S'V

* See also Scalapino's article, Chapter 10 in Ref. 1.


Thus the equation of ITlotion of G is siITlply

::: 1'1 Os s'O (l-l')-iL: JdZY(l,Z)Gz(ls,ZSll;l'S"Z+sll). , s II

(94a)

In siITlilar fashion we use equation (9Zb) to obtain an alternative but equivalent equation of ITlotion for G:

::: 1'1 ° 0 (l-l')-iL: JdZV(l Z)G (l's' Z-s"'ls ZS") s s' , Z ' " • , S II

(94b)

We have now introduced the three key ITlatheITlatical eleITlents needed for the Green's function analysis of any ITlany-body system: the boundary condition, analytic continuation, and the equations of ITlotion. Moreover, we have begun to see that the technique of cOITlplex tiITle Fourier transforITlation constitutes the connecting link tying together these three eleITlents. We can gain a greater appreciation and understanding of these three key eleITlents and the way they are linked together by considering two siITlple exaITlples.

Non-Interacting Particles

For non-interacting particles V::: 0 and the equation of ITlotion (94a) reduces to

[ i1'l_O- -:Ke(l)] G(ls,l's')=1'I0s,s'O (1-1'). ('It 1

(95)

Upon Fourier transforITling in both space and tiITle, we obtain

G ,(p,z):::o J(Z-E). 55 ~ 55 ,g (96)

Clearly then

A ,(p,w):::Z7T6 ,O(w- t p); ss ~ s,S

(97)

ASSI{p,w) has delta function peaks at the allowable excitation energies, E. p.

".,

109

The result (97) can also be obtained directly from the definition of A. Equations (75) and (79) lead to

i(I?:~_gl.~I) A{ls,l'sl)== ~ e -----

f,f' V 0

<

Since linear momentum is conserved for a translationally invariant system, the only allowable value of p' is p. Now suppose "i" is a state of the system with energy E. "Then

which implies

= e -iEpt/ f'lc (O) c,gs (tl ) ..., ,gs

Similarly

t t iE t J f'l (O)

c (tl')= e .E 1 c ns • ps ~

So we find that

A{ls, l's')== ~ p

from which (97) immediately follows.

Quasiparticle Approximation

The equation of motion, equation {94a),may be written as

[ if'l.L -:K (l)J G{ls, lISI)= 0\ e

= f'l Os slo{l-l')+~ Jd ZJ){ls, 2S") G{2SI,1"SI). , S II

(98)

The quantity~is generally referred to as the self-energy. The point of defining a self-energy becomes apparent when we note that upon Fourier transformation equation (98) yields

G(p,z)=l/ [z-Ep-~(p,z)1 • ~ ~ ~

(For the remainder of this section we assume that Ke andY contain no spin-dependent interactions, so that all quantities are diagonal in spin.) Thus the poles of G occur not at the €lnergies € l" but at the roots of the equation

(99)

The only general comment that can be made about the roots is that they must be real - a fact which is dictated by equation (87).

One can show, in complete parallel with the derivation of equation (87), that

r(w) z-w

where we have suppressed position and spin variables and

r (w) = 1; >(w) + D «w) , <

analogous to equation (84) defining A in terms of G>.

As a consequence of equations (88), (99), and (100),

A(p, w) = r (p,w) ~

~

where

E(p, w)=e + ReDp, w), ~ p ~

and

00

RelXg,w)=Pf dw' r(w') -00 27T W_W'

(100)

(101)

(102)

(103)

(104)

Thus A(,E, w), viewed as a function of w for fixed p, has a peak not at £ , but at e .e+Re%X,E,w); physically then,"Re !X,E,w) represe~s an energy shift - the self energy. Moreover, r (p,w) is the width of the peak at half maximum.

It is this sort of a finite peak with a non-zero width that we interpret as a quasiparticle.

To continue this example, suppose that E and r are independent of w. If we put equation (102) for A into equation (87) we obtain

G(p,z)=1I{z-(E(p) + ir(p)/2]}, for Imz ~O, ~ ~ ~

(105)

in apparent contradiction to the implication of equation (87) that the poles of G(J2" z) occur for real z. More careful inspection, however, reveals no contradiction. When G(£" z) is defined for Imz ~ 0, the pole occurs for Imz ~O. We say that the poles occur on the second sheet. Thus poles on the second sheet represent quasiparticles.

The meaning of r can be understood by noting that

(106)

is the probability that one can add a particle with momentum 'h .£ to the system at time 0, remove a particle with momentum 'h p at time t, and return to the original configuration. For the simple case considered above, provided r ~)< < 11 fJ , we find

> 2 2 I G (f' t>l '";;: [ l-f(E(p» ]

-r(p) It I l'h e (107)

Thus 'h/r (p) is the lifetime of the quasiparticle state of momentum 'h p.

We thus see that peaks in the spectral function are to be associated with quasiparticles. The center of the peak occurs at essentially the quasiparticle energy and the width of the peak is inversely proportional to the quasiparticle lifetime.


MICROSCOPIC THEORY OF SUPERCONDUCTIVITY

Introduction

We are now in a position to develop the microscopic theory of superconductivity without further ado. Though we have not fully proven it, equations (94) for G are adequate even for strongcoupling materials, provided -V- is interpreted, according to equation (93), as a sum of screened Coulomb and phonon-mediated interactions; the only physical possibility not included in these equations is a spin-dependent interaction. The equation of motion technique furnishes a powerful and convenient means for calculating the one-body propagator, G. Once we have obtained G, we can calculate the spectral function, A, according to equation (88). In turn, A yields the excitation spectrum. Moreover, from G we can calculate the thermodynamic properties and the response to a magnetic field for a superconductor. In brief, using thermodynamic Green's functions and the equation of motion technique of calculation, we can work out those properties of superconductors needed to check agreement between theory and experiment for our list of the six main facts which a theory of superconductivity must explain.

G ' .. 27 The or kov Factorlzatlon

To proceed we must decide what to do with the G2 in equations (94). For a normal metal the lowest approximation to G 2 is the Hartree-Fock approximation (HFA):

G2 (ls, 24-';l's', 2',4,') ::::: G(ls,l's')G(2.A-, 2'4..J)-G(ls,2'.4..')G{2.v, l's').

Diagrammatically, this approximation appears as shown below.

:is J'S'

2: I G~ I :2'4' =

1'S J.15 l'S'

2-.4.--cc-----'2~.4' - 2~ ~ ~ ~

15

This approximation amounts to neglecting all dynamical (as opposed to statistical) correlations among the electrons. For the superconducting phase, the lowest approximation to G2 is to take account of just those correlations that are relevant for the formation of Cooper pairs. Both the HFA for the normal state and the Gor'kov factorization (GF) for the superconducting state neglect scattering.

One way of obtaining a handle on the GF is to consider the superconducting phase at absolute zero. If we denote the T= 0 state of the N -particle system by I N> • then

Gz (lS)2..6..> i/S')2~/) = l:i)~<NIT 'Ysl.t) ~(2) 1/ffcz.I) lI;!t1/J1N)

= (. ~ t [(N IT Y$(1) l/ts~(1'J'N> <NI T 1JA(2)~~{2/) IN) -.J,.

+ much smaller terms,

or

G 2 (ls. 2..4-;1' s I. 2!4-')= G(ls. l' s I)G(2...4-,21..4;') -G(ls. 2IAJ)G(2~ 1's 1)+

+(1) 2 < N I 'I[' ~ s (1)~)2)1 N+ 2> < N+ 2\ 'I['tJ\21)~S: (11)1 N> +

+ much smaller terms.

T he remaining terms are much smaller because

< 'I[' ~ ~ t > ~ the particle density.

and

113

t t < N I 'I[' P'P I N+ 2> < N+ 2 I'I['~ ~ IN > '" the Cooper pair density.

Now what about non-zero temperatures? We note that for every state 1 ", N> • there is a corresponding state I ~ • N+ 2> , which is the same as , ~ • N> except that it has two more particles. We also note that for any local operator

<~.NI~ l~,N> = <~,N+2'¢ I ~.N+2>+O(1/N). (108)

Further

E -IJ. N= E -IJ.(N+ 2) + O(l/N). SN ~ ,N+ 2

(109)

114

where El" N is the energy of the state ,~, N> • lead us t't> write

W. N. MATHEWS Jr.

These considerations

G 2(ls, 2.4-;1'5', 2'.4.,) ~ G(ls, l's')G(2.b, 2'4..')-G(ls, 2'.&)G(2..v, 1'5')+

+F(ls,2~Ft(2'..4,.,,1's')' (110)

with

1 -P(E -I-I.N) F(ls, 24-0)= i ~ e ~N <~, Nl 'Jf~ s(l)~ (2) I ~, N+ 2>

~, N Z (111)

and

We note that

t t F (15, 2~ f. [F(ls, 2 ...... ) ] ~ ~

We often simply use the shorthand notation

1 tIt t F(ls,2~=i <'Jf~ s(1)~...z.-(2»,F (15,2·4= r <'Jf¢ s(1)~..J2) >. (113)

Diagrammaticall y IS) 2,4

(

2.A

is

F(ls, 2 A..} =

and

t F (15,2-4 =

Thus the GF, equation (nO), appears diagrammatically as

1S l'S' is 1~' E

.~ I G~ I :~~ 2.4- 2'J.' X' 2 2'4+ ~ 1S) c::: +

2.4

We may think of F and F t as the probability amplitudes for the breakup and formation, respectively, of a Cooper pair.

The whole point of the GF is that the primary processes in the superconducting phase are

and

Breakup of a Cooper pair

".

The pair is lost in the sea of Cooper pairs

><

Formation of a Cooper pair

A second view of the Gor'kov factorization is also instructive. We ordinarily decompose l/> s according to

~ (i) = L: cp (l, s) c • s a a a

(55')

For the sUPz9"conducting phase, however, we can follow Bogoliubov28

and Valatin in writing

~ (l)==~ [u (l,s)y +v (1 s)*y t] s aa a a' a·

(114)

Th t d t f· t e opera ors y an 'Yare ermlon opera ors, a a

(115)

and satisfy the grand canonical average rules,

t < 'Y a 'Ya ,> == 0a, a ' f(E a),

t t <'Y 'Y ,>= <'Y 'Y ,>=0. a CI (Y a

(ll6)

The Gor'kov term in G Z then stands on the same footing as the Hartree and Fock terms.

If we define a unitary operator R by*

then equations (ll3) can be written as

1 ttl t t F(ls,Z.4.)= i <1f~ s(l)¢.c1 (Z)R >, F (ls,ZA.)=i<1fR¢s(l)~ (Z»

(ll3')

Sometimes equation (114) is also written as

)/\ (l)=~ [u (1, s)y + v (1, S)*R Y t ). IS a a a a a

(ll4')

Our previous conclusions about the structure of a Cooper pair are verified by equations (lll) and (11Z). The important point is that F and Ft are sums of matrix elements which are diagonal in all respects but particle number. Thus, as long as spin is a good quantum number, the two quasiparticles making up a Cooper pair have opposite spin. For a pure system the wave vector is a good quantum number, and so Cooper pairing occurs between quasiparticles with opposite wave vector. The result is the usual Cooper pairing, (kt, -kl), for a pure system. For a system with no spin-dependentint;-ractions, the proper pairing follows from the requirement of no net current in the equilibrium state. This

,~ See Ref. 4, p.SZ


is most readily assured by taking the current associated with each Cooper pair to vanish .... which is most generally achieved by pairing time reversed states. c.l Thus the (nt , li~) pairing again follows.

Eq uations of Motion for G, F, and F t

If we use the Gorlkov factorization, equation (llO), the equation of motion of G, equation (94a), approximates to

[itr.£ "1Ce(l)+i~ Jd2Y(1,2)G(2SIl,2+SIl)]G(lS,PSI) = (ll7) ct l s II

= f! 0 10 (l-P)+i~ Jd2V(1,2) [G{1s,lSIl)G(2S Il , P SI) -S,S S"

- F(ls,2s") F\2+S Il , PSI)] •

There are three possible simplifications of this equation:

(1) The Hartree term, the third term on the left, may be included in the chemical potential.

(2) In the absence of spin-dependent interactions, which we henceforth assume,

G(ls,2~=G(ls,2s)0 = G(1,2)0 ~, s .4/, s

(3) Equations (lll) and (ll2) and the correspondence between I C,N> and I C,N+2 > imply

t t F(ls,2~= F(ls,2-s) 0 F(ls,2N)=F{ls,2-s)0

~ -s, ~, -s,

and

With these simplifications equation (117) reduces to

(118)

(119)

(120a)

(120b)

[ if! L -:Ie (1)] G(l, P)= tr 0 (1-P) + (121a) o tl e

+ i Jd2V(l, 2) [G(l, 2+)G{2, P) -F(l, 2)F t (2 ~ P)] •

Similarly, equation (94b) leads to

- [i~ _0 - + JC e*{l)] G{l' ,1)= ~o{l-l') +

otl (121b)

_ _ t + ifdZV{l,Z)[G(l',Z)G{Z ,1)- F(Z ,l')F (l,Z)] •

Equations (IZI) couple G, F, and Ft. Thus we need equations of motion for F and Ft as well as for G.

We first work out the needed equation for Ft. From the equation of motion for ~s t • equation (9Zb). we readily obtain

_ t~_o_+ JC *(l)]Ft{I.ll)=_iL: JdZY(I,Z)(})2<'Jf~t{l)¢ (Z-)¢ t{Z)~~{l'» LotI e s ~ s s

In complete analogy with the Gor 'kov factorization of G2 , we write

( 1) Z t t t i <'Jf~s (1)~.4..-,(Z')~AY)~S' (l') > ~

(lZZ)

~ G(ZIAJ, Z.4J F t {Is, 1's ')_F t (Is, ZAo1G{Z!.4J. lis I) _

- G{ Z !.d.,1 , Is) F t (ZA/, I I S I) •

which appears diagrammatically as

lSi 1s I

2~' 24 2~'

'""- 24

1S IS

2'.4.' jS'

c:' c

C2~ •

2!4.' is ~ .is


1£ we absorb the first term on the right of equation (122). into the chemical potential. which is consistent with what we did to obtain equations (121). then we find

[0 '!.]t - iii otl +3{e (1) F (1.1') =

= iSd2V(l,2) [Ft (1,2) G(2-,1')+G(2-,1)F t(2.1 ' )]. (123)

To obtain an equation for F, we note that equation (92a) for p s leads to

Again, in analo gy with the Gor 'kov factorization of G , we write 2

(124)

~ G(2b, 2'.4.-')F(ls, l's')-G(ls, 2'b')F (2,v, lIs') -

- G(l' s', 2 '.a.)F(ls, 2.A1 )

which appears diagrammatically as

.is J_

.--- :1S

lA~---I

1 'S' ... f-----i

1::) ~ ~~' ;;) 2~' 1£ we again absorb the first term on the right of equation (124)

into the chemical potential, which is consistent with what we did in working out equations (121) and (123), then we obtain

[ i1'l-O- - JC (1)] F(l,l') = (It 1 e

(125)

= iJ d2"V(l,2>(G(l,2+)F(2,l')+F(l,2)G(l',2+)] •

Equations (121)., (123), and (125) are the desired equations for G, Ft, and F in a superconductor. Together these equations are known as the Gor'kov equations.

The Narn.bu Forrn.alisrn.

If we adopt the convention that by G(l, 1) we rn.ean

G(I,I+)=G(l-,l)= - ~ < ¢ t(l)¢ (1» , (126) 1

then the four equations of rn.otion, equations, (121), (123), and (125), can be written as a single rn.atrix equation:

h1'l.L7"o-Ke (1) 7" 31 g(l,l')= lotI

=1'I6(1-1')r o+iJ d2V(1,2>r3g(1,2)7"3g(2~11). (127)

Here the rn.atrix Green's function is

g(l,2)= (G(lt , 2t )

F t (1,2)

F(1,2) ).

-G(2l,U)

T he one -particle spinor Harn.iltonian is

where

7" = 3

(128)

(129)

More generally, the equations of motion before the Gorlkov factorization can be written as

[11" fz,. 1:'0 - Xe (1l1:',] f/' ( 1 J 1') = t ,f{t-l'J1:'o +

+ 1.J J2 V(l,Z)(l( (T '1:'3 ~(1) (!f!"1(2)1:'a hJ) or"'i1'») where

'It (1) = (~Jf (1)) , \l( t (1) = ( III t (1), II) (1)) ~ t (1) T t r ~

(130)

121

are known as the Nambu spinor fields. We note that

g(1,2) = .!- < 'lI' w (1) ~t (2) > • 1

(128 1)

This matrix formalism was introduced by Nambu 30 with the object of handling F and Ft on exactly the same footing as G. Nambuls scheme provides a very convenient technique for dealing with a retarded effective interaction, which is essential to a proper treatment of strong-coupling superconductors. *

Our purpose here, however, is merely to note that within the framework of the Nambu formalism the GF has a very simple interpretation. One can readily show that

<'lI'7" 3~(1<i'\2)7"3'1!(2))'lrt(1'b =

= 7" 3 <'lI' (~(l)\l(t (2)) 7"3(i'(2 hT,t(11))> •

Thus the simplest factorization, which is a Hartree-like factori~ation, /., ) t (f) <'lI' 7"3'1' (\vt (2)7"3'l'<2) 'i' (llb.z

1 tIt ~ 7" 3 i < 'lI''i' (l)~ (2» 7"3 i <'lI'i' (2)'.JI (lIb

'" 7" g(1,2)7" g(2,1'), - 3 3

is in fact precisely the Gorlkov factorization (presuming that the usual Hartree terms have already been incorporated into the chemica 1 potential).

* See Section 7 -2 of Ref. 4 and Section IV of Rev. 25.


Boundary Conditions on F, F ~ and g

With the use of the definitions (111) and (112) and the correspondence between the states II:, N> and I ~ ,N+ 2> furnished by equation (108), it readily follows that for a time -independent situation F and Ft satisfy the same kind of boundary condition as G:

~ t t

F(l, 2)~ .= -F(1,2) , F (1, 2)~ =.-F (1, 2)~ =t-ljjP =t =t-ljjj1 =t

10 10 10 10

(131)

Consequently F and Ft can be complex time Fourier transformed according to equations (72) just as G.

In addition, G also obeys the boundary condition

G(l, 2 t = -G(l, 2)1

t =t.ijjp t =t 2 0 2 0

It thus follows that the matrix Green's function, g, satisfies the same boundary condition as G,

g(1,2)1 = -g(1,2~ =t ~ =t-i ll P ~

1 0 1 0 (132)

and can thus be complex time Fourier transformed according to equations (72).

The BCS Spectrum

We now specialize to the system considered by BCS; i. e., we assume:

(1) no external fields; (2) translational invariance in space and time; (3) an instantaneous effective interaction, as in equation (90).

The approximation of an instantaneous interaction Simplifies our problem considerably. This comes about because for tl=t2

*

That is,

F(l, 2)= - [Ft (2,l)J * for tl=t2 • (133)

Consequently equations (121a) and (123) are two coupled equations for the two functions G and Ft; equation (125) for F is unnecessary.

Spatial Fourier transformation of equations (121a) and (123) results in

IL/i.£... _ €,,)Gq')tC~11) = Jr S(t1-:tj ,J - (134) ~ Ot1 "" I'V

- .1. L V f}J:,-r,/) F(p')t=O) F ~ /> ,xi -tt ,)) 1" e:z tv IV

"...,

and

where

(P-pl)l ( = 6. - ~ V ff .......... _ < n(pl» £, p pi e 2 .....

is the normal phase quasiparticle energy in the HFA. In obtaining these results we have assumed

and

< n(-pb = < n(pb • ..... Equation (133) implies

F~,t=O)=- [Ft ~,t=O~ *. We also note that

* Y(l,2) = "'V(2, 1) = V(l,2)

implies

* VeffUs) = Veff (!9.

in. Ix -x ) e Ii." ~l -2 1 <n(pb for t = t ,

'Z'..... 1 2

(136)

Thus equations (134) and (135) can be written as

( ih..2.... -E )G(P't1-t1')=hO(t1-t1')+~(P)Ft(P,t -t ), at p ~ ~ ~ 1 I' 1 ~

(137)

and

(138)

where

~(p)=-i~ V ff(P-P') F(p',t=O). ,....... e ~~ ~

p (139)

The result of complex time Fourier transforming equations (137) and (138) is

(z-E jG(p, z)_~p)Ft (p, z)= 1 p ~ ~ ~

* t -~p) G(p, z)+ (z+E p)F (p, z)= o. ~ ~ ,....,

Solution of these two equations yields

Z+E.l2 =_~ G<.B, z)= 2 2

z -Eg 2Ep

and

where

Ep= [E E 2+ I 6(:} I 2] 1/2.

__ 1 __ ) , (140)

z+Ep

_1 ), z+Ep

(141)

(142)

The self-energy, defined according to equation (99)' is

2 E(E,z)= E -Ep+ I ~p)1 l(z+Ep ). (143) p ,....,

~ ~


The spectral function, defined according to equation (88), is

A(p,w)= W+E.e. Ep

Consequently the BCS quasiparticles have energies ±E£ and infinite lifetime.

A means of calculating .6.{p) is necessary to complete our understanding of the BCS quasiparticle spectrum. From equations (85), (138), and (144) we obtain

FT§( p)W) =7T ~(J?J'Ir{f(W) 1. [[(W-£"J -S(w+£,,)]. '" E., l-.f(WJ! IV IV

'"

then equation (139) leads to

(144)

l1(p)=-N

L p'

~ (.e-;eJ) t()'hh(~E~'/l) ~(pJ) (145) efT ~ )

'" 2. Epl tV

which is a nonlinear integral equation for the gap function, .6.{p).

We can easily understand why .6.{£) is called the gap function. Suppose, for simplicity, that Ep is an increasing function of I p-p F I and .6.{£) depends ong~ only via its magnitude. Then

Ep;?: E "" = 1.6.{£,) I I - I.6.{PF) I . ~ PF P p=p

F

The quasiparticle spectrum contains a gap of width 21~(pF) I centered about the Fermi surface - as shown in Fig. 2a, -and in accord with the second i.tem in our list of the main facts to be explained by a theory of superconductivity.

BCS originally solved the gap equation, equation (145), for the model potential

-v V eff (p-p') ==

for I E I, I E I I :s; wD p p ~ ~ (146)

o otherwise

where wn is a more or less typical phonon energy. The simplest solutionto the gap equation is then

6(p) = o

for I E,e' :s; wD

otherwise (147)

Since wD < < IJ., tl, taken to be a real non-negative quantity, is given as a function of temperature by

WD

j = N(O) V ~ de -WD

i-a.nhQf.2. + 6(T)2./Z k8 T) • 2 i~~ +,1(1")2'

The resultant form of 6(T) is in excellent agreement with the experimental results for weak-coupling superconductors.

(148)

The model of superconductivity following from the BCS theory and equation (146) is referred to as the BCS mod~ Equations (142) and (147) give the BCS quasiparticle spectrum, 31 and equation (148) is known as the BCS gap equation. Rothwarf has pointed out that it is the form of the gap equation, equation (145), rather than the form of Veff' which is most Significant in determining many features of superconductivity.

At absolute zero the BCS gap equation reduces to

1

N(O)V

This equation is readily solved to give

i::l.(O)=wD/sinh (l/N(O)V). (149)


The critical temperature, T c , is given by setting ~T c) = 0 in equation (148):

1

N(O)V

Since wD» 2kB T c' this gives

where

W e D

Y = 1.781072 ••• E

-1/N(O)V

is related to Euler f s constant,

C= O. 577215 ••.

according to

Y E

C = e •

(150)

-1/2 For weak-coupling superconductors, wD ex M , which explains the isotope effect.

From equations (149) and (150)

2~0)/k T = (2fT/Y )/Il - e -2/N(0)V) B c E\:'

In the weak-coupling limit, N(O)V « 1,

2~0)/k T ~ 2fT /YE= 3. 52~, B c .

which is in excellent agreement with experiment.

Meaning of the BCS Model

(151)

The BCS model form of the effective interaction, equation (146), can be written as

(152)


where

(153 )

The basic meaning is that only those electrons with energy within a shell of width ZWD centered about the Fermi surface are affected by the interaction which gives rise to superconductivity. On physical grounds then, it would appear that equation (15Z) is equivalent to using

V 3 t H. (tl ) = - -'62: fd x Iii (l),/i t (1),/. I (1) II. (1)

lnt Z I lT s T s I '/' s T s s, s (154)

with

~ (1) = 2: s

ip.x

p

e c ~ pSG (155 )

Clearly the interaction described by these two equations involves only those electrons with energy in the desired range. Here we shall prove that the model described by equations (15Z) and (153) is completely equivalent to that described by equations (154) and (155). *

We begin by noting that if we use equations (75) and write

ike (x -x ) V (xl' XZ) = 2: V flls,) e ~ ~ 1 ,~Z

eff ~ ~ kef (156)

then the interaction part of the Hamiltonian (57) becomes

1 t t H = - 2: 2: V (p -p) c c c c

int Z s Sip P P P eff~3 ~l p s P4 s " PZSI _~ls , ~ l,~ Z ,~3 ,~4 ~ 3 ~ ~ .-

x

,.~ See section 3Z. Z of Ref. 6.


With V eff given by equation (152), this becomes

If we write

1 J 3 -ipox 8 = _ d xe "" "" 8 (x), p V ""

"" 0

and note that

8p 8 8p a. • ",,1 £.2 ",,3~4

and use the inverses of equations (75), then

Hint=- ~ "10 ~SISdSr~d3~1Sdg~;tS dg~Si3S+ ~ J Yo Vo Vo Vo

X el!-EJ) e(,t-~) e(t- ~)€(~-~J"I-

X l/t?;&) ¥;~~) l/; {!t) lfs(£tJ· But now, suppose that

ip.x f(x)=2: f(p) e "" ""8 (p).

"" "" .E

Then

J d3~ Vo

Thus equation (158) reduces to equation (154) if (155) holds.

129

(157)

(158)


If we use the BCS model form of the effective interaction from the outset, then we find for the Green's functions equations of motion

(159)

= tz 6 (1-1')'T -iVV 'T g(l,lh g(l,l'). o 0 3 3

These equations are an appropriate starting point for the consideration of space and time dependent effects in superconductivity.

Thermodynamic Properties

The thermodynamic properties of a many-body system can be entirely determined from the grand potential, .0.. The calculation of .0. is facilitated by writing the Hamiltonian of the system as

H=H + ~ U. o

(160)

For)" = 1 H is the Hamiltonian for the actual physical system, while for X = 0 H reduces to Ha, which we take to be appropriate for a simpler system. If we define

r -P(H +,\ U -~N)] then

and

.n.A=-kBTlnZ).=":'kBTln [I're 0 , (161)

= Tr [e -'(Ho + A U-PN) uJ Tr[e-~(Ho+ AU-f'N~

.0. =.0. + 1 0

(162)

Thus we have the grand potential for the actual physical system, .0. l,in terms of .0.0 ' the grand potential appropriate to Ho ' which presumably is easy to calculate, and an integration of the interaction energy with respect to a variable coupling constant.

The interaction energy is readily expressed in terms of the single-particle Green's function, G. From equation (92a) we easily obtain,

For a space- and time-independent system this reduces to

.l.L; 00 < <V>::~ f dw (w-£)G (p,w).

p, s 2 'IT P ss ~ (163)

-00

If, in equation (160), we include the Hartree-Fock part of the effective electron-electron interaction in H , and if we assume no spin-dependent interactions, then we may ~rite

(164)

from which it follows that 1 00

\,-1 r dw J1.:: J1. o+f dl\l\. L;J_ - (w-€ )f(w)A,(p,w).

-00 2'IT p 1\ ~ o P ~

(165)

~

For our purpose we use the HFA for the normal phase and the BCS spectral function, equation (144), for the superconducting phase. Moreover, we as sume that the contribution of the lattice to J1. is the same in the two phases. We thereby obtain

1 2 J1. -J1. :: -fdAA -1L; I L::,. (p) I tanh (PE 12)

s n ~.E (166) o p 2E

P ~

For the BCS model this simplifies to

J1. -J1. :: S n

where

V 2 - f dV' (L::,.I IV') ,

o

L::,.I,! L::,. (V I),

(167)

we have replaced A. with the physical coupling constant, V, and we have used the BCS gap equation, equation (148).

More generally, if we use the HFA for the normal phase and the BCS model for the superconducting phase, but do not assume translational invariance, equation (162) leads to

V 2 3 .n.s - ~ = - S dV'(V')- S d xl (168)

o Vo

Here the space and time dependent order parameter is defined by

.6.(l)=iVV F(l,l), o

(169)

which is consistent with equation (139). Obviously equation (168) reduces to equation (167) for a homogeneous system.

Equation (167) can be re-expressed as

.6. 2 .n. s - .n. n = S d.6.' (.6.') d(l/V')

o d.6.'

If we use the BCS gap equation, equation (148), to evaluate d(l/V') / d.6.', then integrate by parts, and carry out a small amount of algebra, we obtain

Within the HFA, the electronic part of .n.n is given by

.n. ~ 4N(0)SWDdE~.!.. ln f(E)+ E] ~ - ('IT 2/ 3)N(0)(kBT)2, (171) en- ~ -

o

where, in the first approximate equality, we have assumed P wD»l. Thus the electronic part of .n.s is given by

.n.es :::: N(O)SWDdE{.6.~ [l-2f(E)] + ; ln f (E)+ 2(E+ d} , (172)

o and, after a small amount of algebra,

A es '" -IN(O) { d< l <; + L"z £(E) -N(O)"'n t 1+ (LV "'n) l],,-1. (173)

This result is equivalent to equation (3.37) of BCS.

Since .6.-40 and E ~ lEI as T-7T c' equation (170), together with the as s umption that the lattice contribution to .n. is the same in the normal and superconducting phases, has the immediate consequence

ELEMENTS OF THE THEORY OF SUPERCONDUCTIVITY iS3

(.11.. -.11..) I =0. s n Tc

(174)

From equation (172), the entropy,

S --es (0.11.. /dT)

es V~, 0'

(175)

is given by

S es:::::::" - 4N(0)kB jdE{f(E) In f(E)+ [l-f(E)] In [l-f(E)]}. (176) o

This is what we should expect for a collection of independent quasiparticles with the BCS spectrum. Obviously

(177)

so that the normal-super transition with no applied magnetic field is of the second order. Yet another differentiation with respect to temperature shortly leads to

(Cs-Cn)I T = IN(0)/VoJkBfJc2(0.6.2/0~)IT • (17S) c c

Differentiation of the BCS gap equation, equation (14S), with respect to ~ yields

(0.6.2 /o~) IT = ~ c -3 STl/7 , (3), c

(179)

where ~ is the Riemann zeta function. The last two equations then yield an equation which is equivalent to equation (3.50) of BCS,

(C -C ) /yT I = 12/7'(;. (3) = 1.426, s n -Tc

(ISO)

which is in reasonable agreement with experiment.

In the presence of a uniform applied magnetic field, ~, the grand potential is given b y 32

.11.. (Ha)=.n.(0)-Sd3xS1JadH' .M(x,H' ), ""-' ~a ~ ""-' ""-'a (lSI)

o where M is the magnetization vector, and T, V , and ~ are held constant. For a bulk sample with demagnetiza'fion coefficient


D, the magnetization vectors in the normal and superconducting phases are

M = ~n

~n -1 M = -, ~s (182)

4 '" (l-D)

where ~n is the permeability of the normal phase. For such a situation, the field within the sample reaches the critical value, H , when c

Thus, if we neglect the changes in V and ~ at the phase transition, and neglect penetration e~fects, the requirement

.n. ( HI ) =.n. ( HI ) S c n c

results in 2

sn (0) - .n. s(O) = V oiJ.n Hc 18," •

Using equations (171) and (173), we find i

~n Hc 2 18," ~ [N(O)/V oJ {WD 2 [1+ (~wD) 2] -wD 2

[

00 2+ 1:::.2 } - (-rr2/3~2 1 - (6 P2 /-rr2){ dE 2E E f (E)]

(183 )

(184)

(185)

(186)

which is equivalent to equation (3.38) of BCS. (The normal phase of most superconducting materials is non-magnetic, i. e •• ~ = 1.) The resultant curve of H versus T IT agrees reasonably weff with equation (5a) and, inc fact, yields getter agreement with experiment in many cases.

At absolute zero (and in the' wD » 1 limit), equation (186) gives _ c ....

Ho~[(4,"/iJ.n)N(0)Vo] 21:::.(0). (187)

If we calculate ')1, defined by equation (2). from equation (171). then we find that the theory predicts a universal constant,

2 2 2 'Y T c 7~ nHo = 'Y E /6," = 0.16~, (188)


independent of the material in question, and again in satisfactory agreement with experiment.

From equation (186) we find the t,.::::.O and t =.1 approximations to Hc to be

Hc(t):::::: Ho ~_(YEZ/3)t2]: Ho (1-1.06 t 2),t:::: 0 (189a)

H (t) ~ H Y [8/7 S (3)]:2 (l-t)::: 1.74 H (l-t), t,.::::.l. c - 0 E 0

(189b)

We note that the slope of the Hc versus t curve vanishes at t::: 0, in accord with the third law of thermodynamics, and is finite and negative at t::: 1, in agreement with experiment.

Current Density-Vector Potential Relation*

The task remaining bef ore us is to derive the current densityvector potential relation for a superconductor and to show that this relation implies the Meissner-Ochsenfeld effect. It is possible to carry out this calculation in a fully gauge invariant manner, 33t and strictly speaking this is what we should do. However, the gauge invariant calculation is rather complicated and its details tend to obscure the comparatively simple physical results. We thus choose to follow BCS and work within the London gauge,

'il • A ::: O. (190)

The full Hamiltonian given by equations (57) and (89) may be written as

(191)

where Ho is the field independent part and

H~:::~ Sd3xl{2:Uci[l/,ts(1)'ill~S(l) - ('ill ~t(l)~s(l)] .~(l) +

+ e 2 [A(l)l 2 IjJ s t (l)~s(l)l. 2mc2 ~

(192)

* The calculation given here is essentially that in Section 52 of Ref. 7.

t See also Chapter 8 of Ref. 4 and Section 4.7 of Ref. 16.

Using the rule for expressing a one-particle operator in second quantized form, the electric current density operator,

J(l) = - ~ L; [v. 6(x.-r.) + 6(x.-r.) v'1 ' "-' 2i ~ ~~ ~~~

rna y be written as

e 2 ;!,(l) = 1 0 (1) - ~(l)n(l),

mc

where

= -~ L;[~t (1) 'i11~ (1) -('i11,1J t (1) ) ,I. (1)] 2ml s s s Ts is

and

t n(l) = L; I/J s (l)~s(l) •

s

(193 )

(194)

(195)

We wish to calculate < 2:,(1) > 1::;' the grand canonical ensemble

average of J in the presence of the vector potential. It is readily shown 34* for any number conserving operator, e (1), that

. \ [ ] 2 < e(lb~ = < e (1» - ~ f dt' < e(l), HA (t') > + 0(1::; ),

-00

(196)

where the unlabeled brackets on the right denote a grand canonical average in the unperturbed (~= 0) ensemble and the operators on the right are in the modified Heisenberg picture defined in terms of H. With the observation that o

< J (1» = 0, ~

equations (192)-(196) lead to

e 2 < J(lb = - ~(l) <n(lb +

"-' A mc (197)

+ ~c /1 dt2 fd 3x z < [1.0(1),20(2) . ~(2) J > + 0(~2). -00

Thus the part of the electric current density which is linear in A may be written in the form of equation (6a), or more conventio~ally

,~ See also Section 32 of Ref. 7 and Appendix Al. 5 of Ref. 33.

<dPbA = - ~ 'rT f d21t (1,2) • ~(2),

with

~(l, 2) =

where

... < n(l) > 6 (1-1I)'ll

137

(198)

~R P (1,2), (199)

(200)

Thus our problem reduces to the evaluation of the grand canonical average of the retarded current commutator in the unperturbed ensemble.

The retarded commutator is inconvenient for our purposes. We prefer to work with a time-ordered product. Hence we define

"it (1,2) = - < ']I' J (1) J (2» • -"'0 -.10

Obviously

~ (1,2) = - i [~> (1,2) -1 «1, 2)J e (tl -t2 ),

and so it is clear that we can obtain ~ from ~.

(201)

(202)

A less direct but more useful connection between ~and~R can be obtained as follows. With the use of equation (194) we may write

1(1,2) = _(eli) 2~ r('i7 l -'i73)('i7 2 -'i74) G2(1s,2SI;3s,4SI~ 2m s, Sl L :1

It is easily established that

,(203)

3= 1+ 4= 2+

G2(ls,2SI;3s,4SI),' = - G (ls,2S I;3s, 4S I)1 ' t t 2 t· =t -ilifJ i= 0 1 0

where i= 1 or 3. This implies that .. ~

6>(1,2)ltl

__ to=@(1,2)\, tl =to-ilifJ

(204)

138 W. N. MATHEWS Jr. ~

which in turn implies that tP can be Fourier analyzed according to equations (72), but with

~ = ( 1T / -i~ )2\1 (205) (J\I .

appearing rather than z\l as given by equation (73). Then, with a small amount of algebra, we obtain

(206)

with

p=@> ~ , (207)

where 03 ~ (c:,.!)S';w) are defined analogously to equations (7l), but without the (+). In addition, equations (202) and (207) lead to

00

~ ~ ) = f dw ' P-- X x"w ,~ ,

-00 21T

~ f (x, x' ;w ' ) ~~ (208)

w-w'+ if]

where we have time Fourier transformed ~R accowng to equations (7l) without the(±). Thus, once we obtain UJ (~,~'; r\l) ~ need only make the replacement d\l -" w+ iT/to obtain

(~,~' ;w).

To proceed with the calculation of 'St , we make the Gor'kov factorization in equation (203), thereby obtaining

1(1, 2)= -2(·~)2«\ll-\l3)(\lZ-\l4) [G(l, 4)G(2, 3) -F(l, 2)Ft (4, 3)111 , 2ml l ~3=H

4= 2+

(209) and

~ .) 21 1 If' (q, "') = -2 ( eni'i1 -V ~ --:--~ ~ (p+ q/ 2)(p+ q/ 2) x

\I D -1 I ~ ~ ~ ~ o~ \I .

(210)

xrG(p+q,z I +~)G(p,z ,)+F(p+q,z I + ~)Ft(p,z I)] . l ~ ~ 'J (J'J ~ 'J ~ ~ 'J (f'J ,..., 'J

In evaluating equation (210) we use equations (140) and (141) for G and Ft, respectively. We also note, say from equation (125), that for the BCS model

F (,e, z) = Ft(,e, z). (2ll)


The frequency sum may be evaluated with the use of the equation

which follows from

lim ~ e

B

z - C" v

= ---17~0+ V z-x

V

= AB; [f(e ) -f( e.') ] , e -e -tv (ZlZ)

In equation (Z12) A and B are any two quantities independent of v. It is also convenient to note that

~ -(p (:!" - lv) = ~ (:!,. 7\;)'

which can be seen from equation (ZlO) with the aid of the transformation

p~ _p_q, Z I ~ Z I - "J-. • ~ ~ ~ V V (Iv

In this way we quickly obtain

Qj(~::7)=-i(e!)2~O ~££ (u+u_+v+vJ2(f+-f_) x p ~

x[;,+ (E+ -E J -'r, -(E+ -E J]- (u+ v - - u_ v + )2(l_f+ -f_) x

x[~,+ (E++EJ - ~,-(E++EJ 1 where p has been replaced by p-q/Z,

~ i ~ ~ ! ~ = ~EE+ E"£)/ZE£] ,vE= [(E:e,-Ej»/ZE,g]

E = E ± p± q/Z

~ ~

* See Appendix A of Ref. S and Section Z. S of Ref. 7.

(Z13)

(Z14)

(21Sa)

(21Sb)

(Z16)


and

(Z17)

We~ow make the replacement"}' -+ w+ i'1 in equation (Z14) to obtain pR (q,w), and then insert the "i-esult in equation (199) to obtain ~

k(q,w)= 47Tne2 ! _ 87T(~) Z_I_ ~ P P L (w, E'_, E'+), mc l mc Vo p ~ ~

(Z18)

where

L(w,E' _,E'+)=i(l+ E'+C+6.Z ) (f+- C )[ 1 _ 1 J E+E_ w-(E+-EJ+i77 w+(E+-EJ+i~-

(Z19)

_!(l_E'+E'_+6.Z

)(l_f+_fJ r 1 1] 4 E+E_ Lw-(E+ +EJ+i71 w+(E+ +E_)+i71 •

The funct~g~ L is precisely the function defined by Mattis and Bardeen 'in their theory of the anomalous skin effect in super-conductors. Thus the remainder of their results, notably equations (3.3)-(3.5), follow. Moreover, equations (198), (ZI8) and (Z19), in the w~O limit, are completely equivalent to equations (5.14)-(5.16) of BCS. Following Section 5 and Appendix C of BCS, the explicit form of the kernel in the w ~ limit is found to be

H K (1, Z)= J(R, T) () (tl-tZ)' (ZZO)

A (T) = _m ___ ",-----

ns (T)e Z (221)

where the "superelectron density" is given by

Z n (T)=n+ 'h_...,..-_

s 37TZ m (22Z)

* In comparing equation (3.1) of Ref. 35 with our equation (ZI9), we must note that our wand 71 are equivalent to - 'h wand 'hs of Ref. 35.


and the coherence distance is given by

and

n J(R,T)=

ns(T)

~T)

~O)

2 co - - ~T) f d(

'TT 0

. (2R ) l-2f(E) ] sln -- ( llv ( E

F

This result for ~ is in close agreement with equations (6).

141

(223)

(224)

To demonstrate the existence of the Meissner-Ochsenfeld effect, we consider the long wavelength Fourier components of the current density, or equivalently consider 1b.e vector potential to be slowly varying compared to the kernel R. With the use of the relation

co

f dR J(R, T) = ~ , o (225) o

which follows from equation (224), we obtain the London equation,

C A (T) :[ = -~, (226)

which is known to lead to the Meissner-Ochsenfeld effect. 36

In the Pippard limit (~o » A. ), which is the usual case except very near T c' the penetration depth is found to be .

[~T) ] -1/3 A. (T) = A (0) ~O) tanh (~T)/2kB T) •

Except near T , this result is very close to the empirical behavior as gi~en by equation (3).

Conclusions. BCS and Beyond

(227)

We have now completed the program which we initially laid out. The results of the BCS theory are in complete agreement with the list of the main experimental facts known about superconductivity midway through the decade of the fifties. Of course, considerable experimental and theoretical progress in


understanding the superconducting phase has been made since then. Moreover, the list of properties and effects associated with the superconducting phase is now much more diverse. Consequently, as a field of research and a source of technological applications, superconductivity is now much more complicated and much richer than it appeared to be at the time of the initial development of the microscopic theory.

We cannot hope to discuss or even enumerate all the various aspects of the superconducting phase which have entered the scene in the last decade and a half. We shall content ourselves with merely mentioning a few of those aspects which appear to be of importance for the fu ture.

The largest critical magnetic fields and current carrying capacities are to be found in the Type II materials. 37* Thus these materials are of technological importance, in making magnets for example, and perhaps eventually for use in power transmission. Among their distinctive characteristics are: (1) the ratio of the penetration depth to the coherence distance, A / ~, is greater than l/..['[; (2) between Hcl and H 2 (the lower and upper critical fields, respectively) the normal cJ=nductor-superconductor phase transition is of the second order; (3) between Hcl and Hc2 the magnetic flux penetrates the superconductor in the form of individual flux quanta or vortices. The latter condition is known as the mixed state.

The occurence of gapless superconductivity has led to the realization that a non-zero superconducting order parameter is not necessarily accompanied by an energy gap in the quasiparticle spectrum. The implication is that superconductivity is characterized by pair correlations rather than a positive minimum quasiparticle excitation energy. Among the pair-breaking mechanisms which lead to gapless superconductivityt are magnetic impurities, an exchange field, a mag~stic field (applied to a small specimen), and an electric current.

The behavior of a superconductor when subjected to a strong magnetic field has been the subject of recent intensive investigation. For Type I materials a macroscopic arrangement of alternating normal and superconducting domains - the intermediate state - results. 39:t. The mixed state is the analogous configuration

* See also the articles by Fetter and Hohenberg, Chapter 14, and Serin, Chapter 15, in Ref.l.

t See Maki l s article, Chapter 18, in Ref. 1. * See also the article by Livingston and DeSorbo, Chapter 21, in Ref. 1.


for Type II materials. The transport properties under these circumstances, most notably flux flow in the mixed state, * are of particular interest.

The classical isotope effect (a = 1/2) has been found to be in no sense universal. Values of a ranging from approximately 1/2 for the strong-coupling materials, for which the Coulomb interaction is least important, down to about -2 for a -uranium have been observed. The theoretical understanding of the isotope effect requires a realistic treatment of the electron-electron interactions. t

With the exception of the isotope effect, the properties of the strong-coupling materials differ measurably from the predictions of the BCS theory. The theory of these materials has been worked out in considerable detail, resulting in substantial agreement with experiment. 4,25,26:1: The role of phonons in superconductivity has been strikingly confirmed by the results of tunneling experiments with strong-coupling superconductors.

Tunneling itself has come a long way since the early days of Giaever tunneling. It has developed into a sensitive tool for determining the properties of the superconducting phase. Moreover, tunneling has recently been used to probe the excitations of the barrier region and the metal-insulator interface. 40 There are even some strong indications that tunneling into bulk single crystal superconductors can be developed into a tool for probing the details of the F~ii~surface and the interaction giving rise to superconductivity. '

Perhaps the most exciting '\rd far reaching development concerns the Josephson effect. The revolution in the voltage standard brought about by the Josephson effect will undoubtedly be complete in a very few years. Moreover, the "weak link" behavior characteristic of the Josephson effect is central to a number of devices that are now coming into use.

One of the main thrusts for the near future will be directed toward understanding the influence of specific material properties on superconductivity. On the one hand, such investigations are the prime hope for finding superconducting materials with higher critical temperatures, critical fields, and current carrying

* See the article by Kim and Stephen, Chapter 19, in Ref.l. t See the articles by Rickayzen, Chapter 2, and Meservey and

Schwartz, Chapter 3, in Ref. 1 • . * See also Scalapino's article, Chapter 10, in Ref.l. I!! See Mercereau's article, Chapter 8, and Josephson's article,

Chapter 10, in Ref. 1.

capacities. On the other hand, only a materials approach can lead to an understanding of how to make materials with desirable superconducting properties in a form such that they can be made into usable wire and otherwise fabricated.

On the theoretical side, there are four additional calculational tools that should prove to be of considerable use in studying the effects of external fields and investigating space and time dependent phenomena in superconductivity.

The Ginzburg-Landau theory, ':' though derivable from the Gor'kovequations, must really be considered a distinct technique. It is the one calculational method in the theory of superconductivity which is closest to passing into engineering practice. A time dependent version of this theory has been worked out only for T -:: 0, T ':::' T c' and for very dirty materials. Obtaining a more general time dependent Ginzburg-Landau theory or its equivalent is a task for the future.

The self-consistent field method t together with the JWKB approximation offers a usable tool for dealing with spatially inhomogeneous systems 39 , 43-45 and will undoubtedly find wide application in this connection. This method may even be developed into a usable technique for time varying situations.

Eilenberger has derived a set of transport-like equations which are valid at all temperatures and from which the superconductor order parameter and the magnetic field can be found. 46 This approach was motivated by the observation that the forbidding complexity of the Gor'kov equations stems, at least in part, from their generality. Essentially, Eilenberger integrates the Gor'kov equations with respect to the energy variable, and so obtains somewhat simpler equations. He also shows how to calculate the grand potential in terms of the solutions of these simpler equations. This method has been used to investigate the structure of an isolated vortex, 47,48 and will undoubtedly find wide application in the theory of inhomogeneous superconductors.

Recently a constrained density matrix method, which has previously found wide application in molecular physics, has been utilized to obtain the properties of the BCS ground state, 49 This method offers considerable promise for dealing with spatially inhomogeneous systems, and for incorporating experimental results directly into a calculation of the properties of the superconducting phase.

'!< See, for example, Werthamer's article, Chapter 6, in Ref. 1. t See Ref. 19, Chapter 5.


Many of these problems and techniques, and many more as well, will be dealt with by the subsequent writers in this volume.

Thus our long chapter on the microscopic theory of superconductivity draws to a close. We have tried, on the one hand, to present a clear physical picture of the basic properties of the superconducting phase. On the other hand, we have attempted to provide an understandable introduction to the use of thermo-dynamic Green's functions in the microscopic theory of superconductivity. To the extent that we have succeeded, we have laid a foundation for the careful reader to build upon in broadening and deepening his understanding of the phenomenon of superconductivity. At this late date, a decade and a half after the creation of the microscopic theory of superconductivity, numerous obituaries have been written on the demise of superconductivity as an active and ongoing field of research. Nevertheless, there remains much to be done. The study of space and time dependent and nonequilibrium effects is well along, but by no means complete. Our understanding of the Josephson effect and its many ramifications leaves much to be desired. Our comprehension of the role of specific material properties in determining the critical temperature, critical fields, current carrying capacity, and mechanical workability of superconductors is little more than primitive. Consideration of these and other open questions will contribute to our understanding of the basic physical phenomena. In addition, and perhaps more importantly, such considerations will playa decisive role in determining whether or not the budding promise of technological applications of superconductivity comes to full £lowe r •

REFERENCES

1. For a comprehensive treatment see Superconductivity, edited by R. D. Parks (Marcel Dekker, Inc., New York, 1969).

2. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev.-108, ll75 (1957).

3. For a historical survey and a status report on superconductivity written shortly after the original work of BCS, see J. Bardeen and J. R. Schrieffer, in Progress in Low Temperature Physics, Vol. 3, edited by c. J. <::r"orter (North-Holland, Amsterdam, 1961), pp. 170 ff.

4. J. R. Schrieffer, Theory~ Superconductivity (W.A. Benjamin, Inc., New York, 1964).

5. L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics (W. A. Benjamin, Inc., New York, 1962) •.

6. A. A. Abrikosov, L. P. Gorkov, and 1. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Prentice:-Hall, Inc., Englewood Clufs, New Jersey, 1963).

7. A. L. Fetter and J. D. Wallecka, Quantum Theory of ManyParticle Systems (McGraw-Hill Book Co., New York-;- 1971).

8. D. Bohm and D. Pines, Phys. Rev. E, 609 (1953).

9. D. Pines, Phys. Rev. E, 626 (1953).

10. A. Rothwarf, Phys. Rev. B2, 3560 (1970).

ll. E. Maxwell, Phys. Rev. 78, 477 (1950).

12. C. A. Reynolds, B. Serin, W. H. Wright, and L.B. Nesbitt, Phys. Rev. ~, 487 (1950).

13. H. Frohlich, Phys. Rev. '!.1.., 845 (1950).

14. J. Bardeen, Phys. Rev. '!.1.., 167 (1950);~, 567 (1950); ~ 829 (1951).

15. L. D. Landau, JETP ~, 920 (1957).

16. D. Pines and P. Nozieres, The Theory of Quantum Liquids, Vol. 1 (W. A. Benjamin, In~New York, 1966).

17. W. A. Little, J. Polymer Sci., Pt. C, No. 29, 17 (1970).


18. R. S. Schneider, J. Polymer Sci., Pt. C, No. 29,27 (1970).

19. P. G. deGennes, Superconductivity ~Metals and Alloys (W. A. Benjamin, Inc., New York, 1966), pp. 99-102.

20. L. N. Cooper, Phys. Rev. 104, 1189 (1956).

21. P. W. Anderson, J. Phys. Chern. Solids & 26 (1959).

22. Gordon Baym, Lectures on Quantum Mechanics (W. A. Benjamin, Inc., New York, 1969), Chapter 19.

23. D. Falkoff, in The Many-Body Problem, edited by Christian Fronsdal (W. A. Benjamin, Inc., New York, 1962), pp 1-35.

24. Kerson Huang, Statistical Mechanics (John Wiley and Sons, Inc. New York, 1963), Appendix A. 3.

25. Leo p. Kadanoff, in Lectures on The Many-Body Problem, Vol. 2, edited by E. R. Caianie110 (Academic Press, New York, 1964), pp. 77-112.

26. D. J. Scalapino, J. R. Schrieffer, and J. W. Wilkins, Phys. Rev. 148, 263 (1966).

27. L. p. Gor'kov, Soviet Phys. -JETP 7..., 505 (1958).

28. N. N. Bo~oliubov, Soviet Phys. -JETP 7,41 (1958); Nuovo Cimento !.XJ 7...,794 (1958); Soviet Phys. -Uspekhi ~,236 (1959).

29. J. G. Valatin, Nuovo Cimento [X] 7...,843 (1959).

30. Y. Nambu, Phys. Rev. 117,648 (1960).

31. A. Rothwarf, Phys. Letters 24A, 307 (1967).

32. L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media (Addison-Wesley Publishing Co., Inc., Reading, Mass., 1960), Chapters IV and VI.

33. G. Rickayzen, Theory of Superconductivity (Interscience Publishers, New York;-r965), Chapter 6.

34. p. Nozieres, Theory ~Interacting Fermi Systems (W. A. Benjamin, Inc., New York, 1964), Chapter 2.

35. D. C. Mattis and J. Bardeen, Phys. Rev. Q!,412 (1958).

36. Frtiz London, Superfluids, Vol. 1 (Dover Publications, Inc. , New York, 1960), Section B.


37. D. Saint-James, G. Sarma, and E. J. Thomas, ~.!!.. Superconductivity (Pergamon Press, Oxford, 1969).

38. J. Bostock (private communication).

39. ReinerKumme1,Phys. Rev. B3, 784 (1971).

40. J. Lambe and R. C. Jak1evic, in Tunneling Phenomena ~ Solids, edited by Elias Burstein and Stig Lundqvist (Plenum Press, New York, 1969), Chapters 17 and 18.

41. G. 1. Lykken, A. L. Geiger, K.S. Dy, and E. N. Mitchell, Phys. Rev. B4, 1523 (1971).

42. W. D. Gregory, R. F. Averill, and L. S. Straus, Phys. Rev. Letters 27, 1503 (1971).

43. W. N. Mathews Jr., Ph.D. Thesis, Univ. of Illinois, 1966 (unpublished)

44. Reiner Kumme1, Phys. Kondens. Materie 10, 331 (1970).

45. John Bardeen, R. Kiimme1, A. E. Jacobs, and L. Tewordt, Phys. Rev. 187, 556 (1969).

46. Gert Eilenberger, Z. Physik 214, 195 (1968).

47. Gert Eilenberger and Helmut Buttner, Z. Physik 224,335 (1969). -

48. M. C. Leung and A. E. Jacobs (to be published).

49. W. L. Clinton (to be published).

JOSEPHSON EFFECT IN A SUPERCONDUCTING RING>\F. Bloch Department of Physics Stanford University Stanford, California

I. INTRODUCTION

It was pointed out in a preceding letter' that the Josephson effect2 •3 can be interpreted as the direct consequence of general principles if one considers the geometry of a superconducting ring interrupted by a barrier. In close analogy to the earlier explanation' of quantized flux trapping, the periodic dependence of the current upon the flux through the ring was found sufficient to account for the essential features of the effect. The considerations were restricted to the simplest case where the flux through the ring has a single well-defined value. This situation is encountered if the ring is sufficiently thin to neglect the variation of the flux caused by penetration of the magnetic field into the material or, equivalently, if the shielding by the supercurrent is sufficient to prevent any appreciable penetration. It was shown in particular that one obtains in this case the Josephson relation between voltage and frequency of the current.

A rigorous and general derivation of this relation will be presented in Sec. n, including the previously omitted proof of the periodic flux dependence. lnstead of demanding a single value of the total flux, it merely demands this property be satisfied for the part which arises from external sources. Irrespective of any specific assumptions concerning the ring, the Josephson relation will be seen to strictly apply to a reversible process and to refer to the voltage which is induced by a time-dependent external flux through the opening surrounded by the material. Section ill deals with additional conSiderations concerning the effects of pairing, the presence of a barrier, branching, and deviations from reversibility.

A more detailed discussion for the case of a thin ring will be presented in Sec. IV. It shows that the presence of a barrier is essential for the pos-

94305

sibility of reversible alternations of the current and includes the criteria for the occurrence of irreversible processes, accompanied by hystereSiS, as well as the consideration of dynamic effects. The effects of a barrier are further discussed in Sec. V and extended to take field penetration into account.

While the considerations presented here do not lead to results other than those originally obtained by Josephson, they differ in regard to their derivation and interpretation. Instead of using the specif. ic theories of Ginzburg and Landau, or of Bardeen, Cooper. and Schreiffer. the treatment is based upon some fundamental facts of electrodynamics and quantum mechanics which bear upon the characteristic properties of the superconductive state. It is not possible without specific reference to the microscopic explanation of these properties to evaluate the coefficients which determine the magnitude of the supercurrent. While their change upon the introduction of a barrier will be investigated, there remains a factor of proportionality for which no more than qualitative arguments can be offered within the framework of the present treatment. Since the consideration of a closed ring is essential, the method, furthermore, is not directly applicable to the effects of a voltage across the barrier in an open-ended geometry. The inclusion of this case requires as a separate assumption that the conditions at some distance from the barrier are immaterial to the manifestation of the Josephon effect.

On the other hand, the conclusions reached here are not affected by the unavoidable approximations inherent to the phenomenological or microscopiC approach in the theory of superconductivity. They are particularly suited, therefore, to clarify the reasons for the exact validity of the Josephson relation.

>~ork supported in part by the Office of Naval Research, Contract No. Nonr 225 (75). Reprinted by permission from Physical Review B, Vol. 2, No.1, pp. 109-121 (July 1, 1970).

149

150

II. GENERAL PROOF OF JOSPEHSON RELATION

The application of a dc voltage V results according to Josephson in an alternating current of frequency

v=2eV/h

across the partitioning barrier of a superconductor. Whereas an open-ended geometry permits the use of a battery as voltage source, it is necessary in the case of a superconducting ring to deal with the voltage induced by a time-dependent flux of the magnetic field tl>t"ough the ring. In order to investigate the effect of this voltage, one has to consider that the electromagnetic field arises from the superposition of two parts. One of them is the field contributed by the charged particles in the ring. In addition to the particles themselves, this field is to be regarded as a constituent of the system formed by the ring and is to be described by a separate set of dynamical variables. The other part consists of the field due to external sources which can be arbitrarily controlled so that It enters into the description of the system through a set of adjustable parameters rather than of dynamical variables.

Accordingly, the Hamiltonian representing the total energy of the system in a given external field is to be considered as a function of the dynamical variables which pertain to the particles as well as of those which characterize the field contributed by their charges. While a partial elimination of field variables permits one to express electromagnetic interactions In terms of particle variables, It is neither necessary nor convenient to assume that such an elimination has been carried out. Similarly, it is possible to partly eliminate the vlLriables pertaining to the ions with the result of an effective additional interaction between the conduction electrons. This interaction is essential for the pairing process which leads to the superconductivity of electron systems, and will later be taken into account to obtain the Josephson relation in the form of Eq. (1). The deeper roots of this relation are more eVident, however, if the ions are treated as constituent particles of the system on the same basis as the electrons and without explicit reference to their role in the pairing process. It is sufficient at this stage to assign to each of the N particles a definite charge according to its indivdual characterization as an ion or an electron.5

Retaining complete generality, the Hamiltonian of the system shall be denoted by

:JC=:JC[Pr e IA(rl )Ie, r l 1 (2)

in its dependence on all the momenta PI and coordinate vectors rl of the particles with charges

F. BLOCH

el (j ~ I, 2· •• N) In the presence of the vector potential A( r). Although the notation only emphasizes this particular dependence, a further dependence on spin variables of the particles and on field variables shall be understood without being explicitly indicated. The total magnetic field H = curIA derived from the vector potential includes the contribution due to external sources. Writing

A=Ao+A.,

this contribution shall be given by

HI = curiA.,

and it is essential that it only enters Into the Hamiltonian through the combination of A and ~ which appears in Eq. (2).

(3)

It will be assumed for the purpose of this section that HI is a variable field which vanishes in the whole region R occupied by the ring but contributes the amount .1 to the flux through the opening 0 (Fig. O. Such a field is obtained, for example, from the variable current of a long solenoid passing through 0; any other field HI is equivalent, however, provided that its penetration into the region R is of negligible significance. As the line integral of the external electric field, the applied induced voltage

(5)

has under this condition the same value for any closed path in R which surrounds the opening o. The same condition will be seen to lead to an otherwise entirely general periodicity of the free energy and of the current circulating through the ring in their dependence upon 4\. The following proof is based upon the method used earlier' to explain flux quantization with the difference that It refers to the external flux .1 rather than to the total flux • through the opening.

Inserting A from Eq. (3) Into the Hamiltonian given by Eq. (2), one obtains the energy levels of the system by solving the equation

(6)

with the condition that the eigenfunction ,p(r/) is

FIG. 1. Superconducting ring R with opening 0, barrier B, and circulating current 1.

JOSEPHSON EFFECT IN A SUPERCONDUCTING RING 151

single valued in all particle coordinates r J irrespective of its implicit dependence on other variables. Whereas the magnetic field H, due to external sources can be arbitrary everywhere else, it was assumed that H, = 0 in the region R which contains the particles of the rtng, so that in view of Eq. (4) one can write A, = gradXI in R. Considering that 1jJ(rJ) has to vanish if the location r J of any particle is not inside that region, and noting further that Ii J = (h/21ri ) grad J' the gauge transformation

1jJ(rJ)="'0(rJ)exp[2m~eJ XI(rJ)/he) (7) J

leads from Eqs. (3) and (6) to

(8)

where :JC 0 is the Hamiltonian obtained by letting AI = O. Since 1jJ is single valued and since the line integral of AI around 0 increases XI by the amount

.AI·4S=~, 1jJo is multiplied by the factor exp(- 2m eJ ~/he) when the particle j is brought around the ring. With all the charges eJ of the particles given as positive or negative integer multiples of the elementary charge e, this factor repeats itself whenever ~ changes by the amount

(9)

The same repetition occurs in the solutions 1jJo and, hence, in the set of energy levels obtained from Eq. (8). Since the latter uniquely determine the partition function Q, it follows that the free energy F = - kT In Q is a periodic function of <1>, with period he / e. Granting the system under consideration to be invariant against time reversal, it is further seen that F remains unchanged if the sense of rotation around 0 and, thereby, the sign of ~ is reversed so that F must be an even function of <1> ..

The combination of these two properties permits the free energy of the system to be written as a Fourier series of the general form

F= ~F.cos21Tna" (10) •• 0

(11)

One should observe that the preceding proof of this important conclusion is entirely based upon some of the most fundamental principles. In fact, its validity requires no more than to accept invariance under a gauge transformation and under time reversal, together with the requirement of singlevalued wave functions and the elementary nature of the charge e. In particular, the effect of electromagnetic interactions is fully taken into account so that renormalization cannot alter the result.

Furthermore, the conclusion holds irrespective of any specific properties of the ring and thus remains valid in the presence of a barrier as a special feature concerning the potential energy of the electrons. Such properties will later have to be taken into account, however, to discuss the hitherto arb'trary magnitude of the coeffiCients F •.

The free energy refers to the thermal equilibrium c.1 the system at a fixed value of the flux <1>"

bllt retains its significance for variable values p, ovided that the variation is sufficiently slow to permit at any instant the establishment of equilibrium. One deals in this case with a reversible process and can use the thermodynamic relation for constant temperature that the rate of change of the free energy represents the work per unit time delivered to the system. Under application of the external voltage V and with a total current I circulating around the ring, one has dF/dt =1 V. With F given as a function of <1>1 and in view of Eq. (5), it follows that the current is obtained from the free energy by means of the relation

dF I =-e d<l>l ' (12)

which leads through Eqs. (10) and (11) to the result

1= E I. sin21Tn al , ... with I. = 21Tn e F./h •

(13)

(14)

This result has the same general validity as that derived for the free energy and likewise refers not only to constant equilibrium of the system but also to reversible changes under the influence of a time-dependent flux ~.

In particular, the application of a dc voltage corresponds to the linear time dependence

(15)

as the result of Eq. (5) for constant V with <1>1 = 0 chosen at the time t = O. By inserting the corresponding value of al from Eq. (11) into Eq. (13), the current

l=-E I. sin(21Tne V/h)t (16) •• 1

is seen to exhibit a periodic variation with a spectrum of frequencies

lI=neV/h (17)

which confirms Josephson's result in a generalized form. Indeed, the preceding proof did not specify the coefficients I. in Eq. (16) or the integer n in Eq. (17). It requires further considerations, presented in Sec. m, to arrive at the particular choice n = 2 for the Josephson relation in the form

152

of Eq. (1). Only the general form of Eq. (17) is needed, on the other hand, to allow the highly precise determination of elh from a measurement of frequency and voltage.' In fact, the fundamental character of this relation demands n to be an exact integer and the knowledge of e and ~ is sufficiently accurate beforehand to recognize the one and only integer which is compatible with the measurement.

The dc current caused by the simultaneous application of a dc voltage V and an ac voltage [V'cos(2I1v '( +'1'») represents another manifestation of the Josephson effect which can be confirmed with equal generality. In analogy to Eq. (16), one obtains from Eq. (13)

1=- ,EIn sin['if (2I1VI <: sin(2I1v 'I+<P») l (18)

A finite time average, and hence, a dc component of the current demands that the relation

n' v' =ne Vlh, (19)

with exact integers n and n', is satisfied' so that this effect provides an equally fundamental method for the precise determination of elh.

Ill. SUPPLEMENTARY CONSIDERATIONS

A. Off·Diagonal Long.Range Order and Pairing

In order to emphasize their fundamental character, the results of Sec. II were derived in such generality as to require no assumption whatever about the nature of conduction so that it seems irrelevant whether or not one deals with a superconductive ring. The basic equations (10) and (13), however, already anticipate the characteristic property of a superconductor to maintain a finite current in the absence of an applied voltage. Indeed, according to Eq. (13), any set of finite coefficients In allows the existence of a circulating dc current for constant a , or <1>, and hence, in view of Eq. (5), for a vanishing voltage V.

The fact that the coefficients Fn for n " a in Eq. (10), and therefore the coefficients In in Eq. (13), can have finite values has been recognized earlier to rest upon a special condition. The resulting flux dependence of free energy and current in the thermal equilibrium of a macroscopic ring requires, in fact, that a superconductor exhibits offdiagonal long-range order (ODLRO)' or, equivalently, a singular velocity distribution' of the particles responsible for conductivity. Applied to the conduction electrons it was recognized, furthermore, that the exclusion principle prevents this requirement from being met if each electron is considered to move as an individual unit and that the

F. BLOCH

supercurrent has to be attributed to the common motion of electron pairs. This circumstance is taken into account through the replacement of the charge e by 2e or, in view of Eq. (11) for a" through the specification that only coefficients Fn and In with even index n can be different from zero since they always appear in combination with na,.

The preceding conclusion was reached without particular reference to the microscopiC origin of the pairing process. Bardeen, Cooper, and Schrieffer'o have shown that it can be explained by an effective attraction between the electrons which arise from their interaction with the ions. Their theory can, in fact, be used to obtain definite values for the coefficients Fn and In. While it would be prohibitively difficult to calculate these values from a rigorous treatment of microscopiC processes, it must be remembered that any effect of the ions was included in the developments of Sec. n. The restriction to even indices is thus fully consistent with these developments and imposes no further specifications upon the coefficients as long as the ring is considered under quite general conditions.

B, Current Reduced by Interruption

Among such further specifications, those arising from the interruption by a barrier B (Fig. 1) are of particular importance. An uninterrupted ring has been understood in connection with flux quantization' to exhibit a pronounced dependence of the free energy on the flux or, correspondingly, to permit a sizable circulating supercurrent. Any such current is prohibited, on the other hand, in the limit in which the barrier acts as a complete interruption. A continuous variation of the coefficients In from relatively large to vanishing values must be expected in the gradual transition from the first of these two extreme cases to the second. In particular, the case of complete interruption can be asymptotically approached by a progressive widening of the barrier, since the transmission coefficient e of an electron decreases exponentially with increasing width. It will be shown in Sec. V that In is proportional to en if e« 1 so that for a wide barrier the dominant contribution to the current arises from the term with the smallest even index n = 2. Neglecting higher terms, one thus obtains from Eq. (13)

(20)

and from Eqs. (16) and (17), the Josephson relation in the form of Eq. (1). It is of interest to note that the corresponding expression for the current density, derived by Josephson [J. Eq. (3.11)) from the theory of Ginzburg and Landau, is obtained by replacing 411a, in Eq. (20) by the phase difference


of the order parameter on both sides of the barrier. ll

It is further to be remarked that an increasingly effective potential barrier does not offer the only possibility to cause a gradual reduction of the current. Such a reduction can also be achieved by a progressively narrowing constriction at some location of an otherwise uniform superconducting ring since such a weak link likewise leads to the ultimate prevention of a circulating current. Another practical method consists of the replacement of the barrier by an interrupting layer of normally conducting material. The transition from vanishing to large thickness of this layer similarly results in a gradual reduction of the supercurrent so that in the end merely an Ohmic current is permitted to pass.

C. Branching

The results of Sec. II can be extended to include the novel features which appear if the ring divides into several branches. For Simplicity, it will be assumed that there are only two branches s and I, joined by the portion r of the ring (Fig. 2). The external magnetic field will again be considered to vanish in the whole region R, composed of the sections r, s, and I, with the difference that besides the flux cI>, through the opening 0, it may have a finite flux cI>a through the opening Oa surrounded by the two branches.

In order to be single valued, the wave function 1/1 must remain unchanged whether a particle is brought around the ring passing through the branch s or the branch I. Upon elimination of the external

vector potential by a gauge transformation and considering that its line integral is equal to cI>, or cI>, + cI>a, depending upon whether one chooses a path of integration through r and s or r and I, respectively, the further arguments of Sec. II remain unchanged. It follows that the free energy is a periodic function of cI>, as well as of cI>, + cI>a (or .a) with period he! e. Taking time invariance into account, one thus obtains in analogy to Eq. (10)

F=L: F ... cos211[mOl,+n(OI,+OIa»), (21) .... where OIa=.a!(he!e). (22)

The current I. and I, through the two branches are obtained from Eq. (12) for the total current •

1=1. +1, (23)

and f rom the analogous relation dF

I,=-e dcI>a ' (24)

which results from considering the work per unit time in a reversible change of cI>a. With F given by Eq. (21) one finds that

I.=(211e!h)L: mF ... sin211[mOl,+n(OI,+OIa») , (25) .... I,=(211e!h)L: n F ... sin21l[mOl,+n(OI,+OIa)J. (26) ....

As discussed before, the pairing of conduction electrons demands the summations to be extended only over even indices m and n. In the presence of wide barriers B., B, with small transmission coefficients 0., 0, in the branches s and I, respectively, F ... is proportional to O~ O~. Retaining only the lowest powers with m = 2, n = 0 and m = 0, n = 2, Eqs. (25) and (26) reduce to

I. = f.a sin4"OI, ,

I, = I .. 9in4,,(OI, +OIa)

(27)

(28)

as an extension of Eq. (20), obtained for 012 = 0 with Ia = I .. + f'2' For a given time dependence of a, and, hence, for a given applied voltage according to Eqs. (5) and (11), the maximum of the total cur-

B~ rent 1= I, + I, through the circuit is given by

FIG. 2. Superconducting ring with opening 0, which contains two branches sand t with barriers B. and Be, joined by the portion r. The total circulating current I divides into the currents 1. and 1, in the branches. surrounding the opening 0,.

IIal=U:a+I:a+ 21 .. 1 .. cos4"OIa)'/2 (29)

as a special case of the interference phenomena discussed by Josephson (J. Sec. 3.2.2). The periodic dependence of the maximum current on a static flux .a, provided by a solenoid through the opening O2, has been experimentally established. ,a

D. Deviations from Reversibility

One is led to an additional consideration by recalling that Eq. (20), as well as the more general equation (13), refers to the ideal limit in which any

154

change within the system is completely reversible. For a given rate of variation of the external parameters, this limit represents the better appro xi -mation the more rapidly the system is able to adjust itself to the instantaneous equilibrium. In orde r to estimate the effect of small deviations, it will be assumed that the adjustment takes a certain relaxation time T such that the state of the system at the time t is that of the equilibrium at the slightly earlier time t - T. Accordingly, Eq. (20) will be modified in the sense that atl =e4\(t )/kc is replaced by atl + t.CYI = e 4\ (t - T )/kc. To first order in T, one thus obtains from Eq. (5) t.al = e V T Ik and to the same order from Eq. (20)

1 = 12 [sin4lTal + (4lTeVT Ik)cos4lTal]' (30)

Application of a dc voltage V still leads to an alternating current with the frequency v given by Eq. (1) but with a small phase shift for VT« 1. The presence of an Ohmic resistance Ro causes another deviation from reversibility and calls for an additional correction proportional to V. Combined with Eq. (30) and using the notation

(31)

one has then

1 = 12 sin4lTatI + (V cos4lT al)/R2 + V IRo, (32)

in agreement with the corresponding expression for the current density given by Josephson [J. Eq. (3. 10)]. A more pronounced manifestation of irreversibility is associated with the appearance of hysteresis and will be discussed in Sec. IV.

IV. THIN RING

The rigorous conclUSions of Sec. IT were reached under the condition that the external magnetic field HI vanishes in the region R occupied by the ring so that its flux through any closed curve around the ring has the same value "'I' Irrespective of the magnetic field, this greatly simplifying property of the flux can be used for a sufficiently thin ring and applies in this case not only to the part "'I contributed by external sources but also to the total flux "'. It can likewise be used for a ring of sizable thickness provided that the penetration of the magnetic field into the region R is of negligible significance. A major exception, to be considered in Sec. V, can arise through the presence of a barrier in such a ring since the magnetic field in the barrier may Significantly contribute to the flux through a closed curve and lead to differences of '" depending upon where the curve traverses the region of the field. For the purpose of this section, it will be assumed that no such differences are encountered and the abbreviating nomenclature of a

F. BLOCH

"thin" ring is meant to characterize this assumption.

Since the field contributed by the particles has to be taken into account in order to obtain the total flux, it is indicated to separate the energy stored in this field from the total energy of the system. Denoting the corresponding term by Je", one has for the total Hamiltol\ian 3('of Eq. (2)

:1('=3(" +Je" (33)

:1(''' depends only upon the field variables and the dependence of the term Je' upon the particle variables can again be expressed in the form

:J("= 1("[P/-e I A(i\)/c, r/], (34)

used to indicate this dependence in Eq. (2). Whereas the consideration of quantum effects and a statistical treatment are essential in dealing with the particles, it is permissible with entirely negligible errors to describle the field in classical terms and to ignore its statistical fluctuations. In particular, this allows to regard the total vector potential :A( r) as being uniquely determined and to obtain from the eigenvalues of Je' the free energy F' of the particles under the influence of the total magnetic field H = curtA. Under the conditions of a thin ring, the arguments, used in Sec. IT to derive Eq. (10), again apply if one replaces Al by

A, or "'I by "'= fA' ds so that

F' = t F~ cos2lTna (35) •• 0

with a = "'/(kcle). (36)

Upon the further replacement of F by F' , the derivation of Eq. (12) for the total circulating current 1 likewise remains valid, thus leading to

dF' 1=-c# (37)

or from Eq. (35) to

(38) •• 1

(39)

in analogy to Eqs. (13) and (14). While Je'leads to the free energy F' of the par

ticles, the other term:J(''' in Eq. (33) is responsible for the free energy F" which is stored in their accompanying field. Since thermal properties of the field can be ignored, no distinction between free energy and energy is here required so that the latter can likewise be denoted by F" It is further sufficient for the present purpose to consider only the magnetic field Ho caused by the Circulating current 1 through the ring. The energy stored in this field is given by LJ 2/2 where L is the self-induc-

tance of the ring. Using the relation ~o = LeI between the current and the flux ~o of the field Ro, one thus obtains

(40)

Considering that F' appears through Eqs. (35) and (36) as a function of the total flux

(41)

it is more convient by means of this relation to express F"likewise in terms of ~ and to write in analogy to Eq. (33)

F(~)=F'(~)+F"(<I» ,

where F"(~)= (~_ ~1)2/2Le2

in view of Eqs. (40) and (41).

(42)

(43)

Although F( ~) can In some sense be interpreted as the total free energy of the system, it is important to distinguish this quantity from the actual free energy F, given by Eqs. (10) and (11). Whereas F Is uniquely determined by the external flux ~l> it is seen that, given this part, the total flux and, hence, F(~) depend upon the variable value of the part ~o. The distinction arises from the fact that F refers to thermal equilibrium of the system where ~o has the definite value demanded by the equilibrium current. This value has to be inserted into Eq. (41) for in order to obtain the argument of the function F(<I» at which it is equal to F. Equivalently, F is to be characterized as the absolute minimum of this function. In fact, 

plays the role of a coordinate and F( <1» that of a potential energy for the dynamics of the system so that it will be in equilibrium at the absolute minimum of F(<I». 13 It will be necessary, however, to also consider the conditions of stable equilibrium which correspond to other minima of F(~).

With the flux measured in units of hc/e by means of the dimensionless quantities of Eqs. (11) and (36), one obtains from Eqs. (35), (42), and (43)

F(a)= ~ F~ cos2"na +h2(a - al)2/2Le2, (44) ... and an extremum of this function requires in view of Eqs. (38) and (39) that

I(a)=q(a- al)'

where q=h/Le .

In addition, one must have

dl da <q

(45)

(46)

(47)

in order to deal with a minimum of F(a) and, hence, with a stable equilibrium. The significance of Eq. (45) for the comparison of flux quantization

with the Josephson effect was previously discussed by means of a graphic representation. I It was shown, in particular, that the Josephson effect is to be understood as a consequence of a sufficiently reduced current I(a). Indeed, Eqs. (45) and (47) permit in this case only a single solution for a in the vicinity of a h corresponding to the existence of a single minimum of F(a) and, hence, to a definite equilibrium of the system for every value of a 1" The maintenance of this equilibrium represents the condition for a reversible change which was seen in Sec. II to lead from Eq. (13) for the current to the Josephson effect as the result of a linear variation of al. 1<

A different situation arises, however, if 1(1l) is large enough to allow several solutions of Eqs. (45) and (47), thus indicating besides the absolute minimum the existence of other minima of F(~). 1n order to study the transition to this case, it will be assumed that the reduction of the current is caused by a sufficiently wide barrier to result in a transmission coefficient 9 «1. 1n analogy to Eq. (20), only the term with n = 2 in the sum of Eq. (38) is then required so that

and from Eqs. (45) and (47)

I; sin4"a = q(a - al)'

4" I ;cos4"a < q .

(48)

(49)

(50)

Since the absence of an external flux ~I obviously permits a stable equilibrium with vanishing total flux~, the solution a = 0, obtained from Eq. (49) for a l = 0, must satisfy Eq. (50) for any value of q. By going: to the limit q - 0, it follows therefore that I; < O. Given a finite (negative) value of I; and a finite (positive) value of q, it now depends upon the ratio I;/q whether Eqs. (49) and (50) permit one or several solutions. The transition between these two cases can be seen to occur when I/;/ql = 1/4" so that one obtains the Josephson effect for II:!ql< 1/4".

It will now be assumed, instead, that I/;/ql > 1/4". Starting with the solution a = 0 for a l = 0, Eqs. (49) and (50) permit a continuous reversible increase of a with increasing a l until a has reached a value a. such that 4"/; cos4"a. =q. At that point, the system is in a state of indifferent equilibrium but the same value of a l permits one or several solutions of stable equilibrium, depending upon the magnitude of I/~/q I. The transition to such a new equilibrium with a at a new value a. will take place in an irreversible process whereupon a again increases reversibly with a further increase of al. This alternation between reversible processes repeats itself periodically upon a

156

a

FIG. 3. Circulating current I in a thin ring versus nux Ct, measured in units he/e. The straight sections (a,b), (c,d), (I,m) and (e,/), W,h), (I,k) with slope q indicate irreversible transitions at the values of the external flux "I in units of hc/ e, given by their intercept with the Q' axis I for slowly increasing and decreasing values of O'h respectively.

monotonic increase of a, and an analogous alternation with the accompanying hysteresis takes place upon the reversed change of a,. Both are illustrated in Fig. 3 with 112'lql chosen such that for each irreversible transition only a single final equilibrium is available and Fig. 4 represents the corresponding variation of a with a,. It is to be noticed that successive irreversible transitions occur upon an increment or decrement of a, by the amount! or according to Eq. (11) upon a change of the external flux 4>, by the amount of the flux quantum

4>* =he/2e • (51)

The observation of these transitions thus serves their well-known use for the measurement of small changes of a magnetic field.

Considering ever larger values of IIU q I, irreversible transitions can end up in an incre~sing number of stable equilibria and it then depends upon the dynamics of the system which of them will actually be reached. Accordingly, the change of the flux through the ring, undergone in such a tranSition, can assume an increasing number of values and it can be seen that these values amount the more closely to Integer multiples of the flux quantum the larger 11;lql. The experiments of Silver and Zimmerman's clearly demonstrate such irreversible transitions under different conditions which are essentially equivalent to different magnitudes of I: although the reduction of the current Is achieved by a variable weak link instead of a barrier,

In order to Investigate dynamic effects, It Is necessary to add to the magnetic part of the field energy the contribUtion stored In the electric field. With the voltage due to the flux 4>0 given by

v: =-! ~ o e dt (52)

this contribution can be written In the form

F, BLOCH

~ cvg, where C Is the effective capacity of the circuit. '8

Equation (40) for the energy stored in the field of the ring is thus replaced by

F" =C ~) 2/2e2 + 4>~/2L e2 • (53)

The rate of change of this energy is given by - Vol, the work per unit time performed by the current against the voltage Vo. Therefore,

dF" 1 d4> d1=c~1

and with F" from Eq. (53)

(lle)(C .j;o + 4>o!L) = I. (54)

In order to account for damping effects, an Ohmic term will be added to the expression for the current of Eq. (48) so that one has for the right side of Eq. (54)

1=1: sin41Ta+ VIRo' (55)

With V=- U/e)(d4>ldt) and using Eqs. (11), (36), (41), and (46), one thus obtains from Eqs. (54) and (55)

q(LCa +LaIR 0+ a - LCa,- a,) = I: sinha (56)

as a differential equation for the total flux a( t ) at arbitrarily varying external flux a,(t), both measured in units of he Ie,

If a, varies sufficiently slow to neglect its der�vatives' Eq. (56) reduces to Eq. (49) provided that the derivative of a can be assumed to be lIke-

a,

FIG. 4. Total flux " versus external flux "10 both measured in units he/e. The plot is obtained from Fig. 3 by the following construction: A straight line with slope q through a point with the abscissa" on the curve In Fig. 3 Intercepts the " axis at the corresponding value of "I'

wise negligible. The single solution obtained for II~ /q 1< 1/411 is compatible with this assumption since it can be shown to be stable against small perturbations. In view of the preceding considerations, Eq. (56) thus correctly describes the Josephson effect under the appropriate conditions. For II~/ql> 1/411, the derivates of a are no longer negligible, however, when the solution of Eq. (49) has reached the value, previously discussed, at which equilibrium ceases to exist. Indeed, a small deviation from this value can be seen to first build up exponentially with the subsequent time dependence to be obtained by integration of Eq. (56). Because of the damping term, proportional to 1/ Ro, a will finally reach a new value corresponding to stable equilibrium and the magnitude of the damping coefficient determines which among several such equilibria will actually be established.

The application of an external dc voltage corresponds to a linear variation of a, and, hence, to the absence bf the term with &, in Eq. (56). The solutions can in this case be demonstrated by the mechanical analog of a pendulum with viscous friction, connected by an elastic spiral spring to a coaxial shaft which rotates with constant angular velocity wP Denoting the angular deviation of the pendulum from the vertical by fJ and the angle of rotation of the shaft by fJ, with the spring unloaded for fJ, = fJ, 47ra and 47ra, are in the analog to be replaced by fJ and (3" respectively, so that a full turn of these angles represents in view of Eqs. (11) and (36) an increase of the corresponding flux by the flux quantum 4>* of Eq. (51). It can further be seen that the correspondence of the applied dc voltage V to the frequency v = w/211 with which the shaft rotates is that of the Josephson relation given in Eq. (1). The conditions for the Josephson effect and for the irreversible processes, considered above, are reproduced by choosing for the pendulum a relatively small and large mass, respectively, with the damping term adjusted by means of the coefficient of friction. '8

V. EFFECTS OF A POTENTIAL BARRIER

It was remarked in Sec. III B that continuity demands a gradual reduction of the circulating current with increasing width of a barrier. In order to investigate this effect in greater detail, it is sufficient to represent the barrier by the constant potential energy U of an electron between two parallel planes, separated by the width w. In the vicinity of the barrier, a coordinate system will be used with the z axis perpendicular to the two boundary planes, located at z = 0 and z = w (Fig. 5).

Starting with the case of a thin ring, discussed in Sec. IV, the magnetic field in the region R, in-

\ \

\

\ \

\ \ ,

w--------~-----

I I

I

f I

X,Y

I I

FIG. 5. Vicinity of a barrier with width w. In the absence of field penetration, the flux can be obtained from the line integral of the vector potential around the ring starting from a point with coordinates x, yon the positive side of the plane z == 0 and ending at the same point on the negative side.

eluding the barrier, will be assumed to be negligibly small. The total vector potential in this region can thus be written as A = grad X and is eliminated from the partiele Hamiltonian:JC' of Eq. (34) by means of the gauge transformation

I/I(rl)= I/Io(rl) exp[27ri ~ e I x(r I )/ke 1 (57) I

Analogous to Eq. (8), one has then

:JC~ 1/10 =E' 1/10' (58)

where JG: is obtained from 3(" by letting A = O. The notation E' instead of E is used to distinguish the eigenvalues of JC' from those of the total Hamiltonian JC. Considering a particular electron and demanding 1/1 to be single valued, it follows with a given by Eq. (36) that 1/10 is to be multiplied by e·"·" when the electron is brought around the ring. This can be done by starting from a point x, y on the plane z = 0 in the direction of positive z so that after going around the ring one returns to the same point from the side of negative z. The two sides of the plane shall be indicated by z = O. and z = 0.. Omitting the dependence of 1/10 on all variables except the z coordinate of the electron in

158

the vicinity of the barrier, one can write 'Po = 1/Jo(Z) so that one has to demand

(59)

Since instead of the plane z = 0, any neighboring plane could equally well have been chosen, the same relation has to hold for the Z derivative with the result

(60)

to be noted for the following purposes. The potential energy U in the barrier will be

conSidered sufficiently large to cause the dominant z dependence of I/Jo for 0 < z < w so that in this range

l/Jo<Z) =ae" + be ... •

with a and b independent of z and with

K = (2m U/Ii"l112 (61)

.po and SI/Jo/Sz at the two boundaries of the barrier are thus connected by the relations

0~o+~)\o+ = (KI/Jo+~le-Kw

~.po-~)\ w = (KI/Jo-~l+e-KW , or in view of Eqs. (59) and (60) by

(KI/Jo+ ~)\o

(KI/Jo-~)I W =

(KiPO +~w e-Kw-h 'a

~I/Jo- ~l e ... w+2.'a

(62)

(63)

Considering that the side z = O. of the plane z = 0 is reached from the plane z = w by going around the ring, the relation of Eqs. (62) and (63) represents boundary conditions for the solutions of Eq. (58) In the open- ended part of the region R which remains upon exclusion of the part occupied by the barrier. These conditions must be satisfied for each electron at all pOints x, y on the two boundary planes and determine the complete set of eigenvalues E' to be admitted in Eq. (58). In the limit K- 00 of an infinitely high barrier, they reduce to the familiar condition that the wave function has to vanish at the boundaries.

Generally, the barrier width as well as the total flux 4> through the ring enter only through the exponentials on the right sides of Eqs. (62) and (63). Under otherwise given conditions, the eigenvalues E' therefore only depend upon these expontials and result in the free energy F' of the particles as a function of 8 e2.'a and 8 e~·'·, where

F. BLOCH

(64-)

is the transmission coefficient. Expanding in powers of 8, one has

F'(8e z• tOl, 8e-2I'ia);;;; L; e'm eZI'UI-m)ael+ m ',," .. 0

... c

or F';;;; ~ eZI"nQ ~ dnfj el"I+2~ • .0

(65)

with fixed coefficients c,,, or d... Considering finally that F' must be a real even function of a, one obtains Eq. (35) with

F; = 2:; do. 82• ... and F: = 28" 0 d •• 82•

• .0

(66)

(67)

for n> O. For a sufficiently wide barrier with correspondingly small transmission coefficient 8, only the terms with Ii = 0 need to be retained so that F;-8"andfromEq. (39)1;-8·. Withpairing taken into account, the dominant contribution to the sum in Eq. (38) arises in this case from the term with n = 2 so that one obtains Eq. (48) for the current with 1~ - 8 2. Considering that 8 represents the transmission coefficient for a single electron, it is plausible that the current will be proportional to 82 since it is maintained by the simultaneous tunneling of electron pairs through the barrier. Neglecting higher terms in 8, it can further be shown that" I. =l: so that the results obtained above lead at the same time to the properties of the coefficients I. used in Sec. III B.

The preceding discussion was based upon the assumption that the penetration of the magnetic field into the ring is negligible. Because of the Meissner effect, this assumption can be safely made, except in the vicinity of the barner since the current across the barrier may be too small to provide suffiCient shielding. One deals in this case with a magnetic field not only inside the barrier but also in the region extending beyond its boundary planes to a distance comparable to the London penetration depth. It will be assumed that this region lies between the planes z = - ~ and z = w + ~ (Fig. 6), such that the magnetic field at greater distances can be conSidered to be vanishingly small. In accordance with the corresponding treatment by Josephson (J. Sec. 3. 1) it will be further assumed that the sideways dimensions of the barrier are sufficiently large compared to wand .5 to neglect edge effects and that the penetrating field has a vanishing z component.

Denoting the part of the vector potential responsible for the flux through the opening 0 again by gradx and the part due to the penetrating field by ii, one has thus inside the region R

I I

z

+8

+

8

\ ~ P* \

~.

~P

X,Y

.P

J J

I I

FIG. 6. Vicinity of a barrier. The field i. assumed to penetrate into the region between the planes II: = - 6 and z = w + 6. In tbis region. the curve C ••• around the ring is chosen to run parallel to the z axis and to pass the plane z = 0 at a point with coordinates J<. ". One has to consider line integrals of the vector potential along tbis curve. starting at a fixed point P' outaide the region of penetration and ending on the variable point P. Two alternative positions of P are chosen to indicate how they are to be reached without passing through the plane II: = O.

(68)

where a. =a, = 0 and where a. "# 0 only for - 0 < z < w + O. In contrast to the case of negligible penetration, it is not possible to rigorously eliminate the vector potential by a gauge transformation. An approximate elimination is achieved, however, by replacing X in Eq. (57) by X + X., where

x.=f~a.ds (69) p

is defined as a suitably chosen line integral within R from a fixed point p* in the region of vanishing magnetic fie Id to a variable point P. The path of integration is chosen to pass through the plane z = w + 0 or z = - 0 if P is a point between these planes with z > 0 or z < 0, respectively, and to traverse the intermediate space in both cases parallel to the z axis. X. thus becomes a uniquely defined function of the coordinates of P which satisfies the relation a X. laz = a.. Whereas a. = a, = 0,

one finds, however. that

f:= -/ H,dz and ~= / H.dz,

where H. anI!. H. are the components of the penetrating field H = curIa. Consequently. the vector condition it = gradX.. required for elimination of the vector potential A, is satisfied only in regard to its z component.

It is permissible, nevertheless, to maintain the validity of Eq. (58), based upon complete elimination of A, by assuming that the x and , derivatives of X. are sufficiently small to neglect their appearance in the expression for the transformed Hamiltonian. Considering that thel are obtained '?Y integrating the components of H over intervals of z no larger than w + 0, this assumption imposes nb severe limitation upon the magnitude of the penetrating field. It can be shown to merely imply a negligible effect of the Lorentz force caused by field penetration and corresponds to the assumption, made in Josephson's treatment, that the magnetic field does not appreciably affect the magnitude of the order parameter. Although the x and y derivatives of X. are thus assumed to be sufficiently small, it should be noted that this does not exclude an appreciable variation of X. over the relatively large sideways dimensions of the barrier.

By extending the integral in Eq. (69) to the point P with coordinates x, ", O. from the side z> 0 and x, y, O. from the side z < 0, it is seen that the corresponding values of X. differ by the contribution

~a.ds

of the penetrating field to the flux through a closed curve C ••• around the ring. This contribution arises only from the vicinity of the barrier which is to be traversed on a straight line parallel to the z axis with coordinates x and ". In view of Eq. (68), the total flux

~X.ds

through e •.• is therefore given by

4>(x,,,)=aX+ t.: a.(x,,,,z)dz, (70)

where aX represents the contribution of the field through 0 and is Independent of x and y. The fact that the wave function I}J must be single valued. combined with the effect of the barrier. thus leads again to Eqs. (62) and (63) as boundary conditions for l}Jo. As an Important difference from the preceding case, however, a is no longer a constant but a function of x and y, given by

( ) 4>(x,,,) a x,y = hcle (71)

160

Instead of a function of the variables ge,2<Ia, F' becomes a functional of 9 e,2rl.(x •• , or, considering pairing, of 9 2e,"Ia(x •• ,. The coefficients of an expansion in powers of 92 are here multiple integrals of the form

J K(xlYt. X2Y2"') exp(411i[± a(x,y,) ± a(x2Y2) ± ••• ]l

X dx,dy,dx.ftY2· ..

so that it is not possible, in general, to express F' as the surface integral of a definite free energy per unit area of the barrier. This is pOSSible, however, if the transmission coefficient is sufficiently small so that all higher terms in 92 can be neglected. As a real quantity which must be even in ll, O'le has in this case

F' ~ F~ + f; J cos411a(x, y) dx dy , (72)

where f: =K9 2

and where, because of the omission of edge effects, the more general kernel K(x, y) is replaced by the constant K.

The fact that a. differs from zero only in the vicinity of the barrier shall be formulated by writing

a. =g(x,y)h(z), (73)

where h(z) is essentially constant for 0 < z < w and proportional to e-I.I/'o for z > w and z < O. ).0 stands here for the London penetration depth so that 0, while small compared to the sideways dimensions of the barrier, has to be chosen several times larger than ).0' It follows then from Eq. (70) that

(74)

In analogy to the derivation of Eq. (13), one obtains relations for the current density J in the vicinity of the barrier by letting .it and thereby ~)( and g(x, y) depend upon the time. The isothermal work per unit time done upon the system is given by

dF' j(- -dt = j·E)dT, (75)

where E = - [gradrp + (l/e) (aA/al») represents the electric field and rp the scalar potential. With divi =0 and using Eqs. (68), (71)-(74), comparison of the terms with a~"yjat and [a.,,(x,y)Jldt on both sides of Eq. (75) can be shown to yield the relations

IF. Bloch, Phys. Rev. Letters 21. 1241 (1968). 'B. D. Josephson, Phys. Rev. i:;Uers I, 251 (1962);

Rev. Mod. Phys. ~, 216 (1964). -

'B. D. Josephson, Advan. Phys. li. 419 (1965); ref-

J j.dxdy~j: J sin411adxdy and

[jh(z)j.dz)/[jh(z)dz) =j; sin411a,

respectively,2° where

j:=411f~ elh .

F. BLOCH

(76)

(77)

These relations are not independent since multiplication with h( z) and integration over z on both sides of Eq. (76) gives the same result as integration over x and y on both sides of Eq. (77). The left side of Eq. (76) represents the total circulating current I and includes Eq. (48) as the special case of constant a with I; obtained by multiplying j; with the area of the barrier. Since I~ was found to be negative it follows that j2' < O.

In order to arrive at a differential equation for a, one has to take the line integral over the closed curve Cx • on both sides of the Maxwell equation curm- <ile)(aE/8t)=411ilc. Noting that

.. (Eod5)~ -.!~ j cat'

ii ~ curia, and that i as well as a differ from zero only in the vicinity of the barrier where C x,. is parallel to the z axis, one finds that

2f 1 a24> 411/. -v a.dz+?8fI~~ J.dz, (78)

2 _ a2 a2 where V - a:? + ar . In view of Eq. (77), the right side of Eq. (78) can be expressed in terms of a if one defines an effective width d of the barrier by

d=[j h(z)dz)[J j.dz)/[j h(z)j.dz). (79)

It reduces to d ~ w if ).0 «wand if h(z) is constant for 0 < z < w, but otherwise depends upon how far a. andj. extend beyond the barrier. Since~)(

is independent of x, y, one obtains from Eqs. (70), (71), (77), (78), and (79)

2 1 a2a 1 . v a-?&? =-w sm411a, (80)

where ). = (- hc2/1611 2 j: ed)'/2

in agreement with the corresponding result derived by Josephson2' [J. Eqs. (3.12), (3.14»).

erences 10 this paper will be preceded by the letter J. 'N. Byers and C. N. Yang, Phys. Rev. Letters 7, 46

(1961). -

'One may go further and include in the particles the

neutrons and protons constituting the nucleus of the ions or even any virtually present mesons. The following proof merely requires that all charges are positive or negative integer multiples of the elementary charge e.

6W. H. Parker, D. N. Langenber, A. Denenstein, and B. N. Taylor, Phys. Rev. 177, 639 (1969).

'The expansion of the right side of Eq. (18) in powers of V' shows that the current consists of a sum of terms proportional to exp {2,,/t I± meV/h) ± n' (v')1) with integer values ,(. Equation (19) expresses the condition that a conjugate complex pair of these terms has a finite time average.

'c. N. Yang, Rev. Mod. Phys. M, 694 (1962). 'F. Bloch, Phys. Rev. 137, 787 (1965). 'OJ. Bardeen, L. N. Cooper, and J. R. Schrleffer,

Phys. Rev. 108, 1175 (1957). "Applied to a closed ring with single-valued order

parameter, the equivalent relation between phase difference and enclosed flux can be shown, conversely, to be a consequence of the Glnzburg- Landau theory. It should be observed, however, that the derivation of Eq. (20), presented here, does not hinge upon this theory.

I2R• C. Jaklevic, J. J. Lambe, A. H. Silver, and J. E. Mercereau, Phys. Rev. Letters 12, 274 (1964).

13The exact definition of F by mean7 of the partition function Q requires the consideration of the integral jexp[-F(~)/kT]d~. As a macroscopic quantity, F(~) causes a sharp decrease of the integrand as soon as ~ deviates even slightly from the value at which F(~) has its absolute minimum F .... so that Q - exp[ - F_/kT).

'4The dependence of the equilibrium current on a" given by Eq. (13), is obtained by inserting the solution of Eq. (45) for a into Eq. (38). This allows one to express the coefficients I" in terms of the coefficients I~; in particular, one finds for a sufficiently small current that I" '" I~ since it follows in this case from Eq. (45) that a" a,.

15A. H. Silver and J. E. Zimmerman, Phys. Rev. ill, 317 (1967).

'6in the presence of a barrier, the electric field may be assumed to be negligible everywhere else so that C represents In that case the capaci ty of the barrier acting as a condensor.

l1H. E. Rorschach and J. T. Carter (unpublished). The anslogy of a pendulum has been previously noted by P. W. Anderson, in Lectures on the Many-Bod" ProiJlem, Ravello, 1963, edited by E. R. Caianello (Academic, New York, 1964), Vol 2, p. 126.

'''with the mass M of the pendulum concentrated at a distance D from the axis, one finds the equation of motion MDii+~+/<p-P,) =-MgslnP, where/and K are proportional to the spring constant and the coefficient of friction, respectively. The anslog Is verified by comparison with Eq. (56) for Q, = 0 whereby the constancy of a, corresponds to letting p, = - wi. In particular, the ratio III q is seen to be replaced by - Mg/ (if so that increasing values of L (with Ro proportional to L) and, hence, of Illlq I are Indeed reproduced by choosing an increasingly larger mass M of the pendulum.

"See Ref. 14. 20The proof of Eq. (76) requires consideration of the

integral jdiv[j(8x/8t))dT. Dividing the connected region R by a plane z = const and applying the Gauss theorem, one obtains (8C.x/81)fj.dxdy, since the values of X on the two sides of the plane (e. g. the sides O. and 0- of the plane z = 0) differ by the amount c.x.

2' Analogous to the remark in Sec. III B, the correspondence requires the replacement of 4"01 by the difference of the order parameter on hoth sides of the barrier. In Josephson's notation, the positive constant -His further replaced by it and the identification of v with c follows from his Eq. (3.13) if one inserts for the capacity C per unit area of the barrier that of a plane condensor with the two plates separated by the effective distance d. The introduction of d by Josephson in Sec. (3.1) is essentially equivalent to the definition given here through Eq. (79).

* TIME-DEPENDENT SUPERCONDUCTIVITY

D. J. Scalapino

University of California, Santa Barbara

Santa Barbara, California 93106

ABSTRACT

The consequences of charge conservation on time dependent phenomena in superconductors are explored. combined with the simple relaxation form of the time dependent Ginzburg-Landau equations, the demand of charge conservation implies that there is a charge buildup and a difference in the chemical potential of paired and unpaired electrons when there is a divergence of the supercurrent. As examples of the theory we will discuss:

*

(1) the breakdown of the Josephson frequency condition,2eV = hv;

(2) current flow across a normal-super interface:

(3) the dynamics of superconducting weak links.

Research supported by the U. S. Army Office, Durham, North Carolina.

163

164 D. J. SCALAPINO

I. INTRODUCTION

The purpose of this lecture is to discuss a simple phenomenological theory of time dependent superconductivity and develop its physical content by applying it to some specific problems.

The theory takes the relaxation form of the time dependent Ginzburg-Landau equation~ such as developed by Abrahams and Tsuneto,l the two fluid description of the current as a superposition of normal and super flow, and demands that the charge deviation satisfy the continuity equation. This demand of charge conservation leads to two new relationships which we believe play a central role in the phenomena of time dependent superconductivity. The first of these relates deviations in the local charge density to the divergence of the supercurrent. The second one introduces the notion of a difference between the electron electrochemical potential and the pair chemical potential and shows that this difference is also proportional to the divergence of the supercurrent.

This last idea, while perhaps new in this context, is well-known in chemistry. Consider the chemical reaction

(1.1)

In equilibrium the chemical potentials of the atomic and molecular species are related by

lLa = 2lLa 2

(1. 2)

Now, if H2 molecules are added, the reaction is forced to proceed as follows:

H2 - H + H , (1. 3)

and one knows that

(1.4)

In a superconductor an analogous situation occurs in which pairs dissociate and quasi-particles recombine~

TIME-DEPENDENT SUPERCONDUCTIVITY 165

2e ;;;! e + e • (lo 5)

In equilibrium the chemical potential of a pair, denoted by convention as 2~ , is equal to twice the electrochemical potential ,p~ ,

2U = 2~ , P

(lo6 )

so that, in fact, only ~ is usually considered. However, in a non-equilibrium situation such as pair tunneling across a tunnel junction or current flow across a normalsuper interface or weak link, the reaction (l.5) may be driven in one direction so that locally

2e -+ e + e , (lo 7)

or vice versa. In this case, to the extent that the pairs and quasi-particles are separately in internal thermodynamic equilibrium it is natural to expect that

(lo 8)

The work that I will describe in these lectures has been carried out in collaboration with Dr. Tom Rieger and Professor James Mercereau. The reader interested in more details should see References 2 and 3.

II. A PHENOMENOLOGICAL THEORY OF TlME-D~PENDENT SUPERCONDUCTIVITY

Originally Ginzburg and Landau4 (GL) introduced their phenomenological theory of superconductivity in order to describe thermal equilibrium properties. The superconducting part of the free energy was assumed to depend upon a complex order parameter,¢{x),which vanished in the normal state. Near T , GL expanded the free energy density in powers of ¢cand its spatial derivatives:

F = Jd3x{al¢{x) 12 + ~I¢{x) 14 + ~~I (V7 - \2ce A) ¢12'}. (2.l)

Here b is a positive constant and, as usual in Landau's theory of phase transitions,

a = a (T-T) o c (a > 0) • o (2.2)

166 D. J. SCALAPINO

The vector potential was introduced in analogy with quantum mechanics. The supercurrent density associated with this part of the free energy is

-OF j =

s OA - iefl ['/'*(x) (11 - 2ie A-+ ),/, (x) ] (2 3) m 0/ flc 0/ - C.C. • •

The equilibrium state is determined by making F stationary,

o OF 01/;*

2 = [ a + b I ,I, I 2 - ~ ( 11

0/ 2m

A natural length,

~2 (T) fl2

2mlal

.2 2 ~A"') ] fI c I/; •

enters the theory. This is known as the temperature dependent coherence length.

(2.4)

(2.5)

In addition to the superconducting part of the fluid, there is a normal part. Phenomenologically, the two fluid picture has proved useful. Here the current consists of a normal component,

Q. 11 J..1.

e (2.6)

in parallel with the supercurrent,j ,Eq. (2.3). In (2.6), a is the conductivity and J..I. Sthe electrochemical potential. The total charge density including the positive ionic background is

p Po + Ps + P. ~on

In equilibrium P vanishes, of course.

(2.7)


with the success and the relative simplicity of this approach for time independent phenomena, interest turned toward the development of a time dependent Ginzburg-Landau (TDGL) theory. Near T , the exponential decrease of the spatial correlatioRs on a scale ~(T) allowed a local spatial description. S Unfortunately, as emphasized by Gor'kov and Eliashberg. the time correlations exhibit a long time oscillatory behavior reflecting the singularity in the single particle density of states at the gap edge. This prevents the development of a local time dependent description, except for the case of a gapless material where Gor'kov and Eliashberg showed that a relaxation TDGL equation similar in form to that of Abrahams and Tsuneto was valid. Here the relaxation rate of the order parameter is taken as proportional to the deviation in F!

2 =1.[l+11I!J>1

T

{2.8}

The relaxation time T is related to ~(T) and the diffusion constant by

{2.9}

The parameters 11 and the temperature dependent coherence length.~(T}.can be determined from the microscopic theory although, as previously noted, the simple relaxation form of the dynamics. {2.8}. has only been derived for the case of a gapless superconducting alloy.S Here we treat these parameters phenomenologically. The electrochemical potential.~.is given by

~ = ~ o

+ 2~ o 6P 3p + eV • (2.l0)

o

Here 6P represents a charge deviation and V is the scalar electric potential.

168 D. J. SCALAPINO

NOw, to complete this theory, an expression for the charge deviation,op, must be constructed. In order to assure charge conservation we begin with the continuity equation,

p - 'V . j

From Eq. (2.10) and Poisson's equation,'V 2v it follows that

o

2f..L 2 3p o 'V oP - 417'eOp •

o

substituting this into (2.11) one finds

with As the Thomas-Fermi screening length:

f..Lo 2 617'ne

(2.11)

- 417'OP ,

(2 . 12 )

(2.13)

(2.14)

Since T» 1 and 417'0' (2.13) reduces to

6P -1 ...,

'V . J S 417'0' (2.15)


This is the first of the two new relationships which we will be discussing. It implies that in a region in which pairs are dissociating there is a net negative charge.

In normal metals one is used to thinking that a charge density involves large Coulomb energies, because the characteristic distance over which the charge deviation occurs is the short Fermi-Thomas screening length, A. However, in a superconductor the coherence length, e~contro1s the variation of the order parameter and hence j. Therefore, the char~e density described by Eq. (2.rS) is spread out over ~ and not A. The Coulomb energy per unit volume associated with th~s 6P is in fact small compared to the superconducting condensation energy density, so that Eq. (2.9) can adequately account for the dynamics.

In order to discuss the relationship between the chemical potentials it is necessary to define the pair chemical potentia1,~. For this purpose we use the Josephson relation, p

(2.16 )

Operationally the pair chemical potential of a metal could be measured relative to a reference pair chemical potential by determining the frequency of the Josephson ac current across a tunnel junction formed between the metal and the reference material. Thi9 definition can be extended to a local part of the surface by imagining a small tunnel junction. Inside a metal the measurement of the local pair chemical potential can only be made relative to other parts of the metal. We have already introduced the electrochemical potentia1,~, and operationally it can be measured relative to the u of a reference metal using a standard cell in a null potentiometric circuit.

From the TOOL Eq. (2.8), it follows that

.., -ieflT 0 2iU V.J = [11>* (at + fa ) II> - c.c. ] • (2.17)

s me2 (T)

170 D. J. SCALAPINO

Now using the definition, (2.16), for ~ ,Eg. (2.17) becomes P

A divergence of j occurs whenever pairs are dissociating or quasi-particle~ recombine to form pairs. The fact that

(2.18)

this implies a difference in the chemical potentials is analogous to the chemical reaction discus sed in Section I.


III. APPLICATIONS

A. A Correction to the Josephson VoltageFrequency Relation

To show in more detail the physical consequences of these ideas, we discuss three problems: modification of the Josephson voltage-frequency relation, current flow across a normal-superconducting interface, and the dynamics of a proximity effect weak link. These illustrate in succeeding stages the two chemical potentials, the two chemical potentials and the charge deviation, and the time dependence of ~,respectively.

6 As discussed by Bloch, the Josephson voltage-frequency relation,

hv = 2eV , (3.1)

is a thermodynamic equilibrium relationship. However, in an actual measuremen~ the system is not in thermal equilibrium. Consider the e/h experiment where radiation is shown onto the junction and current is driven across the insulating barrier in the presence of a bias. An auxiliary set of normal leads is used to compare the electrochemical potential established across the junction with the chemical potential difference, /::"1.1. 11' of a standard cell. In a separate measurement,CfioE associated with superconductivity, 1::. LLcell is calibrated in terms of eV so that .

t11J = ~lJcell = eV . (3.2)

If this electrochemical potential difference, 61.1., is then used in the Josephson relation,

hv = 2.61.1. = 2 8/J.cel l = 2eV, (3.3)

then e/h can be determined from the voltage V and the frequency v.

Since the driven system is not in thermodynamic equilibrium, the problem is to focus on that aspect of the non-equilibrium situation which is most important. The experimental configuration is such that energy, quasi-particles, and pairs can be transferred at the junction. Since the chemical potential is conjugate to

r-~~

(

i I I i I

\ I

I I

N

: IS

" I

S

I N

I I I I I

r.::..

..::.~

:J <

FIG

. 1

. S

ch

em

ati

c

dia

gra

m o

f a

tun

nel

jun

cti

on

to

d

isp

lay

th

e co

rrecti

on

to

th

e

Jose

ph

son

v

olt

ag

e-f

req

uen

cy

re

lati

on

. T

he

jun

cti

on

is

b

etw

een

th

e la

rge

vo

lum

e o

f su

perc

on

du

cto

r o

n

the ri

gh

t an

d th

e

sm

all

er

vo

lum

e o

f su

perc

on

du

cto

r o

n th

e le

ft.

[F

rom

Ref.

2J

... ;j

? ~

CII

()

> > ." Z o


the particle number, one would expect that the latter two effects would dominate. Since, in principle the quasi-particle transfer could be reduced by limiting the rf voltage excursions, by lowering the temperature, and by eliminating shorts, we turn to an examination of the effects of the pair transfer on the frequency voltage relation,Eq. {3.1}.

In Figure I we simplify the geometry of the problem by treating a junction in which one side is thin compared to a coherence length. With the current flowing from left to right, pairs enter the thin superconductor and quasi-particles leave along the normal lead. Integrating Eq. {2.18} over the volume enclosed by the dashed surface, which extends into the normal lead to a point where the current is dominately quasi-particle, we obtain

fJ - fJ p

m~2 {T} I = ____ -=s..,....

41 e I 7' Ollt> I 2 {3.4}

Here 0 is the volume inside the dashed surface and I is the supercurrent flowin~ a~ross the junction. UsingSthe equilibrium value for 1lt>1 and the microscopic forms for the parameters, this can be written as

I T S {3.5}

where w~ have introduced the free electron Fermi energy, fJo = PF /2m.and n is the electron density.

In this case, where the pair chemical potential.fJ , differs from the electrochemical potential,fJ,it is p important to recognize that the radiation couples the pairs so that

hv 2 /). fJ p {3.6}

The null measurement relates ~ fJCel1 to the electrochemical potential so that

{3.7}

174

t 111'1

0.5

-3 -2

--......... , 1 " "-'\.

"-. J

-4 -3 -2

-1

" "-\

\

o X/~

\

" " -1 0

X/~

" ,

D. J. SCALAPINO

2 3 4 •

--In --- js

2 3 4 •

FIG. 2. Schematic illustration of the variation of the magnitude of the order parameter and the supercurrent and normal current densities in the vicinity of the N-S interface.


Now since the deviation,Eq. (3.5), depends inversely upon the volume of the superconductor, the shift in the large superconductor can be neglected. Combining Eqs. (3.5, 3.6, and 3.7), we find a correction to the Josephson voltage-frequency relation~

~ I T o s

2eV - hv = 91elnO (3.8)

Expressing 0 as a thickness d times the junction area A we consider a strongly coupled junction with

d 103 1\. and I /A '" 1 amp/ ern 2. For the typical values -11 s 22 3

T '" 10 sec, n '" 10 /cm, and ~ o

on the right-hand side of Eq. (3.8)

'" 10 eV, deviation -9 7 becomes 10 eVe

B. Current Transport Across a NormalSuper Interface

Current flow across a normal-superconducting interface remains a problem of continuing interest. Here we will not discuss the effects of the gap potential in scattering quasi-particles as they enter the superconducting region~ but rather focus on the aspects of this problem directly related to the theoretical ideas presented in Section II. One might expect that this treatment would be appropriate for a gapless superconductor or a superconductor near Tc .

In going across the interface, the order parameter and currents vary as illustrated in Figure 2. In the region of the interface the current has both normal and superconducting components. An electrochemical potential gradient is required to drive the normal current. If this chemical potential also controlled the rate of phase change of the order parameter,

cp = 2 tJ ~/fl (3.9)

then the rate of phase change would vary with position across the interface. Eventually this would necessitate a phase slip process and associated time dependent phenomena. In addition, there must be an excess positive charge buildup at the interface in order to produce an electric field in the normal metal.

176 D. J. SCALAPINO

I I

--J.1.p ---- JJ.

-

I

I I I I I I I I I

x

81['"

21C

FIG. 3a. Magnitude of the order parameter as a function of position in the vicinity of the N-S interface. Spatial grid marks are one coherence length apart in each graph.

FIG. 3b. Quasiparticle electrochemical potential u and pair electrochemical potential U as functions of position. Deep in the normal material the Elope of U becomes ej/a.

FIG. 3c. Charge density as a function of position.[Ref.2]


Experimentally there appears to be evidence for two types of behavior. Below a critical current a time independent current flow exists,while above a critical current radiation can induce steps indicative of time dependent superconductivity. The time independent low current transport can be understood in terms of the present theory. In the vicinity of the interface, there is a divergence of j. According to Eq. (2.15) this divergence induces aScharge deviation which gives rise to the electric field in the normal metal. Furthermore, although the pair chemical potential remains constant, the divergence of j leads (Eq_ (2.19» to a shift of ~ away from U which ~hen drives the necessary normal current. p

In Reference 2, the current flow across the normalsuper interface of a thin wire was investigated. The diameter of the wire was taken less than ~(T), reducing the problem to one linear dimension. This was divided in a spatial grid and the TDGL equation, (2.8),transformed into a set of time differential-difference equations. A few coherence lengths inside the normal metal,thermal fluctuations in ~ become important and eventually wash out the order. To partially take this into account, one-sided differences were used deep in the normal metal. Deep in the superconductor the current is carried by the pairs, so that this boundary condition sets the phase gradient, while deep in the normal metal it sets the gradient of the electrochemical potential.

The results of the numerical solution are plotted in Figure 3. The order parameter, 1~I,is shown at the top. The electrochemical potentials are plotted in the center, and as discussed,u remains constant while U changes continuously acro~s the interface. The charge density determined from Eq. (2.15) is shown at the bottom of the figure.

c. Time Dependence of ~ Across a Proximity Effect Weak Link

Presently, weak link devices play an essential role in superconducting quantum electronics. Of these, the tunnel junction first analyzed by Josephson remains the most thoroughly investigated and theoretically understood. Many of these junction theories have been usefully extended to describe point contacts and other weak links. However, it is important to keep in mind that

(V)

//

//

//

//

jc

//

//

//

//

7'

// h

//

//

//

//

//

//

//

FIG

. 4

. C

urr

en

t-v

olt

ag

e ch

ara

cte

risti

c o

f a

pro

xim

ity

eff

ect

wea

k li

nk

. E

xp

eri

men

tal

ch

ara

cte

risti

cs ap

pear

basic

all

y id

en

tical

to th

is

alt

ho

ug

h th

is

fig

ure

is

actu

all

y th

e th

eo

reti

cal

resu

lt o

bta

ined

in

R

ef.

3

usin

g th

e so

luti

on

~(x,t)

dis

cu

ssed

in

th

is

secti

on

.

... ~ ? ~

CI> ~ >

."

Z o


the tunnel junction dynamics describes a strictly phase dependent phenomena with the magnitude of the order parameter on each side of junction fixed. Energy is exchanged between the electromagnetic fields and the phase dependent coupling energy of the junction. This energy exchange is usually only weakly damped by quasiparticle processes.

From a practical standpoint, the chemical and mechanical stability of proximity effect weak link structures9 would appear to offer a considerable advantage. These structures consist of a thin film of superconductor which is locally weakened by depositing a narrow strip (_ I ~) of non-superconducting material across it. This structure exhibits the I-V characteristic shown in Figure 4. Above a critical current, I , a dc voltage,(V),is developed. However, as shown, c there remains an excess supercurrent,~ I. Experimentally, the voltage spectrum across the wegk link has a peak at w = 2e(V)/~ and weaker structures at harmonics which rapidly disappear as the dc current becomes large compared with I . c

Recent work by Rieger, et al.: has shown that these phenomena can be explained in terms of the time-dependent theor¥ of Section II. The time variation of the magnitude,l~I,as well as its phase,~,play an important role. Further:more, the energy exchange involved in the basic phase slip process is dissipative rather than oscillatory as in the tunnel junction.

As a final example of time dependent superconductivity, we discuss the space-time dependence of ~ across a proximity effect weak link. A detailed discussion of this problem and a comparison of the experimental and theoretical I-~, I-V and vet) spectrum can be found in Ref. 3.

In this analysis, the proximity effect section of the weak link is represented by a region with a reduced value of T. As for the N-S interface, the weak link was dividea into a spatial grid. At the edges of the grid, the magnitude of the order parameter was maintained constant and the phase gradient adjusted so that all the current was carried by the superfluid.

For a constant drive current greater than the critical current, the solution,~(x,t) ,is shown at various times in Figure 5. In these figures, the order parameter

180

, \

\ \

I I ,

I

/ I

/

\ \

/

\ \ I I

I , I

I

, \

\ \ \ \

I I

I

I I I

./

( a )

( b )

I I J I

I I

( e)

D. J. SCALAPINO

FIG. 5. Each frame shows a plot in cylindrical coordinates of the magnitude and phase of the order parameter as functions of position along the weak link for various times during one cycle of phase slip. I~I is the radius vector; ¢ is the phase angle; X is plotted along the axis. (See the text for a detailed discussion of each frame.) [From Ref. 3J

TIME-DEPENDENT SUPERCONDUCTIVITY

is plotted using cylindrical coordinates in which the magnitude, II/> I, is the radius, ¢ is the phase angle and the spatial coordinate,X,of the weak link is measured along the axis of the cylinder.

181

In Figure Sa, the phase difference across the link is small and the variation of the order parameter with position arises simply from the presence of the reduced T section of the weak link region. As time proceeds, tfle voltage developed by the normal current increases the phase ~radient along the link. The phase gradient decreases 11/>1, reducing the density of superconducting electrons. Initially the supercurrent increases as the phase gradient. However, the reduction in 11/>1 leads to a maximum in the supercurrent,beyond which an increase in the phase gradient reduces j. Figure Sb shows the situation as it developed from Figure Sa.

There are two time scales in the problem, by the Josephson frequency,2eV/h,and the other the relaxation time,T,of the order parameter. typical operation conditions are such that

liT » 2eV/h

one set set by Present

(3.10)

The ini-tial time development of I/> occurs over times set by h/2eV. However, once the critical supercurrent is exceeded in a given region, it becomes unstable and the order parameter collapses on a time scale set by T. As the order parameter collapses, thermodynamic fluctuations become important. The left-hand part of Figure Sc is an enlargement of the center section of Sb. On the righthand side a thermal fluctuation has given rise to a phase slip.

This phase slip was put into the numerical calculation by the following ansatz. The free energy, 6F, of the region within a coherence length of the centerOof the link was calculated as a function of time. This was compared with the minimum free energy, 6F2",which this region would have if the phase was slippea by 2" across the section. When .6F2" became less than 6F , the time evolution proceeded from the phase slipped co~figuration Figure Sc (right-hand side). In general, to determine the dynamics of the phase slip, all possible fluctuation configurations weighted by the appropriate Boltzmann factors should be included. However, because the dynamics are developing so rapidly at this point, our simple

182 D. J. SCALAPINO

ansatz for treating the phase slip gives a useful description of these devices when Eq. (3.10) applies.

Following the phase slip, the order parameter grows and the phase "untangles" on a time scale again set by T. Figure Sd shows ~ at a later stage, and finally the system reaches a state, Figure Sc, identical to the initial state,Figure Sa. The process repeats periodically at the Josephson frequency. This last feature is a direct consequence of the basic equations and is independent of the details of the phase slip process. The instantaneous rate of phase change across the weak 1ink,~,is simply

d ~¢ = dt

2eV -~- (3.11)

because (V • J )/,~,2 vanishes outside the weak link region. In on~ period,T, ~¢ goes through 217' so that

2"e (V)T, (3.12)

and consequently the basic period is set by the dc bias (v) :

h T = 2e(V> (3.13)

TIME-DEPENDENT SUPERCONDUCTIVITY

REFERENCES AND FOOTNOTES

1. E. Abrahams and T. Tsuneto, Phys. Rev. 152, 416 (1966).

2. "Charge Conservation and Chemical Potentials in Time Dependent Ginzburg-Landau Theory", T. J. Rieger, D. J. Scalapino, and J. E. Mercereau, Phys. Rev. Letters!:2, 1787 (1971 ).

183

3. "Dynamic Behavior of Weak Superconductors", T. J. Rieger, D. J. Scalapino, and J. E. Mercereau, to be published.

4. V. L. Ginzburg and L. Landau, Zh. Eksp. Teor. Fiz. ~, 1064 (1950) •

5. L. p. Gor'kov and G. M. Eliashberg, Zh. Eksp. Teor. Fiz. 54, 612 (1968) [Soviet Phys. JETP !:2, 328 (1968)] •

6. F. Bloch, Phys. Rev. ~, 109 (1970).

7. This is a factor of 10 larger than the estimate given in Ref. 1, because there fJ. 0 was taken as 1 eV. For Pb or Sn 10 eV is more realistic.

8. R. Kummel, Z. Phys. 218, 472 (1969).

9. "Ac and dc Potential in Superconducting Phase Slip Structures", J. E. Me rcereau, H. A. Notarys, and R. K. Kirschman, Phys. Letters 34A, 209 (1971).

REFRIGERATION FOR SUPERCONDUCTING DEVICES

Robert W. Stuart

Manager, Research and Development

CRYOGENIC TECHNOLOGY, inc.

Drs. Gregory and Mathews, in their invitation to me to give this talk, suggested that my discussion should be reasonably general, in that my talk is the only one having to do with refrigeration. We also felt that most of the people attending this course would be more familiar with the physics and theoretical aspects of superconductivity and less familiar with the practical applications aspect. Therefore, my talk should emphasize the practical problems.

Many people involved with superconductivity would consider refrigeration to be one of the greatest problems ,associated with the application of superconductivity. I'd like to counter that impression by describing what can, in fact, be done now or in the not too distant future. In my talk I plan to describe 1) the ways by which refrigeration can be produced, 2) some generalized refrigerator characteristics, and 3) some examples of refrigeration systems which would be appropriate for superconducting devices.

WAYS OF PRODUCING REFRIGERATION

The basic problem with producing refrigeration for superconducting devices starts with the Second Law of Thermodynamics, which in one of its forms, is as follows:

(1) W Th-Tc - = -----Q Tc

185

186 ROBERT W. STUART

That is, the ratio of the work required (W) to the refrigeration produced (Q) is related to the temperature at which the refrigeration is produced (Tc) and the temperature at which the heat must be rejected (Th). Since we typically are talking about temperatures of the order of 4°K, the ratio of work required to refrigeration produced is typically 50 to 100. This is the theoretical minimum. As a practical matter, because of the compounding effect associated with the extremely high temperature ratio over which we are operating, small losses very severely penalize performance, so that practically realizable efficiencies are of the order of 5 to 10%,leading to work requirements of 1000 or more watts per watt of refrigeration.

Let's look at the most common and useful cycles and see how they are used.

Joule-Thomson Cycle

The Joule-Thomson (J.T.) process is by far the simplest in terms of the complexity of the equipment required for producing the refrigeration. The process involves expansion of a highpressure, non-ideal gas, in conjunction with a counter-current heat exchanger to conserve the refrigeration produced. The system requires a compressor to recycle and recompress the gas to high pressure. See Figure lAo A First Law analysis, (Fig. lC), of the simple J.T. process shows that the refrigeration available is equal to the difference between the exiting and entering stream enthalpies. The enthalpy of an ideal gas is not a function of the pressure~ therefore, we must utilize non-ideal gases to effect cooling and refrigeration. Table 1 presents the characteristics of the fluids which are useful for refrigeration systems for superconducting devices.

Inversion Boiling Triple Critical Material TemE' oK Point oK Point oK TemE' oK

Nitrogen (N2) 621 77.3 63.1 126.0

Hydrogen (H2) 204.6 20.4 13.96 33.2

Helium-4 (He4) 50.5 4.2 5.19

Helium-3 (He3) 3.19 3.32

TABLE 1 RELEVANT TEMPERATURES OF REFRIGERANT FLUIDS


J T VAllE

IA SIM?LE J T

Ti.

P RECOJLlNG

IB PRE-COOLED J.T.

FIGURE 1 JOULE THOMSON CVCLES

Q = H our-H IN

I C ENERGY BALANCE

The inversion temperature is the temperature at which the fluid starts to exhibit a cooling effect with pressure reduction. Above this temperature the fluid warms rather than cools. It is important to note from the heat balance shown in Figure Ie that the temperature of interest is the temperature at the warm end of the heat exchanger. Namely, the refrigerant fluid must be initially cooled below its inversion temperature before it can

187

be used to produce refrigeration at lower temperatures. The Joule-Thomson effect is the degree of cooling which occurs, whereas the refrigeration available is equal to the amount of energy which must be added to the gas to return it to its initial temperature (minus heat exchanger losses if any).

To use helium for the production of refrigeration at 4°K with a J.T. process, the gas must first be cooled below SO.SoK. As a practical matter, the refrigeration effect at SO.SoK is vanishingly small, so that the gas actually must be cooled to the region of IO-ISoK to be very useful.


The Boiling Point is the temperature at which the fluid vaporizes under a 1 atm pressure and is, therefore, the most common temperature for the refrigeration use of the fluid. The Triple Point defines the minimum freezing point and, therefore, the minimum useful temperature of the fluid. Everything except helium is frozen below l3.96°K and so helium is the refrigerant fluid for all but a few applications of high critical temperature superconducting materials.

As noted above, helium requires precooling. Early refrigeration systems used cryogens such as liquid nitrogen and liquid hydrogen to provide precooling of the helium. Later the cascade system used nitrogen tY1 hydrogen J.T. systems to effect the necessary precooling. Because of the problems associated with a multiplicity of fluids and circuits and the high pressures (2-3000 psi) necessary for efficient cascade refrigeration systems, current practice is to use an engine pre-cooled J.T. system for the lower temperatures and an engine alone for the higher temperature cooling requirements.

Let us now discuss expansion engine cycles. I would like to classify the cycles into two categories, as a function of the heat exchanger used for conservation of the refrigeration effect produced by the engine. The first category contains those that use a counter-current, two-passage, heat exchanger (recuperator) and the second contains those utilizing a reversing flow, singlepassage heat exchanger (regenerator).

Work Extraction Cycles Using Recuperators

The basic difference between the work extraction cycles and the J.T. process is that the high pressure gas, after compression, aftercooling, and precooling is allowed to expand against a moveable member in such a fashion to do work against it. Typical expansion devices are reciprocating plastic or metal pistons or high speed rotary expansion turbines. The First Law(energy balance) analysis of an expansion engine process (Figure 2C) shows that the refrigeration capacity is equal to the work extraction less any heat exchanger losses. Since all gases cool as they do work, it is not necessary to have a non-ideal gas nor do inversion temperature considerations play any significant role. If the gas is initially precooled below its inversion temperature, the J.T. effect may be additive to the work-related refrigeration effect. The amount of refrigeration produced by expanding a quantity of gas with work extraction is considerably more than the refrigeration effect associated with isenthalpic (J.T.) expansion.

(1) p. K. Lashmet and J. M. Geist. Advances in Cryogenic Engineering, Vol. 8, Plenum Press, (1963), p. 199.


The lack of inversion temperature considerations makes work extraction processes more broadly applicable. This, in conjunction with the higher efficiency of such processes, leads to the preference of work-processes and justifies the additional complexity of the cold work-extraction mechanism.

189

The simplest version of such a cycle is illustrated in Fig. 2A. The gas, after compression, is introduced into a countercurrent heat exchanger and, after being cooled to a low temperature, is further cooled in the process of doing work against a piston. The gas then cools a load, is warmed in the heat exchanger, and returns to the compressor. In a reciprocating engine, cold valves control the flow of gas into and out of the engine. The turbine nozzles and blades carry out the work extraction process by using the available energy to accelerate the gas which then turns the turbine wheel.

In the Joule-Thomson process the fluid is usually liquefied. This liquid is then vaporized in the cooling process. Since the liquid is boiling at a constant pressure, it remains at a constant temperature. Varying the low pressure in the J.T. process, therefore, provides a very convenient means of adjusting and controlling the temperature. In the expansion engine process, the gas is usually not liquefied (although such processes are

COMPRESSOR AL ES

AF ERCOOlER

-- J TH ExPANSION

~"IGINE

~E GINE g-ZtALVES

I '----- --' ~.' \---.J

, ,Tc LOAD

lc 2A - SIMPLE WORK 2 B -WORK EXTRACTION

FXTRACTION CYCLE PLUS J. T. ( BRAYTON CYCLE ) (CLAUDE CYCLE)

Hcm HIN

Q = W - (H 1"1 - H au!)

2C - ENERGY BALANCE

FIGURE 2 WORK EXTRACTION CVCLES WITH RECUPERATORS


now starting to be used).(l) Rather, the engine exhaust stream is used to precool a portion of the gas which is expanded in a Joule-Thomson circuit (See Fig. 2B). It is also possible to use a number of expansion engines in conjunction with the J.T. system to enhance the efficiency of the process.

Cycles similar and related to those shown in Figure 2 form the basis for most refrigeration systems intended to produce relatively large quantities of refrigeration at temperatures around 4°K.(2,3) They are efficient, versatile, and flexible. However, for miniaturization, a different class of cycles finds greater favor.

Work Extraction Cycles Using Regenerators

A regenerator is a reversing flow, single passage heat exchanger wherein the inlet gas flows in and is cooled by a matrix of material within the flow passage. The heat removed from the inlet gas is stored in the matrix during one half of the cycle and then returned to warm the exiting low pressure gas during the second half of the cycle. Regenerators can be very efficient,in that the hydraulic radius can be made very smalL resulting in very efficient heat transfer, high surface area per unit volume and small size. Such heat exchangers are also tolerant of a certain degree of contamination, in that the exiting low pressure gas tends to purge the regenerator of material carried in during the inlet half-cycle. Regenerators suffer from the requirement to not only transfer the energy, but store it within the matrix. Therefore, the regenerator must have a reasonable heat storage capacity. The heat capacity of materials is a function of the temperature, as described by the Debye relationship, falling off at low temperatures, and it gets increasingly difficult to provide adequate heat capacity as the temperatures are lowered. For this reason, the heavy metals (especially lead) are the preferred regenerator matrix materials at low temperatures and even these materials lose heat capacity rapidly in the 10-15°K region. This becomes a serious limitation.

(1) R. W. Johnson, S. C. Collins, J. L. Smith, Jr., Advances in Cryogenic Engineering, Vol. 16, Plenum Press, (1971) p. 171.

(2) S. C. Collins, Science, Vol. 116, (1952), p. 289.

(3) W. Baldus and A. Sellmaier, Advances in Cryogenic Engineering, Vol. 10, Plenum Press, (1965), p. 13.


PRESSOR

CO".PRESSOR VALVES

___ ,TH AF TERCOOLER

l.J"',./l-.r-..--"1.J REFRI(,

VALVES

~QLlc LOAD

3A GIFFORD-McMAHON

WORK CYCLE

ExPA OER +QL J Tc

38 GIFFORD -Mc MAH()\J NO-WORK CYCLE

3C STIRLING CYCLE

FIGURE 3 WORK EXTRACTION CYCLES WITH REGENERATORS

The other problem associated with regenerators relates to the fashion in which they are used. The regenerator is usually incorporated within the active portion of the system, so the regenerator void volume must be pressurized and depressurized on each cycle. This increases the gas circulation and losses, and reduces efficiency.

191

Figure 3A presents the Gifford-McMahon Work Cycle. (1) This cycle can be best understood by comparing it to the Brayton Cycle, Figure 2A. Two significant changes are: 1) the engine valves have been moved from the cold engine to the warm side of the heat exchanger, and 2) a regenerator is used instead of a recuperator.

Figure 3B shows a system wht~~ is designed to operate on the Gifford-McMahon "No-Work" cycle. ( In this system the warm end

(1) H. O. McMahon and W. E. Gifford, Advances in Cryogenic Engineering, Vol. 5, Plenum Press, (1960), p. 354.

(2) W. E. Gifford and H. O. McMahon. Advances in Cryogenic Engineering, Vol. 5, Plenum Press, (1960), p. 368.


of the cylinder is closed over. creating a warm volume which is connected to the warm end of the regenerator so that the volume is pressurized and depressurized in phase with the cold volume. The piston now becomes a displacer. in that no significant pressure differential acts across it. and. therefore. no work can be transmitted out of the system. Thermodynamically. work is extracted from the cold region (allowing refrigeration production) and dissipated in the warm volume. Because the work is 'dissipated within the system. no heavy force bearing mechanisms or flywheels are required. The small forces allow one to very conservatively design the drive mechanism. This contributes to long mechanical component life. The self-cleaning nature of the regenerator and the use of warm valves (which are less subject to problems. especially trace contamination freeze-out) also help extend system life potential. Such a system can, when properly designed~ be used to build long-lived. reliable. refrigeration devices. \1) This system has found broad application for the development of refrigeration for small systems.

Figure 3C illustrates one version of equipment designed to operate on the Stirling Cycle. In this system. the compressor and refrigerator valves are eliminated. The pressure variation necessary for operation of the refrigerator section is obtained by moving the refrigerator and compressor pistons in and out-ofphase relationship. The compressor and refrigerator pistons must operate at the same speed. which is usually set reasonably high. Such a system can be quite simple in mechanical concept and also quite small. The cycle is, in theory, reversible and the practically realizable efficiency figures are good. For these reasons, the Stirling Cycle ha~ found considerable application in cryogenic refrigeration. (2)

Other Refrigeration Means

There are other ways of producing refrigeration. which are generally of less interest for superconducting applications. The Vuilleumier cycle uses thermal energy input to operate a thermal compressor in conjunction with a cold expansion piston. somewhat similar to a Stirling cycle. Because of the inherent low pressure ratio associated with thermal compressors, this cycle

(1) w. H. Hogan. Microwave Journal. December. 1968.

(2) A. Daniels and F. K. duPre, Advances in Cryogenic Engineering, Plenum Press, Vol. 16, (1971), p. 178.

REFRIGERATION FOR SUPERCONDUCTING DEVICES 193

is not well suited to low temperature. superconducting applications. The Pulse Tube and Ranque (Hilsch) Tube eliminate cold moving parts -- however. they are only of interest for high temperature applications. Magnetic Refrigerators utilize a paramagnetic material as the refrigerant medium and through magnetic cycling produce refrigeration at temperatures of the order of thousandths or millionths of a degree. Fortunately, superconductive devices dOQ't require such low temperatures. Finally. the Helium3/Helium4 dilution refrigerator dissolves Helium3 in Helium\which produces cooling in the region of 10 to 50 millidegrees. Again, most devices apparently do not require such low temperatures.

GENERALIZED REFRIGERATOR CHARACTERISTICS

We've talked about the most important ways of producing refrigeration for superconducting applications. I'd like. now. to present in a general fashion, refrigerator characteristics so that one can estimate roughly the impact of adding a closed cycle refrigerator to a system using a superconducting device. Fortunately, a number of authors have conducted surveys of existing refrigerator characteristics, from which I will draw heavily in this presentation.

Efficiency

The first factor to consider is the matter of efficiency. Often. efficiency is overstressed, since as a practical matter power is generally available and inexpensive. For very large systems the refrigerator power is important -- however, for small systems, the refrigerator power input, although it may be of the order of a few Kilowatts, is usually not important. Efficiency does have bearing for a few specialized requirements such as military and space applications. Efficiency also has an impact insofar as it determines the size of the required compressor, which pretty much determines the size and weight of the system. Figure 4 presents W/Q, the work input power requirements per unit refrigeration as a function of temperature. Figure 4 is from a survey by Daunt and Goree. (1) For superconducting systems which will operate in the 4 to 20 0 K range we are talking about 500 to 5000 watts of power input per watt of required refrigeration.

(1) Daunt and Goree, ONR Tech. Report No. I, Contract No. NONR-263 (73), August 1969.

194

" .. a: J ... .0; a: .. .. :Ii ... >-

'00 = 50

'0

3 OS

ROBERT W. STUART

2l ~ I I I I II I 26 ' 6 3J _ 2 20 '9 35

h ........ 1- ,. " '\ "

, • 3' 30 '0 )2 :J8 " 11,28 '8 , '5

~ 39 ,....- ./37 20 \ 29 i'" ~

,~ e ,. J6 7 / 3 "

~ '29 .......... '2 " :J.<

~~ ..,. r r-.. ..-- ~

" ..... GOOO PEA FQFoMA NCE FOR M 'lIo1ATUAE SYSTEMS ...........

I " 1 ...........

'-0 50 '0 50 '00

SYSTE M WEIGHT PER WATT REFR 1GATION - Ib /w"l1

FIGURE 4 INPUT POWER PER WATT OF NET REFRIGERATION AS A FUNCTION OF REFRIGERATION TEMPERATURE

o • 1.8 -9K .& 1 0 -3 0K • 3 0 -90K

Open S y mbo ls Indicate U n der Devel opment 10~ ~--~----~-----L----~----~--__ ~ __ ~

1 0- 1

Capacity. Watts

FIGURE 5 EFFICIENCY OF LOW TEMPERATURE REFRIGERATORS AND LIQUEFIERS AS A FUNCTION OF REFRIGERATION CAPACITY

, • Ii \

SOD

I

I

'000


There is a very significant size effect associated with cryogenic refrigerators. The small amount of useable refrigeration required by small superconducting devices is often greatly overshadowed by some of the peripheral losses associated with the refrigerator, so that small system efficiency is typically poor. Figure 5 presents results of a survey by Dick Strobridge of the NBS (1) on the efficiency of refrigerators as compared to the Carnot ideal described in equation (1). Small system efficiency is of the order of a few percent versus the 10 to 20% attainable with larger equipment.

195

Survey type figures such as Figures 4 and 5 are useful for order of magnitude approximations, and as a means of estimating trends. Obviously, there is considerable scatter in the performance data. Some of this comes about because of the way of reporting data. Manufacturers often times rate the capacity of a refrigerator at less than the actual capacity to provide a safety margin. Many machines included in such surveys are of an outmoded design, or were designed without efficiency as a primary consideration. Therefore, whenever you are conducting studies on the impact of refrigerators, it would be very wise to solicit the cooperation of a refrigerator supplier to insure that the data used are appropriate to the requirement under consideration.

Size, Weight, and Cost

One is also usually concerned with the size, weight, and cost of the refrigerators. Figures 6, 7, and 8 present the results of Strobridge's survey(l) regarding the size, weight, and cost of refrigerators. Again, I must qualify the data in these figures by noting that there is an order of magnitude scatter in the data, because of reasons similar to those given above, relevant to the efficiency figure scatter. It is possible in the design of a system to favor particular requirements. The most difficult figure to drive down, unfortunately, is the cost, of which we will speak more later.

Other Characteristics

Next to the ability to initially cool the device to the required temperature, the most important characteristic of a cryogenic refrigerator is the ability to keep the device cold

(1) T. R. Strobridge, IEEE Trans. on Nuclear Science, Vol. NS-16, No.3, (June, 1969).

M E

oj

E

~

(5 >

10

3,

~

o

10

21

e1 ;

8-9

K I

fe

10

1,

I-/1

_

j,.~

o

10

°1

_

_ Ie

-A

I -I

>-':

/

j

~/

•• Y

-

IOp

en

Sym

bo

ls I

nd

ica

te

Un

de

r D

eve

lop

me

nt

10

1 1

02

10

3 1

04

10

5

Ca

pa

cit

y.

Wa

tts

FIG

UR

E

6 V

OL

UM

E O

F LO

W T

EM

PE

RA

TU

RE

R

EF

RIG

ER

AT

OR

S A

ND

LIQ

UE

FIE

RS

A

S A

FU

NC

TIO

N O

F R

EF

RIG

ER

AT

ION

C

AP

AC

ITY

10

6 e

1.8

-9

K

10

5 I

I t;

A""

I

10

4 ~

.H:.

J

10

3 1

:Y

Cl

7/ ~

10

2 ,

Y

-I

I ,/ /~"1I----+----I

10

1

'" ~ °

vi

10

II

I "' ~ 2

10

10

1

10

°

.30

-9

0K

10

1 I

-~

IOp

en

Sym

bo

ls I

nd

ica

te

Un

de

r D

eve

lop

me

nt

10

° 1

0-1

1

0°

1

01

10

2 1

03

10

4

Ca

pa

cit

y.

Wa

tts

FIG

UR

E

7 M

AS

S O

F LO

W T

EM

PE

RA

TU

RE

R

EF

RIG

ER

AT

OR

S A

ND

LIQ

UE

FIE

RS

A

S A

FU

NC

TIO

N O

F R

EF

RIG

ER

AT

ION

C

AP

AC

ITY

10

5

~ '" o D

> m ~ ~ en

-t

C > '" -t


~ -VI 0 U

107 ~ ____ ,-____ -. ____ -. ____ -, ______ ~L---,

106 ~----4------r-----+----~~~--r-----1

105

104

• 1.8 - 9K .10 - 30K .30 - 90K

103

...

10-1 10° 10 1 102 103 104

I nput Power, kW

FIGURE 8 COST OF LOW TEMPERATURE REFRIGERATORS AND LIQUEFIERS AS A FUNCTION OF INSTALLED INPUT POWER

105

~/SUlZER 3500 WATT HELIUM REfRIGERATOR

-0 • • •

FIGURE 9

197


for the desired period. Reference(l) provides data on the inservice reliability of one 4°K refrigeration system used for cooling a satellite communications receiver amplifier. Basically, the data show that in this particular system an initial mean time between failure (MTBF) of 1550 hours was exhibited. However, over a period of six years, with continuous redesign to eliminate the sources of failure, the system MTBF was extended to 13,000 hours. Two lessons are to be learned from these data. 1) It is possible to get good life out of cryogenic refrigeration systems, and 2) good life characteristics are best achieved when a reasonable number of units are operated over extended periods of time to allow for elimination of weak links. Unfortunately, the quantities of refrigerators in use are still limited, and the design of refrigeration is in a continual state of flux. Therefore, typically the accumulated running time on a particular system is limited. I would, however, quote from one refrigeration system user: "Today,cooled parametric amplifiers are more reliable than uncooled ones!" It is possible to provide adequate refrigerator system life and, particularly as the quantities in use cooling superconducting devices increases, the refrigerator life should not be a limitation.

It is also important that the superconducting device user consider other characteristics of refrigerators which may affect device performance. Vibration is one. Some manufacturers data shows actual vibration measurements on the refrigerator with peak accelerations of the order of 1 g. There is usually vibration associated with the compressor section, which must be separated in a vibration sensitive application.

Since the refrigerator often has a motor and/or moving magnetic parts, it can have a magnetic signature. Some measurements were run on one particular system which showed that the magnetic field induced in the area adjacent to the cold tip was of the order of 10-4 gauss.

Temperature control of the order of hundredths and even thousandths of a degree is possible, particularly with the JouleThomson process producing a boiling liquid phase which provides a stable sink. Temperature regulation with an expansion engine system will usually require an active controller, typically an electric heater with temperature feedback control.

(1) W. H. Hogan, Microwave Journal, December, 1968.


SUPERCONDUCTING DEVICE REFRIGERATORS

Let's now look at some actual refrigerators used for cooling superconducting devices. I'd like to separate the applications into two categories - Megasuperconductors, and Microsuperconductors.

Megasuperconducting Applications

The first category contains those systems wherein large amounts of power are being handled, typically large in size, requiring large refrigerators. Power generation, transmission, magnets, motors, accelerators, energy storage devices, etc., usually fall into this category. Generally these systems capitalize on the high current carrying capacity of the superconductor, and typically use the hard superconductors. Although such materials exhibit high transition temperatures, the current carrying capacity is very low near the transition temperature. Therefore, one normally wishes to operate at temperatures well below the critical. The magnetic field and the peak current which a conductor can carry, as a first approximation, follow a parabolic temperature relationship as in equation (2),

(2)

where Hc and Tc are the critical field and temperature. Therefore, it is important to reduce the ratio of operating to critical temperature. Especially with some of the hard superconductors with relatively high critical temperatures, lowering the temperature to 4.2°K makes most of the current carrying capacity available. Since refrigerator cost and risks increase below the 4.2°K level, there is a tendency for megasuperconductors to be operated at 4.2°K.

There are some instances wherein it does become important to go lower, as in the superconducting accelerators. Such systems presently operate at 1.8°K and studies are looking lower.

The capacity requirements vary from 10 watts for small magnets and motors to hundreds of watts for a good sized motor. Even larger system, into the thousands of watts of refrigeration, will be required for power transmission. The range of interest for megasuperconductors is about tens to hundred of watts at 4.2°K (or less).


FIGURE 10


The refrigerators used for such requirements are fairly large with moderately good efficiency (10-15% per Fig. 4). There are incentives to maintain high efficiency because it effects overall system size and cost. The cycle of preference is usually a version of the Claude Cycle (Figure 2). For larger requirements, turbines are the first choice for expansion devices, whereas for smaller cooling requirements, reciprocating expanders still enjoy favor because of greater availability and efficiency.

Figure 9 is a photograph of a turbine expander, Brayton cycle plant. This particular plant is designed for 3500 watts at 17°K, which would be roughly equivalent to a 1000 watt, 4.2°K' tefrigerator. Figure 10 is a photograph of the cold box for the 300 watt, 1.8°K refrigerator installed at the superconducting acce11erator at Stanford. This unit uses reciprocating expanders. In addition to the cold boxes shown in Figures 9 and 10, there also is a compressor (and possibly vacuum pumps) to recirculate the gas. As can be readily appreciated, such systems are large, custom designed, and use a large number of proven industrial components. However, they are stable and capable of long operation.

Two areas of interest to mega superconductivity should receive work in the next few years. Firstly, continued work on turbines to improve their utility in smaller systems,extending the lower range of applicability down to the 100 watt 4.2 0 region. The second is in the development of systems in the smaller size ranges, tailored to specific requirements on a quantity basis. This should reduce the cost of such systems.

Microsuperconducting Applications

The other major area of interest for superconducitivity is in the small electronic device category. Most of the.interest focuses on application of Josephson Junction devices as detectors, amplifiers, filters, etc. Work also is directed towards resonant superconducting cavities. For such devices, the refrigeration requirements are small, typically one watt or 1ess,and the temperature of interest may be somewhat higher than the 4.2°K level so common with larger systems.

I believe that reciprocating expansion devices have important advantages over rotary expansion devices in the smaller size applications. The efficiency of small turbine expanders is very poor which makes for very large systems, even for a small


N ... w a: ;:)

~ II.

... w a: ;:) (!I

II.


requirement. The efficiencies of the compressors are even worse than the turbines, and a large number of stages of compression will be required to attain superconducting temperatures. The cost of such systems would be extremely high, which I think will rule them out of contention.

Small system requirements, in general, are better served by the regenerative expansion engines rather than engines using recuperative heat exchangers. This is because of the greater mechanical simplicity which leads to higher reliability and lower cost. Such systems are less subject to faulty operation caused by minor amounts of contamination which can be detrimental to miniaturized Claude or Brayton Cycle equipment.

203

Figure 11 is a three stage Stirling Cycle system which achieved 7.8°K.(1) Figure 12 is a three stage Gifford-McMahon refrigerator combined with a J. T. System. This system has been used in a number of ways. The engine alone achieved a terminal temperature of 6.5°K and was used to cool a superconducting lead cavity to 6.7°K. (2) The J. T. system was also used in a supercritical mode to produce 6° refrigeration and in a normal mode to produce 4°K refrigeration. Both of these systems are prototype units, and as such, do not represent the best design with respect to miniaturization or other system characteristics. They do, however, define the limit of our present capability with respect to the attainment of low temperatures with regenerative engines alone. Note that the engine terminal temperatures were 7.8 and 6.5°K. The problems, as noted above, associated with low temperature regenerator operation (heat capacity limitations and void effects) both become severe at this temperature level. Also, significant lowering of the temperature will require lowering of the low side compressor pressure to provide for the non-ideal gas characteristics and also vapor pressure considerations. Such lowering of the pressure will increase the compressor size and pressure ratio, making the compressor less attractive. For these reasons, I believe that lowering the temperature to the 4 to SOK region with a regenerative engine is a task that will not be accomplished for years. For the near term future, a regenerative engine, combined with a J.T. system similar to the unit shown in Figure 12. will still be the preferred means of producing a watt or so in the 4° region.

(1) A. Daniels and F. K. duPre, Advances in Cryogenic Engineering, Vol. 16., Plenum Press, (1971), p. 178.

(2) R. W. Stuart. B. M. Cohen, and W. Hartwig. Advances in Cryogenic Engineering. Vol. 15, Plenum Press, (1970) p. 428.


The incentive to eliminate the Joule-Thomson circuit is that the addition of the J. T. system to the engine increases the size, complexity, and cost of the equipment. Present engine plus J.T. systems are 3 to 4 times as expensive as comparable engine only systems. However, the major part of this high differential is associated with the small quantities in which the 4°K systems are used. Assuming comparable quantities, the cost of a J.T. plus engine system should still be higher, but only by a factor of 1.5 to 2 rather than 3 to 4.

Cost Considerations

Note that Figure 8 presents the cost of refrigerators as a function of power input. We've talked about power inputs, from Figs. 4 and 5,of the order of .5 to 5Kw per watt of useable refrigeration at temperatures useful for superconducting materials. Therefore, for small systems which might require 1-2 watts of refrigeration, and 1-10 Kw of power input, the present cost of such refrigerators is in the 104 dollar-range. Larger systems, requiring 102 watts of refrigeration,will require 102 Kw of power and cost of the order of 105 dollars. One must take into consideration the way in which such regrigerators are presently developed and sold. There are no applications for closed cycle refrigerators requiring refrigerators in mass produced quantities. The largest quantities are sold for cooling infrared detectors in military reconaissance aircraft. Such systems require a watt of cooling at 20 0 K or 77°K. These systems sell for $5, 000 to $10, 000, Part of the high cost is associated with the very severe environmental requirements and the quality assurance programs required for such applications. However, even without such requirements, the cost of such refrigerators will remain high until the quantities increase significantly. Only if we can start manufacturing refrigerators in quantities of thousands will be be able to bring the cost down dramatically. When that occurs, however, we will be able to project refrigerator costs almost an order of magnitude under the figures shown in Figure 8.

CONCLUSION

I've described ways of producing refrigeration, refrigerator characteristics, and some specific machines built and operating, mostly in superconducting applications. I've also inserted some projections along the way which I'd like to collect and repeat here.


For large power type applications, industrial, Claude type, refrigerators exist, with turbines for large systems and reciprocating expanders for the smaller sizes. Future work will be directed towards lowering the applicable range of turbine type systems to 100 watts in the 4.2 oK region.

For small (1 watt) application, regenerative systems find their greatest applicability. Neither turbines nor Claude cycle equipment has proven, or is likely to prove useful. Above 7°K regenerative engines function well -- at 4° or below, regenerative engines plus a J.T. system is the preferred approach.

The transition region still requires a J.T. system although future work may lower the applicable range.

The field demonstrated reliability of existing systems is already in excess of 10,000 hours MTBF. However, attainment of this figure in a new system may entail considerable life testing and redesign.

The char~cteristics of existing refrigerators may not be tailored to specific superconducting application system requirements. Therefore, when the system requirements are important, it is wise to check with a manufacturer.

The cost of present refrigerators is of the order of 104 to 105 dollars, depending on size. Significant reductions in cost figures are potentially realizable -- however, realization of lower costs will require that the refrigerators be produced in quantities of a thousand or more.

Part II

Superconducting Materials

EXPERIMENTAL ASPECTS OF SUPERCONDUCTIVITY: EDITORS' NOTE

Several of the sessions of the course were devoted to discussions and demonstrations of laboratory techniques and the experimental aspects of superconductivity. During the day set aside for laboratory visits, Rolfe E. Glover III and his group at the Universi ty of Maryland demonstrated thin film techniques; Robert A. Hein, John E. Cox, Donald U. Gubser and Russel A. Meussner of the Naval Research Laboratory discussed preparation of alloys and demonstrated high critical field measurements using both adiabatic de-magnetization and dilution refrigerator cooling; and the Low tempera~ure group at the National Bureau of Standards (James F. Schooley, Robert J. Soulen, Jr., and Jack H. Colwell) demonstrated heat capacity and d.c. magnetization measurements,and Josephson effect thermometry. In addition, during a lecture demonstration at Georgetown, Sidney Shapiro demonstrated an analog computer device built by Clark Hamilton that "solves" the Josephson equations in graphical form, for a variety of experimental conditions. During other laboratory demonstrations at Georgetown, a.c. susceptibility, d.c. magnetization, Giaever tunneling and Josephson effect detectors and radiators were demonstrated by William D. Gregory. Also, some actual experiments were shown using video tape recordings.

Many of those who demonstrated experimental techniques also lectured during the course, and the techniques they demonstrated are discussed in their lectures. Also, there are a number of lectures in this series dealing with the experimental aspects of Josephson effect and flux quantum devices. Thus the discussion of Josephson effect experiments will not be repeated. A portion of the demonstrations dealing with phenomena found in very pure single crystal superconductors is not otherwise available in these volumes and is reproduced below as a representative sample of the lectures delivered on experimental subjects during the course.

209

SUPERCONDUCTIVITY IN VERY PURE METALS

W. D. Gregory

Georgetown University Department of Physics Washington, D.C. 20007

I. INTRODUCTION

Recently it has been possible to obtain very pure (99.999+%) metals at a very reasonable cost due tQ the development of improved large scale purifying processes tlJ • Studies may now be performed cheaply on pure single crystal specimens of a type approximating the "ideal" specimen assumed in most theories of normal and superconducting metals. In this lecture we summarize the results obtained from experiments involving the superconducting properties (and some normal state properties) of such materials. These results are indicative of the new and interesting phenomena one might observe in very pure single crystal specimens. We will discuss only our own results, for reasons of time limitation and familiarity, although similar data are now found with increasing frequency in the literature.

These experiments were begun at M.I.T. about 1964. The first work was done with 99.9999% pure gallium single crystals. A very sharply defined superconducttn~ transition with a transition width of about 10-5 K w~s observed 2. The critical field curve of gallium was measured t3 ) versus the T-62 3He vapor pressure scale in order to establish gallium as a secondary thermometric substance in the region of 1 K. These results, along with a detailed analysis of the a.c. susceptibility method used to obtain them, are discussed in Section II.

The discovery of a very well defined superconducting transition in gallium made possible a search for very small effects on the superconducting transition temperature. One such study, com-

211

212 W. D. GREGORY

pleted at M.I.T., involved observing the effect of ~Qundary scattering on the superconducting critical temperature( J. (One would expect boundary scattering to average the anisotropic pairing interaction and lower Tc ') )A technique was developed for making single crystal Ga plates(5 as thin as 25~ and the critical temperatures of twenty-one such plates, ranging in thickness from 25~ to 250~, were measured. No shift in Tc was observed. However, a similtt)study of pure single crystal In plates, completed at Georgetown ,showed that boundary scattering does affect the critical temperature in that material (a 25~ In plate has a Tc reduced below that of bulk In by 2-3 mK). The difference in behavior of Ga and In in these experiments will be discussed. Measurements such as these may be useful in the future for mapping the anisotropic superconducting p~r1ng interaction as a function of crystal orientation. These results are discussed in Section III.

In order to compare estimates of the anisotropic pairing interaction obtained from the Tc measurements on thin plates to other superconducting state properties, we(h~ve performed tunneling experiments on bulk Ga single crystals 7J. In the process, a somewhat general technique of producing tunneling barriers on bulk single crystal materials was developed that does not require the use of samples formed in a vacuum, or ones that are heavily etched, to obtain clean surfaces. This method has been successfully extended and adapted to perform tunneling into bulk In and Al samples. Combining the technique for making tunneling barriers with the method of producing thin single crystal plates, it should be possible to look for quantum interference effects (the Tomasch effect) in single crystal tunneling data, from which the Fermi velocity, as a function of crystal orientation, might be obtained. These experiments are discussed in Section IV.

II. A.C. SUSCEPTIBILITY EXPERIMENTS

A. The Superconducting Transition in Pure Galli~ Sinale C!1stals

A study has been made of the suP{rco~ducting transition of 99.9999% pure gallium single crystals 1,2. In this study the superconducting transition temperature Tc of gallium was very accurately measured, the variation of Tc with magnetic field (critical field curve) was measured in the range 0.82 to 1.083 K, and effects on the transition width were investigated. Transitions of a number of pure gallium single crystals indicated t~at they all had the same superconducting transition temperature to within 10-3 K, that a supercooling of Tc was present, and that the phase change for the best samples was 90% completed within a temperature interval of about 2 x 10-5 K. An ideal transition, such

SUPERCONDUCTIVITY IN VERY PURE METALS 213

as this, is useful for the study of the superconducting phase transition. It may also be possible to use such a well defined transition as a secondary temperature standard in the region below its critical temperature where a measurement of the critical field would suffice to determine the temperature. (A more complete description of the possible thermometric uses for superconductors will be given in another lecture during this course by Dr. Schooley).

In our studies, the transition was identified by observing the change in the depth of penetration of an a.c. magnetic field from the normal state skin depth (6) to the superconducting penetration depth (>.) which occurred as the material transformed into the superconducting state. This effect was observed by placing the gallium sample in the core of a pair of coils driven at 23 Hz and noting the change in mutual inductance (M)between the coils which occurred during the transition. The temperature was measured using the resist~~ce of a carbon resistor rigidly epoxied to the sample via a support rod. When M was plotted as a function of resistance of the carbon thermometer, the transition was observed as a sudden change in M, as shown in Figure 1.

The interior of the cryostat used during the gallium study is shown in Figure 2. The samples in the core of the coils were epoxied to an oxYgen-free high conductivity (OTIIC) copper rod. This sample holder was firmly clamped to a supp0t;, which was screwed tifihtlY into the base of a conta~ner of He. The vapor above the He liquid was pumped, lowering the temperature to the transition te~perature of gallium (1.08 K) or less. The figure also shows a He vapor bulb, used to calibrate the carbon thermometer, which was epoxied to the support rod. The section shown in the figure was contained in a vacuum can kept at a pressure of 2 x 10-5 torr. This jacket was immersed in liquid helium. The specimen holder and vacuum can system were designed to isolate the inner parts of the cryostat, thereeY reducing temperature fluctuations,by thermally connecting the He pot, the carbon thermometers, and the specimens within the can at one point in order to eliminate temperature gradients. Not shown in Figure 2 is a second resistor epoxied to the support which served as the heater to warm the specimens through the superconducting transition.

The electronics used to measure the mutual inductance and thermometer resistance are shown in block diagram form in Figure 3. The resistance of the carbon thermometer was measured using a 33 Hz Wheatstone bridge. The decade box resistance was set to balance the carbon thermometer at the transition temperature. The off-balance bridge signal, which occurred when the temperature of the sample differed from that at the transition, was amplified, rectified in a phase sensitive detector, and the d.c. signal ap-

214

If)

!::: z ::;)

> a: <I: a: I-CD a: <I:

~ a: <I: a: I-CD a: <I:

~ <I

SUPERCONDUC TI NG

SUPERCONDUCTIN G

SUPERCONDUCTING

D. M MAX - 2-

D.MMAX -2-

NORMAL

NORMAL

NORMAL

w. D. GREGORY

(A)

lIT(OK)

(8)

lIT(°K)

(C)

l!. T (OK)

Fig. 1. The superconducting transition for a 6-in.-long by 1/16-in.-square gallium single crystal. In traces (A) and (B), a O.012-G field was used in the primary coil. In trace (C), a O.003-G primary field was used.


11: POT PUMP LINE

HI VAPOR PRESSURE

HI VAPOR BULB

CARBON RESISTOR

, ...... _- SPECIMEN SUPPORT ROD

PRIMARY COIL ----_~II

SECONDARY COIL ----1111ii

r-IWt--_ SPECIMEN

Fig. 2. Inner parts of the cryostat used in the a.c. susceptibility studies.

215

216 W. D. GREGORY

23 C P.S OSCILLATOR

MUTUAL TUNED PHASE INDUCTANCE AMPLIFIER SENSITIVE

BRIDGE (23C.P.S) DETECTOR

'I EXTERNAL .-'1 MUTUAL RECORDER __

INDUCTANCE

J J J ,-- ~ ~ ~..::r- SAMPLE COILS I I

....,) "- : 4704 SPEER . I .AAA ...J.-"" I '" '" '" - I CARBON L ______ J RESISTOR • RESISTANCE TUNED PHASE

BRIDGE AMPLIFIER SENSITIVE (33C.P.S) DETECTOR

33 C.PS. OSCILLATOR

Fig. 3. Block diagram of the electronics used in the a.c. susceptibility stUdies.


plied to the x-axis of an x-y recorder. The change in mutual inductance, which occurred during the phase transitt·g~, was detected using a bridge described by Pillenger, et al., ) and the 33 Hz signal amplified, phase sensitive detected, and plotted on the yaxis of the x-y recorder. Thus, the variation of mutual inductance with temperature could be automatically traced out as the specimen was warmed or cooled through the super conducting transition in a constant magnetic field.

The gallium samples used in this work were made from very pure (99.9999%) Alcoa gallium ~s~ng a technique similar to that described by Yaqub and Cochran l9J • The samples were grown in lucite molds by injecting liquid gallium into channels of appropriate size and initiating the growth of the solid phase with a piece of solid single crystal gallium used as a seed. The single crystal quality of the samples was checked using Laue back-reflection X-ray photographs.

The superconducting transition temperature of pure gallium single crystals was measured as 1.0833 ± 0.0005 K on the T-62 3He vapor 9cale. This result basically agrees with data of Seidel and KeesomllO ) once their data were corrected for earth's magnetic fields. The critical field curve was measured in the range 0.820 to 1.083 K. This is shown in Figure 4. Deviation of this curve from that predicted from the BCS theory (see Figure 5) indicates that the gallium superconducting energy gap may be up to 20% anisotropic. However, this is subject to considerable uncertainty insofar as the value of H is uncertain. (See also section III and Reference 34)

o

Under proper experimental conditions, the transition from the superconducting to the normal state can be used with the critical field curve to obtain temperatures on the T-62 3He vapor pressure scale* with an error of less than 0.0005 K, in the temperature range 0.820 to 1.0833 K. Since this range of temperatures is a very awkward region in which to calibrate apparatus lacking a 3He vapor bulb, the critical field of gallium provides an excellent secondary standard.

The effects of several parameters on the transition width were investigated. The transition width showed a correlation with the length of the specimens in a fashi9n that cannot be explained by simple intermediate state theories lll ). Specimens longer than the five inch primary coil exhibited the narrowest transitions (~T ~ 10-5 K) while specimens shorter than five inches had transition widths on the order of 10-3 K (see Figs. 1 and 6). It is possible that

* While the temperature m~ be related to the T-62 3He scale to ± .0005K, the absolute accuracy of the T-62 3Ue scale is probably not this good near lK. Thus, the experimental error should be less than the systematic errors in the table.

218

en en ::;)

<I: (!)

" J:

w. D. GREGORY

1 1 o 1

0

201- ... -

0 .. 0

15- 1 -

0

.. 101- 0

.. 0

51- -

o Tc =

~ 1.083°K

OL-__________ ~~I----------~~I---------~~!~, 0.8 0.9 1.0 1.1

T (OK)

Fig. 4. The critical-field curve for gallium. The various symbols used for the data points refer to different samples measured on different days.

N

I

-.0.06

Fig. 5. The deviation of the critical field from a parabolic dependence on temperature, h=Hc(T)/Ho ' t=T/Tc • Our ma~netic data lie close to the calculated curve of Clem for <a >=0.4, while Phillips calorimetric data lie closer to the BCS curve. (See Ref. 3.)

(/)

I-Z :::J

>Q: « Q: lii) Q: « ::2' -Q: « Q: t:: CD Q: « ::E -Q: « Q: !:::: CD Q: « ::2' <I

SUPERCONDUCTING

2.8xI0-3 OK

SUPERCON DUCTING

L::.MLONG

SUPERCONDUCTING

NORMAL

(A)

NORMAL

(8)

L::.T (OK)

NORMAL

(C)

3.8·10-3 OK L::.T (OK)

Fig. 6. The superconducting transition for three Ga single crystals with different sample shapes. (A) The transition in a sample 6-in. long by 1/16-in. square. (B) The transition in a 2-in.-long piece cut from the 6-in.-long sample. (C) The transition in a 7-mil-thick by l-in.-long and 1/4-in.-wide plate.

219

220

~ a-

.s :E <I

'it .... I-e ::)

.Q a-

S ~

Superconducting 1/4"'1

Tch/4"'1) J t

w. D. GREGORY

Normal

~ T. (1/8"J c

--------~1~.O~X~10~-~4~K~---4-T-(-K-)

Superconducting Normal

<I~--___________ =N~~~~ ______________ _

6.2X10-4 K 4T(K)

Fig. 7. Comparison of the zero field superconducting transitions in single crystal and polycrystal gallium samples. (A) The transition in sample 1/4 #1 (single crystal) compared to that in a 1/8" square by 2" long polycrystal. (B) The transition in sample 1/4 #1 compared to that in a 1 mil thick by 1" long by 1/4" wide polycrystal plate.


TABLE I

The effect of annealing

on the T of single-crystal gallium plates. c

liT a c llTca 112TCb

Annealing Total

221

Sample thickness

(mil) (m K)

run No.lc (m K) d

run No.2

(m K) (run No.2)(run No.1)

time beyond annealing

7 +0.100

1 +0.076

5 +0.114

1 +0.078

2 +0.016

2 +0.088

0.00

-0.084

+0.066

+0-.070

+0.018

-0.027

-0.100

-0.160

-0.048

-0.008

+0.002

-0.115

----------------- _._._-_._-a liT =T (thin plate) -T (bulk sample). c c c

b 112T =llT (run No.2) -liT (run No.1). c c c

40 h (h) time (h)

31

116 156

c Run No. 1, liT measured after 40 h anneal at room temperature. c

d Run No. 2, liT measured after extra annealing time beyond 40 h, as indicated;

222 w. D. GREGORY

the presence of the blunt ends of the shorter specimens in the primary coil resulted in a complicated nucleation of the superconducting state which smeared the transition over a 10-3 K temperature interval.

Polycrystalline structure and impurities produced the expected smearing of the transition (see Figure 7). Bent and annealed single crystals showed no residual effects on the transition provided the annealing took place at room temperature for 40 hours or more (see Table I). The pressure of an epoxy bond at one end of the sample to insure thermal contact with the cryostat was shown to broaden the transition but it did not shift Tc substantially (see Figure 8). A typical superconducting transition for these specimens in a magnetic field is shown in Figure 9. Note that the supercooling phenomenon as well as intermediate state effects are seen.

B. Analysis of the A.C. Susceptibility Technique

The observation of a very ideal superconducting transition in gallium using the "a.c. susceptibility" technique suggested an experiment "in the reverse". That is, the use of very pure materials to study the a.c. susceptibility method itself.

The a.c. susceptibility method has found widespread use in studying superconductors, primarily because it is a convenient compromise between accuracy and utility. Unfortunately, the problems regarding the accuracy of interpretation have become obscured in recent years as more and more workers have employed this technique. Of particular concern is the faith placed in a.c. susceptibility results obtained on alloys, deformed samples, small particulate samples, etc. All of these systems have the potential for exhibiting superconductivity in very small volumes which might be interpreted as a full participation by the whole sample. Even if the volume of the sample participating is left an open question, the critical temperature obtained from these measurements is often different from that obtained with better measurements, such as d.c. magnetization. Even worse, there is an effect often seen in the a.c. susceptibility data that looks like superconductivity but is entirely a normal metal phenomenon. It is this latter phenomenon that we came upon during this inVestigation. It might be correctly termed a .2.!!!. depth effect. At least some of the detail, Of this effect were investigated earlier by Strongin and Maxwellll2 •

The problem with the skin depth effect comes from a qualita-

SUPER CONDUCTING NORMAL

(j) t: z ~

>-'" <t 0 '" t: co

'" ~ ::. <l

3.8 x 10 · ' OK l!. T (OK)

SUPERCONDUCTING NORMAL

l!.T(OK )

Fig. 8. The superconducting transitions in a I-mil-thick by l-in.-long by 1/4-in.-wide single-crystal gallium plate before and after coating with epoxy resin. The change of mutual inductance from 0-1 corresponds to a 100% complete transition. (A) Before coating with epoxy. (B) After coatin£ with epoxy.

He :l9.SG

Fig. 9. The superconducting transition in a field of 19.5 G for a 1/8-in. square, 2-in.-long Ga single crystal.

223

224 W. D. GREGORY

tive criterion for superconductivity developed over the years(13). If one examines both the change in mutual inductance and the change in the resistive component of the sample coil impedance during an a.c. susceptibility experiment, it was quite often found that the resistive component peaked at Tc, with a corresponding discontinuous change of mutual inductance. (Separate examination of these two signals is possible because the output of the secondary coil is observed witn a phase sensitive detector, so that shifting the reference phase by 90° is all that is required to observe these two components.)

The jump in mutual inductance is readily explained as a change in the effective depth of penetration from the normal state value, given by 2 1

t'i = (--=:1L)2 , \J ow

in the classical limit, to the superconducting state value, given by the penetration depth A. For sufficiently low a.c. frequencies (w) used in the primary coil, t'i»A and the amount of flux penetrating the sample and secondary coil changes abruptly at Tc. In equivalent circuit language, this change of flux penetration is the same as a change in mutual inductance.

The explanation for the resi9tiye componen~ phenomenon is not as obvious. The general thinking\13J appears to be that near Tc the N+S phase change involves an intermediate or mixed state that has normal and superconducting regions mixed together, and it is the energy lost by "Shaking" these regions about in an a.c. field that produces a maximum in the loss, as measured by the resistive component.

The difficulty with this interpretation is illustrated in the next few figures. Figure 10 shows a plot of the resistive component of a.c. susceptibility versus temperature as measured at 300 Hz for a 3/16" diameter Ga-In( .03)sample. A rather distinctive peak is noted at approximately 25K. Resistive component measurements on several other Ga-In samples varying in In concentration from 1.5% to 3% also showed a resistive peak at temperatures between 25K and 35K. Figure 11 is a plot of the inductive component versus temperature for the same Ga-In(.03) sample. In addition to the general change in skin depth (resistivity) expected in this temperature region, there is a discontinuity or change in slope at a temperature near 25K. The combination of an anomoly in the inductive component and a peak in the resistive component at the same high temperature would seem to indicate, by the above criterion, that there might be superconductivity occurring in some portion of the material at a temperature substantially higher than that observed so far in any other material.


i .. c5 >-~

1 -------------------------T~(~K~)-

25K

Fig. 10. The resistive component of a.c. susceptibility plotted versus temperature for a Ga-In(.03) sample measured at 300 Hz.

-en .~ c

::J .Q 6K

10. « -.... := c. .... ~

0 > 10. ca

"C 40K C 0 CJ Q)

U)

T(K)

Fig. 11. The inductive component of a.c. susceptibility plotted versus temperature for a Ga-In(.03) sample measured at 300 Hz.

225

226 w. D. GREGORY

Figure 12 shows a trace of resistive component versus temperature for a single crystal gallium sample. The sample was a 3/16" diameter cylinder I" long with the B axis along the cylinder. The measurements were made at 300 Hz. The resistive component peak occurs at 11K, a temperature in excess of Tc of all reported phases of gallium. (Gallium has several known phases - a, S, y, 0, etc. with various Tc values of 1.oSK, 4.7K, 7.2K, etc.\14}). In addition to observing the resistive peak at 11K, it was observed that the peak temperature, Tp, could be driven to lower temperatures by the application of a d.c. magnetic field. It was also observed that the resistive peaks were not as large and were noticeably flatter as the applied field was increased. The plot of the a~plied field He versus Tp 2 in Figure 13 was in agreement with the T dependence for He of a superconductor.

In spite of the interesting comparison to superconducting features, other data (heat capacity and d.c. magnetization) showed that these samples were not undergoing any substantial volume changes to the superconducting state at Tp' In fact, an investigation of well known normal material shows that this is just a normal state effect. Figure 14 is a plot of the resistive component versus temperature from measurements on copper that exhibits a resistive peak at approximately 61K. Since pure copper has not yet been identified as a superconductor even at the lowest temperatures measured (T<.lO mK) under any conditions, it was obvious that the resistive peak has its origin in some normal state phenomenon unrelated to superconductivity.

1. Isotropic resistivity. In order to understand the phenomena observed about 25K in the Ga-In alloys, it is necessary to analyze the normal state electromagnetic response of the samples under the assumption of little or no bulk magnetization, i.e., all response is due to normal state shielding currents ("eddy" currents) only. To do this, we must calculate the m~nitude and )hase of the voltage induced in the secondary coil see Figure 3. The first step is to calculate the B field penetrating the specimen. The starting point for this analysis is the set of Maxwell equations and constitutive relations:

-+ aB V x E = - at

-+ B IlH V . B = 0 = -+

-+ =1 ...m. D e:E V x H + = at (1) V • D = p r = aE

where our assumptions of a good metal with no bulk magnetization reduce to


-., .~

:5 .d ~ 12K

= a. a ~

l~ ___ ---=--;--:-

T (K)

Fig. 12. The resistive component of a.c. susceptibility plotted versus temperature for a cylindrical B axis gallium sample measured at 300 Hz.

O+-----r---~----~----~--_,----~-o 20 40 60 80

Fig. 13. Plot of He' the applied field, versus Tp2 for B axis gallium sample measured at 300 Hz.

227

228

-~ ·S ::;)

J:i a-S i ~ ~

61K

l~ ____________________________ ~~ T(K)

W. D. GREGORY

Fig. 14. The resistive component of a.c. susceptibility plotted versus temperature for a copper sample at 300 Hz.

X=R/&

10

Fig. 15. Plot of secondary output of phase sensitive detector, Vout, versus R/6 for various phase angles e: where Vout is calculated from simple resistivity model.


e: = e: o

-7 p = 0, ~ = ~ = 411 X 10 , o = 4 -12 8.85 x 10 •

(2)

The a.c. field generated by the primary coil will be sinusoidal so that

and

Thus

But

-+ iwt 13(;) B = e

-+ iwt E(;) E = e

~ -+ ~ -+ B = iwB, E = iwE.

-+ 'V x H =

for even

a WE

o » 1

'V x

the

-+ B -+

= E(a + ~o

worst case

iWE 0) •

we will encounter,

That is, for copper at ~ temperature,

a = 108/n-m,

so that for a 300 Hz a. c. current in the primary coil,

Thus thisJ4axwell equation reduces to -+ B -+ 'V x - = aE

~o

Taking the curl of both sides, -+ -+ -+ -+ -+ V x (V x B) = ~ a'V x E

0 -+

v(v i3) 2 -+ aB . -V B = -~oa at But

-+ -+ 'V • B = 0,

so that -+

v2i3 = aB (3) ~oa at

The solution to equation 3 for a semi-infinite plane with the B field parallel to the plane is x

B(x t) = B e-x/ o ei ('8 - wt) (4) , 0

229

230 W. D. GREGORY

where 1 o = (_2_?

1l00W

is the normal skin depth and x is the distance from the surface into the sample. Note that in addition to an eddy current ~, there is also a different phase shift at each·point x in the interior sample. It is this phase shift, which is often ignored in simple treatments of the eddy current decay, that is responsible for the resistive peak effect discussed above.

To compute the emf induced in the secondary coil having Ns turns, we simply obtain the net flux cutting the secondary coil,

CP = NsjB(X) • a.A(x) ,

and calculate the emf from Faraday's law, i.e., dCP

V = - - . dt

For a cylinder of radius R we may use our one dimensional solution for B (in equation 4) provided R»o. The result (with phase shifts taken properly into account) is given by

V = 2nwB e -iwt R2 i!. R3 1 in o ["2 e 2 - 6/2 6" e 4J . (5)

In the experimental arrangement, the secondary coil emf is phase sensitively detected. The output of the phase sensitive detector is proportional to the magnitude of the emf times the cosine of the phase angle between the emf and the phase detector reference oscillators, i.e.,

V t = Ivi cos ~ ou

~ = tan -1 (Re V lIm V) (6)

for the case when the reference oscillator is maintained in phase with the primary coil current. If one shifts the phase of the reference oscillator by an amount E, this is equivalent to examining the emf induced at a phase E away from the primary current and the output is given by

V t = Ivi cos(~ - E) • ou

For example, setting E=O corresponds to zero shift from the phase of the primary current, which is the signal expected for a pure resistance, hence we call this the resistive signal component. A phase shift of 900 would give the signal expected from a reactive circuit element (capacitor or inductor), so this output is called the reactive component, or since we are working with inductors, the inductive component.


Figure 15 shows a plot of the expected phase detector output vs. t~e quantity Rio for a series of phase shifts from £ = 0 to £ = ±~. These plots were obtained by programming equations 4 through 7 on a Hewlett Packard model 9100B desk computer with a plotter attachment. Note that the resistive component (£ = 0) has a peak while the reactive components (£ = ±~) are monotonic. The peak in the resistive component (£ = 0) comes at a value of ~ = 2, but for slight phase shifts aw~ from £ = 0, the position of the peak is shifted dramatically. For example, note that the peak for £ = 0.1 radians is at ~ = 1.

The predictions of the normal state response calculated above agree quite well with the data obtained on copper. For example, using the criterion that the peak in the resistive component occurs for Rio = 2, ~e would estimate that copper has a resistivity of about 7.5 x 10- n-cm at 60K. This compares favorably to a value of 6-8 x 10-8n- cm at 60K for commerical grade copper found in the WAD~ technical report of low temperature properties of materials t 15 ) •

To compare these calculations to the gallium data, it is necessary to make some assumption about the resistivity in the plane of the sample in which the eddy currents are being conducted, 9 since the resistivity in single crystal gallium is anisotropic( ). We found in our earlier work that one can predict the experimental results reasonably well by using the average resistivity in the plane. Recently, we have reformulated the calculation above for an anisotropic material and obtained a much better comparison to experimental results.

2. Anisotropic resistivity. Calculation of the resistive component behavior of a.c. susceptibility can be extended to the case of an anisotropic material if certain assumptions are made concerning the sample geometry and crystalline orientation with respect to the magnetic field direction. Restrictions of this type are necess&rl since the general diffusion equation pV2B = ~ oB. from which B inside the sample is calculated, is no longer

at valid. For resistively anisotropic materials p is a second rank tensor whose product with B is a vector. which. in general. complicates the solution for B. However. the problem can be made much more tractable if one is able to work with a system of coordinates in whic~ the resistivity tensor is diagonal. Consider the calculation of B fOfl~)long sample of square cross section with the principal axes along the edges of the sample. as shown in Figure 16. Furthermore. assume that the external B field has only a Z component along the sample length and that aB = O. Starting then with Maxwell's equations ~ az

232 w. D. GREGORY

Direction

of 1 B Field

Direction of Principal Axes Fig. 16. Orientation of edges of long, square cross-section sample with respect to principal axes of sample.

Fig. 17. Plot of the resistive component of Vout versus temperature for C axis gallium at several frequencies where Vout is calculated from anisotropic model.


and

-+ -+ al 'i7xE=-at

::t- -+ -+ an v x B = IlJ + \l at #

Rote that Ohm's law takes the form, in this case, of

(8)

E i P Jij i = EJ • PiJ = 0, i#j (10)

the resistivities along the principal , axes, x,y,z respectively.

(11)

and -+ -+ -+ -+ 'if x B = IlJ - iwe:,E1l J (12)

if it is assumed B(t)-= B e-iwt • Assuming typical values for components of P and e: foroa metal, even at high temperatures,

P - 10-8 ~-m and e: = dielectric constant - 1, so

it is clear from the discussion above that for frequencies ~«1~5. wp« 1, so the second term m~ be neglected and j = 'il x =F". Substituting into Eq3. 8 and 9,

\l ~ VxB aB" v x (g • -Il-)= - at (13)

Assuming i is Il is close to

V x (g •

not a function of distance along any Il (non-magnetic materials), then

o -+ aB V x B) = -Il at

one axis and

(14 )

aBz Since Bx = By = az = 0, only the Z component equation is non-trivial, i.e., 2

a2 a aBz - "2 (BzP22 )- "2 (BzP ll )= Ilo at (15) ax ay

Calling B = B then z P2 a2B Pl

--+ Ilo ax2

(16)

233

The coefficient for each second derivative term contains the resistivity component of the other direction in the xy (or 1,2) plane.

Equation 16 is basically of the form of a heat conduction equation. Since this type ~f-+equation has been solved for a myriad of boundary conditions, B(r,t) can be determined once the boun~a~ conditions have been established. We seek a solution for B(r,t) in the interior of a long sample of square cross section of side 2a, given that the field intensity on every ~oint of the bounding surface is B (o)e-iwt • Carslaw and Jaeger~l7) have solved the heat conduction e~uation under equivalent boundary conditions

234 w. D. GREGORY

Transcribing their solution, we find 00

16 B(r,t) = 2" L 1T £',m=O

t: [ (2m+1 )1TY ] x LCOS 2a

Cl£. m(_l)l+m (2£'+1) (2m+1)

[ (2£'+1)1TX] cos 2a

Cl >.. 1 e lm IP ( >.. ) d>" J

where IP(>") is the time dependent form of the external field, B(>..), and

2 Px 2 (2£'+1) + -- (2m+l) ] where Plap • P2=P .(18)

llo x Y

= B sin(w~ and solving the integral we obtain o B Cllmt

sin (w~d>" = -2~o~-~2 [e (Cllm sin (w~-w cos (w~)+w] Cllm + w (19)

Rewriting

Cllm sin(w~w cos wt as

Rlm sin (wt+n.em)

where

no- = tan-l (- ~) and .un Cl,lm

R = 2 2 1/2 .em - (Cl.em + w )

the last part of Equation 19 can be written as

B o 2 2

Cl.e.m + w

The first term is the steady state solution and the second term is the transient solution. Because of the exponential decay, the second term is significant only for the lowest order term

-10 Cl.em = Cl • Assuming P ~ 10 n-m, typical for metals at low tem-00 -2

perature and a ~ 10 m; Cl O ~ 1 sec. While all higher order terms decay more rapidly and mayObe neglected, the ClOO term is comparable to the steady state solution. However, by using phase sensitive detection techniques, the effect of the transient solution is reduced. This has been tested by varying the rate of temperature change and no contribution due to the transient term is noted. Continuing the calculation with the steady state term and solving for cjI = Ns J ~·dl, we have


-en .. . -c ~ .d .. ct -.. ::I Q. .. ::I 0 > .. " " c 8 G)

en

••

4 8

• •

12

• Experimental Data -Anisotropic Model

Theory

T(K)

16 20 Fig. 18. Comparison of the theoretical and measured resistive component of a.c. susceptibility plotted versus temperature for a long, square cross-section, C-axis gallium sample at 200 Hz. The theoretical curve is calculated from the anisotropic model.

Frequency

33

100

150

200

250

300

400

Table II

Calculated and measured values of the resistive peak temperature

(Hz) T (measured) (K) T (calculate.i.) (K) -p -p

11.1 11.1

13.6 13.9

15.0 15.1

16.1 16.1

16.5 16.8

17.5 17.4

18.6 18.5

235

236

 by averaging E with a reference signal sin(wt+£) of arbitrary phase £, we find for each term in the series

w J27f /w =<Et,~ 27f cos (wt+n.em) sin(wt~£)dt (22)

o

or E 1. ( ) < ~ - - Sln no_ - £ -t.,m 2 oUr!

(23)

Thus the secondary output is given by the series solution

128a2N B 00

V =<E> L cx.em sin(n.em -E) = s 0 (24) out 4 l,m=O 7f 2 w2 )

(2l+l)2(2m+l)2 (cx.em +

The resistive component (£=0) can be plotted versus temperature for a sample of a given size and a given measuring frequency by inserting the proper temperature dependence of the resistivity of the material for the appropriate principal axes. Assuming th~ ) temperature dependence for gallium as given by Yaqub and Cochran t9 , Figure 17 shows a series of curves plotting <E> versus T for a C axis gallium sample (cross section,AB plane) of side 0.239 cm for a number of different frequencies.

Figure 18 shows a comparison between the measured and theoretical resistive component output as a function of temperature. It is evident from the comparison of the theoretical and measured curves that T coincides exactly at 16.1K for this case, where w = 200 Hz. FurtEermore. the comparison between measured and theoretical values of T in Table II for seven different frequencies shows excellent agreemeRt. In no case is the difference in Tp larger than O.3K and in general the two values of Tp coincide within O.lK or less.

This agreement is far superior to the deviation by as much as lK to 2K observed when Tp was calculated by the method using sim-

pIe resistivity averaging. Mathematically, this difference in Tp is due to the fact that the more rigorous solution contains terms where l and m are not equal. Physically, this corresponds to averaging of the axial resistivities Py and Px for eddy current conduction at angles to both the pure x and y axes. This type of averaging over various angles ~;m)is a more accurate model of the true physical situation than simple resistivity averaging.

An experiment was performed to determine if the C axis gallium sample was in the anomalous skin depth ti~t at these temperatures. (From the data of Yaqub and Cochran 9) one might expect the mean free path to be larger than 6 at these frequencies if T<lOK.) In this experiment the resistive component was measured as the frequency was varied. The results were compared to the theoretical calculation of the anisotropy model for the two cases of normal and anomalous behavior. Figure 19 shows a plot of the experimental results and the theoretical curves calculated from the anisotropy model. The theoretical curves were plotted from/ the expression.for ~£> for th~ anisotropic model assuming o-w-l 2 for the nQrmal case and 6-w-1/3 for the anomalous case. Due to an uncertainty in the direction of a correction term in the data, there are two experimental curves plotted. Assuming either direction for the correction term, the data clearly fits the theoretical curve for the anomalous case much better than the normal case. The conclusion is that the gallium sample is in an anomalous type limit at these low temperatures.

It is likely that this anomalous behavior is responsible for the difference in the shape of the theoretical and experimental curves (Figure 18) at low temperatures. While the general shape of theoretical and experimental curves agree quite well, the curvature begins to deviate substantially at lO-12K, with the theoretical curves being shallower than the experimental data. To show this, we must examine the method for converting the normal to anomalous calculation with a bit more care. In the anomalous region thelexpression for the skin depth changes from 6 = (2P)~ to

2P bl 3 ~w 6 = (~ wK) where Pbl is a temperature independent quantity and K

o is a constant whose value depends on the type of surface scattering. Assuming that the skin depth depends on the resistivity term to the one-third power but that Pbl is more like p(T)d, where d is the length of the side of the sample, then 6 = (2P(T)d)t. Re-

1.1 wK o writing the expression for <£> for the anisotropic calculation as

alm/w sin(nlm-E ) <£> = Const. 1: 1

l,m=O [1+(alm/w)2]Z (2l+1)2(2m+l)2

00

238

-U) .t: c :;:)

.c .. <C -..

::::s Q. .. ::::s 0 .. G)

~ Q. E <t

x X

Xx )(

Normal Skin Effect

X X

200 300

w. D. GREGORY

X Correction Added • Correction Subtracted

X

400 500 Frequency (Hz) Fig. 19. Comparison of theoretical and measured dependence of Vout on w for a C axis gallium sample. The two theoretical curves are calculated from the anisotropic model theory assuming 6-(w)-1/2 for the normal skin effect and 6-(w)-1/3 for the anomalous skin effect.

-en .t:: t: :l .t:l -. <C -..

::::s Q. .. ::::s o > .. ca

"C t:

• Experimental Data - p -" ~rr}d) 1/3 " Anisotropic

Model Theory

8 G) T(K) 0~~--~--~--r-~--~--'---r-~-----

6 10 14 18 22 Fig. 20. Comparison of theoretical and measured shape of resistive peak for a C axis gallium sample at 200 Hz. The theoretical curve is calculated from anisotropic model assuming (p)1/2 ~(p (T)d)1/3

K

with

and

then form

n2 2 2 2 alm = -:2 [oA(2!+1) + o~(2m+l) ]

8a 1

2p A B(T)d '3 °A,B = ( ~'wK )

o alm one may define a quantity Sim:: -w which is very to (R/o)-l of the isotropic model calculation.

similar in

The resistive component of <E> is plotted in Figure 20 as a function of temperature for C axis gallium ~th d = .239 cm and

239

f = 200 Hz. This curve is compared to the resistive data for C axis gallium. The data agree with the plot of the resistive component both in value of Tp and shape of the curve down to 8K if one assumes that K = 2.78. This value of K is very close to the K = 2.80 value calculated by Reuter and Sondheimerll8 ) for the anomalous skin effect limit assuming specular reflection of scattered electrons from the sample surface. Under this assumption the curvature of the theoretical and experimental resistive components agree much better than the case assuming non-anomalous behavior.

The isotropic resistivity model predicted that Tp occurs when Rio = 2. Since 2 1 for the normal resisti-:e region then

o = (~)2 ~ow

2 R ~ w (-T-)PEAK = PpEAK (26)

and In W ak = In P ak within a constant additive factor. The pe pe validity of this relation for an anisotropic material was checked by comparing the experimental and theoretical plot of log w versus T with a plot of log P versus T for <ph of gallium. Figure 21 s~ows a comparison of these curves. It is clear from the excellent agreement that the relation In w ak ~ In P ak is valid for pe pe an anisotropic material. Thus, measurements of the variation of Tp with frequency can be used to measure the temperature dependence of the resistivity in this temperature region. In the case of an anisotropic material the temperature dependence of PA, PB, and Pc can be determined from the temperature dependence of In w ak for different sample orientations. pe

The agreement between the observed val~es of Tp and the values of Tp estimated using the Rio = 2 criterion for the gallium, copper,and Ga-In samples proved that the resistive peak was not a phenomenon associated with superconductivity. The question then arises as to why the resistive peak has been observed at the su-

240

Q.. C ..... .. a c .....

15

W. D. GREGORY

p Resistivity • fA) Anisotropic Theory x fA) Experimental Data

T(K)

20 Fig. 21. Comparison of In(p) versus temperature and In(w ak) versus temperature for a C axis gallium sample. pe

Frequency (Hz)

T(K)

o Fig. 22. A plot of pe.ak frequency versus temperature for a indium sample.

perconducting transition temperature byStrongin and Maxwell(12).

This question can be answered by considering the Rio = 2 cri-terion in the form R2 For a sample of radius R and a

V w *=1 measuring frequency w, the resistive peak occurs when p(T) becomes small enough to satisfy the Rio = 2 criterion. p(T) does not decrease as rapidly with decreasing temperature in an impure sample as it does in a pure sample. The peak will occur at a lower temperature for an impure sample. For samples that are Y!!:IL impure, Tp predicted by the Rio = 2 criterion may be below Tc of the material. In that case the peak is not observed above Tc but at Tc , where p decreases rapidly due to the onset of superconductivity. Early in the superconducting transition p becomes small enough co satisfy the Rio = 2 criterion and the resistive peak occurs in the midst of the superconducting transition. This explanation is consistent with the observation of the resistive peak at Tc by Strongin and Maxwell in GdxThl_xRu2 samples(12), since resistance ratio measurements indicated that these samples had a large resistivity just above Tc'

R2v w The relation ___ 0 __ = 1 also indicates that the resistive

peak can be forced ~oPoccur at Tc even for pure materials by making Rand w small enough that Tp should occur below Tc' This observation was tested by making resistive peak measurements on very thin pure indium samples at various frequencies. Figure 22, a semilog plot of frequency versus observed Tp for a sample of indium, shows that Tp is lower at lower frequencies. The log plot indicates that Tp for 33 Hz should be below Tc = 3.4K. The observed Tp for 33 Hz was 3.4K, with the resistive peak occurring early in the superconducting transition. Thus, the resistive peak will be observed at Tc for cases when the calculated Tp<Tc'

III. BOUNDARY SCATTERING IN SUPERCONDUCTORS

It is evident from the above discussion that, properly interpreted, the a.c. susceptibility technique can be used to study very minute effects on superconducting properties, at least in pure materials. We have pursued one such study to determine the effect on Tc of reducing the mean free path (mfp) in single crystals by means of boundary scattering. The purpose of this investigation was to observe thE; sQ called "mfp" effect first noted by Lynton, Serin, and Zucker. t19 ) This effect, a consistently observed reduction of Tc with reduction of mfp by the addition of impurities, was explained theoretically as the result of averaging the ~isQtropic pairing interaction present in real superconductors. t20 )

242 W. D. GREGORY

Two materials were investigated in this study - Ga and In. In the gallium study, 21 single-crystal plates of gallium with thickness ranging from 25~ to 250~ were used. The crystals were divided into three approximately equal groups with the crystals of each group having one of the principal axes of gallium perpendicular to the surface of the thin plate. Shifts in critical temperature were measured with reference to bulk single crystals of identical purity, using the change in mutual inductance of a pair of coils containing the sample as an indication of the phase transition.

The experimental results are shown in Figure 23, where the shift in Tc of the thin plates from the Tc of bulk gallium samples is plotted versus lid, where d is the sample thickness. One can see that for samples of all orientations, the shift in Tc is negligible. The data gave the best least-squares fit to a function of ~he form ~Tc = A/d+B and it was found that A = (-O.28±0.059) x 10- K cm and B = (0.lo±o.o4) x lO-3K. The constant term B is of the order of the scatter in the data and only represents the slight uncertainty in the measurement of Tc' Associating the A term with the mfp effect, we find that this shift of Tc with specimen thickness,when converted to ~Tc versus mfpi is much smaller than that reported earlier for the mfp effect.\ 9,20) In fact, the shift of Tc may be zero within the uncertainty of the data. This apparent lack of a mfp eff~ct)is surprising since the Fermi velocity of gallium is typically\21 6 x 107 cm/sec, so that scattering from the boundaries of a film 25~ thick will take place every 4 x to-ll sec. The lifetim~ of superconducting pairs has been calculated 22) and measured l23J to be of the order of 4 x 10-5 sec; hence about a thousand collisions should occur during the typical pair lifetime and anisotropy averaging should be complete.

When the results shown in Figure 23 were found, the effect of boundary scattering on another superconductor was investigated. Five micron thick pounded and annealed indium foils were fouyd jO exhibit the same mfp effect as indium with impurities added. 2(4 ) This same result was obtained earlier by Lynton and McLachlan 25 using similar foils. However, in both this investigation and in Ref. 25, the super conducting transition widths were considerably broadened compared with the gallium data, presumably because of the less desirable metallurgical condition of the indium samples. Preliminary data obtained by this author and C.A. Shiffman indicate that gallium with silver impurity added exhibits a shift of Tc comparable with that found in other metals with the impurityaddition technique for limiting the mfp.

In order to remove the uncertainty in the In data, the experiments were repeated with In single crystals.

The methods of making specimens(5) and measuring their criti-


~

M o ... X t-Y cO

12,16 e e13,17

A2 LEAST SQARES FIT

79 158 236 315 l/d (cm-1)

394

--........ ---___ ---..' lid (Mils-1) .2 .4 .6 .8 1.0

Fig. 23. The shift of Tc with specimen thickness in singlecrystal gallium plates. Open triangles, A-axis samples; open circles~ B-axis samples; solid circles, C-axis samples. Numbered points correspond to data from damaged samples which were excluded from the analysis.

. e •••

Gato,

I 0

• w E -0.4 • ~-0.6

-0.8

• -1.0 • -1.'2

Fig. 24. 6T versus lid for single crystals of Ga and In where 6Tc=Tc (thin plate) - Tc (bulk sample) and d is the specimen thickness.

243

244 W. D. GREGORY

III no' cm- 1, o ' • 11 16 10 ,.

Fig. 25. Solid line -LIT versus 1/R... for In where R... is the boundary-limited mfp gi%en by Eq. 29. Dashed line -- the fit of lITc to lid from Fig. 24, showing the difference between a "11R..." and "lid" effect.

TABLE III

Comparison of the Ga-energy gap anisotropy measured by various techniques, showing the contrast in estimates

obtained from "surface-" and "bulk-type" measurements .

l>1ethod measuring bulk properties Method

Mean-free-path-effect -impurity scatteringb

Specific heat c Critical fielde Ultrasonic attenuationg Thermal conductivityh Nuclear-spin relaxation

time i

. -----.--.-.-----.- -.----

0.02

0.02 0.04 0.075 0.017 0.011

Methods measuring f t ·" 2)a sur ace proper les -... a

~1ethod

Mean-free-path effectb -boundary scattering

Tunnelingd Tunnelingf

0.0025

0.0025 0.0020

._._--_._----- --.- .. -----a (a2) defined as in D. Markowitz and L.P. Kadanoff, Phys. Rev.

131, 563 (1963). b ---

W.D. Gregory, Phys. Rev. Letters 20, 53 (1968). c T.P. Sheahen, J.F. Cochran, and W.D. Gregory (unpublished). d K. Yoshihiro and W. Sasaki, J. Phys. Soc. Japan 24, 426 (1968);

?.~, 860 (1969); 28, 262 (1970). e W.D. Gregory, T.P. Sheahen, and J.F. Cochran, Phys. Rev. 150,

315 (1966); W.D. Gregory, ibid. 165, 556 (1968). f J.C. Keister, L.S. Straus, and W~ Gregory, Bull. Am. Phys.

Soc. 15,321 (1970); J. Appl. Phys. (to be published). g B.W. Roberts and H.R. Hart, Bull. Am. Phys. Soc. 1, 185

(1962); also (private communication). h N.V. Zavaritskii, Zh. Eksperim. i Teor. Fiz. 37,1506 (1959)

[Soviet Phys. JETP 10,1069 (1969)]. i R.H. Hammond and W.~ Knight, Phys. Rev. 120,762 (1960).


cal temperatures relative to a bulk single crystal(2,3) in zero magnetic field were essentially the same as for the Ga work. Twelve randomly oriented 99.999% pure In single crystals that

245

were grown from seed in molds as flat plates 6 mm wide, 25 mm long, and ranging in thickness from 23~ to 250~ were used. Figure 24 shows the difference of the Tc of the thin plates and that of a bulk sample (called ~Tc) plotted versus lId where d is the specimen thickness. The previous results for Ga single crystals are also plotted for comparison. While no shift of Tc within the scatter in the data is seen for Ga, the In data fit an expression given by

~Tc 6 lId = -(2.81 t O.15) x 10- K -cm (27)

Lynton and McLachlan(25) previously measured a finite shift of Tc in pounded and annealed plycrystalline In foils 5-6~ thick. If these data are plotted along with the sin~le-crystal In data and our data on annealed foils (see Figure 25), all of the data can be fit to the expression

~T

l/! = -(2.94 t o.15) x 10-6K -cm (28)

where i is the mfp due to boundary scattering(26), i.e.,

i = ~ d in (i/d) (29)

where i is the bulk mfp, taken as 350~ for all of the data. (Residual-~esistivity measurements made during this work and that of Ref. 25 confirm this value of i o ') Although polycrystalline specimens exhibited slightly broader superconducting transitions than the single crystals, it would seem that, since the bulk mfps io were the same for all specimens, the reduced values of ~Tc seen for the polycrystalline samples are due to the logarithmic factor in i and are not the result of an accidental compensation of part of the mfp effect by residual strain or damage left after annealing. (The fact that Tc and io are the same for the polycrystalline and single-crystal samples of comparable thickness, while the transition width is larger for the polycrystalline foils, is not inconsistent. For example, the transition width is known to be more seriously affected by imperfections than Tc (27».

Thus, the use of both polycrystalline and single-crystal data allows an evaluation of ~Tc over a sufficient~y large range of specimen thicknesses to distinguish between a true mfp effect and a "1/d(20jfect. Comparison of these data with the Markowitz-Kadanoff theory . for the mfp effect shows that the free parameter in this theory A~ (the ratio of the characteristic tim~ to destroy anisotropy to the transport-9011ision time, i.e., A~ = Tanis/Teoll) is very nearly unity (A~=l), implying that the scattering process

246 w. D. GREGORY

is "dif'fuse", accordt·ll£ to the theory of' Ref'. 4. Using this same theory, we concluded J that the null result f'or Ga could be explained, within the mf'p theory, by assuming that boundary scattering in Ga is "specular" or correlated, at least f'or those electrons af'f'ecting Tc' Several possible rea~ons why correlated scattering would occur were proposed in Ref'. 4, with the contrast to In somewhat uncertain because of' a concern about the origin of' a shif't in Tc in the polycrystalline samples. Now that the shif't in Tc in In is conf'irmed with single-crystal samples, one of' these postulates deserves further consideration. That is, Ga has a highly anisotropic Fe~mi)surf'ace while In ha~ an almost isotropic Fermi surf'ace. Price l28 and Ham and Matti s l 29) have shown that f'or materials with highly anisotropic Fermi surf'aces, the conditions of' conservation of' energy and momentum during the scattering at a boundary would severely limit the range of' scattering angles (producing correlated or specular scattering) while a material with an isotropic Fermi surf'ace would have many allowed scattering angles (producing uncorrelated or dif'fuse scattering). Our results f'or Ga and In would agree with this observation.

On the other hand, a more basic reason f'or a lack of' a mf'p ef'f'ect in Ga would be a lack of' an energy-gap anisotropy in the f'irst place. Since several authors estimated large values of' gap anisotropy f'or Ga (see Table III), this did not seem a likely argument previously and, theref'ore, was not presented. In(,~ct, we had obtained preliminary measurements of the anisotropy ) using the mf'p ef'f'ect produced by addition of' silver impurity to Ga and f'ound a reasonable anisotropy (-15%). Recently, however, direct measurements of' the gap energy ~y tunQeling into bulk Ga single crystals by two dif'f'erent groups(30,31) has yielded estimates of' the gap anisotropy in Ga no larger then 5%. If' one now compares all of' the methods f'or measuring gap anisotropy, one f'inds that surf'ace-type measurements (such as tunneling or the boundary-scattering shif't of' Tc) produce low estimates of' the anisotropy and bulk measurements (such as ultrasonic attenuation, thermal conductivity, etc.) produce high estimates of the anisotropy. This comparison is given in Table III. Although it must be cautioned that all of the anisotropy measurements are subject to some criticism (see Shepelev32 ), the fact remains that a very interesting connection appears to exist between the surface measurements and negligible anisotropy effect. Since Ga is the only superconductor that expands 9n f'reezing and is also known to collect impurities at the surface,l33) it is entirely possible that the Ga surface is strained or impu~~ and does, locally, have most of the gap anisotropy removed.l3 J

For those single-crystal materials that do exhibit an mfp effect due to boundary scattering (such as In), we note that application of the theory of' Ref. 4 shows that the shif't of Tc will de-


pend on the orientation of the crystal with respect to the surface of the plate and that a study of the mfp effect for samples of varying orientation m~ be used to unfold the anisotropic-pairing interaction. In fact, some of the scatter in the data in Figure 23 seems to be related to variation in crystal orientation. Using Ref. 4, the angular dependent part of 6Tc would seem to be about 30% of the total. This part of 6Tc should be measurable, to 10% accuracy, if the scatter in the data can be reduced by a factor of 3.

Finally, we note that Naugle and Glover(35) have recently reported a lid-type depression of Tc ' for thin films of amorphous materials. The total 6Tc ' as well as the slope [6Tc/(1/d)] that they observe, are in good agreement with the Markowitz-Kadanoff(20) theory, leading to the interesting speculation that they are seeing the mfp effect. However, it is difficult to see how the amorphous materials they use could have any of the gap anisotropy required to observe an mfp effect. Also, they note that the resistances per square (Ra) of their films are proportional to lid and that the films appear to have no ordered structure beyond tens of angstroms. These results imply that the mfp is much smaller than their film thicknesses so that the sample boundary would not determine.t. We must conclude at the present time that, although similar phenomena, our data and those of Naugle and Glover are not necessarily due to the same effect.

IV. TUNNELING INTO BULK SINGLE CRYSTALS

Since the early work performed by Giaever,(36) there has been a wealth of tunneling data reported on film-type samples. Film samples are easy to make and a great many of them can be made in a short period of time. However, fllms are often dirty and usually pOlycrystalline(37) and the basic orientatio~ of the film crystals cannot, in general, be controlled very well. l38 ) In addition, various crystalline phases of metal can appear in film sample~ which do not appear in bulk material under normal conditions. l39 ) Therefore, if one wants to determine the energy gap in a singlephase material, particularly as a function of crystal orientation, one must turn to bulk tunneling in single crystals.

Unfortunately, the difficulty involved with making junctions with bulk materials has greatly hampered progress in bulk tunneling work and it has only been recently that much of this effort has been reported. In particular, the results published to date have depended40n extensive surface preparation such as vacuum crystal growth,( 0) polishing and etching,(41) and oxidation 9( ~he tunneling surface under carefully controlled conditions. l 2) It was, therefore, considered desirable to determine if such surface prep-

248 w. D. GREGORY

aration conditions could be relaxed to increase the ease in fabrication. The elimination of some of the reported preparation procedures would be essential in those instances where the procedures themselves might destroy the properties of the material being investigated.

Gallium was chosen as the material to be used in developing a new fabrication procedure because it melts near room temperature (29.8°c), thus, permitting the fabrication of many samples in a short time. Also, several properties of gallium insure that any simple preparation technique would be put to a severe test. For example, gallium expands when it freezes, unlike almost all other materials. Therefore, the tunneling surface is more apt to be rough. G~ltum also has a very anisotropic thermal-expansion coefficient(43) and, as a result, stresses at the tunneling surface (which might affect the energy gap) would be more complex than for most other materials. Furthermore, because of its low melting point, gallium must be cooled during the film probe evaporation process, increasing the possibility that the tunneling surface will be contaminated with condensed gases. Gallium is also interesting to investigate as a material in its own right. It has very anisotropic properties and was explored for possible anisotropy in the energy gap. However, the boundary scattering experiments discussed above seem to indicate that gallium might not be very anisotropic after all. A recent review article(32) further points out the discrepancy in the measured gallium energy-gap anisotropy between tunneling studies and other methods of determining the energy gap. This discrepancy makes the further study of this metal a matter of some urgency. To make such a study, bulk gallium must be used in lieu of films because of the appearance of several phases of gallium, each having different superconducting chft~~teristics, whenever films are plated onto a cooled substrate. )

The results of this investigation showed that it was, indeed, possible to fabricate tunnel junctions, using bulk gallium, with less restrictive surface preparations than previously used, provided one was willing to tolerate junction impedances ranging from lOOO~ to 20M~. Since little information was available in the literature on the effects of high junction impedance on energy gaps obtained from tunneling investigations, a study was undertaken using the well understood and easily prepared Al-Al203-Pb film junctions to determine the effect of sample impedance on the energy gap of aluminum films. The results of the high-impedance study showed that there were no effects of impedance on the measured energy gap in the range of lOl~l07~. Results of the gallium tunneling work yielded energy gaps, 2~(O), ranging from 2.9 to 3.8kTc (which are comparable to energy gaps obtained by others for gallium), and some preliminary estimates of the gap anisotropy.


A. Sample Preparation and Experimental Procedures

The problems involved with obtaining bulk tunneling singlecrystal samples center on the following: obtaining single-crystal specimens having highly polished surfaces, creating a very thin, tough, insulating layer on the polished surface, evaporating film probes on top of the insulating layer, and connecting wires to the film probes. A comprehensive report of the samplE; mt3king and experimental procedures is given in the literature. l45J

B. I-V Tunneling Characteristics

Figures 26A to 26n show (solid lines) the actual X-Y recorder plots of four different types of I-V curves observed in this work. In spite of the large variation in the character of these I-V curves, all were obtained from samples exhibiting negligible amounts of current at zero voltage, ruling out the possibility that the less ideal curves are the result of damaged or shorted samples. Further evidence of this was obtained by repeating these measurements with several different samples and it was always found that the I-V characteristics were uniquely related to the sample orientation perpendicular to the tunneling surface. We concluded that the I-V characteristics were in some way related to the properties of gallium and we considered several possible correctioQ~6to the theoretical calculations of I-V curves by Shapiro et al. l I) that would explain these characteristics.

The most obvious possible correction wo~ld arise6from the high impedance of the tunneling barriers (10 to 4xlO ohms). A study of the effects of the sample impedance, discussed in Ref. 42, showed no difference in the data obtained from high impedance and low impedance samples.

A second correction, indirectly related to the tunneling barrier thickness, is the effect of sample impedance on the electronic circuitry. This effect was analyzed using a load line technique. The current flowing through the specimen, II' is a function of the actual voltage drop across the specimen, V', i.e., Il=f(V'), where f(V') is obtained from Ref. 47. At the same time, the current flowing through the circuit, 12, is determined by the expression, I2=V-V'/Rs' where Rs is the voltage source resistance. The actual current flowing in the circuit is obtained by plotting both II and 12 as a function of V' for various values of the internal source voltage, V. For each value of V, the current that flows is that for which the two curves intersect, i.e., where 11=12 , (see insert, Figure 26A). The most significant correction obtained with this procedure was to produce a somewhat shallower negative resistance

250

Ga-GaXOy-Pb

: S.oxl0S ohms ~ T-0.92 K

8-6S" > .. ~ -Corrected ] Theoretical Curve

~ .5

~ .. = u~ __ ~~~ ____ .-~~~~~~~

Ga-GaxOy- Pb

III 7.0Xl0S ohms .. T-0.92 K ·c 8.70" :I

.. - Corrected ~ Theoretical Curve .. :s ~ .5 .. ! .. = in mV U~~~~~~~~~~~~ __

l4S

Ga-GaxOy-Pb 2.Sx 104 ohms

: T-0.9K ·c 8-7S.5" :I > 'Corrected e Theoretical Curve .. :e ~

.5 c: f ~ u

1.2 1.3

Ga-GaxOy-Pb 4.4Xl0S ohms

: T·0.93 K § 9:32

> .. ~ :a

(BI

W. D. GREGORY

Voltage in mV 1.4 I.S

~ -Corrected .5 Theoretical Curve

i 1:(It.A+91,~.v1O .. 2AA(O)=7.2 kTc J~~ __ ~ __ ~ __ ~2~~)=~1»~k~~~

1.1 1.2 1.3 1.4 1.5 1.6 Voltage in mV

(0)

Fig. 26. IV curves obtained from electron tunneling into singlecrystal gallium in the A-B plane at various crystal orientations. The crystal orientation refers to the gallium A-axis. The solid lines are experimental data traces and the dots are IV curves calculated under various assumptions about the anisotropy and multiplicity of the gallium energy gap, discussed in the text. (A) Almost ideal IV curve measured at an orientation near which the energy gap did not vary appreciably. (The inset is a sketch of the "load line" method used to correct the IV curves for effects due to the im-pedance of the voltage source.) (B) IV curve obtained in a region of crystal orientation where the energy gap varies substantially. (C) IV curve obtained at orientation where the energy gap changes discontinuously. (n) Strongly per-turbed IV curve obtained consistently at orientations of 32° and 36°.


region than that calculated using the equations of Shapiro et ale

According to the calculations of Clem,~7) the tunneling current will sample the energy gap structure over a cone of angles of 2 to 3 degrees from the primary tunneling direction. In regions where the energy gap varies considerably over this cone of angles, the measured I-V curve will be some average of the I-V curves associated with all of the energy gaps in that cone. The net effect of this correction is to round off the sharp cusp at the voltage corresponding to the difference of the gaps, (~Pb-~Ga)/e, and to produce a more gradual rise in the current at the sum voltage, (~Pb+~Ga)/e.

Finally, there is increasing evidence seen in tunneling data i~ the literature fo: mUlr~Pte energy gaps, particularly in pure s1ngle crystal mater1als. 8 In our previous work (Ref. 45), we saw I-V characteristics in the C-axis direction in gallium that looked very much like the data shown in Figures lC and lD. This curve could be explained empirically as resulting from the sum of two separate I-V curves having two separate gallium energy gaps. In the present work, the sums of theoretical I-V curves were fit to such data.

C. Comparison of Theoretical and Experimental I-V Curves

The corrections described in the last section were applied in varying degrees to the four curves shown in Figure 26. In Figure 26A a very small correction for local gap anisotropy (averaging gallium gaps varying by t~) and the load line correction were used. The final theoretiEal curve (the dots in Figure 26A agrees quite well with the experimental curve.

In Figure 26B, in addition to the load line correction, a more substantial local anisotropy correction was used, averaging I-V curves with gallium energy gaps differing by t2.5%. Again, the agreement is good, although the experimental negative resistance region is still slightly shallower than predicted theoretically.

Figure 26c and 26D illustrate the use of a multiple gap correction to the theoretical I-V curves. Figure 26c was fit using the load line correction plus the assumption of two gallium energy gaps differing about 4% and each of these energy gaps was averaged about 1%. Figure 26D was fit assuming two energy gaps of 1.6kTc and 7.2kTc and assuming a ratio of the currents of the small to the large gap of approximately 9 to 1, with tunneling dominated by the smaller gap. It should be noted that, with regard to the multiple gap assumption as an empirical fitting technique, the size of the energy gap regulates the position of the bumps in the I-V

252 W. D. GREGORY

curve as a function of voltage and the ratio of the tunneling currents for the two gaps regulates the strength or predominance of these bumps. As a result, the size of the gaps and the relative tunneling currents are independent parameters in the fit, making it possible to obtain more information from straight I-V data than with derivative (dI/dV versus V) techniques, particularly with high impedance junctions where electronic noise reduces the information available in the derivative curves. For the multiple gaps, other than at 32 degrees and 36 degrees (see dashed line~, Figure 27), the current ratios were close to 1:1 for the two gaps. The accuracy of the gaps for the latter case was as good as in single gap measurements, i.e., about ±~. The current ratios were good to 5% to 10%. The best fit for2the two cases of largely varying gaps at 32 degrees and 36 degrees was noticeably worse if the current ratio was changed from 9 to 1 to 7 to 1 or if either of the gaps was changed by about 10%.

D. Anisotropy of the Energy Gap

Figure 27 shows the reduced energy gap, 2~(0)/kTc' as a function of the crystal orientation perpendicular to the tunneling surface, obtained from I-V data such as that discussed above. The scatter in the data points obtained in this work (solid dots in Figure 27) was quite small and the best fit curve to these data was simply the solid line connecting the points. Places where multiple gaps were observed are designated by dashed lines.

The labels on each of the I-V curves of Figure 26 in-dicate the crystal orientation for which the energy gaps were measured and plotted in Figure 27. The amount of energy gap anisotropy averaging necessary to fit the I-V curves of Figure 26 is comparable to the local variation of the energy gap over the 2 to 3 degree cone calculated by Clem at the crystal orientation at which these curves were measured, such as at 75.5 degrees (Figure 26B). I-V curves such as for Figure 26A (65 degrees) occurred at points where there is very little variation of the energy gap within 2 to 3 degrees of this crystal orientation and indeed very little averaging of gap anisotropy was required to fit the I-V curve. It appears, then, that the empirical correction for the local anisotropy made above is some indication of the extent to which the energy gap varies within 2 to 3 degrees of the sample orientation.

The data analysis presented above shows evidence for multiple energy gaps in gallium in at least five different directions in the (001) plane (i.e., at 32 degrees, 36 degrees, 61.5 degrees,


3.7

~u ;;:.= ......... 0 3.6 Ci c-4

3.5

•• 7.2 I I I

•• 1.6

253

(l00) 40° 60° 80°(010) Crystal Orientation

Fig. 27. Gallium energy gap vs crystal orientation, in the A-B plane, referred to the gallium A-axis. The data are from Ref. 7.

254 W. D. GREGORY

70 degrees, and 82 degrees from A axis). In addition, there is a sharp change of gap at 43 degrees and evidence for a multiple gap in the C axis direction was presented in Ref. 42. Roberts and Hart(49) have also seen some evidence of multiple energy gaps in gallium in ultrasonic attenuation measurements. The presence of these multiple gaps and perhaps the ratio of currents tunneling to each gap can be explained by comparing the data to the gallium Fermi surface in the extended zone scheme ~si~g the selection rule proposed by Dowman, MacVicar, and Waldram.~50) In fact, this comparison is the first definitive evidence for the validity of any selection rule and as such will be discussed in the following section. We will restrict our comments here to the significance of this multiple gap structure.

Except for the possible multiple energy gaps, there does not appear to be a large anisotropy to the energy gap in the gallium (001) plane. This is somewhat surprising in view of the large amount of anisotropy in other metallic properties of gallium. At least one other measurement, involving the effect of boundary scattering on the superconducting critical temperature of gall~~discussed above, might possibly lead to this same conclusion.~4,34) The present results indicate yet another possible explanation for this apparent discrepency beyond those discussed in Ref. 4. If the multiple energy gap effect is taken into account, methods which probe anisotropy in the bulk and over a large range of orientations would tend to yield large values of the anisotropy, while techniques such as electron tunneling, which probe over a small cone of angles, would show correctly that large variations of the gap occur primarily where multiple energy gaps are possible.

The experimental I-V curves had somewhat shallower negative resistance regions than even the corrected theoretical curves. This may be due to either a.c. effects (in the load line correction no effort was made to use complex impedances), or to the fact that the densities of states for both normal and superconducting metals were assumed to be theoretically ideal. The data itself indicates that in many directions the density of states in the superconductor was very complex.

Finally, we wish to point out that in all the discussions above, that only anisotropy in the (001) plane is used to analyze the results. Clearly, the effects of the anisotropy in a 2 to 3 degree wide cone off the (001) plane should also be included in the analysis for completeness. In light of the discussion of the selection rules that follows, it is probabably because of the symmetry properties of the (001) plane that no serious error is caused by this omission, i.e., the anisotropy is related to the Fermi surface structure in the extended zone scheme and these properties are relatively constant a small distance above and below the (001) plane due to reflection symmetry.


E. Selection Rules for Tunneling

We will concentrate here on the matter of interpretation of bulk tunneling data, such as illustrated in Figures 26 and 27.

255

The problems involved in interpreting tunneling data obtained from single-crystal spe9im~ns were summarized recently by Dowman, MacVicar, and Waldraml50 } (DMW). There are two questions involved: (1) What tunneling direction dominates across the barrier? For many reasons pointed out in Ref. 50, the barriers produced on single-crystal substrates might have a highly ordered structure, giving rise to an anisotropic tunneling probability that is not a maximum perpendicular to the barrier surface, as is usually assumed. (2) Which electrons are associated with the tunneling process? The usual rule has been to choose electrons with a group velocity perpendicular to the tunneling b~rier1 but application of this rule has met with little success. l51 ,52

In this section, we will examine these problems using our( ) data on gallium and similar data of Yoshihiro and Sasaki (YS). 53

1. Effect of barrier structure. Figure 28A shows the reduced energy gap, 2~(0)/kTc' as a function of crystal orientation in the (001) plane of gallium obtained by our group and YS. (Note that the gallium phase we are dealing with, the one stable near room temperatures and atmospheric pressure, is orthorhombic.) In several directions, indicated by the dashed lines, we have identified multiple energy gaps. This identification is not made by YS, probably because of a difference in data analysis.---

Aside from the difference of interpretation of multiple gaps, our experiments and those of YS were essentially the same with one exception: the tunneling barriers used in our work were untouched, naturally grown oxides while YS bombarded the single-crystal surfaces with ions, t9 r~duce barrier impedance, before the tunneling probe was applied. l54 ) As a result, their barriers were undoubtedly disordered while the barriers used in our work had some chance to develop an ordered structure. (However, both barriers were probably somewhat disordered.) As one can see from Figure 28A, there is good agreement between the two sets of measurements, with only slightly more scatter in the data obtained by YS. Some scatter in their data might be expected since the bombardment process would also produce shorts and damage in the barrier region and this might give rise to slightly imperfect tunneling characteristics.

The fact that two types of barriers yield the same energy-gap data is an indication that tunneling occurs in the direc-tion perpendicular to the barrier for all orientations measured. This point can be checked further by assuming perpendicular tunneling and comparing features of the energy-gap curve with the features

256

3.7

(l00)

"7.2 , , , ' , ,

!l:rr'\~ ~ ~~ : I 0

¢: :p , ,

20° 40° 60° 80° (01 0) Crystal Orientation

( a)

kx (A axis)

(b)

W. D. GREGORY

Fig. 28. (a) Reduced energy gap, 26(0)kTc ' as a function of crystal orientation in the gallium (001) plane. (The crystal orientation is the direction perpendicular to the tunneling surface.) Open circles are dat a point s from Ref. 30 and solid circles are data points for this work. (b) Extengedzone-scheme plot of the Fermi surface in the gallium k =0 plane using the nearl§-free-electron approximation (Brillouin zones are numbered). (c) Same extended-zonescheme plot as in (b) with both the nearlyfree-electron and WOOdIS augmented plane-wave calculations included in the fifth zone near 61.5°.

82°

32°


of the Fermi surface.

2. Fermi-surface selection rules. It has been commonly assumed that the tunneling electrons come from pieces of the fe~i surface with a group velocity perpendicular to the barrier. l5lJ DMW point out that a WKB calculation shows that, aside from problems associated with barrier structure examined above, the tunneling probability is maximum for electrons with k vectors entirely normal to the tunneling barrier (zero transverse wave vector, kT=O). These two selection rules are at times the same, but for a realistic Fermi surface the group velocity cannot always be in the normal k-vector direction. To determine which of these rules is valid, it is necessary to compare all the features of the energy-gap curve with the Fermi surface using each rule. Only the valid selection rule can be expected to explain all of the features of the data.

Group-velocity rule - Since the electron group velocity is given by the gradient of the electron energy in k space, the group velocity is perpendicular to the Fermi surface in k space. Using this criterion, one must identify all groups of electrons where the normal to the Fermi surface is parallel to the tunneling direction. This is usually done by calculating the Fermi-surface contours in the Reduced-Zone-Scheme (BZS). Typical contours for gallium have been calculated by Wood,l55) using the augmented plane-wave method. There appears to be no correlation between the Reduced-Zone-Scheme contours, given in Ref. 55, and the energy-gap features, such as multiple energy gaps, observed in our data. A similar lack of agreement with the group-velocity ru~e has been found in previous investigations in other materials. (51,52)

~=O rule - In addition to establishing somewhat firmer physical reasoning for using the ~=O rule, DMW also argued that this rule must be applied in the Periodic-Zone-Scheme (PZS) (sometimes called the Repeated-Zone-Scheme) rather than in the RZS in order to obtain all of the possible correlations to the Fermi surface. (Recall that, because of the Bloch theorem, each piece of the Fermi surface may be "repeated" into all of the Brillouin zones.) However, a construction involving all of the repeated zone pieces of the Fermi surface would be horribly complex, so we have compared our data only to those pieces lying close to the nearly-freeelectron sphere. The assumption here is that for nearly-free-electron materials, these portions of the Fermi surface would dominate, and for other materials, any correlations observed are at least an allowed subset of all the possible correlations.

As a result, we have calculated an approximate extended-zone model for the gallium (001) plane by drawing the Brillouin zones for this plane, drawing in a free-electron sphere with a radius of

258 w. D. GREGORY

1.17 a.u.-l that is suggested by Wood's calculations, and perturbing the free-electron sphere at the zone boundaries in the usual fashion; i.e., there is an energy gap at the zone boundary and the Fermi surface should be normal to the boundary. The results of this calculation are shown in Figure 28B. The position and the shapes of several pieces of the Fermi surface calculated in this fashion were compared with Wood's calculation by remapping some of Wood's data back from the first zone.

If we now examine the Brillouin-zone diagram in Figure 28B as a function of the angle from the kx axis (which corresponds to the A-axis direction in real space), we find that there are several places where the k vector crosses zone boundaries and may even have an overlap with two zones at a boundary. In order of increasing angle these are at 17° (several possibilities), 32°, 36°, 43°, 64°, 70°, 82°, and 84°. We have data available at all points except near 17° and 84°. We have observed multiple energy gaps very near 32°, 36°, 64°, 70°, and 82°,at which directions there is an overlap of two zones in Figure 38B. At 43° we observed a sharp change in the magnitude of gap and in this direction there is a change in the contribution to the Fermi surface but no overlap of the zones in Figure 28B. The multiple gap measured at 61.5° is undoubtedly associated with an overlap of the fifth and sixth zones occurring at 64° in the NFE calculation. The slight disagreement in the angle at which this multiple gap occurs might be due to the approximation in the NFE model. A piece of the fifth zone, calculated by Wood, was remapped from the first zone and better agreement was found. Figure 28c shows the Fermi-surface diagram in the EZS with the lines marking the zone boundaries removed.

F. Discussion

The ~=O rule used with the EZS explains all of the features of the energy gap versus crystal orientation curve in the gallium (001) plane that we have observed, while the group-velocity rule is deficient. Multiple energy gaps occur at or near zone boundaries where a group of electrons from each zone may have a k vector perpendicular to the sample surface. The energy gap also changes as the dominant contribution of electrons change from one zone to another. All of this analysis presumes that the dominant direction for tunneling in real space is that perpendicular to the sample surface. The good comparison of our data and those of YS, using different barriers, further suggests that barrier structure does not affect the tunneling direction in this case.

It would seem possible in principle to compare quantitatively the features of the curve of the gap versus orientation with the Fermi surface (such as the magnitudes of the gaps and the ratio of of multiple-gap tunnel currents). However, such analyses are fruit-


less at the present time since the NFE model obviously does not yield enough information about the Fermi-surface shape within a zone, and in any case, is only a subset of the total correlations expected using the periodic zone model.

V. Conclusions

The results discussed above indicate that a great deal can still be learned about the basic interactions causing superconductivity by estimating the natural variations in the strength of these interactions from measurements of the anisotropy of superconducting properties. The recent development of refined experimental techniques, coupled with the availability of the necessary pure materials, at moderate cost, now make it possible to conduct such experiments on a sufficiently large scale that detailed models for superconductivity in real metals m~ be tested. This information would be a valuable guide in the quest for better materials and devices, to be discussed in the remainder of this course. Before the experimental data can be put to good use, however, it will be necessary for progress to be made on the theoretical side. Hopefully, the exposition given above has served the dual purpose of encouraging such theoretical studies, as well as introducin~ those of you who have attended this course to the details of some of the more refined experiments recently performed on superconducting materials.

VI. Acknowledgements

As the references indicate, a large number of workers, particularly present and past graduate students in the Georgetown Low Temperature Group, have contributed to the work discussed above. The author is deeply indebted to these students for their painstaking efforts and for numerous enlightening discussions. This work benefited from the support of the Atomic Energy Commission at Georgetown University and the Advanced Research Projects Agency at f-1. 1. T.

References ------1. One of the reasons pure metals are now available seems to be

due to the growth of the semiconductor and integrated circuit industries. Extremely pure metals are required as dopants for many semiconductor devices and for components, such as thin film resistors, in integrated circuits. Host of the elemental superconducting metals are used in one of these capacities.

2. W.D. Gregory, Phys. Rev. 165, 556 (1968). 3. W.D. Gregory, T.P. Sheahen-Bnd J.F. Cochran, Phys. Rev. 150,

315 (1966). 4. W.D. Gregory, Phys. Rev. Letters 20, 53 (1968).

260 W. D. GREGORY

5. W.D. Gregory and N.A. Superata, J. of Crystal Growth ~, 5 (1970).

6. W.D. Gregory, N.A. Superata and P.J. Carroll, Phys. Rev. B3, 85 (1971).

7. J.C. Keister, L.S. Straus and W.D. Gregory, J. of Appl. Phys. 42,642 (1971); W.D. Gregory, R.F. Averill and 1.S. Straus, Phys. Rev. Letters 27, 1503 (1971). Similar data for In may be found in R.F. Averill, L.S. Straus and W.D. Gregory, Appl. Phys. Lett. 20, 55 (1972).

8. W.L. Pillenger, P.S. Jostram and J.G. Daunt, Rev. Sci. Inst. 29,159 (1958).

9. N. Yaqub and J.F. Cochran, Phys. Rev. 137, A 1182 (1965); J.F. Cochran and M. Yaqub, Phys. Rev. 140 A 217l+ (1965).

10. G. Seidel and P.R. Keesom, Phys. Rev. 112,1083 (1958); G. Seidel and P.H. Keesom, Phys. Rev. Letters S 261 (1959).

11. D. Sh8enberg, "Superconductivity," 2nd. ed., Cambridge, New York (1952).

12. 1.1. Strongin and E. ~1a.xwell, Rev. Sci. Intr. 34, 590 (1963). 13. J.F. Schooley in "Proceedings of the 2nd Symposium, 1969

Spring Superconducting Symposia," (NRL, 1969) NRL Report 6972, p.7; R.A. Hein and R.L. Falge, Phys. Rev. 123,407 (1961).

14. L. Bosio, A. Defrain and I. Erdboin, Compt~end. 250, 2553 (1960). -

15. WADD Technical Report PB17l6l9, Clearinghouse, Springfield, Va. (1960.

16. J.E. Neighbor, J. Appl. Phys. 40, 3078 (1969). 17. R.S. Carslaw and J.C. Jaeger, "Conduction of Heat in Solids,"

Clarendon Press, Oxford (1959). 18. G.E.R. Reuter and E.R. Sondheimer, Proc. Roy. Soc. A 195, 336

(1948). 19. E.A. Lynton, B. Serin and M. Zucker, J. Phys. Chem. Solids 1,

165 (1957). 20. D. Markowitz and L.P. Kadanoff, Phys. Rev. 131, 563 (1963). 21. R.J. von Gutfeld and A.H. Nethercot, Jr., Phys. Rev. Letters

18,855 (1967). 22.

23. 24. 25. 26.

J.R. Schrieffer and D.M. Ginsberg, Phys. Rev. Letters §., 207 (1962). D.M. Ginsberg, Phys. Rev. Letters 8, 20l~ (1962). G. Chanin, E.A. Lynton and B. Serin, Phys. Rev. 114,719 (1959). E.A. Lynton and D. McLachlan, Phys. Rev. 126, 40-rl962). The expression for the mean free path associated with current conduction along the lon~ direction of a thin plate, as calcu-lated by Fuchs [K. Fuchs, Froc. Cambridge Phil. Soc. 34, 100 (1938)] is l=t~ In(l !d). Equation (29) was obtained--irom Fuch's model, but sigce the free paths, in the present case, were not weighted in favor of the lonr, dimension, as they are for current conduction, the mfp was found to be smaller. This reduction amounts to replacinG the factor 1. in the Fuchs expression bY!. The details of this calcul~tion are contained

2


in Appendix 5, lv.D. Gregory, Ph.D. thesis, MIT 1966 (unpublished) •

27. J.F. Cochran, Ann. Phys. (New York) 19, 186 (1962). 28. P.J. Price, IBM J. Res. Develop. 4, 152 (1960).

261

29. F.S. Ham and D.C. Mattis, IBM J. Res. Develop. 4,143 (1960). 30. K. Yoshihiro and W. Sasaki, J. Phys. Soc. Japan-24, 426 (1968);

860 (1969); 28, 262 (1970). -31. J.C. Keister, L.S. Straus, and W.D. Gregory, Bull. Am. Phys.

Soc. 15, 321 (1970); also see Ref. 7. 32. A.G. Sheplev, Usp. Fiz. Nauk 96, 217 (1968) [Soviet Phys. Usp.

11, 690 (1969)]. --33. E~ Papp and K. Soymar, Tsvetn. Mettal. 5,147 (1963). 34. Recently, D. Gubser (Phys. Rev. §" 827(1972)) has shown

that a combination of anisotropy and deviation of the pairing interaction strength from weak coupling could cause the observed deviation of the critical field curve from the BCS weak coupling limit.

35. D.G. Naugle and R.E. Glover, III, Phys. Letters 28A, 611 (1969). 36. Ivar Giaever, Phys. Rev. Lett. 5,147 (1960). ---37. L. Holland, Vacuum Deposition of Thin Films (Wiley, New York,

1961), p.2. 38. D.M. Evans and H. Wilman, Acta Cryst. 5,731 (1952); see also

G.I. Lykken, A.L. Geiger, and E.N. Mitchell, Phys. Rev. Lett. 25, 1578 (1970).

39. R.W. Cohen, B. Abeles, and G.S. Heisbarth, Phys. Rev. Lett. 18_, 336 (1967).

40. N.V. Zavaritskii, Sov. Phys. JETP 18,1260 (1964). 41. A.F.G. Wyatt, Phys. Rev. Lett. 13,160 (1964). 42. M.L.A. MacVicar and R.M. Rose, J. Appl. Phys. 39,1721 (1968). 43. H.Yaqub and J.F. Cochran, Phys. Rev. 137, A1l82 (1965). 44. ll. Wuh1, J.E. Jackson, and C.V. Briscoe, Phys. Rev. Lett. 20,

1496 (1968). 115. J.C. Keister, L.S. Straus, and W.D. Gregory, J.Appl. Phys.

42, 642 (1971). 46. ~ Shapiro, P.ll. Smith, J. Nicol, J.L. Miles, and P.F. Strong,

IBN Res. Develop. 34, 34 (1962). 47. J .R. Clem, Am. Phys. (New York) 110, 268 (1966). 48. See, for example, J.W. Hafstrom -and M.L.A. MacVicar, Phys.

Rev. B2, 4511 (1970). 49. Private Communication. 50. J .E. Dowman, r-i.L.A. NacVicar, and J .R. Wa1dram, Phys. Rev. 186,

452 (1969). 51. N.V. Zavaritskii, Zh. Eksp. Teor. Fiz. ~, 1839 (1963) [Sov.

Phys. JETP 18, 1260 (1964)]. 52. B.L. Blackford and R.H. March, Phys. Rev. 186 397 (1969). 53. K. Yoshihiro and W. Sasaki, J. Phys. Soc. Jap. 28, 452 (1970). 54. K. Yoshihiro and W. Sasaki, private communication. 55. J.H. Wood, Phys. Rev. 146, 432 (1966).

T 's--THE HIGH AND LOW OF IT c

Bernd T. Matthias

* University of California, San Diego , La Jolla, Calif.

and Bell Telephone Laboratories, Murray Hill, N. J.

Preface: The following manuscript was adapted from a tape recording of the lecture presented by Professor Matthias at the Conference, "The Science and Technology of Superconductivity" held at Georgetown University, August 1971. His hand drawings (blackboard drawings during lecture) are included here for clarity.

I would like to show you how to discover superconductors, how to predict the transition temperatures, and how one can reach an understanding of superconductivity in a way that is different from the way taught to you by the theorists. Any real understanding always leads to a procedure which will enable us later on to predict. Only once a problem is understood properly can it be predicted, and therefore prediction is an extremely good criterion. The people who don't seem to be able to predict anything may also not understand anything. And that will be the essence of my talk.

First of all, let me ask you, don't wait until the end with questions. You're not going to rattle me, and I wish you would ask me immediately whenever something doesn't seem clear to you, or you don't agree with what I have to say. Speak freely. It is much better for me when you do that, because then I don't have to talk into a vacuum.

* Research in La Jolla sponsored by the Air Force Office of Scientific Research, Air Force Systems command, USAF, under AFOSR grant number AFOSR-63l-67-A.

263

264 BERND T. MATTHIAS

Superconductivity is a collective phenomenon. A collective phenomenon, or a cooperative phenomenon, is a phenomenon that happens only when there is a sufficient number of individuals participating in it. In other words, one tin atom is not going to be superconducting, or--as you are told now and then--one molecule of benzene. And similarly, one atom of iron is not going to be ferromagnetic when it's by itself. It takes a certain number-then suddenly the collective phenomenon occurs. Now this is a very crucial feature of superconductivity--much more crucial than you have been led to believe until now. Because, you see, only if you view superconductivity in the framework of collective phenomena in general can you get even a vague idea what the whole thing really is about, and what its limitations are. Because, as we all hope, the superconducting transition temperatures will get higher, these things will get more and more useful, and new materials will come along that will help the whole field. But in order to do this, one must understand superconductivity in the framework of collective phenomena in general.

Now let me tell you just a few of the collective phenomena that do exist. The oldest collective phenomenon that is known to mankind, in physics that is, is melting. Three more recent ones are ferroelectricity, ferromagnetism, and superconductivity. Now these are just a few of the collective phenomena that occur in nature. One of the common features of all collective phenomena is! nobody can predict them, nobody can calculate them, and from a mathematical point of view they are totally intractable. As a consequence, most theories of superconductivity are somewhat unrealistic, since they deal with superconductivity as sort of a degenerated resistivity, concerning one electron only. That of course leads you to Green's functions and the absence of any further predictions.

Actually, collective phenomena are very important when you consider them in the framework not just of physics. The fact that we can't deal with collective phenomena can be seen everyday around us. When there are too many, people can't handle things any longer, whether it's traffic jams (cars), or the economy (dollars), or light bulbs (like the black out of New York)--the moment there are too many, the lightsgo out, the dollar goes down, and the traffic stops--collective phenomena are all around us. Politics is another one which people cannot predict, which people cannot deal with, and therefore, you may ask--or you should have asked then-what is going to be done about it.

I'll tell you what I try to do about it. It was an old idea, a friend of mine once suggested many years ago. Let's look at so many instances of one given phenomenon that at least we can get a rule--a feeling for what the crucial conditions are. If we do this, then relying on the correctness of these conditions, we then

Tc's-THE HIGH AND LOW OF IT

can predict them. This is what I did in superconductivity, and formerly in ferroelectricity. And the fact that all these compounds become superconducting is--in a way--a justification for this approach.

265

But mind you, it is not a mathematical approach. The mathematics, if it comes at all, comes afterwards; in nothing of what I'm going to tell you will Green's functions ever be mentioned. As a matter of fact Green's functions have been abused to an extent which is unbelievable, in my opinion. Therefore I will talk to you,at the extreme, only about the number of electrons or concepts like this. And I will not talk to you about the collective phenomena which occur outside of this building.

Let me just mention to you here: the collective phenomena I have just mentioned in physics are all interconnected. And only once it has been realized that these interconnections are crucial has it become possible to understand the limitations of superconductivity. As you know, for the last fifteen years or so the promises as to how high the transition temperatures will go--room temperature, far above, and all that--nothing has ever come true. And one might ask why. with all the theories around us, with all the explanations, not a single superconductor can be predicted, not a single temperature can be guessed--and I don't mean 10%, I mean orders of magnitude. How come all the predictions concerning the elevation of transition temperatures are wrong? Well, part of what I'm going to talk to you about today is that only by considering all collective phenomena in general will you see why the transition temperature is limited. The limitations imposed by collective phenomena other than superconductivity are the crux of the matter. It is the reason why we cannot go very easily to higher temperatures, why the elevation of superconducting transition temperatures is such an extremely difficult affair. In the very end it will all center around the melting point. And that is not a facetious statement. People might say "Good heavens, the melting point is a high temperature phenomenon, and superconductivity is a low temperature phenomenon." But you see, that was the mistake of the past. People tried sort of to put things in little boxes without ever looking left or right. And then of course you can't get anywhere at all.

Part of my talk will go to show you why, for instance, in many-valley semiconductors, the superconductivity is limited, not by the valleys or anything of the sort, but strictly by ferroelectric behaviour. There are other superconductors which are limited by their ferromagnetic behavior, and most superconductors today--the high temperature ones--are limited by their melting point. Namely, they just won't form. The compolliLd forms only after it has frozen, and those superconductors which would have been very high, according to the rules which I am going to show

266 BERND T. MATIHIAS

you now, won't form. And that is an ominous feature. Whenever we can predict a compound that would have a very high superconducting transition temperature it won't form, or it won't form correctly, or it will form only approximately. I'm going to show you all that right away. But I just wanted to tell you what I'm going to talk about. I'm going to talk about empiricism, and the experimental approach to superconductivity, and purely on a statistical basis from this point of view. We are looking at so many instances that we can't help noticing what's going on.

Let me now give you in one sentence the crucial conditions for superconductivity. It is a very simple concept--the number of electrons outside of the filled shell. This applies to elements, compounds, alloys, whatever you want. There are a certain number of exceptions, and a certain number of deviations, but in essence, it is always the number of unpaired electrons outside of the filled shell that determines the superconducting transition temperature. In one sentence you can really say: if one has a maximum number of unpaired electrons one has a maximum transition temperature. Of course, there are many facets to this. But maybe I should show you now how this looks in nature, i.e. in the periodic system. Solid state physics in my opinion always has to deal with the periodic system, and when I show you the periodic system you will probably understand much better the approach I am talking to you about. You can see already that essentially the majority of elements undergo a phase transition of one kind or the other. When' the square is only half shaded, it means you have to use certain tricks. That is not particularly astonishing--100k at phosphorous, for example. Phosphorous is a very good insulator. It has a resistivity of about 10150hmcentimeters in its natural state. Obviously, this will never become superconducting. What you have to do, therefore, is to squeeze it into a metal. We compressed it, [1] and the more we compressed it, the more metallic it got, until it finally became superconducting at 5Kor so. Now, that is quite a change in resistivity, particularly considering that the superconducting resistivity is not known, except that it is at least below 10-25 ohm-centimeters. So just with a little bit of pressure, 110 ki10bars or so, we managed to change the resistivity by 40 orders of magnitude, which is quite a change.

When Geba11e and I started out, many years ago, the periodic system didn't look anything like this. There were a lot of holes. For instance there was a hole where molybdenum is now, it wasn't then superconducting. Tungsten wasn't superconducting. Iridium-the whole place was full of holes. Now, nature just doesn't make any holes and neither does the periodic system. So we realized that these elements must be superconducting. And over the years almost all elements were found to be superconducting where one had expected them to be. For instance, iridium became superconducting

H

I F

ig.

1 T

he

Peri

od

ic

Sy

stem

Li

~Be0-i

I B

I

C

I N

I

0 I

F

Na

Mg

f:l,A1~ S;~ P~

"""

,,~

/ /

S

C1

K

I Ca

Se

I

Ti

I V

rr

~r.rm

:::~~

.r::mf.

~I:r

:::CoJ

lt~;J]

Cu ~Z

n~.G

.a.%

1 .~e

.%1

,A,s

)j1

.S.e

.%1

Br

Zr

I Nb

M

o Te

Hf

Ta

I . W

Re

Th

Pa

LANT

HAN

I DES

(R

ARE

EAR

THS)

TRAN

SITI

ON

EL

EMEN

T SU

PERC

OND

UCTO

RS

TRAN

SIT

ION

EL

H1E

NT

SUPE

RCO

N-~

DUCT

ORS

(ONL

Y UI

1DER

PR

ESSU

RE)

~

NO

N-TRA

I~SI

T IO

N E

LEllE

iH

SUPE

RCON

DUCT

ORS

~

NO

N-T

RAN

SITI

ON

EL

EMEN

T S

UPE

R-

COND

UCTO

RS

(ONL

Y UN

DER

PRES

SURE

)

Ru

.Rh

Pd

Ag

t%Cd%ln%t/Sn~ Sb~

Te)

j I

Os

I I r

P

t Au

V

-HQ

'l:}'

lTlI'

A/:

Pb

'i1

Bi'i

1

Pn

I A

t

~

SUPE

RCO

NDUC

TING

(O

NLY

UNDE

R PR

ESSU

RE)

-I~

AGNE

TI C

TRA

NS IT

I ON

EL E

MEN

TS •

nl~f~

RARE

EA

RTHS

AN

D TR

ANSU

RANI

C E

WIE

NTS

D

NOT

SUPE

RCO

NDUC

TING

~

EXTR

APO

LATE

O

.....

He

n ",- L

Ne

::r

m

::r

I G

) A

::r

>

Z

K

r 0 ,....

0

Xe

~

0 "TI

Rn

=l ..., 0- "


literally overnight between Bob Rein and us. The moment we knew how these things would happen, we could do it through themail.by phone; it was no problem any longer.

One of the most vexing holes was at the site of rhodium. So rhodium had to be superconducting; only we didn't know where. Now, we can tell where and I'll show you how this was done. [2] We made rhodium-iridium alloys, we made rhodium-osmium alloys--and as temperature is a logarithmic concept, one has to plot it logarithmically. When plotted vs concentration of rhodium, one has a very acid test. If the two lines will intersect at the site of pure rhodium,--there is only one temperature for rhodium, after all--if they intersect properly, we've got it made. If not, we don't believe it.

100~--.---.---.---.---'---rr--'---.---.--.

10

1.0

Fig. 2

~ Rh-I~

\ '" Rh - Os '"

% RHODIUM

Extrapolation to the superconducting transition of rhodium.

Tc's-THE HIGH AND LOW OF IT 269

What are you shaking your head about? SCHOOLEY: Well, that curve is very reminiscent of molybdenum and niobium. MATTHIAS: That was a linear curve if you remember correctly, not a logarithmic one, and it wasn't mine. The extrapolation isn't so big, look at the scale. I mean, we are extrapolating from 5mK to .2mK. That isn't so far, considering we are coming from 600mK. At any rate, we can extrapolate that rhodium would become superconducting at .2mK since the two curves intersect beautifully-without any swindle. Now the question is, will it really become superconducting at .2mK? Well, that depends. If we could get to .2mK - - and some people say they are close to it-- even then you may not see pure rhodium because iron impurities are magnetic and we know how it goes. Iron sometimes is bad for superconductivity, sometimes it is harmless. As you go over to rhodium, it gets worse and worse. And, so we can predict: rhodium will become superconducting under the conditions that the combined im19urities of manganese, iron, cobalt and nickel must be less than 10- • Well, I'm quite safe right now because they can't make rhodium that pure today. SCHOOLEY: I remember in the proximity effect studies, people can predict for one of the members of a sandwich whether or not it has a positive transition temperature. Can that experiment be done for rhodium? MATT HIAS: Yes, but right now they are just clinging to copper and gold. They haven't touched rhodium yet. Well, that's the point of course. They get positive transition temperature for everything. After the fiasco with cesium, nobody ever dares to say again that something won't become superconducting ••• SCHOOLEY: Then how can you say organics won't become superconducting? MATTHIAS: Oh, you just wait! I am talking about physics, metals, and the truth! No, what Schooley means is the proximity effect, which is very interesting. You have two metals, one is superconducting and one isn't. If you look how the non-superconducting metal effects the transition temperature of the super conducting metal you can then roughly predict--roughly, within an order of magnitude or two-- where the other metal might become superconducting. Unfortunately, so far it hasn't worked very well since nothing has been predicted. But, they will try rhodium eventually.

Now, this is the reason why I mentioned cesium. You know, I'm sorry to bring the theory in again, but once upon a time there was a conference in Colgate--a conference on superconductivity. And, I said,according to the third law, I think every metal must go through a phase transition at a sufficiently low temperature. I can show you right away why I think so. You know what the third law is. It means for an equilibrium state, the entropy must go to zero. Now, if you have the temperature here, and the entropy here, I just don't believe that it will go to zero with a finite


tangent. Everything vanishes at absolute zero including its derivatives. So, therefore, it should go like the figure on the right hand side.

s f

T

If this is indeed the way, then one can predict right away that somewhere there must be a phase transition. Well, I was asked at that time, by Morrel Cohen, about the alkalis. And I said that the alkalis too will become superconducting if it is just done right. He assured me that I shouldn't worry about it. None of the alkalis would ever become superconducting. Well, cesium became superconducting. SCHOOLEY: As it happens, I read the quote in the---MATTHlAS: He changed it afterwards to sodium. Don't worry--SCHOOLEY: Actually he must have changed your words too. MATTHlAS: In a little way. SCHOOLEY: That quote refers very specifically to sodium. MATTHIAS: No, I spoke about the alkalis. He said sodium never will be. It will. So far, if one column shows a superconducting element, eventually all elements in that column will be superconducting. That's how we found all these. Here everything was going to be superconducting and we knew it. SCHOOLEY: How about carbon? MATTHlAS: Well, just give us time. I'll talk about carbon later on if I ever get to it. Cesium is superconducting. Rubidium hasn't become superconducting yet, and for potassium the pressure will be much higher. And just you wait, sodium will become superconducting. Unless you want to dispute it--it's on video tape~ SCHOOLEY: I would like to point out that an element under a sufficient pressure changes its state completely---MATTHlAS: Not completely, just a little bit. SCHOOLEY: Well, like an insulator to a metal is, I would say, a rather drastic change, wouldn't you? MATTHlAS: It's a change, let's not quibble about it. It is a change but it is nothing to worry about. And I can see in my friend Bob Hein's face, he doesn't like the fact that we say hafnium is superconducting. Well, I am personally convinced from many experiments that hafnium is a superconductor. Everything in


a column always is. And if one element is, it doesn't take much-or it takes a little now and then--to make the related ones too become superconducting. But superconducting they will become-including metallic hydrogen, but not at room temperature. About carbon, I will come to that in a little while.

So, you see, since almost everything is superconducting we have only to explain what determines the transition temperature. And, that I told you already. It is the number of electrons. Now, if you go through the periodic system, the transition temperatures for the transition elements go like this,

Tc

ex electrons

and the sp metals are all superconducting to begin with. For the sp metals the maximum temperatures are around 8K or 9K. For the early transition metals the maximum transition temperatures are between 12K and 13K. and for the later transition metals they are around 15K and 16K.

I want to show you now how these two maxima come about. The maximum number of unpaired electrons is at the 6th column. Namely, five d-electrons, one s-electron. This is like the melting point of the elements which,with a certain amount of imagination, shows siTI1ilar behavior to the superconducting transition. As a matter of fact, when the magnetic elements of the 3-d series are considered, then you even get a minimum melting temperature there in the middle. Now why is there a minimum of the transition temperature in the middle? If we said the maximum number of unpaired electrons should give an optimum number for the transition temperature, then we would have the explanation. You see, to explain two maxima in nature is always awkward. It is much easier to explain two maxima with the superposition of one maximum and one m~n~mum. That's what we are doing here. We have one maximum here, the number of electrons. and one minimum which we have to explain. And that minimum is something which is an anathema to the band theorist, and one of the hobbies of the people who believe in bonds inside of metals rather than bands. Bands are true for the one electron picture. But, bonds are something which can be handled from the periodic system point of view. And if you look at it froTI1 a bond picture, then in the TI1iddle there are five d electrons.


Available d electrons

Reduced sd scattering

2 4 6 8 10 std electrons

Fig. 3 The half filled d-shell, explanation of the two maximum temperature regions.

Five d electrons is a half filled d-shell. A half filled d-shell is quite stable. Not as stable as a filled d-shell, but close to it. Therefore, molybdenum, tungsten, and elements lying at this minimum, behave not very much as transition metals, as far as superconductivity is concerned, and as a consequence the temperature is very low; as my friend Bob Hein has found, tungsten is the lowest elemental superconductor (with the exception of rhodium) that is known today, namely between 11 and 15rnK. And that is due to the fact that the electrons do form a half filled d-shell. So we had to break it up. Every way it's done, it works.

Now, when I had that picture, I wasn't really quite sure whether it ever could be shown. Well, in the meantime it has been shown in a very beautiful way by Hammond in Brewer's lab in Berkeley. [3] What they did was the following: They made amorphous films of all these elements here in the middle row of the periodic system, going through the periodic system, but making sure there was no longer any well defined crystal structure or,in a wa~ any directed d-bond. When this is done, suddenly the two maxima no


longer exist and have been replaced by one curve that shows one maximum only at 6.3 electrons/atom, pretty much where we think it should have been, only now the maximum temperature is lower. The maximum is now again near 9K. In other words, by destroying the nature of the o-bond, by making amorphous samples, by making them more analogous to sp elements, we get the same behavior as the non-transition metals. Everything is superconducting, only one maximum, rather low, and superconductivity does not show the two maxima any longer. So you see, we do need the directed d-bonds in order to get to high temperatures. But at the same time if we do this, we have the trouble that we form a half-filled d-shell, and we are out of luck. Now this in essence is really all I have to say about what determines the transition temperature. Because, as in every empiricism, if it is true, it will work invariably-wherever you look at it.

And now I'm going to show you how it works also in compounds. In compounds we just take the average number of electrons, you no longer look at the individual atoms alone. Take the primitive average, the arithmetic average, and that is all. I'm not going to talk much about the compounds because I'm sure you have heard quite a bit about them. There are three structures which have shown to be very good for superconductivity, for reasons which nobody knows, but that's just the way it is. I'll give them to you by their 8eneric term, which means those were the crystals that gave the name to the structure. Sodium chloride: the best known compound in the structure is, of course, niobium nitride. The next one is the S-W structure: there are many well known ones-V3Si, Nb3Sn--are the best ones known in this structure. Nb3(Al,Ge) has the maximum transition temperature known today. Now the third structure is a very weird one. It is called the plutonium sesquicarbide structure, because that's where Zachariasen first determined the structure. It is a cubic structure which is quite complicated and has carbon pairs, carbon molecules in this lattice, and the best known superconductor in this system is (Y,Th)2C3' which again gets us to 17K. Now the sesquicarbide structure was discovered about four or five years ago, [4] and only these three structures at present get us to these highest temperatures.

Now what do these structures have in common? They have in common the argument I gave you about electrons. But there is something else I haven't mentioned yet. There is a whole complex of new compounds where the maximum transition temperature occurs for 3.8 electrons. Don't ask me for an explanation- since I can't really explain the maximum at 6 or 5 and 7, how could I explain the maximum at 4? But we can use it, and so we could make many compounds which are superconducting between 12 and 14K, if we look at metals with electron concentrations at four electrons. But these are details. They really shouldn't bother you very much


because they are there only to be used to make superconductors and to look for new ones. At our present level of understanding, they really don't have any profound meaning.

Again, what do all these crystals have in common? The one thing that I really know for sure is that they are all cubic. The sodium chloride lattice is a simple lattice. S-W is a good deal more complicated, and the plutonium sesquicarbide is an extremely complicated structure, but they are all cubic. In other words, all high transition temperatures which we have today will be found in cubic lattices--not one dimensional, not two dimensional. You see, we need cubic lattices for the following reason. A collective phenomenon will never occur in a chain, as you know, because if you have one dislocation, you're dead. In a plane it's debatable, according to the Ising model it should work, according to others it shouldn't. I personally believe that a real phase transition never will occur in a two dimensional array, for the same reason. In other words the optimum conditions are for a three dimensional array, because you get the optimum interaction conditions. Now one of the most three dimensional arrays in nature is a cubic lattice. You can see this right away. Any kind of structure can be explained by superimposing a preferred orientation over a cubic lattice. Only one thing is more cubic than a cubic crystal lattice, and that is an amorphous lattice. A liquid. In a cubic lattice, at least you have CII' C12 and C44 ' but in an amorphous structure you don't even have that. So the only arrangement more three dimensional than a cubic lattice is an amorphous structure and that is the reason why so many of the amorphous structures show a pronounced superconducting behavior. You see why we have a contradiction here. For the high temperatures we want directed d-bonds in a matrix where we don't want any direction. And you see the contradiction: we do want directionality and yet we don't want it, because we want a completely three dimensional array. And that is why it is so difficult to make all these high temperature superconductors.

Now, what happens when we write down compounds which could occur and which should become superconducting, and suddenly they don't. For instance, Nb3Si, I'm sure nobody will argue with me, would be a much higher superconductor than Nb3Sn, but it doesn't form. Or take for instance, Nb3Ge. Nb3Ge for a long time was superconducting at 6 K. That didn't fit this scheme at all. Then Geller found out what the composition really was, namely, Nb3.3Ge. So we decided we have to force nature in order to make 3.0. We just splat cooled it. In other words, you have an arc furnace with an arc, there is a melt, then you blow a shock wave in, and you condense the melt rapidly. Before nature has time to make 3.3:1 we have cooled it already, and we essentially freeze in the composition of 3:1. And in fact we got this way from 6 K to 18K. But it was disordered, as you can very well imagine. Well we can


order it, but then we give it time to form 3.3 again. So it's very difficult. Now if we could make Nb3Si, and that seems to be not possible right now, it would be higher. But it just won't form. When I say it won't form, you see right away where the melting point comes in, because what forms is what happens below the melting point. You might say "What does the melting point really have to do with it? After all there are bonds." There aren't bonds. That is the crucial thing. The melting point has very little to do with the bond structure in a metal, because the melting point is controlled by the same feature as superconductivity: all the electrons outside of a filled shell; you don't destroy any bonds when you melt a metal. In the liquid the atoms are still together. The coordination may have sunk or risen a little bit, but the bonds are still there, only it has become more isotropic than before. Again, all the electrons outside of a filled shell are responsible for it. And now we can understand, or at least sort of understand, why at or below the melting point some of the things which would suit us just won't form.

Alright, now let me show you then in this slide what will happen if these things don't form. On the right hand side, you see S-W. It is a beautiful cubic lattice. If we are at a

0,

Fig. 4 Structure of cubic S-W vs. tetragonal Ni3P or Ta3Si.


composition which would have been at the higher transition temperatures, suddenly you get a mess like the structure on the left which is no longer superconducting, and yet it is almost the same. It is tetragonal with a cia ratio strictly of two; the a axis is exactly what we would expect from the cubic one. This is the Ta3Si structure, which Nb 3Si also forms. But we don't want this because it is not cubic and these tetrahedra are not even at a correct symmetry, they are at an off angle. A long time ago, I found a new superconducting compound which was very intriguing, M03P, It was the only A3B compound in this category that formed and was superconducting above 5Ko And only recently have I found out why this is so. It is for the reason that the tetrahedra are not in the Ta3Si configuration, a mess, but are exactly aligned along the diagonal axis. It is still tetragonal, with a cia ratio of two. But at least we have an added symmetry in as much as the tetrahedra now are exactly at 45°. And so we are at 10K instead of at 20K o But we can tell from the structure pretty much what the chances are; we always need the synnnetry.

Let me show you the next slide, which I think is very interesting.

FiB- 7. ~tnP

Z' ~, \' l:.~ o,!- .:!-

o '0 j . '

"'{o 0 ,:{o yo . ' o '0 0 I • I •

• L - ~ - ~ 0 •

Fig. 5 Variations on the MnP structure.


These are many different structures, all variations of the manganese phosphide structure. And only in the manganese phosphide structures do we get superconductivity. And you see right away why. Because in one case we have a fairly well connected array, but in the others things are interrupted and don't work very well. Therefore, no superconductivity appears in any of these structures. So you see,when we have a cubic structure, the right electron configuration, and all this, we are in luck and we can.make higher temperatures. But nature usually does not want to form high temperature superconductors. And either we have to use tricks in order to make them form or we have to look for new structures. That's all. Nobody could have predicted S-W before Hardy and Hulm found it. And PuZe3, well that was sort of predicted, but it still was strictly a matter of hope and luck. There are still a large number of cubic structures nobody knows--that haven't been found yet. And they will help us to find new superconductors.

Now, I have shown you that these things sometimes form and sometimes don't. When we want them to form, when we sometimes very much want them to form, they just won't do it. And now comes something very weird. Very often when they do form, when we get the structures which we want, they don't even stay that way. They are cubic when we look at them. And suddenly at low temperatures, they just don't behave right. And we found during the last few years that the reason for this is that they transform not at the melting point or at high temperatures, they transform at low temperatures. Now in S-W this had been known for a long time: that at very low temperatures--Batterman and Barrett found this[5]-that the crystal structure which is cubic ~ - W, V 3Si, suddenly transforms slightly, but ever so much, into a tetragonal structure. The cia ratio is still very close to one, but yet they are no longer cubic. Very often I wondered if that could be sort of a general thing. Once upon a time, I found a crysta~ ZrzRh,which was extremely high, near 11K. So we tried to modify it. Whatever we did, we lowered the transition temperature. We just couldn't raise it. So we ground it up in order to see what would happen, and suddenly we got an enormously broad transition temperature. [6] When we ground up an isomorphous compound, ZrZlr, which wasn't quite that high, we found two transition temperatures. And that suddenly made it clear to me, maybe this compound also transforms at low temperatures. Only these transformations being very sluggish, quite frequently don't take place, so when you cool the crystal, it is no longer in the equilibrium state. But if you grind it, if you introduce shear strain, you facilitate the transition which is usually a shear transformation to begin with. So you enable the second modification, the less symmetric one. Now you can understand immediately why mistreating of a metal very often broadens the transition as much as it does. For the simple reason, when you cold work a metal you introduce a range of shear strains. And when you introduce a range of shear strains,


you also introduce a range of transition temperatures according to the distortion the lattice will take on at a lower temperature. Now that, if we were right, should have been a general phenomenon. And so it was.

Zr2Rh is called the Cl6, the CuAl2 structure; it is a complicated structure, it is tetragonal. Then there are the Cl4 and Cl5 structures, the Laves phases. Again, by grinding and maintaining the pressure, we managed to resolve the two transition temperatures. We did the same with V3Si. Until now people have looked for martensitic transformations in superconductors and just hoped for the best. Sometimes they transformed, sometimes they didn't. Well again in V3Si the two transitions are now beautifully' resolved.

(J)

I-Z :::;)

>-a:: e:( 10 a:: I- V3Si CD a:: e:(

z >-I-..J CD ~ 5 CL. UJ (J)

:::;) (J)

uJ > l-e:( ..J uJ a::

06 10 12 14 16 18 TEMPERATURE (K)

Fig. 6 Transitions of cubic and tetragonal V3Si.


Now we can predict something. As I said before, whenever you understand something, even vaguely, you can predict it. We predicted that all experiments that had been done in the past with pressure on V3Si were wrong. Because people had squeezed V3Si and found that the transition went down. They did not do it hydrostatically. So, what we needed was a hydrostatic pressure on V3Si. You see usually when V3Si is squeezed, the temperature drops like a rock. But now T. F. Smith [7] in La Jolla has squeezed it hydros tatically, and got from 17K to quite a bit

279

above 18 K. Literally on the basis of these lattice instabilities and assuming that they effect the transition, the transition temperature was raised. You realize, of course, then there was no longer any instability--hydrostatic pressure prevents the transformation, and finally, we even are at higher temperatures than before. It works often but not always. The sodium chlorides are very hard to squeeze, because I think there the carbon or nitrogen is in its metallic form. The superconducting crystals in the sodium chloride lattice are interstitial compounds: nitrides, carbides, even oxides. And I think this is where we find carbon, oxygen and nitrogen in their metallic forms. And to squeeze them is now an extremely difficult thing. So you see, when we have a lattice that is good for superconductivity, even then we aren't safe yet because at lower temperatures nature still might ruin things.

Finally, I'm going to show you another slide, a very depressing story from a paper by Shaun McCarthy. [6] The higher the transition temperature, the stronger the depression if these phase transitions are induced. Here we are plotting the depression of transition temperature versus the original transition temperatures for all these structures which you see here, a-Mn, Laves phases, S-W, and so on. For all of them, the highest superconductors are the most sensitive ones. And when we mistreat them, we lose. So you see, these built-in instabilities are very difficult. High transition temperature superconductors, I have shown you, sometimes just won't form. And even when they are formed, we're not even safe yet.

280

8

6

2

o

o

Fig. 7

/ /

/ /

/ /

/ ;'

;' ./

• ...... ./ C)

~-6 .0 __ 0 ... ..-

-~o

4 B 12 Tc(OK)

/ /

/

/ /1.

/ /

/

16

BERND T. MATIHIAS

I A I

I I

I I

LEGEND

I

A I

I

20

Depression of T with strain as function of T . c c

Here now are some other stumbling blocks: ferroelectricity and ferromagnetism. A long time ago, we found a system of superconductors which were called the tungsten bronzes. It looks something like this.

FE sc FE


We start with W03 and now begin to dissolve sodium in it. Finally we come to the perovskite lattice, NaW03' What happens in the meantime? The first range is ferroelectric and it remains ferroelectric for quite awhile. Then when we exceed about .1 sodium, there suddenly comes a superconducting phase, the tetragonal phase. If instead of sodium there is potassium,then there is another phase here, also superconductin&which is the hexagonal phase, until at about .3 and .4 sodium atoms one gets into a cubic structure. And that cubic structure suddenly is no longer superconducting. Now that at first stunned us because, after all, we said cubic structures should be superconducting and the electron concentration around here doesn't chang~ So why do we suddenly lose the superconductivity in such a cubic structure? But it isn't really cubic, it is almost cubic. It has a slight distortion to the tetragonal phase, and the distortion is so slight that it can be seen only optically, not even with x-rays. And I suddenly realized that must be the cause--this distortion is due to ferroelectricity since it is the most famous group of ferroelectrics, the perovskites, and apparently a ferroelectric metal can't be superconducting. A ferroelectric metal is a hard thing to swallow. How are you going to measure spontaneous polarization, or piezoelectric effect, or anything like this in a metal? You can't. Fortunately, just recently [8] the quadrupole spli4ting of these bronzes was measured, something that had been done in sodium niobate for quite awhile. And, let me show you now the next slide.

..... N

:I: ~

~---I.c. N ..

1.5

0.5

Fig. 8

300

1- 11=0.517 NaxW03 0 II = 0.72

D 11=0.855

0 \DD -.\

,\ \ -' \

e\ \0 ,

\ \ \

" \

1\ \ \

'00 500

Quadrupole splitting in N~W03 [Ref. 8].

T (eK)

They found a quadrupole splitting which shows exactly the same behavior below a certain temperature as in the ordinary ferroelectrics and antiferroelectrics. In other words, this really was the cause, the ferroelectricity that prevented superconductivity. Now why would ferroelectricity prevent superconductivity? Let me show on a purely dimensional basis why one can't have a ferroelectric superconductor. I told you before, we need a three dimensional array. But there has never been a ferroelectric yet which was ferroelectric in more than one direction only. Usually in Rochelle salt and similar compounds, the direction is determined and limited by the crystal orientation. In barium titanate one can switch the orientation. Sometimes the crystal switches itself. But there is always only one direction in which the crystal is ferroelectric, and in the plane perpendicular to that direction there is never any ferroelectricity. So from the purely dimensional argument, the crystal becoming ferroelectric, having a preferred orientation, will destroy superconductivity.

Now in organic compounds like the TCNQ's and so forth, compounds once considered to become organic superconductors, one certainly has a preferred orientation. I remember Kapitza had introduced the Low Temperature Conference in Moscow with the words, "We are on the threshold of room temperature superconductivity." Since then in Russia it has been discovered that TCNQ's have dielectric constants around between 500 and 1000, are ferroelectric, and of course there isn't even a trace of metallic conductivity left below 501<. [9] So you see the ferroelectricity, the tendency to give a preferred orientation, is why we cannot reach superconductivity on this basis.

Let us now go to two dimensions. A long time ago, some of us, Hannay, Geballe, and other~ [10] found that when alkali metals are intercalated in graphite superconductivity results. That was a very interesting result. Because the moment we had more than one layer of graphite between the alkalis, we would destroy the three dimensionality and with it the superconductivity. The aaxis remains the same, but the c-axis changed and always approached that of graphite much·more than before. The moment the c-axis was no longer electronically connecting the alkali layers superconductivity disappeared. Well, some of my friends said "Yes--it has just become lower. You just don't see it any longer." I said "Nb,there just is no two dimensional superconductivity; at leas t not in that sys tem. II Now during the las t few months, the Russians again published something very intriguing. [11] They discovered a ferromagnet at 221< that has the formula, MoC15' Now with ferromagnetism, in contrast to superconductivity, you can always anticipate by the Curie Weiss Law the transition temperature. When something goes wrong with a straight line of l/X vs T, you can anticipate from much higher temperatures that the nature of the interaction has changed.


Now let me show you what happened with Mo-pentachloride, when they intercalated it in graphite,because this is too good an analogy to miss

Graphite I II III N

------- ~ ------- ~ ·u ------- ------ ,...,

, --_·-z There is the phase I, where graphite layers and Mo-pentachloride layers alternate. Then there are two, three, and four graphite layers in between. And in Fig. 10 there is the Curie Weiss behavior for these intercalated compounds. When the graphite layers just

J~-------~-------~------~~--

IA .., • ..-4 o c:: ,.... = · e - '" >(00 "U .-4 ....... _ 00

2

1

o Figs. 9 and 10

• y-O ,....

,.... -Intercalations and their Curie Weiss behavior, (a) refers to intercalation II, III, and IV; (b) refers to intercalation I. [Ref. 11]


alternate with the Mo-pentachloride, we still get a clear cut Curie point, no longer at 22 K •. but now it is at 15 K. But we do get a Curie point. What happens when there is more than one layer? And they did it, with two, three, and four layers. Suddenly at 5DK something goes drastically wrong and the Curie Weiss Law is essentially no longer obeyed. The deviation from the straight line has become very pronounced. All one has left is a weak antiferromagnet at 2K. Now, if we are willing to consider this as an analogy to superconductivity, we can see a real pre-cursor an extrapolation towards a phase transition in ferromagnetism. From the moment there is more than one graphite layer, from the moment we have a two dimentional array, we have no longer any phase transition. The analogy to the superconducting alkali intercalation compounds was so beautiful, I had to show this to you.

I want to come to an end, and tell you the rule: A cubic symmetry and many electrons are necessary and we may get to high temperatures. But there is an enormous number of obstacles on and in the way. It won't form, it won't stay, it will form the wrong interaction,and all these we have to ~vercome. And this is the reason why I say, today, the schism between the experiment and the empirical approach on the one side, and the theory on the other side has become pronounced. Of all these obstacles which I have mentioned (with perhaps the exception of ferromagnetism) you never hear much. Green's functions won't help in superconductivity as far as raising the temperature, finding new superconductors--high or low it doesn't matter--and where similar real things are concerned.

QUESTION AND ANSWER PERIOD:

SCALAPINO: What about tantalum sulfide, where they have added a number of tantalum sulfide layers and the transition temperature has not changed? MATTHIAS: My feeling about tantalum sulfide is that it is the reverse of the intercalation compounds, because you add more and more layers of a substance which is superconducting to begin with, and tantalum sulfide is a superconductor. If this were done with something that was not superconducting, I would be astonished. As it is, it is being done with a material where, if you have a slight sulfur deficiency, the temperature goes up immediately. Tantalumsulfur 1. 4 or 1. 5 is already at 5 K. So if you put weird things in between, sure, you can change the temperature. But you see,you don't need tantalum sulfide for this sort of thing. Alekseevskii recently has done a very beautiful experiment. [12] I don't know whether you have seen that. He took beryllium and co-evaporated zinc etioporphyrin with it. Now beryllium films in themselves

Te's-THE HIGH AND LOW OF IT 285

are superconducting at 9. SK. The moment zinc-etioporphyrin, which was also once upon a time considered an organic superconductor, was co-evaporated the temperature rose to 1O.2K. However, Alekseevskii is a good man. He said, instead of zincetioporphyrin, let's intercalate potassium chloride. Then the temperature went up to 10. 6K. And maybe this is the same thing. I don't really know. The moment you change tantalum sulfide, you also change the temperature. What else could you expect, tantalum itself is at SK. You have a whole range to go. It is my personal feeling that if you start out with something that is superconducting to begin with, you can never prove two dimensional superconductivity. SCALAPINO: This isn't a question, it's a comment. I think on a tape, for future students at least, there ought to be some comment that suggests the following. I think that it has been proved that the strong-coupling extension of BCS theory has shown among other things that electron-phonon interactions are solely responsible for all known materials; that given tunneling data, -admittedly this is a tremendous amount of input data-- but given that data, you can calculate transition temperatures to within a few percent. Given additional data along the lines of tunneling data, you can go on to calculate essentially all other dynamic properties, as well as thermodynamic properties, to an accuracy of about the square root of the electron-ion mass ratio. MATTI-llAS: You see, what I am talking about is not what you are calculating after you know the result. That is your privilege, and I am the last to argue with you about that. I won't even argue with you about the phonons because you can't ever stop a crystal lattice from rattling, so this is an almost trivial argument. No, I am talking about something quite different. Now let me be completely specific. From Little to Ginzburg, we have been told on the basis of strong-coupling, of exciton theory, of BCS, all the temperatures that are possible. It isn't what I want, it is what I get~ This is what we have been getting now for essentially 14 years, predictions of things to come. And my argument is that every single one, without exception, every single one of these predictions with respect to the high transition temperatures--I don't argue with tunneling or anything of the sort-every single prediction with respect to the transition temperature has been wrong. Marvin Cohen predicts magnesium should be super conducting between 10 and 12mK, and it isn't. Little predicts TCNQ's or similar things should be superconducting at 1000K, and they are not. We didn't ask for all these predictions--we got them whether we wanted them or not. They clutter up the literature, they confuse the mind, and they give all of us a bad image. Because if they predict, basing it on something, and then fail, we have only two choices. Either they are stupid, which they aren't, or they predicted on the basis of something that isn't true. Now which of the two choices would you choose?


SCALAPINO: I think if one takes a look at the literature, particularly Bill McMillan's work on transition temperatures, one has another choice,which is that one uses the strong-coupling theory in a way based on what one understands about the normal properties of the metal, and then one does very well. It is not that we don't understand the superconductivity, it is that we don't sufficiently well understand the normal metal properties. MATTHIAS: Did you notice how my friend Scalapino completely avoided the answer to the question. However since he mentions McMillan, I will not avoid McMillan. McMillan in his paper talks about superconducting limits for V3Si of 40K and of Nb3Sn of 28K. I asked how could one reach these limits, I mean, tell us a way? At the time of his publication, he was quite specific. He said, Nb thin films must have a higher temperature than the Nb bulk because it is a thin film, it will get softer, and therefore, the phonon spectrum will be modified in such a way that the temperature will go up. I'm completely explicit now. An enormous amount of money has gone into the effort of trying to verify the McMillan prediction that thin films should be higher than the bulk. The result without a single exception again was a complete fiasco. None of his predictions were verified. If I am wrong feel free ••• SCALAPINO: I understand what you say about the Nb case. But, take the Ai case where it has been well verified, I believe, that thin Ai films raise their transition temperatures because of the softening of the lattice, and it fits McMillan's ideas, I think, exceptionally well. The Nb refers back, I think, to the following situation, on which I wouldn't disagree with you. I think that when you ask a physicist, particularly a theoretical physicist, a question that ultimately borders on chemistry and a very local problem, localized bonds, localized magnetic moments, as it approaches the whole question of chemistry, catalysis, chemisorption, we find that we are in a situation where theoreticians have not made great progress yet. And as you start pushing on these situations that are chemical, where the questions are "How do you do something?", "How do you predict that the phenomenon you are after doesn't get prevented by another phenomena taking place, such as the martensitic transformation? ", and "How are the phenomena coupled? II, you are beyond the bounds of what we know. And perhaps the closest case we know about is how virtual spin waves in materials that are nearly ferromagnetic begin to negatively influence the possibility of superconductivity? But you are on the edge there of what we know, and that happens to be the edge that you need to know. MATTHIAS: I'm sorry, Scalapino is a friend of mine, but did you notice he brought up Al films which have been with us since 1947 when Hilsch made them for the first time. Hilsch got to S.8K-nobody has gotten much higher--maybe 6K. So the Al films have been with us for a long time. On the basis of Al films, the soft mode fashion was developed and people were convinced that this was the right approach. As a matter of fact many of them at the time


hoped for up to 40Kwith thin films. Well, of course, this was not true and that had been tried a long time ago. However, MacMillan was quite explicit on Nb, and Nb3Sn films--thin films. That was the only thing he predicted that had not been known. And he predicted it wrong!

287

Now, let me come to something which I mentioned in the beginning. When Frohlich and Bardeen predicted the isotope effect, one over the square root, everybody said "This is the ultimate triumph, now we know it all! We understand everything." I always thought the isotope effect was only as a coincidence near .5, and could take on any other value. Well, Geballe and I spent one year trying to measure the iso,tope effect among the transition elements. And, for the first thing we tried, the very first element, ruthenium, the isotope effect was zero; there was no isotope effect. Well, for a moment there was some consternation from the theoretical camp when ruthenium didn't show an isotope effect. After a while they recovered, and started to predict. Now they predicted that some elements shouldn't have an isotope effect. It should be between 0 and -.5, until in Los Alamos the isotope effect of uranium was measured.

I didn't get to this in my talk: there are many different mechanisms leading to superconductivity. Phonons are always there, but the electron configurations are very crucial and if you want to call it chemistry it doesn't matter, it is still the electrons. And these configurations are essential, and I knew that for U the 5f electrons were crucial and essential for superconductivityand, therefore, the isotope effect there shouldn't be • -5, it shouldn't be 0, it should be positive. In other words, the transition temperature should be proportional to the atomic mas s 0

I'm not going to bore you with the reasons why I carne to this conclusion. If you can understand it, Suhl, Ting and I [13 ] just published an article in Physical Review Letters where we explain this, but it is an old affair now. You cannot just look at the lattice distortion, you also have to look at the electronic distortion. In other words, to treat atoms as rigid spheres is a mistake. You have also to treat them as polarized spheres where you distort the electron sphere and induce the moment. On the basis of this, I predicted the isotope effect had to be positive. It carne out that ex was 2.2. Well, 2.2 is a meaningless number, it could be 2, it could be 1.8 - for that reason if it were 1 that's all irrelevant. All I am trying to say is I predicted, and risked quite a bit because this is an expensive experiment, promising it would be positive. Now to corne back to McMillan. I asked him, what will the isotope effect be in U? He predicted to me in a telegram all three--U has three phases, two can be obtained only stabilized, one is under pressure, ex' ~ , i' --he managed to predict with considerable skill all the isotope effects wrong, sign and otherwise. Now this


is what I am saying. When he promised people that Nb films,--not Al films, we've known this for decades--that Nb films or Nb3Sn films would be higher--and then this rather fascinating comment, that there are limits, 28K and 40K and so forth--when he said the temperature should be raised on this basis, I don't think it would be fair for him to say that Al does it. Nobody is interested in Al. That has been with us for a long time, more than 20 years. And therefore at the Stanford Conference, Strongin got up and said, "I have tried for two years to verify this kind of phonon interaction, that thin films should have higher temperatures, that Nb films should be higher than Nb bulk. It wasn't." I'm asking you now and I want to know, how many of these wrong predictions are we supposed to have?

REFERENCES

1. J. Wittig and B. T. Matthias, Science 160, 994 (1968). 2. A. C. Mota, W. C. Black, P. M. Brewster, A. C. Lawson,

R. W. Fitzgerald, and J. H. Bishop, Physics Letters 34A, 160 (1971).

3. R. H. Hammond and M. Collver, Bull. Am. Phys. Soc. ~, 1613 (1970).

4. M. C. Krupka, A. L. Giorgi, N. H. Krikorian, and E. G. Szklarz, J. Less Common Metals 19, 113 (1969).

5. B. W. Batterman and C.:S. Barrett, Phys. Rev. Letters 11, 390 (1964).

6. S. L. McCarthy, J. of Low Temperature Physics 4, 669 (1971). 7. T. F. Smith, Phys. Rev. Letters ~, 1483 (1970)~ 8. G. Bonera, F. Barsa, M. L. Crippa,and A. Rigamonti, Phys.

Rev. B4, 52 (1971). 9. L. I.-Suravov, M. L. Khideke1', I. F. Shchego1ev, and E. B.

Yagubskii, JETP Letters 12, 99 (1970). 10. N. B. Hannay, T. H. Geba11e, B. T. Matthias, K. Andres,

P. Schmidt,and D. MacNair, Phys. Rev. Letters 14, 225 (1965). 11. A. V. Zvarykina, Yu. S. Karimov, M. E. Vo1'pin~nd Yu. N.

Novikov, Soviet Phys. --Solid State 13, 21 (1971). 12. N. E. Alekseevskii, V. I. Tsebro, and E. I. Fi1ippovich,

JETP Letters 13, 174 (1971). 13. B. T. MatthiaS:-H. Suh1, and C. S. Ting, Phys. Rev. Letters

~, 245 (August 1971).

THE METALLURGY OF SUPERCONDUCTORS

Robert M. Rose

Massachusetts Institute of Technology

Cambridge, Massachusetts

ABSTRACT

The structure and processing of superconducting materials playa dominant role in determining the critical current density, critical field, critical temperature, losses in AC fields at all frequencies, magnetic stability in large DC magnetic devices, reliability and durability of tunnel junctions and other useful and practical properties. Some of these effects are explainable via the Ginzburg-Landau-Abrikosov-Borkov phenomenological theory with flux flow effects included; some are relatively crude matters of thermal and magnetic diffusivity; some are associated with surface topography on a submicroscopic scale; and some, including the critical temperature, are matters of conjecture. What is certain is that whether a superconducting material is useful at all depends crucially on how it is manufactured. The present and past status of practical and potentially practical materials, their properties, methods of fabrication and limitations, are reviewed and discussed.

THE MATERIALS

To be practically useful a superconductor must first of all have an adequately high transition temperature (Tc), and on this basis Table I can be constructed. The table has been subclassified on the basis of crystal structure into four principal groups: body-centered-cubic solid solutions, beta-·tungsten, rock salt, and Laves phases. (Some high-Tc superconductors such as Pb-Bi alloys have been omitted from this table on the rather arbitrary opinion of the author that they are not likely to be useful as

289

290

Table I: Some Potentially (or actually) Useful Superconductors

BODY-CENTERED CUBIC SOLID SOLUTIONS (Tc up to lZ.SO)

IV-V VI-VII

ROBERT M. ROSE

Ti-Nb (to 9.8o){1) Zr-Nb (to 10.8o){1) Ti-V (to 7.So)(1)

Mo-Re (to 11.10){S) W-Re (to So)(l)

Ti-Ta (to 9.0S0)(Z) Zr-V (to S.9°)(4) Hf-Nb (to 9.6°)(1) Hf-Ta (to 6.so)(1)

BETA TUNGSTEN STRUCTURE (Tc up to ZO.SO)

Nb3Sn (18.0°)(6) Nb3Al (18.8o)(7) Nb3Al.8Ge.z (ZQ.So)(8) V3Ga (14.So)(9, V3Si (17.00)(10)

ROCK SALT STRUCTURE (Tc up to 17.9°)

NbN (16.8o)(11) NbC (lO.so)(lZ) NbN-NbC solutions (up to l7.9 o){13)

LAVES PHASES (Tc above 10° and rising!)

VZHf (9.6°)(14) VzZr (8.8o)(lS) VzZr.SHf.s (10.1°)(16)

those listed.) The BCC solid solutions offer critical temperatures up to lZ.SoK (Mo-Re alloys), considerable mechanical ductility and reasonably high (ca. 100 - ISO kgauss) critical fields in some cases (notably the Nb-Ti alloys). The beta-tungsten and rocksalt compounds are quite brittle and require special treatment, but also offer the highest Tcls and critical fields (both Nb3Al and Nb3Al.8Ge.z in the S-tungsten category and NbN-NbC mixtures in the rock salt category. The Laves phases are relative newcomers to the list in Table I, and offer rather high critical fields considering their (to date!) unspectacular but adequate critical temperatures. Figure 1 shows beta tungsten structure.

Thus certain crystal structures f~vor superconductivity. Also, as Matthias pointed out some years ago ll), an examination of the


Figure 1 A schematic drawing of the beta tungsten unit cell, containing six "A" atoms and two "B" atoms (one at the center.) From The Structure and Properties of Materials Vol IV: Electronic Properties, by R. M. Rose, L. A. Shepard and J. Wulff (Wiley, New York, 1966).

References for Table I

1. J. K. Hulm and R. D. Blaugher, Phys. Rev. 123, 1569 (1961). 2. D. A. Colling, K. M. Ralls and J. Wulff, J:-Appl. Phys. ~,

4750 (1966). 3. L. J. Barnes and R. R. Hake, Phys. Rev. 153, 435 (1967). 4. L. Kramer, Phys. Letters 20, 619 (1966).---5. E. Lerner and J. G. Daunt:-Phys. Rev. 142, 251 (1966).

291

6. F. Heiniger, E. Bucher and J. Muller, Phys. Kond. Materie l, 243 (1966).

7. F. J. Cadieu and D. H. Douglass, Jr., to appear in Proc. 12th Int. Conf. on Low-Temp. Physics (paper 8Xa).

8. G. Arrhenius, E. C. Corenzwit, R. Fitzgerald, G. W. Hull, Jr., H. L. Luo, B. T. Matthias and W. H. Zachariasen, Proc. Nat. Acad. Sea. 61, 621 (1968).

9. T. J. Greytak and J. H. Wernick, J. Phys. Chern. Sol. ~, 535 (1964).

10. W. Kunz and E. Saur, Proc. 9th Int. Conf. on Low Temp. Phys. (1965). p. 48l.

11. K. Hechler, E. Saur and H. Wizgall, Z. Physi~ 205, 400 (1967). 12. F. J. Darnell, P. E. Bierstedt, W. O. Forshey,~ K. Waring

Jr., Phys. Rev. 140, A158l (1965).

292 ROBERT M. ROSE

13. O. I. Shulishova, Inst. Metallofiziki AN (1966g); also Neorganicheski Mat. ~, 1434 (1966).

14. V. Sadagopan, E. Pollard and H. C. Gatos, Sol. St. Comm. l, 97 (1965).

15. B. T. Matthias, V. B. Compton and E. Corenzwit, J. Phys. Chem. Soc. 19, 130 (1961).

16. K. InOUe, K. Tachikawa and Y. Iwase, App. Phys. Lett. 18, 235 (1971).

superconducting elements on the periodic table will show that there is systematic behavior. Table II shows tnat the highest transition temperatures are in groups V and VII of the transition series. This general behavior extends to alloys and compounds as well. For instance, the highest Tc in the Nb-Zr alloy' system, nearly 11°, occurs near the 20 - 30% (atomic) Zr composition and, as shown in Fig. 2, the Tc maxima of most V-IV transition metal alloy systems tends to be located in that vicinity. Then we have the beta-tungsten superconductors which can be 25% Group IV and 75% Group V CV3Si, Nb3Sn) or 25% Group III and 75% Group V (V3Ga, Nb3Al). If we use (as Matthias did more than 14 years ago(l» the column number in the periodic table as a parameter to be averaged in alloys in compounds, distinct maxima appear at 4 1/2 -4 3/4 and also 6 3/4 , as shown in Fig. 3, from Matthias' original plot. You will find that just about every useful superconductor we have has effective electron-to-atom ratios near those numbers. For molybdenum (Group VI) - titanium (Group IV) alloys, for instance, the magic number is 4 1/2 - 4 3/4 and this corresponds to 10 - 15% molybdenum. Indeed, the highest Tc falls in this range of compositions.

Although the empirical situation is, as you can see, straightforward, theoretical explanation of such behavior is not. Look at the BCS(2) formula kBTc ~ 1.14 KwgexP (-l/N(O)V); kW9 is a cut-off energy for the electron-phonon interaction, N(O) is the density of states at the Fermi surface and V is the electronphonon-electron interaction. You can try to increase the density of states, increase the cut-off energy, or increase the electronphonon interaction, to increase Tc. A good way is by making a softer, more unstable, more easily polarizable material; but, as Dr. Matthias will soon show you, such an approach has its problems.

Since the entire subject of Tc will be discussed later, I will restrict myself to some practical comments on the optimization problem in beta tungsten compounds. In all cases I know of, the key is to keep the "A" chains intact. To disrupt the chains, e.g., by disorder or by putting in substitutional atoms which sit in the

THE METALLURGY OF SUPERCONDUCTORS 293

"A" sites, or to remove "A" atoms by using the wrong stoichiometry (or other means) can decrease Tc, sometimes to less than half its optimal value.

~

~ 0

'-./

J-U

12

10

8

6

4

2

• . []

r:r .. [] ....... t:l .' .

o T~ -Hf ~ V-Ti o Nb-Zr eNb-Ti

o ~---~------~---~------~---~ o 0.2 0.4 0.6 O.S 1.0

Nb,V, To. Ti,Zr,Hf X (GROUP N)-"

Figure 2 Tc vs. composition (atomic percent) for some V-IV transition metal alloys. Data from J. K. Hulm and R. D. Blaugher, Phys. Rev. 123, 1569 (1961).

~

~

Tab

le II

: T

he

Per

iod

ic T

able

, w

ith

Tra

nsi

tio

n

Tem

per

atu

res

for

Sup

er c

on

du

ctin

g E

lem

ents

I II

II

I IV

a V

a V

Ia

VIl

a V

III

Ia

IIa

lIla

IV

V

V

I V

II

0

Li

Be

B

C

N

0 F

Ne

Na

Mg

Al

Si

P S

Cl

A

1.1

8

K

Ca

Sc

Ti

V

Cr

Mn

Fe

Co

Ni

Cu

Zn

Ga

Ge

As

Se

Br

Kr

0.3

9

5.0

3

0.8

6

1.0

9

Rb

Sr

Y

Zr

Nb

Mo

Tc

Ru

Rh

Pd

Ag

Cd

In

Sn

Sb

Te

I X

e 0

.55

9

.5

0.9

2

7.7

4

0.4

9

0.5

2

3.1

4

3.7

2

Cs

Ba

La

Hf

Ta

W

Re

Os

Ir

Pt

Au

Hg

Tl

Pb

Bi

Po

At

Rn

a4.9

0

.17

4

.48

0

.00

3

1.7

0

.66

0

.14

a4

.l5

2

.37

7.

19

86

.3

(ap

pr)

8

3.9

5

Fr

Ra

Ac

Th

Pa

U

1. 3

7 0

.7

a

(Non

e o

f th

e ra

re eart

hs

app

ear

'" to

be

sup

erco

nd

uct

ors

) 0 0

0

m '" .... ~ "" 0 en

m

IEM

PT

Y +-

rei-S

HE

LL:

PAR

TLY

---C

LO

SE

D

cL-S

HE

LL

·1 2

2 •

I~

2...

3 b

4L

iii

o 2L

3/1

. 4a

.

20

.-..

18

~

~

lLJ a:

:::> ~ lL

J a..

::E

lLJ

I-

16

14

12

Z

10

Q

l v;

z ~

I- lLJ

8 6

>

4 ~

r..:

:::i

2 :3

(/) a:

o l v :::>

o z o v a:

lLJ a.. m

o z

TRA

NS

ITIO

N

ME

TAL

SUPE

RC

ON

DU

C TO

RS

,~

-"

(/) a:

o l v :::>

o z o v f5 a..

:::>

(/) o z

NO

N-T

RA

NS

fTfO

N

ME

TA

L SU

PER

CO

ND

UC

TOR

S

(/)

....J ~

lLJ :;: z o z

o 0

~' __

_ -L

___ ~ __

_ ~ __

____

~ __

_ L-

__

_ ~ _

__

~ _

__

~ _

__

~ _

__

-L

__

_ -L

__

_ -L

__

_ -L

___ ~ __

_ ~ __

_ ~ __

____

~~

o 2

4 6

8 10

2

4 6

8

VALE

NC

E EL

ECTR

ON

S PE

R

ATO

M

Figu

re 3

The

var

iati

on o

f Tc

with

pos

itio

n in

the

per

iodi

c ta

ble,

aft

er M

atth

ias

(B.

T. M

atth

ias,

Pro

gres

s in

Low

Tem

pera

ture

Phy

sics

, Vol

. II

p.

138;

C. J

. G

orte

r,

edit

or;

Nor

th-H

olla

nd P

ublis

hing

Co.

, A

mst

erda

m,

1957

). T

he "

elec

tron

-to-

atom

rat

io"

for

allo

ys a

nd c

ompo

unds

as

wel

l as

pur

e el

emen

ts is

sim

ply

the

aver

age

colu

mn

num

ber.

-i

::J:

m

~ ~ r r C

::

0 Q

o ."

VI

C

."

m

::0

n o Z

o c n o ::0

V

I

t.)

~

C\I 2 u ~106

If)

£1.

2 « ... >-?

~

If)

Z ~ 1

05

~

/ Nb

3A

I ?

z W

Nb

3AI.s

Ge.

2 ?

0:::

0::: :::>

U

-.J 1S 1

04

~

Nb

3Sn

, \.

V3

Ga

,

V 2 Z~5

Hf.5

\

0:::

U

L 0 5

0

100

20

0

30

0

RE

SIS

TiV

E C

RIT

ICA

L F

IEL

D,

KIL

O-O

ER

ST

ED

S

Figu

re 4

lc

H b

ands

for

som

e hi

gh-f

ield

, hig

h-T

c su

perc

ondu

ctor

s at

4.2

°K. T

he s

prea

d o

f th

e ba

nds

is d

ue t

o th

e se

nsit

ivit

y o

f lc

CH

) to

pro

cess

ing.

Nb3

Al

prob

ably

ha

s a

crit

ical

fie

ld I

cCO

) o

f abo

ut 3

00 k

Oe

and

Nb3

Al.s

Ge.

2 so

mew

here

abo

ve 3

00 k

Oe.

t.)

~

;;0

o DO

m

;;

0

--t ~

;;0

o CI> m


THE CRITICAL FIELD OF COMPOUNDS AND ALLOYS

The utility of a superconductor is determined by its Tc and also by how much current it will carry in how high a field, i.e. Jc(H). Fig. 4 indicates the sort of data we can expect from the materials known to be in use (past or present) or potentially useful; the applied field at which J c falls off to useless or umneasurable values will be referred to as the "resistive critical field." The primary consideration in determining the resistive critical fie;Ld is the "upper critical field" Hc2 of Ginzburg and Landau(3):

Hc2 =V2KHc All the superconductors in Table I and Fig. 4 are of the

second kind with very large K. For Type II superconductors Hc is defined by the difference in free energy between the super electrons and normal electrons per unit volume:

H2 c - t.Gn-+

87T s (H=O)

Thus Hc2 may be raised in a given material mainly by boosting K, which is given by the Gor'kov-Goodman approximations(4,5,6)

K 'V Ko + Kl

K ~ 1.61 x 1024 (T y3/2/n4/3) (Sf/S)2 o c

Kl ~ 7.53 x 103pyl/2

where y is the electronic heat coefficient, n the valence electron density, Sf the free-electron Fermi surface area, S the actual free Fermi surface area, (all in cgs units) and p the resistivity of the normal state in ohm-cm. The term Ko is for the "pure" material, and suggests that some elements may be intrinsic Type II superconductors. In Group VIa of the periodic table, V and Nb are type II; but Ko for Ta is a trifle low, so that really pure Ta is Type I although the vast majority of "pure" Ta in circulation is type II due to the very small contribution of Kl needed to push K past 1/ 2. In alloys and high-field compounds the main contribution to K is made by Kl and in fact K can usually be neglected in such cases; K is the order of 10 -0100 in the superconductors in Fig. 4, for instance. The easiest way to raise Kl is to raise the normal state resistivity p; an extreme example of this is in the Nb-Ti alloy system where resistivity ratios (p 4.2/ 300) are of the order of unity, and may be less than unity.

However, for high-field alloys the resistive critical field

29B ROBERT M. ROSE

can not be identified solely with Hc2 because at high fields the free-energy difference due to normal state Pauli paramagnetism,

11 G 'V ~ (H2) 2

becomes significant. Due to the formation of Cooper pairs we expect such paramagnetism to be considerably less or even absent in superconductors, so the net effect is to decrease the free energy of the normal state relative to the mixed state, as in Fig. 5, bringing on a "premature" (below Hc2) transition to the normal state. In the absence of spin paramagnetism in the superconducting state, there is an absolute upper limit to the resistive critical field even for K + 00:

'V 'V Hp(paramagnetic limit) = 11(0)/2~B = l8,400Tc

at 0 °K(7), where Mo) is the OaK energy gap ant 11 B the Bohr magneton. However, as Ferrell has pointed out 8), there is spinorbit scatter in superconductors which can contribute a spin

H •

o Hu

f ~-gn(H,t)

g

------------------------- --------

Figure 5 A schematic for the lowering of the mixed-normal state transition field below Hc2 by the Pauli paramagnetism of the normal state. (K. M. Ralls, R. M. Rose and J. Wulff, J. App. Phys. 36,1295(1965).


susceptibility to the mixed state, and this tends to offset same of the normal state Pauli term. Detailed calculations by Werthamer et al.,(9) show that this is indeed the case, so we expect the critical field to be below Hc2 (calculated without considering spin paramagnetism) but above the critical field suggested by Fig. 5. A general way to estimate the degree of paramagnetic limiting is (9)

;'e~p _ a - 5.3 x 10-5

( -:":) T-Te where Hc2 is the usual Ginzburg-Landau critical field with no spin included, and HR is the measured resistive critical field.

299

Even where a critical field Hc2* (Hc2 compensated for spin) can be calculated, we may find further discrepancies because we are dealing only with the critical field for homogeneous nucleation of superconductiVt'ty. However, as Saint-James and deGennes pointed out in 1964 10), heterogeneous nucleation of a thin surface sheath (about ~ thick) can occur at an applied field of

if the superconductor surface is parallel to the field and in vacuum. If the surface is at some angle to the field, the nuc1eatt~y)fie1d is lower, decreasing to Hc2 when it is perpen-dicular • If the surface is coated with a layer of normal metal, Hc3(ti)decreased as the conductivity of the normal layer increases • We can view the heterogeneous nucleation field intuitively as a consequence of the fact that the boundary condition on the Ginzburg-Landau equations

( ~ \7 -~ A ) n ~ = 0

has the effect of locally lowering the free energy of the superconducting state, since

Ui17 -~~ j 2

appears in the free energy. Thus, the surface stabilizes superconductivity, by flattening out the gradient in~. The same sort of thing occurs at temperatures below the Ginzburg-Landau region, but Hc3/Hc2 rises to 1.925(12). Of course, if one surface can stabilize superconductivity above Hc3, two s9rf~ces should be even better. In 1968 van Gelder pointed out~13) that two intersecting surfaces (e.g., the point of a superconducting wedge) will nucleate superconductivity at a field Hc4 which will exceed Hc3 if the dihedral angle is less than 76°. For example, if

10

HR 6 2

6 7

8 T

F

igu

re 6

A

nis

otr

op

y o

f th

e cri

tical

field

of

sev

erel

y d

efor

med

Nb

wir

e as

a

fun

ctio

n

of

tem

per

atu

re.

anis

otr

op

y

dis

app

ears

whe

n ~(

T) ~ cell

siz

e.

From

D.

C.

Hil

l an

d R

. M

. R

ose,

M

et.

Tra

ns.

2,

1971

. -

8 ;10 0 Ill> m

;10

-1

Th

e

~

1433

, ;1

0 0 en

m


the field is parallel to the wedge edge, and the dihedral angle 20. is small,

Hc4

Hc2

With such effects in mind, in 1966 Williamson and Furdyna noted that the discrepancies between the results of "bulk" (e.g., ultrasonic) and resistive methods for critical field measurements on drawn BCC alloy wires agreed fairly well with the anisotropy

301

of the resistive critical field, which in turn looked suspiciously like the result of surface sheath effects at fields between Hc2 and Hc3 (14). However, further work showed that the surfaces of the wires were not invo1ved(14). Since the anisotropy was greatest in severely drawn wires and smallest in annealed material, extended defects introduced by drawing were considered to be the "internal surfaces" at which heterogeneous nucleation occurred. Such defects would be fibre surfaces or cell walls. However, it was not clear at that time why the material at either side of the "internal surface" should not wipe out the nucleation effect in accordance with theory(ll). The answer to the apparent paradox is that no plastically deformed material is homogeneous. The dislocations generated by plastic flow can be as dense as 1012 cm/cm3; they congregate, forming a cellular structure, with the interior of each "cell" relatively free of dislocations. The cell walls, in which the dislocation density is huge, are about 0.1 ~ thick, greatly elongated in the direction of plastic flow, and cell and cell wall shrink and sharpen up as

deformation increases. The physical properties of the cell wall are obviously going to be different; Hc2 will be much higher than that of the cell interior, if only due to the dislocation contribution to p. "Cell wall superconductivity" thus, offers a ready explanation of the disagreement with bulk measurements, and also the anisotropy, since the critical fields of long, thin shapes are naturally anisotropic. How do you prove this picture is true? First, pick a relatively clean material where Pn is relatively low so the effect will be large: Nb, for instance. Now, consider that when the coherence length is larger than the cell size, this whole silly business should disappear. Since

1;0

V1 -t c

1;(T)

we can do this by varying T and by varying plastic deformation. Fig. 6 is a typical case: the an:J.sotropy of the resistive critical field disappears when the coherence length is large enough, resulting in a remarkable critical field curve. We can precisely control the point at which HII~ H~ by controlling the degree of deformation, and produce specimens with ~ anisotropy

302 ROBERT M. ROSE

from 0° to Tc ' or with anisotropy from 0° to Tc ' or any intermediate case. Also, according to classical theoretical pictures of the cell wall, the effective dislocation density is tremendously stress-sensitive, and we should expect the resistive critical field to behave similarly. In fact it does: the stress se~i~) tivity is I - 2 orders of magnitude higher than theoretical , and is an even function of stress, again in agreement with dislocation theory. Finally, as a reductio ad absurdem you can push the critical field up to an order of magnitude higher than the "theoretical" value by very severe deformation, as Fig. 7 shows. A "bulk" measurement or a measurement of Pn would suggest a very normal piece of fairly pure Nb in this case.

A simpler case of an "anomalous critical field" is for heat treated Nb-Ti alloys, which presently constitute the bulk of commercially produced superconductor material. Fig. 8 shows the resistive critical fields for alloys in the system, and Fig. 9 shows what happens to the Jc(R) curve when you heat treat some alloy wire with 60%Ti - 40%Nb and 2400 ppm oxygen added. In the "homogeneous" as-drawn" wire consisted mainly of a single metastable supersaturated bcc solid solution. The heat treatments bring on the precipitation of a metastable (omega) phase and a stable hcp (alpha) phase, both of which are quite rich in titanium. Thus~considerable titanium is dragged out of the bcc matrix, and according to Fig. 8 a drop of slightly more than 5% Ti should be enough to boost the critical field by 20 kOe. A chemical analysis would not, of course, reveal the essential facts; this apparent "anomalous critical field" and many others are really direct consequences of the phenomenological theo~J and the microstructure.


2 10

o 10

-I 10

-z

• 0 • 0 •

0 • 0 •

0

0

1.. 0

• • • • • • 0

0

0

0 0 0

0

II • • • • • •

0

0

0

0

• • • •• ••

•• •• • •• • ••

10 ~~~~--~--~--~--~--~--~--~---10 zo 30 40 so

H Figure 7

Jc(H) in parallel and transverse fields for severely deformed Nb. From D. C. Hill and R. M. Rose, Met. Trans. ~, 1433, 1971.

303

304 ROBERT M. ROSE

'99 % NORMAL RESISTANCE RESTORED I 95 %NORMAL RESISTANCE RESTORED

140 50 %NORMAL RESISTANCE RESTORED

ONSET: 1/4/-,V,/CM. ACROSS SPECIMEN

120

100

f -(!) 80 oX

60

40

Nb-TI

20

O~----~----~------~----~----~ o 20 100

Figure 8 Resistive critical field vs. composition for the Nb-Ti alloy system. Courtesy of K. M. Ralls.


U J

° AS DRAWNx 300°C , I HOUR 6 400°C) I HOUR o 500°C) I HOUR + 600°C) I HOUR

fJ IOOO°C) I HOUR

Figure 9

INGOT

2400-U

305

Jc(H) in transverse fields for a 60%Ti-40%Nb alloy wire (with 2400 ppm oxygen added) with various heat treatments. Courtesy of R. L. Ricketts.

306 ROBERT M. ROSE

THE CRITICAL CURRENT DENSITY

The existence of the fluxon lattice in the mixed state suggests that J c for an ideal, homogeneous Type II superconductor should be zero. A uniform transport current J should produce a Lorentz force on the fluxon of

J<P o

c per unit length, or

, C

J x B

per unit volume, as sketched in Fig. 10. This force produces flux motion which will (see Fig. 10) cause an electric rield with the sense of J; that is, there is a "flux flow resistivity." Although the voltage induced is essentially induc~ive, fluxon motion is accompanied by diss~~ation due mostly to normal currents induced in the vortex corts(~ • A very early and inspired observation by Kim et.al. 17 was that to stabilize the fluxon lattice against the Lorentz force due to the transport current, we need "dirt": holes, inclusions, precipitate particles, cell walls, any heterogeneity with a size of order of ~ or larger. Fig. 11 is a schematic of the situation: the pinning force is due essentially to a reduction (attracting the flux line as in Fig. 11) or increase (repelling the flux line) of the self-energy or line energy of the fluxon. The heat treatments referred to in Fig. 9 increase the level of J c (as shown in Fig. 9) for the Nb-Ti alloy because of the very large numbers of precipitate particles which are brought on by the heat treatment. These numbers are enhanced by the use of relatively high oxygen levels (2400 ppm in Fig. 9): oxy~n is known to stabilize the a

Figure 10 Schematic: Lorentz force and flux flow in the mixed state.


Normal region, not penetrated

by field

(tI) Fluxoid is altracled to normal region which is not penetrated b~ the 'field

Figure 11

(I» Flu~oid sits in normal region _ lowering rrce cnerg~ or the ,uper conduCtor

307

Pinning by a hole or inclusion: superposition puts more electrons into the superconducting state with no change in magnetic penetration. From The Structure and Properties of Materials Vol IV: Electronic Properties, R. M. Rose, L. A. Shepard and J. Wulff (Wiley, New York, 1966).

E

Jc Figure 12

J!.£.. = Pf dJ

Schematic: An ideal flux-flow curve with uniform pinning.

308 ROBERT M. ROSE

precipitate in such alloys. Also, plenty of plastic deformation before the heat treatment increases the number of precipitate particles. The idea of pinning forces suggested a simple force balance:

nVL = f - f L P

where the Lorentz force f is set off by the pinning force f and the produ~t of the flux ftow viscosity n and the fluxon velogity vL' The I-V curve of such an ideal superconductor would resemble Fig. 12, and in the vicinity of the onset of flux flow there would be "flux creep" due to thermally activated breakaway from pinning centers(17). Since the electric field due to the flux motion is just

~ vL E = C B

it follows that

where Pf is the slope of the ramp in Fig. 12, or the "flux flow resistivity." If indeed dissipation takes place in the fluxon cores, then

Pf 'U H Pn H· (0) c2

follows very simply (16) .

The above ideas really are the entire conceptual framework for flux pinning. Using these ideas, it is possible to predict what will happen to the flux-flow properties of a soft Type II foil if diffraction gratings of just the right size are pressed into the foil surfaces. A result of this procedure is shown in Fig. 13. If the grooves caused by the grating are the only "pinners" present (this is possible with extreme precautions), then flux flow along the grooves will be as easy as before, but flux flow across the grooves will be subjected to a pinning force which is calculable from the size and shape of the grooves t18). Because the pinning is highly directional in this case, a Hall effect is also observed.

Howeve~ there is no doubt that nature is meIg) complicated. There should be, as Yamafuji and Irie point out , complications in some cases due to the distortion of the moving lattice by the pinning centers. It is also probable that the entire fluxon


Figure 13 A scanning electron micrograph (above) and schematic (below) of a soft In-Bi alloy foil which has been microgrooved by pressing with a diffraction grating . Flux flow occurs along the grooves with no pinning, but a pinning force is observed for flux flow across the grooves. From D. D. Morrison and R. M. Rose, Phys. Rev. Lett. ~, 356 (1970).

310 ROBERT M. ROSE

lattice does not translate uniformly but piecemeal, by motion of lattice defects in a manner analogous to the plasttzo~eformation of solids. The experiments of Trauble and Essmann , who were the first to directly observe the fluxon lattice, revealed many lattice defects, vacancies, dislocations and stacking faults. Some of these defects were in violation of classical dislocation theory and woult2~tve disappeared except for the presence of pinning centers • Thus~fluxon lattice defects, pinning centers and flux flow have a triangular relationship which has not been completely defined. These and other problems do not contradict the central practical goal: more pinning ~ higher Jcl

GENERAL METHODS OF MANUFACTURE

BCC solid solution technology is similar to that of other ductile refractory metal alloys. Arc melting in inert atmosphere or electron beam melting is followed by mechanical breakdown into wire by forging and/or extruding, then swaging and finally wire drawing. Nb-Ti alloys near the 60Ti-40Nb composition are favored because these alloys are ductile, relatively cheap, and have high critical fields. (By way of comparison, Nb-Zr alloys have limited ductility and require intermediate heat treatments in order to make Wire, have lower critical fields and cost more for raw materials.) High critical current densities are achieved in Nb-Ti alloys by using Ti-rich alloys (to insure more Ti-rich a and W precipitates to pin fluxons), using high oxygen contents (ca. 3000 ppm; oxygen is an a-stabilizer and also tends to form oxide particle pinners) and as much plastic deformation as possible. Whenever possible, cheap Ti sponge is used for raw melting stock since this will lower raw materials expense and insure the high oxygen level necessary to high J c ' High purity melting stock is only used if the ultimate in ductility is needed (dissolved oxygen hardens almost all refractory metal alloys). As we noted earlier, a precipitation heat treatment at 300 - 500 0 (depending on the composition and history of the alloy) will raise J c dramatically. As Fig. 14 shows, too much temperature can "overage" the precipitate; the precipitate particles coalesce and reduce the number of pinning ~enters. Too much time can have the same effect.

The extreme brittleness of the beta tungsten superconductors makes fabrication of such materials into wire a much more difficult proposition. Experimental arc-melted buttons were the first method of making Nb3Sn, but the buttons are quite friable and in no sense fabricable. The first technique for usable wire was developed at Bell Laboratories and is shown at the top of Fig. 15: mixed Nb and Sn powders are tamped in an


(f) 0 w J-(f) 0:: W 0 ~ It) It)

-N E 0 "'-«

0 "?

104

~ ..

50w/o Nb-50w/ol1 510 ppm OXYGEN AGED I HOUR

...... ...... ... -,,' .......

103 '--_---"-__ --'-__ ...l.--__ '--_--'-__ ---L-_---'

o 100 200 300 400 500 600 700

AGING TEMPERATURE (OC)

Figure 14 The effect of heat treatment temperature on the J c (in a transverse field of 55kOe) of a 50-50 Nb-Ti alloy with 510 parts per million oxygen. Courtesy of G. C. Rauch, from Trans. Met. Soc. AIME 242, 2263, 1968, by G. C. Rauch, T. H. Courtney, and J. Wulff.

- Nb

1.

PO

WD

ER

C

OM

PA

CT

ING

-CI2 II

. W

IRE

D

RA

WIN

G

m.W

IRE

D

RA

WIN

G

TIL

CO

IL

WIN

DIN

G

Nb

CI 5

+

S

nC

I 2

TO U

SE

R-

PL

AT

INU

M_

R

IBB

ON

1.

PR

OD

UC

TIO

N

OF

H

AL

IDE

G

AS

ES

. II

. V

AP

OR

D

EP

OS

ITIO

N

NIO

BIU

M

WIR

E

1-. _

_ _ ~-

TO

VA

CU

U:

0 0

0

,-LIQ

UID -TlN~~ --

-..

'-;-0

--;-c

? I.

HIG

H

TE

MP

ER

AT

UR

E

lL H

EA

T

TIN

C

OA

TIN

G

TR

EA

TM

EN

T

(OP

TIO

NA

L)

Fig

ure

15

~.:~AC

KET

~

Nb

3 S

n C

OR

E

Jr.

HE

AT

JZ

I C

RO

SS

-SE

CTI

ON

T

RE

AT

ME

NT

O

F F

INA

L P

RO

DU

CT

f' S, CO

ATIN

, P

t C

OR

E

m.

CR

OS

S-S

EC

TIO

N

OF

F

INA

L P

RO

DU

CT

TO

US

ER

-R

ES

IDU

A L

esn

CO

AT

ING

Nb

N

b3

Sn

LA

YE

R

m. C

RO

SS

-SE

CT

ION

O

F

FIN

AL

P

RO

DU

CT

Sch

emat

ics

for

var

iou

s fa

bri

cati

on

met

hods

fo

r N

b 3Sn

wir

e o

r ri

bb

on

.

Co)

t-.)

;:0

o to

m ~

~

;:0

o en

m


Nb tube, swaged, drawn, insulated and wound into a coil. Th~ entire coil is heat treated to react the wire core to Nb3Sn{ 2). There are problems attached to this procedure. In the first place, getting a really uniform core from a powder mixture is hard to do on a production basis. Second, finding an insulation which will stand up under the heat treatment temperature (which happens to be 900°C) without disintegrating is no mean feat. It was in the end easier to make Nb 3Sn - containing wire that was flexible enough to wind coils directly. This can be done in a number of ways. Fig. 15 shows two. There is the "RCA technique,,(23) , in which the mixed halides are reduced with H2 on a hot substrate. The result is a deposit of Nb3Sn and a halogen acid, usually HCl. This ferocious mixture is cut with a carrier gas, usually helium. The substrate can be anything that can stand up to the temperature and won't deteriorate. Various high temperature alloys, Hastelloys, platinum (if you have the money), quartz and others have been used. Another way shown on the bottom of Fig. 15 is simply to take niobium ribbon and dip it in a tin bath, coating it with tin, and then heating it up to about 900°C, which is again the optimum temperature. A reaction layer of Nb3Sn and maybe a little bit of excess tin on the outside is the result. The essential idea is to make a ribbon with a very thin layer of Nb3Sn which will bend. Ideally, the Nb3Sn should be at the center of the ribbon at the neutral axis where it would never get any stress from bending. Nb3Sn ribbon has also been made by co-evaporation of Nb and Sn in vacuum at high rates by electron beam techniques. Control of stoichiometry is the key to success with co-evaporation, and adequate J c may be obtained through the creation of planar pinning centers by periodically interrupting the ideal stoichiometry and depositing an Nb-rich layer. In all cases, high Tc and critical field are obtained by attaining ideal stoichiometry and making sure the Nb chains in the "A" sites (see section 1) are intact. If temperatures too far below 900°C are used to form the compound, there just isn't enough atomic mobility to do the job. If temperatures too far above (say, above l200°C) 900°C are used, Sn evaporates out of the cOlllpound and Nb atoms pop out of the "A" chains into the vacant Sn sites; Tc can be degraded to 6° or below in this way, although the beta-tungsten structure is still preserved.

Some newer materials have been explored recently. Nb nitride and other rock salt superconductors can be very effectively made by reactive sputtering, and recent work on Nb nitride by Hulm et.al.(24) shows it to perform about as well at 4.2°K as the best Nb3Sn and V3Ga(24). Other methods based on powder technology have not to date been as attractive for the rock salt group. The Laves phases probably have limited ductility, but their practical properties have yet to be explored. They have been prepared by arc melting and by diffusion couples.

314 ROBERT M. ROSE

400r--------------------------------

<f) <f)

~

300

<.:> 200 ~

,., IV 100

- .... .... "

"-

" , , , ,

o SAMPLE

~ SAMPLE 2

, , ,

o ~ ______ ~ ______ _L ______ ~ __ ~~ __ ~~

o 5 10 15 20

TEMPERATURE oK Figure 16

Critical field vs. T for "flexible" Nb3Al on an Nb ribbon substrate. Courtesy J. G. Kohr.


20r-------------------------------~

10

'" I 0 5 ---~ -- ..

tool :E'

~ 0.. :E' 2 ~

>-l-(/) Z w 0

I-z 0.5 w 0:: 0:: ::> u

...J ~ U I- 0.2 0:: u

200 0.1 ~----~------~----~~----~----~ o 160 40 120 80

RESISTIVE CRITICAL FIELD I KILO-OERSTEDS Figure 17

Je(R) for "flexible" Nb3A1 ribbon at 4.2° and -12°k (parallel field). Courtesy J .. G. Kohr.

316 ROBERT M. ROSE

Potentially very attractive as practical superconductors are Nb3A1 and Nb3A1.SGe.2 because they have the highest critical temperatures and fields. The measurements of Foner et.a1. on powdermeta11ur~ Nb3A1 ~gricates a critical field in the vicinity of 300 kOe at 4.2°K( ,and recently we made Nb 3A1 ribbon which has the same properties, at least between 10 and 20 0K, as shown in Fig. 16. Of considerable interest in this case is that Nb3A1 can perform very well at temperatures above 4.2°K. Fig. 17 is the result of a happy accident: we found, while measuring the critical field between 10° and 200K, that the specimens could carry considerable current, and simply went ahead and measured Jc(H) at 4.2° and 12°. (Of course, only parallel field measurements were possible in our apparatusl) It is apparent that this material will perform at 12°K about as well as Nb-Ti alloys will at 4.2°K. A threefold increase in operating temperature permits a threefold increase in the Carnot efficiency of the refrigerator and also, as we shall see, enhanced stability of operation. Until very recently the prospects of reliably producing useful Nb 3A1 were regarded as poor because many investigators had difficulty in consistently attaining high Tc (l8.5 - 18.8°K) for this compound. However, in the last few months a reexamination of the process metallurgy of Nb3Al has cleared up a few things(26). Past problems with this compound stem from the fact that well-ordered, stoichiometric Nb3Al is metastable at room temperature. The stoichiometric composition is stable only at high temperatures ca. 22000K. Howeve~ there is considerable disorder at such temperatures, and after the compound is formed it must be annealed at a much lower temperature to allow ordering of the Nb atoms to the "A " sites. As it turns out, about 10000K will do the trick. However, the stoichiometric composition is not stable at 10000K: the Nb3A1 would like to precipitate out some Nb2Al phase and shift its composition to Nb.88A1.22, which deteriorates Tc considerably. The trick is to turn on the heat long enough to order the stoichiometric Nb3A1 phase but not to allow it to go to equilibrium. We have worked out the heat treatment schedule to do this rod find it works on just about any reasonable configuration(~6. I suspect that, given time and effort, a practical process metallurgy for Nb3Al.8Ge.2 can be worked out as well.

CONDUCTOR MATERIAL FOR LARGE DC DEVICES

High current densities in the materials we have discussed are intrinsically unstable. In any case we expect that the relation

47T J "Ix H = -C-


will hold. In fact, before the idea of flux pinning was born, it was shown by Bean 2 ) that the irreversible magnetization of highJ c ("hard") superconductors could be accounted for very well by setting J = J c in the above relation. Physically, this means that as the magnetic field moves into or out of the superconductor, the maximum shielding supercurrents will flow. On this basis, the above relation can be integrated if Jc(H) is known (or assumed). However, a more intuitive approach is possible. The existence of a large transport current implies (once again via the above relation) the existence of a large gradient in the f1uxon density. However there is a central repulsion between the f1uxons(28) and therefore the f1uxon density gradient creates a pressure gradient, which would be equalized except that there are pinning centers which hold things still. However, a minor fluctuation could set an avalanche loose, since f1uxon motion will mean dissipation and heating, and heating will weaken the pinning, so that more f1uxons will break loose, and the instability will grow. It was this problem which greatly frustrated the early builders of superconducting solenoids, since fluctuations are much more likely in the great length of wire in a solenoid than in the few cm. used for a short-sample test. Consequently, magnets constructed in the early 1960's rarely lived up to the potential of the short-specimen Jc.

The problem has been analyzed extensively by the Rutherford Laboratory Superconducting Magnet Group(29). Central to the problem is that magnetic fields can move through high-K superconductors fairly rapidly, but heat can move through very slowly. This is readily seen by remembering the formulae for magnetic and thermal diffusivity:

and

Magnetic Diffusivity D mag

Thermal Diffusivity D k t

C P

(where everything is ces excepting for p, which is ohm-cm). If the data for Nb-Ti alloys are used, the answer is rather discouraging. Since p for Nb-Ti is very high, Dmag is very high, 104 times that ofnOFHC copper. On the other hand kt is very low in Nb-Ti so that thermal diffusivity for Nb-Ti is probably only about 10-3 of that of OFHC copper. Thus flux may move very rapidly when pinning becomes unstable, but the heat generated by the flux motion remains behind to aggravate the instability because the alloy is such a poor thermal conductor.

The instability headache can be cured three ways. One way is to find a way to bleed off the heat. Another way is to find a

318 ROBERT M. ROSE

way to retard the motion of the magnetic field. The third way is to mak';:; a superconductor that is so fine that you will never be able to generate enough of a heat pulse to get the instability going in the first place. Suppose the Nb-Ti superconductor is surrounded by copper. Copper has a very high thermal conductivity compared to Nb-Ti in the normal state. It has a very low magnetic diffusivity because it has such a high conductivity. Before the stability problem was well understood, it was noticed that copper plating the superconductor helped, but copper-plating metals as reactive as Nb-Ti alloys is almost impossible in ordinary aqueous media. From these early samples, often the copper can be peeled off with the fingernail. On the other hand, if a metallurgical bond is formed by la!ring the wire down on a strip of copper and putting another copper strip on top, and rolling it, then the oxide layer on the surface of the superconducting wire is broken, and you will then find that you have actually stabilized the wire, at least partially. An alternate method is to make a cable, and impregnate the cable with something like indium solder. To play it safe, you can add enough copper so that, even if all the current is shunted to the copper, the temperature will not rise above the Tc of the superconductor. However, this "brute force" approach neglects the importance of the size of the superconductor. The solution of the diffusion equation for an infinite slab (for the sake of simplicity) of 2 thickness 2L shows that the time constant T for diffusion is LID. For example, T is about 50 ~sec for the diffusion of magnetic field through a 2 em. slab of Nb-Ti, but 0.5 sec. if the slab were OFHC copper. Now, we want to get the heat out of the Nb-Ti about as fast as the flux moves out. If we surround an Nb-Ti wire with copper, the motion of the field may be retarded somewhat; but the size of the wire will have to be reduced to allow the heat to get out.

Fine-scale (Nb-Ti) + Cu composites are made by a technique which was actually first used to make Cigarette lighter flints and subsequently used at IBM to study proximity effects on aluminum and lead. The reductio ad absurdum for the technique was done at M.I.T. where we managed to get something like ten million niobium filaments in a 0.006° d~a. copper wire~30). Each of these niobium filaments ,.as 100A in diameter, as verified under the electron microscope. A cross section of the wire before it was reduced to this size is shown in Fig. 18. The way we did this was to put the niobium in a copper tube and draw it into wire, cut the wire up into 56 odd pieces, stuff it inside another copper tube, draw that down into wire, cut that into 56 pieces, stuff it into another tube and draw that. and repeat this five times. Inf Fig. 18 the families of filaments and also the families of families are shown; the filaments at this point were about 2 ~ dia. Commercial versions


Figure 18 A copper-Nb composite. Courtest H. E. Cline from Trans. ASM 59, 133 (1966) by H. E. Cline, B. P. Strauss, R. M. Rose and J. Wulff.

320 ROBERT M. ROSE

of this geometry are produced by assembling Nb-Ti alloy and Cu bars in a Cu shell in as compact an arrangement as possible and then extruding to reduce the diameter, sometimes as much as 16:1. Surprisingly enough, the geo~etry is not destroyed by this severe reduction, and pretty geometries such as Fig. 19 result. It is more efficient to have an even distribution of filaments as in Fig. 19 rather than the "families" of Fig. 18, since the excess copper between the "families" is mostly wasted. Since the magnetic diffusion constant for Cu is about the same as the thermal diffusion constant for Cu is about the same as the thermal diffusion constant for Nb-Ti, the materials are usually scaled near 1:1. On the other hand, if an alloy matrix, say cupronickel is used instead, we have to recognize that cupronickel has about 104 times the residual resistivity of OFHC copper, so Dma is 104 times larger. To keep the thermal time constan~ of ~he superconductor matched we ought to reduce L2/D by 10 , which can be done by reducing L by 100, that is, by using very fine wires.

Since the problem boils down to one of heat transfer, fine wires have intrinsic virtues, and it turns out that a sufficiently fine filament can be completely stabilized even without the copper3l Typically, the maximum size for niobium-titanium, with a critical current density of the order of 3 x 105 amps/cm2 , is about 0.003". Various people have worked out reasonable approximations for the critical size, and have come to the same conclusion (see e.g. Ref. 31):

where C p

d . cn.t.

is the

T -

specific

J c 0 -dJc

dT

heat, and

Commercially available composites can and are being produced in square, rectangular, and hollow shapes. However, without the proper precautions these fancy composites may have the same losses and the same instability as solid superconductors. When the magnetic field is applied to superconducting filaments running parallel to each other in the copper matrix, shielding currents will flow, through and between the filaments. As Smith et.al. have pointed out, the shielding currents will cross over through


the normal matrix and will be very persistent garticularly if the rate of change of the field is high enough~3l). In fact, there is a critical length for the cf~~s~y, of these shielding currents, also found by Smith et.al. ' ,who found that essentially all the filaments should cross through the normal matrix if the length is larger than 1 , where

c

'V 1 'V

C

H

J Ad c

321

(Here, d is the diameter of the filament, H is dH/dt and A a geometrical factor of the order unity.) Now we have trouble. We

Figure 19 A copper - (Nb-Ti alloy) composite, produced by methods similar

to those of Fig. 18, but on a larger scale. Courtesy P. R. Critchlow and E. Gregory.

322 ROBERT M. ROSE

Figure 20 An experimental "metal-insulated" transposed conductor for AC (pulsed) operation, courtesy of A. D. McInturff (Brookhaven

National Laboratory)


could even have two completely isolated wires a mile long and if we connect them at the ends we just have a very long effective length t, much longer than the critical length. The solution to this, again pointed out by Smith and the Rutherford group, is to transpose the wires, so that the field does not see parallel superconductof~90~If any length larger than the pitch of the transposition ' • Initially this was done by twisting and there is one problem with twisting which occurs in large composites: when the composite is twisted the field of the current going through the individual fil~ents tends to overload the outer wires in your composite(32). The self-field effect puts an upper limit on the size of the composite you can make.

To defeat the self-field effect in ~large composite to carry very large currents we have to completely transpose everything, by making braided cable or good old fashioned "litzwire" techniques. A standard twisted composite, for example, would impose a 0.05 - 0.1" diameter limit for 0.0005" Nb-Ti filaments. If more current is to be carried, then litzwire, cable, braiding or the like is the answer, but then there is the problem of mechanical constraint. If the strands of the cable are allowed to flop about, we have problems again. Because really highcurrent (103 - 104 amps) DC or pulsed conductor would be a tremendous asset, various methods are being tried. Fig. 20 shows one current effort, (courtesy of Dr. A. D. McInturff of Brookhaven National Laboratory) Standard extruded and drawn (Nb-Ti) - Cu Composite is tinned and heat treated to form a bronze layer (shown in Fig. 20), then the composite is braided (33 or 132 strands at 0.008" strand diameter, with a 2" pitch); the braid is impregnated with In-lO% Tl alloy (which has a high specific heat) and finally annealed again. In this case, the goal is 3600 amperes and a maximum tolerable dB/dt of 120 kOe/sec.

One obvious question to ask is whether the above technology is at all applicable to beta tungsten or rock salt conductors, since the (Nb-Ti) - Cu composites are created by dint of an enormous amount of plastic deformation. To get well above 100 kilogauss in large-scale devices we have the problem of stabilizing beta tungsten superconductor ribbon, which is a headache. The extruding and forging of copper on top of niobiumtin ribbon would break it to pieces. You can leave a bit of tin on the outside of the Nb3Sn and then copper-plate of silverplate, but this turns out to be unsatisfactory for a number of reasons, i.e. limited adherence, conductivity and thickness, and you still can't develop twisted and transposed multi filamentary composites at a reasonable expense with such a conductor. A partial answer has appeared in several places and that is to put niobium in a tin-bronze matrix, or vanadium in a gallium-bronze matrix, develop the desired geometry with the Nb-Ti alloy - copper matrix

324 ROBERT M. ROSE

technology, and then heat treat to form the beta-tungsten compound at the bronze-niobium or bronze-vanadium interface. The problems are that you have a high resistivity matrix and you have a lot of copper in your beta-tungsten compound.

Thus the answer probably lies with a different composite technology. Since the highest-Tc superconductors are basically Nb-A1 compounds, aluminum rather than copper seems to be a natural choice as a stabilizing matrix, particularly since lower residual resistivities are achievable on a commercial basis with A1 than with Cu, and also since the magneto-resistance of A1 saturates at high fields and that of Cu does not. Perhaps a method involving molten aluminum will be the answer. It is also important to notice that if operation at temperatures above 4.2°K is practiced some of the stability problem may disappear, because the critical size is increased: Cp is higher, and dJc/dT will probably be considerably lower. Thus if the potential of the high-Tc superconductors is exploited fully, composite stabilization may not need to be so sophisticated.

AC LOSSES

In the frequency domain from DC through most of the audio frequency range for a Type II superconductor in the mixed state, magnetic hysteresis (due to flux pinning) is the prime source of loss. (Practical applications at these frequencies nearly always involve fields high enough to ensure the mixed state.) Thus the area of the hysteresis loop described by the material should give the loss per cyc1.~ directly. Sometimes it may be more convenient to calculate E . J. This can be done, for instance, by using the critical state model

4'TT g X H = C J c

and then using Faraday's Law to obtain the electric field. Whatever we use, we will end up with an effective "resistivity" which is proportional to H. Alternative schemes for loss calculation can use the Poynting vector or even the work done by pinning forces. The calculation is complicated by the fact that the surfaces of materials prepared by mechanical means are grossly deformed and have much higher critical current densities than the interior. Thus \7x H shoots up near the surface and as a result most calculations include a "I1H" to compensate for this effect(21). In general we expect loss ~ cycle to increase as the peak field increases but to be independent of frequency. From the materialist's point of view the interesting thing is that AC loss per unit volume always increases with increasing size. For instance, in the critical state model the loss per

unit volume per cycle in the usual plane slab geometry (to make the problem one-dimensional) is, for large peak fields, proportional to the sl~b fhickness, and the same applies to solenoids wound from foi1s l22,. Thus the goal again is fine filamentary

325

or laminar structures. The present composite technology discussed in the previous section fully applies to this problem, and transposition of the superconducting filaments is now quite necessary to avoid additional hysteresis from currents crossing the metallic matrix. A major breakthrough in this area would be the development of electrically insulating (or at least poorly conducting) matrices which could still carry off heat. The technology to do this has not been developed. Another way to reduce AC losses at power and audio frequencies is to increase Jc , even on a local basis near the surface, to such a huge number that field penetration is always restricted to a relatively thin layer.

As the frequency increases from the high audio to the radio range, we will find that at some point, usually below the microwave range, the pinning will effectively disappear. At microwave frequencies in the mixed state the full flux flow resistivity will be observed although J may be well below J c ' This behavior was noted first by Gittleman and Rosenb1um(3S" who then went ahead to determine the nature of the transition from zero (or slightly "creepy") flux flow behavior to full flux flow behavior. They simply summed up the pinning force (in the limit of small amplitude vibrations), the viscous force, and the oscillating Lorentz force, and ended up with the familiar damped harmonic motion equation for a single flux line:

<Po <Po Mit + nx + kx = C J = C J 0 exp (iwt)

where M is now an inertial "mass" for the fluxon; the BardeenStephen expression(36) may be used for M. The pinning force constant k is in some way related to the critical current density. If we notice that the pinning force must have the periodicity of the f1uxon lattice, a sinusoidal ansatz follows, and k is just the zero-displacement limit so that

. . f kd. 27TX p~nn~ng orce = 27T s~n ~

where d is the f1uxon lattice parameter. If we make the naive assumption that the critical current density corresponds to the maximum value of the pinning force we can set

kd 27T

and see that, other things being equal, J c and k are proportiona1(37).

326 ROBERT M. ROSE

This approach has been used in attempts to relate the plastic flow stress to the elastic moduli of single crystals, and it failed spectacularly, because of the existence (unappreciated at that time) of lattice defects. Since we now know such defects to exist in the fluxon lattice some caution is in order. Whatever the relation between k, J c and Q! (the critical Lorentz Force), the solution of the differential equation for x is easily accomplished by setting x = Xo exp (iwt):

x = ¢ oj (iwn + k)

c and the elect tic field obtained from x by

E = ~ B c

iw¢ JB o

c2 (iwn + k)

The complex ac resistivity, E/J follows directly. The dissipation is obtained directly by

D 1/2 Re (E • J)

In Fig. 21 we see a sketch of the result; there is a "depinning frequency" given by

k w

o n

At frequencies below w the pinning forces dominate and at frequencies above w tge viscous forces dominate; the losses we see at the high fre~uencies come from very small-amplitude vibrations about the pinning centers. To raise the depinning frequency seems then to be a matter of stronger pinning again, on the basis of the naive model above or a truly collective model which considers the vibrational properties of the fluxon lattice.

I.O~----:::::=======I

~ 0.8

\N 0.6

~ 04

~ 0.2 N

OL...--~

1.0

Yf. Figure 21

Impedance vs. frequency for the mixed state normalized; J J c ; after Gittleman and Rosenblum, Ref. 35. The "depinning frequency" fo can range from less than 1 mHz to over 200 mHz.


As the frequency rises into the microwave region we can expect other loss terms to appear for the mixed state. Certainly we can expect the fluxon cores to contribute. In the fluxon cores the excitation spectrum consists of closely ~~aced levels with an excitation energy of the order of ~2/EF l3 ; excepting for millidegree temperatures, the cores are therefore essentially normal. There are also significant contributions from fluctuations of the order paramet~r which lead to large anisotropies in the surface impedance(39).

Of course, one way to avoid mixed-state AC losses is to avoid the mixed state, by operation at fields below Hc (of Type II material) or by using Type I materials such as Pb. Then the principal losses should be in the penetration depth (surface resistance), trapped flux (which can be eliminated with careful materials processing and procedures, and losses due to radiation and other experimental or practical problems (which can also be eliminated or minimized). To gain some intuitive feeling for the possibilities, consider that resonant circuits of Pb and Rn have been operated at 20 - 400 MHz with Q's of 106 -l07wO,4l). At microwave (S-band and X-band) frequencies, using Nb as well as Pb, Q's of up to 1011 have been attained(42-44). Such numbers are currently of great interest to the builders of linear accelerators, for cavities, beam splitters et.al. In general the quality factor Q, which is the reciprocal of the fraction of the stored energy dissipated per cycle, is directly related to the real part of the surface impedance:

47T2 1l 1 Q = (geometrical form factor) --C-- ie(Zs)

In a number of cases the Mattis-Bardeen theory for the surface resistance(45) has been confirmed; this theory, like its predecessors, predicts that Re(Zs) should vanish as the temperature approaches zero. If the investigator is careful (and luckyl), he may ~bserve this, but in general, particularly when practical devices are fabricated, there is a "residual" (Le. unaccounted for) loss, which will be discussed below.

At this time Nb appears to be the most likely material from which to construct microwave devices. Although it is easier to electroplate lead on copper (the fluoborate electroplating technology for lead is very well developed and a routine industrial procedure), lead microwave cavities are subject to a multitude of ills, particularly to so-called "aging" process. "Aging" refers to the drastic increases in surface resistance which occurred when the cavities were exposed to air, or even a poor vacuum. Thus the discussion below will concentrate on Nb. Alloys and compounds have not been investigated to any great extent because the very properties (high J c , high hysteresis)

328 ROBERT M. ROSE

which make these materials so excellent for DC applications tend to condemn them in the AC domain. Consider, for instance, trying to eliminate the trapped flux in heat-treated Nb-Ti alloy! Very high chemical homogeneity and an utterly smooth surface (to avoid field concentrations) are the goal. Nb, with a Tc of 9.2 - 9.5°K and an Hcl of about 2 kOe at 4.2°K is the best prospect of the pure elements. The problem of fabrication of Nb cavities for microwave frequencies is not, however, a trivial matter. The cavity itself can be fabricated by machining and welding(44), by deposition on cppper, Nb or other substrates by fused-salt electrodeposition(46) or chemical vapor deposition(47). The surfaces of the cavities must then be outgassed and smoothed. Chemical cleaning and mechanical and chemical polishing (typical components of the chemical polish are HF, HN03 and lactic acid) is followed by annealing at about 2000°C in ultra-high vacua (10-9 - 10-10 torr); the whole cycle may be repeated (e.g. see Ref. 44). The cavity is then transferred (in the evacuated state if possible) to the place it will be used. Exposure to atmosphere invariably degradet ~. Some improvement has been made by chemical anodizing 4 ); it is supposed that the Nb205 layer produced by anodizing protects the Nb surface.

Why is surface condition so important to the performance of Nb cavities? Impurities, imperfections and geometrical irregularities will bring on premature field penetration. Even prolonged annealing will result in thermal "faceting," Le. the rearrangement of the surface into planes of minimum surface energy. Surface roughness will magnify the local electrical and magnetic fields (41 ~d4~tuse drastic deterioration of Q at RF fields far below Hcl '

, • For very well-prepared Nb cavities the breakdown field Hcac can be brought up to about 750 Oe(44). At such high RF fields other effects can be anticipated: the magnetic hoop stress will expand the cavity somewhat, shifting the resonant frequency; field emission of electrons from surface asperities will occur; hard x-rays will be emitted due to the field-emitted electrons. Again, surface smoothness reduces problems; the task of creating a perfectly smooth, perfect surface is not trivial, and many approaches have been used in other industrial contexts. However, Nb, due to its extreme chemical reactivity (the corrosion and oxidation resistance of Nb are due to tightly adherent oxide films) and high melting point has escalated the difficulty somewhat further.

Besides field intensification by surface roughness, there are other possible contributions to the residual surface resistance. The tunneling evidence suggests that for Nb there are two energy gaps, one corresponding to the Tc of 9.2° - 9.~oK, and another one, about an order of magnitude smaller(50,5l). Excitations across the smaller gap are therefore one possible dissipative


me~g~!if' and since the second gap only shows up in very pure Nb ' it is of particular concern in the high-Q case. Another mechanism which explains the observed dependence of residual loss on frequency and mode is the generation of phono~~2tn surface cracks by vibrations induced by the RF field • Again, surface roughness appears as the villainl

329

Progress in the microwave area is apt to come in two steps: (a) improvement of the Nb fabrication technology, both in quality and cost, and much farther off, (b) high-perfection stoichiometric beta-tungsten compounds, with more surface and fabrication problems. The advantage offered by the latter is higher operating temperature than Nb.

MATERIALS FOR TUNNELING DEVICES

Since tunneling has been extensively discussed here before, I can be brief. The many potential applications of tunneling, particularly the Josephson effect will be severely limited in practice unless reliable, long-lived junctions can be produced at reasonable cost. The heart of the junction is the insulating barrier,S - 100 A thick. The barrier must be continuous, with no pinholes; it should be durable. The barrier layer should not coalesce into granules (thereby short-circuiting the junction) if the junction has to remain at room temperature for extended periods. Tunnel junctions are generally made by oxidizing a superconducting substrate to form an oxide insulating barrier, and then evaporating a counterelectrode on top of the oxide. Alternately, semiconductor (e.g. CdS) or insulating (amorphous C) layers may be deposited on the substrate instead, or anodized point contacts (or simply unclean point contacts) used. The latter suffers from unreliability and general mechanical crankiness. To date, non-oxide layers have been plagued w~th pinholes, with the notable exception of amorphous carbon(5 ,54). The "natural" oxides seem to produce the soundest barriers, except in the case of alloys and compounds. Thus the present state of the art is concerned mainly with oxide barriers on pure metals having high transition temperatures, e.g. Pb, Sn, and Nb. The "soft" metals such as Pb and Sn do not produce durable junctions. The high atomic mobility of these metals guarantees coalescence of the barrier into granules if the junctions remain at room temperature for any length of time. (This process is driven by the reduction in surface energy due to the reduction in surface area of the oxide by coalescence.) Thus, thermal cycling of such junctions between 4.20 and room temperature will lead to "aging" effects, i.e. short circuiting.

Again, Nb seems to be the answer. Nb has a high Tc (9.2 -

330 ROBERT M. ROSE

9.5°), low atomic mobility (the melting point of Nb is just above 2750 0 K) and extremely stable oxides. As a demonstration of the stability of Nb-Nb-oxide junctions, I remember some work 5 done by Margaret MacVicar and myself on some Nb single crystals(5 ) junctions were made by oxidation of the crystal surface and deposition of In stripes. (This procedure, by the way, had a yield of 85%, that is, 85% of more than 100 junctions fabricated were "good.") After measurements at 0.8 - 4.2°K were made, the wire connections were stripped off and the junctions stored in a drawer, at room temperature. Several months later, x-ray diffraction pictures were taken of the junctions, and then new connections were made and the tunneling measurements redone. The original curves, made months ago, were reproduced perfectly (that is, well within the precision of our apparatus).

Good tunnel junctions share with good microwave cavities the fact that both must be made on smooth, clean surfaces. In the case of tunnel junctions, asperities in the substrate surface will induce a large Schottky current through the oxide barrier, smothering the tunneling current. Impurities can be expected to increase the oxide conductivity and contribute tunneling events of their own. Thu~ the metallurgy of Nb and the development of superior Nb surfaces is vital to the fabrication of tunneling devices as well as microwave cavities.

REFERENCES

1. B. T. Matthias, "Superconductivity in the Periodic System," Vol. II of frogress in Low Temperature Physics, C. J. Gorter, ed., p. 138 (1957).

2. J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev. 106, 162 (1957); 108 1175 (1957).

3. V. L. Ginzburg and L. D. Landau, JETP 20, 1064 (1950). 4. L. P. Gor'Kov, Soviet Physics - JETP 10: 998 (1960). 5. B. B. Goodman, IBM J. Res. Develop. 6-,-63 (1962). 6. T. G. Berlincourt and R. R. Hake, PhYs. Rev. 131, 140 (1963). 7. A. M. Clogston, Phys. Rev. Lett. 9, 266 (1962~ 8 •. R. A. Ferrell, Phys. Rev. Lett. 3-:- 262 (1959). 9. N. R. Werthamer, E. He1f and P. C. Hohenberg, Phys. Rev. 147

295 (1966). 10. D. Saint-James and P. G. deGennes, Phys. Lett. 2, 306 (1964). 11. J. P. Burger and D. Saint-James, in Superconductivity (R. D.

Parks, editor; Marcel Dekkes, New York, 1969) p. 977. 12. G. R. Hu and V. Korenman, Phys. Rev. 185, 672 (1969). 13. A. P. Van Gelder, Phys. Rev. Lett. 20~435 (1968). 14. s. J. Williamson and J. K. Furdyna:-Phys. Lett. 23, 29 (1966). 15. D. C. Hill and R. M. Rose, Met. Trans. 2, 1433 (1971). 16. see e.g. Y. B. Kim and M. J. Stephen, in Superconductivity

(R. D. Parks, editor; Marcel Dekker, New York, 1969) p.ll07.


17. Y. B. Kim, C. F. Hempstead and A. R. Stnlad, Phys. Rev. Lett. 9, 306 (1962).

18. D. D. Morrison and R. M. Rose, Phys. Rev. Lett. 25, 356 (1970). 19. K. Yamafuji and F. Irie, Phys. Lett. 25A, 356 (1970). 20. H. Traub1e and U. Essmann, J. App. Phys. 39, 4052 (1968). 21. D. C. Hill, D. D. Morrison and R. M. Rose-,-J. App. Phys. 40,

5160 (1969). -22. J. E. Kunzler, E. Buehler, F. S. L. Hsu and J. H. Wernick,

Phys. Rev. Lett. 6, 89 (1961). 23. See RCA Review Vol. 25. 24. J. K. Hu1m, J. R. Gava1er, M. A. Janocko, A. Pat,terson and

C. K. Jones, to appear in Proc. 12th Int. Conf. on LowTemp. Physics (paper 8Xa 3).

25. S. Foner, E. J. McNiff, Jr., B. T. Matthias, T. H. Geba11e, R. H. Willens and E. Corenzwit, Phys. Lett. 31A, 349 (1970).

26. J. G. Kohr, T. Eagar and R. M. Rose, submitted to Met. Trans. 27. C. P. Bean, Phys. Rev. Lett. 8, 250 (1962). 28. See P. G. deGennes, Superconductivity of Metals and Alloys,

(Benjamin, New York 1966) or A. L. Fetter and P. C. Hohenberg, Ch. 14 of Superconductivity (R. D. Parks, ed; Marcel Dekker, New York, 1969) for details.

29. Rutherford Laboratory Reprint RPP?Z 73, November, 1969; by J. D. Lewin, P. F. Smith, A. H. Spurway, C. R. Walters and M. N. Wilson.

30. H. E. Cline, B. P. Strauss, R. M. Rose and J. Wulff, Trans. ASM 59, 133 (1966).

31. P. F-.-Smith, M. N. Wilson, C. R. Walters and J. D. Lewin, Proc. 1968 Summer Study on Superconducting Devices and Accelerators, p.913 (Brookhaven National Laboratory, 1969).

32. A. D. McInturff, to be published. 33. S. L. Wipf, in Proc. 1968 Summer Study on Superconducting

Devices and Accelerators (Brookhaven National Laboratory BNL 50155 (C-55), 1969) p. 511.

34. C. P. Bean, Air Force Materials Laboratory Technical Report AFML-TR-65-431 (March, 1966), Section I.

35. J. I. Gittleman and B. Rosenblum, Phys. Rev. Lett. 16, 734 (1966). -

36. J. Bardeen and M. J. Stephen, Phys. Rev. 140, Al197 (1965). 37. Y. Shapira and L. J. Neuringer, Phys. Rev. 154, 375 (1967). 38. C. Caro1i, P. G. deGennes and J. Matricon, Phys. Lett. ~,

307 (1964). 39. C. Caro1i and K. Maki, Phys. Rev. 159, 306 (1967). 40. C. R. Haden and W. H. Hartwig, Phy~Rev. 148, 313 (1966). 41. J. M. Victor and W. H. Hartwig, J. App1. Phys. 39, 2539 (1968). 42. J. P. Turneaure and I. Weissman, J. App1. Phys.~9, 4417 (1968). 43. P. Kneisel, O. Stoltz and J. Ha1britter, Institu~fur

Experimente11e Kernphyski Notiz Nr. 138, Feb. 1971. 44. J. P. Turneaure and N. T. Viet, App1. Phys. Lett. 16, 333

(1970). -

332 ROBERT M. ROSE

45. D. C. Mattis and J. Bardeen, Phys. Rev. 111,412 (1958). 46. R. W. Meyerhoff, p. 23 of Ref. 33. 47. I. Weissman, p. 32 of Ref. 33. 48. H. Martens, H. Diepers and R. K. Sun, Phys. Lett. 34A, 439

(1971) • 49. M. Rabinowitz, J. App1. Phys. 42, 88 (1971). 50. J. W. Hafstrom, M. L. A. MacVicar and R. M. Rose, Phys. Lett.

30A, 379 (1969). 51. ~W. Hafstrom and M. L. A. MacVicar, Phys. Rev. 2B, 4511 (1970). 52. J. Ha1britter, J. App1. Phys. 42, 82 (1971). --53. M. L. A. MacVicar, S. M. Frea~and C. J. Adkins, J. Vac.

Sci. Tech. 6, 717 (1969). 54. S. I. Ochiai, M. L. A. MacVicar and R. M. Rose, Sol. St.

Comm. 8, 1031 (1970). 55. M. L. A. MacVicar and R. M. Rose, J. App1. Phys. 39, 1721

(1968). --

SUPERCONDUCTING INTERMETALLIC COMPOUNDS - THE

AIS STORY

Robert A. Hein

U. S. Naval Research Laboratory

Washington, D. C. 20390

Twenty years ago the phenomenon of superconductivity was the exclusive property of the low temperature physicist and was regarded as the major unsolved problem of solid state physics. Today one reads and hears that superconductivity is well understood and that most solid state theoreticians regard it as a "solved problem" no longer worthy of their full attention. I believe that, due to the technological promises of applied superconductivity, the re are cur rently more metallurgi st s, engineers and electronic specialists involved with superconductivity than there are physicists, or at least physicists who are engaged in the "physics" of supe rconducti vity.

Major problems concerned with the application of superconductivityare: making liquid helium technology as reliable as the household refrigerator and improving the technologically important parameters of useable superconductors so that liquid helium temperatures are not a prerequisite for the meaningful utilization of the phenomenon. The AIS story is, of course, part of the latter endeavor.

Although the phenomenon of superconductivity was discovered sixty years ago, the AIS "superconducting" story starts with the work of Hardy and Hulm. I Superconducting AIS' s are thus only eighteen years old and have not yet reached the age of maturity. Like the youths of today, the AIS's are much discussed and much has been written about them. Since Prof. Rose will be talking to you in detail about the metallurgy of the AIS' s and Prof. Matthias will be discussing high transition temperature superconductors

333

334 ROBERT A. HEIN

(that meanS the A15's) I view my task as presenting you with an introduction to the A15 story.

Present theory, as detailed as it is, does not furnish workable rules for the fabrication of high T c materials; therefore, one is forced to use empirical rules. Because of this situation I want to breifly run through the historical development of these rules and how they apply to the A15' s of today. Another reason for this approach is that present empirical rules and correlations are rather pessimistic with regard to T c' s above 22K. I hope that by pointing out how these rules have changed over the years some of you will be encouraged to defy these rules and corne up with an honest-to-God 30K supercond\1ctor that will be useable.

Between 1911 and 1923, superconductivity was the exclusive property of the low temperature group at the University of Leiden. During these years they reported superconductivity in Hg, Sn, Pb, Tl, and In. They noted that all these elements were "soft" metals with comparatively low melting points (mp). Hg was low with an mp of -39°C and Pb was the highest with an mp of 327°C. From this observation it seemed as though low melting points were a prerequisite for superconductivity.

Leiden lost its liquid helium monopoly in 1923, and in 1928 the first non-Leiden superconducting element was reported by Meissner and coworkers. This element was Ta, which has a mp of 2996°C, which is the fourth highest mp among elements. The implication that a low mp is a prerequisite for superconductivity was quickly dispelled. Today one knows 2 that the elements with the highest mps are superconductors. They are: W(mp = 34l0°C, T c = O.012K); Re (mp = 3l80°C , T c = 1. 70K); and Os(mp = 3000°C, Tc = o. 65K). Although the correlation between the occurrence of superconductivity and low melting points was short lived, interest in the connection between anomalously low melting points and superconductivity has recently been revived. 3,4

At the start of 1951, which is only two years before the start of the A15 saga, the position of the superconducting elements in the periodic chart appeared as shown in Fig. 1. On the.basis of this figure (those elements enclosed by the dashed lines were"normal" metals at the start of 1951) one sees that superconductivity was restricted to two well-defined groups in the periodic system. This observation lead to the conclusionS that for superconductivity to occur, between 2 and 5 valence electrons outside a closed atomic shell was required. Conventionally one denotes this number of "valence" electrons per atom bye/a. The belief that superconductivity only occurred in metals with 2 < e/a < 5 persisted until the discovery of superconductivity in Ru and Os by Goodman 6

SUPERCONDUCTING INTERMETALLIC COMPOUNDS 335

I n m IV V VI VI VB[

H . . He L1. Be 8 C N 0 F Ne

Na Mg At 81. P S Cl. AT WI'

Ia lla ma IVa Va VIa IVlIa Vlna. Ib fib mb !Vb vb VIb VlIb VIII'a

K Ca Sc Ti V CT Mn Fe Co Ni. Cu. Zn Ga Ge As Se ST KT 1·81 It·3

--: 0·79 1·07

Rb ST Y ZT Cb Mo Pd. Ag Ca. In Sn Sb Te I Xa 0·7 9·22 Tc =RIL=Rh. 0-5'+ 3·37 3-69

Cs Ba La Hf Ta W Re lOs I IT Pl AIL I~.& III Pb Bi. Po Rn 4·71 0'35 ,..3& • • 7-26 • Fr Ra Ac . . . . . . . . . . . . .

Ce, PT, Nd, It, Sm., Eu.,Gd,Tb, Dy,Ho,E.r,Tm.,Vb,Lu.

02, Pa8 Np,Pu, Am, Cm

Fig. 1. Position of the superconducting elements in the periodic chart circa 1950 [Ref. "Low Temperature Physics -Four Lectures", Academic Press, Inc. New York, 1952, p. 97J

F' (n)

o 6 8 9 10 nl NUMBER OF VALENCE ELECTRONS

Fig. 2. Superconducting transition temperature versus the number of valence electrons per atom [Ref. 10 J

336 ROBERT A. HEIN

in 1951. Since these elements have an e/a = S one merely extends the rule to say that for superconductivity one needs 2 < e/a ~ S. The discovery7 of superconductivity in Ir in 1962 raised the upper limit to 9, so that today one believes that superconductivity is not restricted by any e/a considerations, but that for 9 ~ e/a < 2, T c will be extremely low (< O.lK).

The fact that the e/a ratio may be a useful guide in searching for high T c materials was first emphasized by Matthias S who in 1953 called attention to the fact that all known superconducting elements, alloys, and compounds with transition temperature above SK (nine such superconductors were known) had e/a between 4.5 and 5.5. The higher Tc's occuring for e/a values just below 5.

As always happens whenever one makes a generalization, it's no sooner in print than it becomes obsolete. Matthias sent in his report on Aug. 3, 1953 and Aug. 31, 1953 Daunt and Cobble sent in their report 9 covering the discovery of superconductivity in technetium. The inital reported value for T c was 11. 2K, today's accepted value2 is 7. SK. Since technetium has an e/a value of 7 and since at that time none of the elements with e/a = 6 were superconducting, it meant that at least two maxima must exist in the graph of T c versus e/a. Figure 2 is the original version of the Matthias plotlO and depicts the qualitative variation of the transition temperature denoted in the figure as F{n) versus e/a, or n. The graph indicates high transition temperatures occur in the vicinity of e/a = 5 and 7. There is a bit of poetic license here in that one knew that at e/a = 4 there exists low T c elements such as Ti, Zr and Hf. (~O. 40K, O. 55K and O. 35K, respectively) and also that at e/a = 3 one has lanthanum with T c R> 4. 7K. Thus one really had a minimum at e/a = 4 and a third maxima at e/a = 3. In the years that followed, the peak at e/a = 3 was included, excluded, and now finally it is not only back in, but it is quite a high peak with a maxima T c of 17K and possibly higher. 11

Another early restrictive condition on the occurrence of superconductivity was noted12 by Cla'~sis in that while metals have atomic volumes which range from a low 4 cm3/mole for Be to a high of 70 cm3/mole for Li, superconductors seem to require middle-of-the-road atomic volumes of between 9 and 21 cm3/mole. This is shown in Fig. 3, where one has plotted the atomic volume as a function of the atomic numbers for the elements. The dash lines form an envelope which encloses the known superconductors with the exception13 of beryllium (T c = O. 027K). It is of course obvious that molar volume per se is not a sufficient condition for the occurrences of superconductivity as many normal metals are


-;;; 110 E

,0 L. .... on 0'1 100 c: ~

90

80

t 70 ., E :::J 60 0 > u

·E SO 0 .... «

40

30

20

10

0

Rb

K

CI

Co Kr

He A

li H

Be B

C.

Bo Xe

Sr

La

Atomic number Z

• Non-superconducting elemental polymorph o Superconducting elemental polymorph l::. Element exhibiting superconductivity in thin film

form at low temperature or under high pressure

Eu

Yb \ ,

----Pt

Fig. 3. Atomic volume of the elements versus atomic number. Superconducting elements are contained within the two dashed lines [Ref. "Progress in Cryogenics" Vol. 4, Academic Press, Inc., New York, 1964, p.167]

1; OK

~--------~--------.-------~.--------.---------;

IO~--------~--------~~--------1----------+--------~

,-8~--------+---------1----------+~-------+---------1

" "

Too'~-- -- - ---- - - -- -- ----· ... v 4~----~~W-----~~~~-------+---------+---------,

"SII

Ti 2 ~----~~l---------~--~-----+------~T-.+---------,

Hi Cd Zr Zn .. Go AI"

338 ROBERT A. HEIN

enclosed by the dashed lines. When alloy work was begun in earnest in the late 1920' s,

workers were concerned with defining the extent of the phenomena and little attention was paid to how T c varied with composition for any given alloy system. What scanty data there were indicated5 that T c of any alloy system was somehow related to the magnitude of a o ' the lattice parameter. Testardi14 and Testardi et al. 15

have recently emphasized the importance of a o with regard to the A15' s and conclude that in the se compounds T cis a function of the lattice parameter alone. Thus, one of the early correlations, i. e. , between T c and the dimension of the lattice, has taken on a new appeal. However, as we shall see, Smith quickly challenged this belief16 with his high pressure data.

In 1947 DeLaunay and Dolecek introduced17 a new parameter into superconductivity, namely, the Debye characteristic temperature. They presented a graph of T c versus aD which is shown in Fig. 4. Elements belonging to the electro-positive group and with atomic volume s between 14 and 23 cm3/mole lie along the steeply rising portion of the curve while elements with atomic volumes between 9 and 14 cm3/mole lie along the flat portion. Figure 4 emphasizes that electro -positive elements which have atomic volumes between 14 and 23 cm3/mole and which have a low aD are good candidates for superconductivity. It is interesting to note that this early correlation between T c and Eb predicted that protactium would have a T c within reach of helium cryostats. Eighteen years later18 it was found that Fa is indeed a superconductor with T c = 1. 4K. As you have heard in the previous talks, the isotope effect showed the importance of a D with regard to superconductivity. McMillan's treatment of the BCS theory, as you have also heard, tells us that one can indeed enhance T c by lowering aD. So this old and unnoticed correlation isn't nonsensical afterall.l9

Shortly after the discovery of the isotope effect, Daunt introduced20 a second parameter, namely y, the coefficient of the linear term in the specific heat via the plot shown in Fig. 5. Here one has plotted y divided by the atomic volume, V, as a function of T c for the known superconductors. The data clearly suggests that large Tc' s are associated with large ylv. Since ylV changes by a factor of 18 while V changes by only a factor of two or so [See Fig. 3] one see s that large y I S seem to favor large T c' s. The advent of the BCS theory focused attention on y because it is re-1ated to N(O), the density of electronic state s at the Fermi level, which governs T c via the expression T c IW aDe -l/N(O)Vx. From this expression one sees that a large N(O), i. e. , large y,

SUPERCONDUCTING INTERMETALLIC COMPOUNDS

. t i 8

339

10

Fig. 5. Coefficient of the electronic specific heat divided by the atomic volume versus transition temperature for known superconductors circa 1949. [Ref. 20 ]

Fig. 6. The Beta -tung sten or A15 type cry stal structure of the A3 B compounds. Note that not all of the atoms are shown. [Courte sy of Prof. R. Rose of MIT]

340 ROBERT A. HEIN

implies a large T c provided the BCS interaction parameter, V"1f , is approximately a constant.

Thus one sees that the parameters which are discussed today, whenever one talks about superconductivity, namely, e/a, a o ' aD , V, and y have been connected with superconductivity for some time now.

In concluding this brief history of empirical'correlations I want to mention a correlation that one doesn't hear much about. Chapnick in 1962 pointed out that2l most superconductors have a positive Hall coefficient, R, and that at that time the non-superconducting elements with positive values for R were Mo, Be, W, Ir, and Rh. Today one knows that all, except the last, are indeed bulk superconductors. Thus I recommend that you read this paper by Chapnik- -who also discusses how pressure, crystallographic order, etc. , can change R and hence affect superconductivity.

The f3-W or A15 story, as previously mentioned, starts with the work of Hardy and Hulml who, while investigating the silicides and germanide s of Group IV, V, and VI transition metals, found that V 3Si, V 3Ge, M03Si, and M03Ge were superconductors and that V3Si had a Tc of 17.lK. This was a new high surpassing by two degrees the T c of NbN which had ranked supreme since the early 1940' s.

X-ray analysis showed that these silicides and germanides possessed the A15 structure and as such represented the first known supe rc onductor s with thi s crystal structure. The A15 structure is shown in Fig. 6.

For this structure the ideal binary compound is represented by the formula A3B, where A is usually a transition metal of the N, V, or VI group and B mayor may not be a transition metal. Atoms of the minor constituent of the compound, i. e. , the B atoms, form a bcc sublattice, while the A atoms are situated, pair-wise, on the six faces of the bcc unit cell as shown. This Figure does not show all the atoms involved in the unit cell. Each unit cell contains 6A atoms and 2B atoms. (As you may recall atoms on the cube faces are shared by two unit cells and corner atoms are shared by 8 cells). This A15 structure was erroneously attributed to a second or Beta form of elemental tungsten or Wolfram (at. no. 74). Although we now know that this is not the case, early workers were actually observing the structure belonging to W30, the name Beta-tungsten has caught on and one still often refers to this structure as the Beta -tungsten phase.

Attempts by Hardy and Hulml to raise T c by substituting 0.10/0 of the Si atoms by B, C, Al or Ge failed to increase Tc--in fact


T c decreased in SOITle cases by 10K. Substitution for the A atOITl also depre ssed T c • Matthias and coworkers iITlITlediately took the challenge and said that if ternaries do not produce higher Tc values, one should seek high transition teITlperatures in binary A15 cOITlpounds with a favorable e/a ratio (at that tiITle it was felt to be slightly under 5), and a large atoITlic voluITle. This latter constraint follows froITl the beliefS that for a given class of alloys an increase in atoITlic voluITle will increase T c. The Bell Laboratory group looked for and found22 the A15 structure in Nb3Sn and Ta3Sn. Both of the cOITlpounds have the saITle e/a ratio as V3Si (4.75) but the lattice constants were larger by about 12%. While Ta3Sn was found to have a T c of only 6K, Nb3Sn e stablished a new high T c of lS.05K.

After the discovery9 of superconductivity in technetiuITl the Bell Laboratory group focused their attention on finding A15 COITlpounds with e/a ratios in the neighborhood of seven. They discovered ITlore A15 phases of Nb3X cOITlpounds 23 (X = Os, Ir, Pt) but none had a T c in excess of 9.2K. The e/a values ranged froITl 5.75 to 6.25. They also found that V3Sn,which is isoelectronic with Nb3Sn, had a T c of only 7K. Obviously there is ITlore to superconductivity than just the e/a ratio.

By 1956, only a little over two years after Hardy and HulITl' s original work, SOITle twenty A15 cOITlpounds were known and Matthias published24 the graph shown in Fig. 7. This graph was used to argue that the peaks in the T c versus e/a plots for the A15 cOITlpounds are shifted to sOITlewhat lower e/a values than those observed with the transition ITletals. Whenever I look at this early graph and speak of peaks, I cannot help but adITlire Matthias's intuition and his obvious ability to convince the referees that he is correct. My purpose in showing this plot is to encourage you to have faith in your own ideas and concepts.

What is Faith? Since this is a Jesuit University I think it is only appropriate that I anSwer this rhetorical question' by quoting a little scripture. Saint Paul in his Epistle to the Hebrews answered the above question by saying--Faith is that which gives substance to our hopes and which convinces uS of things we cannot see·

Anyone who looks at this graph and speaks of shifted peaks or ITlaxiITla certainly has faith and probably a bit of Divine guidance as well. But as we all have been told faith can ITlove ITlountains and what are ITlountains but peaks; by 1965 the peaks were obvious to all as seen in Fig. S.

Since the early workers in this field were very busy trying to forITl new A15 cOITlpounds and ITleasure their Tc's,little attention

342

Fig. 7.

Fig. 8.

• Nb,sn

V,Go OV,S\

• THE V AND Nb CO""POUND~

CONNECTED ARE NEARLY

IDENTICAL IN VOLUME AND ... AS5

4 THE 0 COWPOUNDS AAE ISOLATED AS

THEIR CORRESPONDING NEIGHBORS FAL.L IN THE NONSUPERCONOUCT'NG RANGE N

2 V,P*

Nb)AU Nb,Sb

/ VJAS

0 V,SO

• Nb'.'! I~ / e VSGe o}-..

~sno r-~ /

~ 4

2

0 V,A5 _,

40 40

~ NblRh

~ ~

_~o.s, / M2JGLe~. NtlJlr

I NblOS

~ 0 ~S SO ELECTRONS PEA ATOM

.0

ROBERT A. HEIN

,

Transition temperature of some A15 superconductors versus the number of valence electrons per atom, circa 1955. [Ref. 24 ]

15

~

'-'0 ~u

··SUPERCONOUCTlIJE TRANSITION TEMPERATLI!E

I-NOT SUPERCONDUCTIVE TO THIS TEMPERATURE

47 ,

r·' II £1

F~ I ...

: ':1 I I : .. : ,'',

I

I' I I I I, I I' I I I I I I I I

\ I

),_ .... / ;. I

, .. , :: 65 :' I :1 ,'-'

... 1 I \ .1 I I

• I I I

'\ ~ 1 •• 1 : ..

I I

.' I I • I '

i: : -a I • ' I' I .1 I

I I : I I

, I ._1 :. \ ,. \ I , 'ir/ • \

rio~"''(' '; .. I •• It ,. i 5 6 7

VALENCE ELECTRONS IATOM

Transition temperatures of the A15 superconductors versus the number of valence electrons per atom circa 1969. [Ref. 2 ]


was given to determining the exact composition of the compound. That is, it was generally assumed, or so one is led to believe, that the chemical composition of the compounds was the ideal A3B composition. However, the work of Greenfield and Beck25

showed that of the thirty A15 phases which they investigated,all of them had narrow composition ranges about the A3B stoichiometric composition. Furthermore, Greenfeld and Beck said that there were considerable inaccuracies in determining the true composition of the A15 phases. Raub and Mahle 26 had also pointed out that in the Cr -Ir and Cr -Os systems single pha se A15 compounds could only be obtained close to AO. 80 B O. 20 and not at the stoichiometric AO. 75 B O. 25 composition. Nevertheless in the early and mid 1950's workers tacitly assumed their samples had the A3B composition even though no chemical analysis had been made.

During the late 1950' s. investigations of the dependence of T c upon composition was begun in earnest. Incidentally. in connection with thi s problem. Wood et al~ 7 pointed out that the compo sition of an alloy prepared by a rc melting is not precisely known. since losses cannot be readily avoided. However. in this case chemical analysis would have been meaningless since the samples were not single phase. Nevertheless, studies with binary A15 systems were performed and it was concluded that the maximum T c in a given A15 system occurs at the stoichiometric composition. A clear cut case of where this is so is depicted in Fig. 9. Electrical resistivity data shown in Fig. 9 are those of Wernick et aJ.28 who noted that such data yielded smaller T c values than those deduced by Blumberg et al?9 from magnetic susceptibility measurements. Wernick et al?8 remarked that this disagreement "is not surprising since the methods of observation are quite different". I feel that the fact that resistance data yielded lower T c values is surprising since dc resistance data do not yield values for T c lower than those deduced from magnetic data. In Fig. 10 one sees that later ac inductance (400 Hz) data of van Vucht et a130 are in excellent agreement with the data of Wernick et al?8 The singularly high Tc point is, I believe,a data point of Blumberg et al. 29 Since Blumberg et al.did not present any details about their V -Ga results, the cause for this discrepancy is unknown to me.

Incidentally, I think it is worthwhile to note that. for the A15 phase of V-Ga. as the Ga content increases from 22% to 36%, a o increases smoothly from 4.l83A 0 to 4. 834A o. Since T c passes through a maximum it is clear that an increase in a o does not per se lead to an increase in T c. The maximum in T c

344 ROBERT A. HEIN

.. II

I.

12

.X 10 !

~ •

•

•

Fig. 9. Variation of transition temperature T c with composition for the V -Ga system. Also shown is the variation of

Fig. 10.

the critical magnetic field (evaluated at T = 0). [Ref. 28 J

18,-------------------------------, • Wernick et al.

E 16 x van Vucht et rtf. ~

~ 14-.a o ~ 12 ~

f:~ i 6~ b.c.c.

4~ \x

2 ' o V

I I I I I I I I I I I I I 1 I

1A1S+ l1; &15

Variation of transition temperature T cm with composition for the V -Ga system. [Ref. 30 ]


is also not what one would expect from the e/a graph of Fig. 8 as e/a decreases from 4.56 to 4.18 over the above composition range and hence T c should always decrease.

The basis for the belief that a given binary A15 compound will exhibit its maximum T c value at the A3B composition is the supposition that for such a composition the degree of crystallographic long range order (LRO) will necessarily be a maximum. The data of Fig. 10 clearly show that in the A15 phase of the V -Ga system T c is a maximum at the A3B (i. e. V. 75Ga. 25) composition, however, no measurement of the degree of LRO present in the samples was reported.

What does one mean by the degree of crystallographic long range order? Fig. 11 once again depicts the A15 structure but in this representation A atoms of adjacent unit cells are included to emphasize the "A-chains". From this figure one clearly sees that the A atoms form a three dimensional network of mutually orthogonal one -dimensional chains of A atoms. Incidentally the interatomic spacing along a given A-chain is actually about 150/0 smaller than in the elemental form of the A element itself. This fact has been used by the theorist to construct a "linear chain model" for the A15' s which will be subsequently discussed.

o~:--I I I I I I I I I I

r~/ .: rly/ 0--- -~----v

Fig. 11. Atomic arrangement for A3B compounds of the A15 type structure. Solid circles donate A-atom sites. Open circles donate B-atom sites. For sake of clarity atoms on three of the six cube faces have been omitted. Note that the extension of the A -chains is emphasized. [Ref. 48 ]

346 ROBERT A. HEIN

When the composition is stoichiometric there will be three A atoms for every B atom. If this is so,then conceivably every A atomic site can be occupied by an A atom and every B atomic site occupied by a B atom. Such a situation is referred to as a perfectly ordered structure. If for some reason A-B atom interchange's are affected,the structure becomes partially disordered. Disorder may also result from a lack of stoichiometry. For example an A rich A15 structure must either have A atoms on B atomic sites or vacant B sites must be present. For a B rich A15 structure the converse is true. Mathematically, one defines the degree of long range order by means of the conventional Bragg Williams Long Range order parameter S. A completely ordered sample will have an S value of unity, and if the site s a re occupied in a completely random (disordered) manner S = O.

For a non-ideal system,van Reuth and Waterstratt3l introduced a modified order parameter for the two atomic sites given by

S = A

(rA -F A)/(l- FA)

SB (rB -FB)/(l-FB )

where:

r A fraction of A sites with A atoms

r = B

fraction of B sites with B atoms

F = A

fraction of A atoms in the phase

F = B

fraction of B atoms in the phase.

Only in the ca se of the ideal A3 B ratio, with no vacancie s pre sent, is SA = SB = S. Values for SA and SB are obtained from x-ray diffraction data. 31

The importance of the degree of LRO present in the sample and its effect on T c was introduced by Hanak et al. 32,33 to account for the data shown in Fig. 12.

Here one has plotted T c as a function of Nb content for the A15 phase of Nb-Sn. The upper curve is the data of Jansen and Saur34 for sintered samples, while the lower curve is the data of Hanak et al. for vapor deposited samples. One sees that there is considerable disagreement in the reported T c' s for the Nb rich A15 phase of the Nb-Sn system. This difference in the behavior of sintered and vapor deposited samples was attributed33 to a low value for the degree of crystallographic long range order present in the vapor deposited samples. That is, while the sintered Nb O. 80SnO. 20 sample was essentially in a state of complete


• JANSEN a SAUR } • RCA SINTERED

t RCA-VAPOUR DEPOSITED

9

BL--L __ ~ __ ~~ __ ~ __ L-__ L--L~

74 76 7B BO B2 84 B6 BB 90 ATOMIC %NIOBIUM

Fig. 12. Transition temperature versus composition for the A15 phase of the Nb-Sn system. Data are shown for both sintered and vapor deposited samples. [Ref. 32 ]

order, the vapor deposited Nb O. 80Sna. 20 sample exhibited extensive disorder. Hanak et al. 33 interpreted their data as indicating that it was lattice disorder of the type where Nb atoms are located on Sn atomic sites and vice versa. They concluded that having Nb on Sn sites was more detrimental to T c than having Sn on Nb sites.

Reed et al~5 quickly undertook a study of the role which crystallographic long range order may be playing in determining T c and reported excellent ag reement with Jansen' s 34,36 re sult namely the A15 phase prepared at 1200°C is stable for N~ _xSnx in the range of 0.72 $ x $ 0.80 and that NbO. 80SnO. 20 has the maximum T c (18.5K). They confirmed Hanak ' s et al. observation33 about the effect of disorder but said it is the occupancy of Nb sites by Sn atoms which causes T c to drop and not vice verSa. Reed et al. concluded that for high T c one must maintain the lIintegrityli at the A -chains. The fact that NbO. 80SnO. 20 has the highest T c clearly demonstrates that having Nb atoms on Sn sites or having vacant Sn sites does not necessarily degrade T c.

Due to the above results and the technological importance of Nb3 Sn, early studies of the effect of ordering on T c tended to concentrate on the A15 phase of the Nb-Sn system. Hanak33 emphasized that the RCA work was the first experimental evidence that microscopic disorder plays an important role in the

348 ROBERT A. HEIN

superconductivity of the A15 phase of the Nb-Sn system. However, the group at RCA were not of one mind for L. J. Vieland37

pointed out that,in view of compositional effects, the conclusion that disorder is responsible for the low T c in vapor deposited Nb3 Sn was not warranted. Hanak 33 also said that since NbO.80 Sno.20 has an e/a of 4.8, the simple e/a consideration,( see Fig. 8,)indicates that the latter composition should have the lower T c contrary to what is observed.

The question of the effect of LRO upon T c was thrown open by later work38 , 39 which showed that the degradation of T conly came about if Sn was volatilized in the high temperature annealing process (1200°C). Such losses change the overall composition of the compound and Sn vacancies are created. Nb atoms may then migrate to the Sn-site vacancies and the integrity of the Nb or A-chain is destroyed. Hence Tc falls. In fact,thevacancies were believed to be ordered and that the low T c was due to a new phase consisting of ordered vacancies. 39 Thus, the situation with regard to the effect of crystallographic order could not be resolved due to compositional effects. In the case of Nb3Al it was also felt39 that Al is volatilized in high temperature anneals and hence vacancies playa role here as well.

In 1965 a program was started at NRL and at Westinghouse, Baltimore,to survey the twenty-six binary A15 systems reported by van Reuth and Waterstratt in their ordering study. 31 The objective was to see if one could detect any Significant changes in T c between samples with different values of the order parameter. Van Reuth and Waterstratt chose40 to work with transitional metal B atoms,as such compounds are more suitable for the determination of the LRO parameter, S, and the ordering could be accomplished with low temperature anneals. Some of the early results of this study are given in Table I. Values for SA and SB were obtained from x-ray diffraction measurements and computer programs. Since the compounds listed in Table I are stoichiometric, SA = SB = S.

Table I

Compound S Tc (K) Lit. (K) % Increase

Ti3lr .91 4.18 Ti3lr 1. 00 4.63 4.5 7.7%

V3Au .92 0.86 0.74 107. % .98 1. 78

Nb3Au .75 8.99 11.5 15.2 % .85 10.60

The behavior of V3Au shown in Table I is of little interest to the technologist, but it is of interest to the physicist who IT1ay happen to be interested in the IT1echanisIT1{s) which give rise to this large percentage change in T co If one could raise the T c of Nb3 Sn by,say 150/0,it would equal the known high41 of 20. 7K. Encouraged by these early data, we set out to see if these increases in T c could be unaIT1biguously ascribed to an increase in S.

To avoid any possible cOIT1plications due to cOIT1position effe ct s, additional IT1ea sureIT1ent s4 2 we re IT1ade on saIT1ple s of V 3Au obtained by fracturing a 20 graIT1 ingot of V3Au which had been subjected to a 1000°C hOIT1ogenization anneal and quenched in order to induce a low value for S. While a segIT1ent of the ingot which had not been subjected to any additional heat treatIT1ent reIT1ained norIT1al down to O. 015K, saIT1ples which were given low teIT1peratures anneals did becoIT1e superconducting. T c for those saIT1ples was deterIT1ined by IT1eans of a dc IT1utual inductance technique. Results of this work are presented in Table II and

TO ( dHC2) Sample Thermal History S dT T

(OK) 0

3 (10 oejdeg)

l' 800°C - 1 hr, slow cool 1.85 -1" l' + 1000°C - 5 day, slow cool 1.79 0.98

2 Leiden Sample - Q <0.015 -3A 1000°C - Quenched <0.015 0.82 ± 0.05

3D 3A + 400°C, 1 hr - Q 0.89 0.92 ± 0.05 25

3C 3A + 600°C, 1 hr - Q 0.90 0.92 ± 0.05 22

3D' 3D + 500°C, 4 wk - Q 1.285 - -3B 3 A + 800 ° C, 1 hr - Q 1.55 0.92 ± 0.05 32

3B' 3 B + 700 ° C, 1 wk - Q 2.48 33

3C' 3C + 600°C, 2 wk - Q 2.88 37

3C" 3C' + 600°C, 2 wk - Q 3.10 38

3B" 3B + 700°C, 3 wk - Q 3.22 16

5 Levitation Melt V. 77 AU. 23 0.71 -

Table II. A cOIT1pilation of transition teIT1perature, To; long range order paraIT1eter, S, and initial slope s of the critical IT1agnetic field curve sand therIT1al history of various V 3Au saIT1ples.

350 ROBERT A. HEIN

I. 0 1-+-< .............. -+-.j_~x:----iP--·}IeIt· t i

30.8 w N

(

0.2

0.0

0.0

Fig. 13.

; * I +

* I * I +

i + ,

01' ~ I

V6T • 3A.2 'IL-. ." *38 -r- '138'

I 638" 6

~ X3C ~ e3C'

j& 03D Q 3D'

'-£.----- -"-5

+ I"

-.{].

1.0 3.0

Normalized initial magnetic susceptibility of various V3Au samples (See Table II) versus temperature.

Fig. 13 where a normalized susceptibility is plotted as a function of the temperature. These data clearly show that T c can be increased by subjecting the sample to low temperature ordering anneals. Because of the small size of the samples involved, Waterstratt could not determine precise value s of S, but the S data did allow one to construct the plot shown in Fig. 14 where T c is plotted as a function of S. Such a graph clearly shows that T c of V3Au can be as high or higher than 3. 2K. (Bear in mind the literature value for the T c of V3Au was O. 74K.)

Fig. 14.

::.:: °

4.0 -

w 3.0 0:: ::::> I-<1 0:: W 0.-

~ 20 I-

Z 52 Iif)

Z ~ 1.0 I-

~ ~ 2.0% 0.5°/,

_I

/~----

/' o. 0 L-_L-.f=I:::::::t==:J:::2..--'-:L-_L-~

0.7 0.8 0.9 1.0 LRO PARAMETER (5)

Transition temperature versus the degree of long range order present in various V3Au samples.

While the magnitude of T c is not very exciting to the technologist, the fact that one can produce an increase in Tc of 3.2K by low temperature ordering anneals, with no changes in chemical composition, is indeed intere sting. Such an effect clearly demonstrates the importance of maximizing the degree of LRO present in samples of the A15 compounds and it is felt that the recent intere st in low temperature annealing of A15 compounds received its impetus from this work which was first reported43 in 1966.

How does the increase in S bring about an increase in T c ?

352 ROBERT A. HEIN

If it doesn't change e/a or a o ' what does it change? On the basis of the BCS expression{T c = const. aD exp [-l/N(O)VX ]lone first thinks of N(O). f

To obtain information about possible changes in N(O) which accompany changes in S, the initial slopes at the critical field curves of the V3Au samples were measured, since

Ny (0) ~ y'" ~ ~ ",::~(Tr 1¥~K2l ",::~(Tr Tc Tc

here V is the molecular volume Hc (T) is the thermodynamical critical field K is the Ginsburg -Landau parameter H c 2(T) is the "Glag' upper critical field = J2KHc(T)

From these expressions one sees that if K doesn't change appreciabily with increase in S (a questionable assumption), then the relative change in initial slope s will be a measure of the re-1ative change in y. Figure 15 shows the re suIt of such an analysis.

Fig. 15.

5.0

4.0

o ~ 2.0

I

V3 Au

% Ib 2D ~O TRANSITION TEMPERATURE (K)

Electronic density of state s derived from magnetic field data versus the transition temperature of V3Au. The different T c values are due to different degrees of crystallographic long range order present in the samples.


Here we see that as S increases so does N~ (0). And we see that as S approaches unity, N~ (0) must increase quite steeply. Well, you can imagine our joy when the Geneva group 44 actually measured the specific heats as a function of S with results as shown in Fig. 16. From the se data one sees that both y and S

increase with increasing T c' One can construct a ')I versus S plot and extrapolate it to S = 1 to yield a value of Yo' The use of Yo as a normalizing parameter facilitates the comparison of various A3B compounds as shown in Fig. 17. These data for the various Nb based compounds and the ternaries [(Nb-V)3Au]compounds show that y does indeed increase with increase in LRO. The Cr based compounds yield the opposite result and the reason for this has been discussed by J. Muller.45 Thus, one sees that evidence does exist which suggests that increasing S increases N(O) hence T c' However, one must bear in mind that the possible role of any effect which increasing S may have on the detailed shape of the phonon spectrum is still an open question.

I want to make one last remark about the use of ordered samples. Berlincourt46 in a recent review article plotted T

c values quoted in the literature as a function of e/a and obtained the results shown in Fig. 18. I believe Berlincourt's intent was to caution people not to concentrate on e/a ratios around 5 and 7 at the expense of neglecting other potentially interesting areas of the periodic chart (note the high T c values at e/a = 3).

The left hand side of Fig. 18 appears to destroy the concept of the Matthias' regularities, however, I believe this is due to the use of poorly characterized samples which leads to a large scatter in Tc for the e/a ratio calculated on the basis of "nominal composition". Our data47 ,48 obtained on well ordered, well characterized samples produces the e/a plot shown in Fig. 19. These data reveal a line structure in the e/a plot which clearly shows that A15 compounds with A atoms from the 4d transition metal series fall along one curve, while those with A atoms from the 3rd serie s fall along a lower curve. This behavior is also reflected48 in that the electron-phonon coupling constant is larger for the 4d based compounds.

Now that one is aware of the importance of LRO, let uS consider the question of stoichiometry and its relationship to T c' The A15 phase was known to exist in the Nb-Ge system but only on the Nb rich side of the Nb3Ge composition, i. e. , from 77% to 84% Nb. The lattice constant, a o , of this phase decreases from 5.l77A 0 to 5.l67A 0 as the Nb content decreases, however T c increases35 from 4. 9K to 5. 5K. A linear extrapolation of these data to the Nb3Ge composition indicated35 that the

354 ROBERT A. HEIN

10

~ Q9 II)

as 07

~12 ":.:: 0

a 11 I

:!. 10 ... E

,";,,9

8 0 2 3

Tc [OK]

Fig. 16. Long range order parameter, S, and electronic specific heat coefficient, y, as a function of transition temperature for V3Au. [Ref. 44 ]

1.5

o. (Cr-X)

1.8

~

'-~ ---__ -0-------0.5 (Nb-V)3Au ---__ -0---- ---

(v-x) Ir

0.0 0.8 0.9 1.0

ra

Fig. 17. Variation of a normalized electronic specific heat coefficient (see text) as a function of the degree of long range order characterized by the parameter ra' [Ref. 45 ]

SUPERCONDUCTING INTERMETALLIC COMPOUNDS 3SS

~------TRANSITIONI---------"I I" NON TRANSITION---j

20 11 1 I 20

18

16

14

12

~ 10

... v 8

6

4

Fig. 18.

Fig. 19.

..

"\ I

18

16

14

12

10

I-- 8

I-- 6 . . 4

I . • I

• L-_li'_~I--4L-_l5-~~_

AVERAGE NUMBER OF VALENCE ELECTRONS PER ATOM

Transition temperature versus number of valence electrons per atom. Ranges of alloy composition are repre sented by solid line s. [Ref. 46 ]

14r--------------~

12

10

8

6

V 0.

4

NtlRh •

2 Nblr ~; I, Au : C~'R:: M\O p,

VP'~ .NbO.

TiPt 0'---- TI.A'!.... }f~~h--CrRh c~:Pt

5 6 ' 7 e/o

Transition temperature of well ordered A3B compounds of the A15 type structure versus number of valence electrons per atom, e/a.

356 ROBERT A. HEIN

stoichiometric A15 phase would have a T c of 6. 3K. Since the stoichiometric composition has an e/a of 4.75, one would expect a considerably higher value for T c. In view of this the Bell Laboratory group attempted4 9 to quench in (splat-cooling) the A15 phase of stoichiometric Nb3Ge with the results shown in Fig. 20

~ ~~ ____ ~ _____ NLb_'7_1G __ ~~2_O ____ ~ ____ L-~_la_)~ ex: 2 ~ W ..J a. ~

"Nb3 Ge li

EQUILIBRIUM PHASE

~ ~O~ __ ~ ____ ~ ____ ~ ____ ~ ____ ~ __ ~L-__ ~

u.

(C)

[[[ I I I I

U. 4 6 --8~----'O~---1~2-----'~4-----16~--~'8

TEMPERATURE,oK

Fig. 20. Superconducting transitions of samples prepared from a. melt of composition Nb O. nGeO. 79: Curve (a) as prepared in the arc furnace, curve (b) rapid quench and subsequent annealing up to 1000°C, curve fc) after annealing at llOO°C for three days. [Ref. 49 ]

The splat-cooled sample produced the data presented in the middle portion of the figure. He re one see s that the magnetic susceptibility of the sample (16 KHz) starts to change at 17K but doesn't exhibit perfect shielding until 5K. This is an extremely broad transition. These data, in conjunction with x-ray analysis of the sample, were interpreted as evi cnce that stoichiometric Nb 3 Ge has a T c of 17K. The broadne JS of the transition was attributed to a variation in the degree of LRO. It was felt49 that the small sudden decrease at 17K is due to well ordered Nb3Ge.

The Bell Lab group predicted41 that if one could obtain the A15 phase of V3Al it would have a very large T c' I personally do not know the basis for this statement, for on the basis of Fig.19, One might expect that since Nb3Al has a Tc of 18.8K, the isoelectronic Vy\l will have a lower T c. How much lower is

SUPERCONDUCTING INTERMETAlliC COMPOUNDS 357

anyone's guess, for Nb3Au has a T c of 11K, while for V3Au it is only 3. 2K. Nb3Sn has a Tc of 18K, while for V3Sn it is only 6K. Why then should V 3Al be expected to have a very high T c ?

V3 A!, when prepared by conventional ITIetallurgical processes, forITIs in a bcc structure. However, the prediction of a high T c by Matthias et a1. 41 is all it took to invoke world-wide interest in V 3A1. Muller in GerITIany tried to obtain inforITIation about the A15 phase of V3Al through a study 50 of Tc and a o of four vanadiuITI based ternaries, i. e., V3Bl_xAlx where B =' Si, Ge, Ga, Sb. MeasureITIents of a o as a function of x showed that c5 a / c5 x>o in the case of Ga, Si, and Ge but was < 0 in the case of 0

Sb. Extrapolation of the se data to x = 1 indicated that the A15 phase of V3Al would have an ao of 4. 87A o. MeasureITIents of Tc versus x showed OTc/Ox > 0 for Ge and Sb, and < 0 for Sb and Ga. These data did not perITIit ITIeaningful extrapolation to x = 1 so that no prediction about the T c of V3Al could be ITIade. Muller eITIphasized that for these vanadiuITI cOITIpounds the T c versus e/a graphs just don't work. Figure 21 shows a plot of T c versus the concentration of valence electrons and one can see the inconsistent behavior of the cOITIpounds.

Fig. 21.

J ~~ f \ ~ VI'· '\ I

! I i

15

1 I

5

I ~ \ I k,.\ I \

I/l \ \ \ ••

~~'-.4 L,f• f I ,,::L.,.t-o 3.0 3.5 Konltn/,,/ion dff V,'rnze/ektronen

lO"/cm J

Transition teITIperature of VanadiuITI based A15 phases versus the nUITIber of valence electrons per cubic centiITIeter. (a) V3Au, (b) V3Pt, (c) V3Sn, (d) V3Pb, (e) V3Rh, (f) V3Co, (g) V3Ni, (h) V3As, (i) V3Pd, (k) V3Cd (1) V3Ir. [Ref. 50J

358 ROBERT A. HEIN

Since the e/a considerations were of little use, Muller resorted to a correlation first used by Pessall and Hulm51 for the superconductors possessing the NaCl structure. Using tabulated value s of atomic radii, one can calculate an a o for a given A3B compound. Dividing this number by the measured a o yields a quotient denoted by R and a plot of T ever sus R appears as shown in Fig. 22. Here one sees that T c peaks in the region of R = 1. 02. Dividing the calculated value by the extrapolated value for V3Al yields R = 1. 02 and hence Tc should lie around 17K. This prediction is in agreement with the Bell Laboratory's group estimate41 of a very high T c .

Asada et a1. 52 in Japan also studied a series of ternaries V3AI-V3X, where X = Si, Ga, Ge, Sn, Sb, and from the measured a o values, they deduced that the A15 structure of V 3Al should have an a o of 4. 84A o. They also plotted T c versus a o for the ternaries and found that in general T c decreases with increasing a o ' On the basis of these data, a value for a o of 4. 84A 0

corresponds to a T of not more than 10K. They concluded that the predictions~l, 50 of a very high Tc for V3Al were unwarranted.

Luo et a1. 53 in 1970 discussed the occurrence of the A15 phases and from their study of V 3 (Al,X) X=As, Ge, Au concluded that V3Al would have a Tc of 17K. This once again is based on an extrapolation of measured ao's for V 3 (Al,X)ternaries. I find it rather noteworthy that the extrapolated value for V3Al is in exact agreement with that of Asada et a1. 52 namely, 4. 84A o.

In the work of Luo et a1. they simply perfo\rm a linear extrapolation of the T c versus a o data to arrive at a value of 17K for V 3A1. Thus we have two-for-one against the high T c prediction.

Hartsough and Hammond54 of the University of California have recently prepared A15 - V3Al. Employing a technique of vacuum evaporation and co-condensation onto a glass or fused quartz substrate held at 350°C these workers found the A15 structure corresponding to the chemical composition V3Al. The measured a o was given as 0.4830 ± 0.003 nm (i. e. , 4. 83A 0). T c for these thin films (3/.l) were determined from de resistance measurements. The highe st observed T c was 8.5K. Being aware of the degradation caused by lattice disorder in these A15 compounds and aware of the problem associated with traces of silicon, they repeated the work using alumina and predeposited V on alumina as substrates. Maximum T c' s of about 9. 6K were then obtained. This value is very close to Asada's et al. extrapolated52 value of 10K. Thus, it seems that V 3Al with the same e/a as Nb3Al, but with a smaller lattice constant, 4. 83A °


::= ~ -t

•

15~----~----~--~----------~---

••

~10~----~-+----~~----------~---

~

-5 .~ -~

5r-----~-+------~~--------~---

1.0 7.05 7.70

359

Fig. 22. Transition temperature of the Al5 phases of various vanadium based compounds (V3X) versus the quotient, R, of a calculated lattice constant divided by the measured lattice constant. [Ref. 50 ]

360 ROBERT A. HEIN

compared to 5.l83A ° , has a much lower T c' Let uS now consider the transition temperature of ternaries.

An overriding theme of the mid-1950's and 1960' s which persisted up until 1967 was1l "All attempts through the formation of mixed phases with Nb3Sn or other compounds crystallizing in the same f3-W structure, to raise the transition temperatures over the values of both of its end points have been unsuccessful." That is, there exists no known ternary which exhibits a T c larger than the T c of the highest binary component.

Well, a not uncommon cry in this city of Washington is "Lets look at the record" and so we shall. In 1962 Reed et al. 35 clearly showed in their study of the Nb3Sn-Nb3Al system that, while their binaries Nb3Sn and Nb3Al had transition temperatures of l8.lK and 17K, respectively, the compound Nb3(SnO.9AlO.l) had the maximum T c of l8.4K.

Now one might sayan increase of 0.3 is not significant, but remember Nb3Sn ranked supreme as the high T c material by beating Nb(C O• 3NO. 7) by only O. 2K. One can also use hindsight and say that one knows that well ordered ordered Nb3Al has a T c of 18. 8K and a T c as high as l8.5K was reported for NbO. 80 SnO.20' To reason thusly, I think, would be falacious, for the data of Reed et al. 35 clearly showed that, starting with two binaries of given transition temperatures, a ternary with about 10 to 20% of the B atoms corresponding to the minor constituent has a T c in excess of the binaries. The same qualitative behavior was also observed with the Nb3Sn-Nb3Ga system. In fact, it seems to be quite a general behavior as seen in Fig. 23.

These are the data of Hagner and Saur55 and one clearly sees the presence 01 a maximum for ternanes. Mother Nature was doing her best to tell us what to do but we were too set in our ways to pay her any heed. Thus while the mystery of the apparent 18 oK limit was very much discussed, the answer was very evident, but no one took heed.

An important piece of work which indirectly led to the demise of the 18K limitation is that of Alekseevskii et al. 56 who showed that in the Nb(All_xGex ) system a sizeable maximum existed for x = 0.2 and that T c was a sensitive function of the degree of crystallographic long range order' present in the sample. What Alekseeskii et al. didn't know was that one obtains the highest transition temperature when the A15' s are subjected to an ordering anneal at temperatures between 750 and 800°C. Their samples were annealed at 600°C for 250 hrs. which our data42 ,43,48 and those of the Geneva G roup57 clearly showed to be too low and to short a time for maximum ordering. A second factor which

'" '" :::> ~ ~

:E I!!

IV

V

m

NU

MB

ER

O

F VA

LEN

CE

ELEC

TRO

NS

PER

AT

OM

N

UM

BE

R

OF

VALE

NC

E EL

ECTR

ON

S PE

R

ATO

M

CO

NS

TAN

T· 4

-750

NU

MBE

R

OF

VALE

NC

E EL

ECTR

ON

S PE

R

ATO

M

'-750

18

-6

OK

4-72

5 H

OO

17-9

1 N

b,T

l"Sn

,."

o a-I

0-

2 AM

OU

NTS

O

F S

UB

STI

TUTE

S

4-67

5 .....

., >S

K .~

.....,. -

~

'5

Nb,

Ge"

Sn,.,

soli

d so

luti

ons

wit

h N

b,Sn

m

ixtu

re o

f pu

re e

lem

enta

l po

wde

rs

pres

sad

wit

h 8

tons

tem

' "n

tare

d f

or 6

hou

rs a

t 12

00 °

C

o

ind

uct

ive

tran

siti

on

4

resi

stiv

e tr

ansi

tio

n

18 3

,0

---~

V

18

Nb,

Pb,.S

n,.,

I

17-9

1 N

b,B

i,Sn,

_,

9 17

-' 0-

3 a

a-I

0-2

0-3

a 0-1

0-

2 A

MO

UN

TS

OF

SU

BS

TITU

TES

A

MO

UN

TS

OF

SU

BS

TITU

TES

Fig

. 2

3.

Tra

nsi

tio

n t

em

pera

ture

of

the A

15

ph

ase

of

vari

ou

s te

rnary

sy

ste

ms v

ers

us c

om

po

siti

on

. [R

ef.

5

5 ]

0-3

en

C ~

m '" () o z o c (

)

-i Z

G) Z

-i

m ~ ~ . .- R

() ~ ~ o c Z

o en

w ~

362 ROBERT A. HEIN

possibly prevented a higher T c was that their starting Nb3Al had a relatively low (16. 5K) T c'

When the Bell Laboratory group reinve stigated41 , 58 the NbGeAl system, the effect of low temperature ordering anneals was known and so they obtained the maximum T c at about the same composition as Alekseevski et al., but it was a good two degrees higher, i. e. ,20.lK. They found that a~ anneal at 750°C produced the highest T c' T. F. Smith16 has attributed the discovery of this compound to a straight forward application of the Matthias plot. I don't know the scientific basis for this statement, as e/a considerations for a ternary give one some degree of freedom as to what composition to choose.

Muller59 has recently completed a detailed investigation of the Nb-Ge -AI system and has shown that the e/a plot yields inconsistent results for the A15 ternaries (see Fig. 21). Muller concludes that its not the e/a value per se, but rather the integrity of the A -chains which is important, and predicts that the compound NbO.75AlO.15GeO.lO should have the maximum T c (2l o K at an e/a of 4. 6); however, such a compound is unstable. Muller cannot account for the fact that the highest T c he observed is almost 10 K smaller than that of the earlier workers. 41,58 He tried to repeat their quick quench result but could never get above 20.1K. The possibility that the presence of two or more phases in the earlier works is responsible 58 for increased T c cannot be ruled out; nor can the effect of some impurity in the starting material. Waterstrat and van Reuth60 have shown that ordering53 among the Ge and Al,which has been suggested58 as the reason for the high T c atoms, does not occur.

Since the e/a plot does not yield consistent results as far as T c is concerned, Muller once again 59 resorted to a plot of T c versus the quotient of the calculated a o divided by the measured a o to obtain the results shown in Fig. 24. In this figure one sees that such a plot yields three curves. However ,if one defines RX as the value of R for which T c reaches its maximum value and then plots Tc versus lOOr(RX-Rl,the results shown in Fig. 25

l Rxj are obtained. Here one sees all A15 data fall along one curve and that hopes of obtaining T > 22 oK seem rather remote. But bear in mind that history tells us,whenever acorrelation like this appears, the exception to the rule is not long in corning.

Although the high T c' s of several of the A15 compounds are responsible for our interest in these compounds, theory seems to have been motivated by their "unusual" normal state properties. A listing of these properties,such as the large y values, temperature dependent Knight Shift and magnetic susceptibi1ity, low


Nb3X.R' V3Z.R' Nb3V.R'

~ ~ ~ 20 IAI.GY

I : "7 \ "V GO\

I ~ t Ga .. ::J I' ii ~ °Au D.

~ 10 PI

0- il" II £ u ~ ";: ... .

jPI .,~ ~ +Au +

_~I:-:,-'_++ I I 6.1.. __ 0------:-=1 ,-----_ 0.98 1.00 1.02 1.04 1.06 1.08

Fig. 24. Transition temperature versus the quotient R (see

r u

text) for various niobium and vanadium A3B compounds of the A15 type structure. [Ref. 59 ]

• I

•

O~--S~---4~~-~J--~~--~1--~O--~1~~~~~ ( R~R )100/R·

Fig. 25. Transition temperature of various niobium and vanadium based A3B compounds of the A15 type structure as a function of a normalized quotient, see text. [Ref. 59 ] Dots are Nb3 X with X equal to Al (Al,Ge) Ge, (Al,Si), Ga-, Au, pt, Os, Rh and Ir. The triangles are Nb Y with Y equal to In, Sn, Sb, Pb and Ri. The cr~sses are V~Z with Z equal to Si, (Si, AI), Ge, (Ge.Al), Ga, (Ga, AI) Sb, (Sb, AI), Au, Pt, Sn, Rh, Co, As. Pd, Ir.

364 ROBERT A. HEIN

temperature structure transformation, etc. , are to be found in several publications. 61

Early theory (1960-65) was concerned mainly with supplying a theoretical basis for the large N(O) and the increase in N(O) with decreasing temperature required to fit Knight Shift and susceptibility data. Clogston and Jaccarin06la emphasized that these data required a high, narrow (0.04 ev) peak in the d-band density of state s of these A15 compounds. Furthermore, the Fermi energy must be located within this peak. A width of 0.04 ev is a much finer structure than that which is normally talked about. In general, transition metals have d-band widths of the order of several ev, and detailed structure is only known to .... 0.1 ev. Thus, it was felt6la that if such fine structure exists, it must be associated with the peculiar structure of the A15' s.

Weger6lc ,i agreed with this general concept and said the fine structure is due to the essential one dimensionality of the A -chains in the A15 structure. That is, the distance between A atoms along a given chain, Fig. 11, is considerably smaller than that between A atoms belonging to different chains. (Interchain spacing is a o /2, intrachain spacing is 1. 23 (a o /2).

Weger used a tight binding approximation and considered only nearest neighbor interactions to show that E(k) along a given chain was a function of only the component of k along that chain. The total d-band of the crystal was simply a superposition of three, one -dimensional bands. This is the birth of the socalled linear chain model.

For a one-dimensional system, one has E(k) = Em cos kd where d is the interatomic spacing, k is the wave vector and Em is the band width (overlap integral). From this it follows

dE/dk .... Em sin d = Em 'Jl-cos 2 kd. Since N(E) .... * -;;;;; ,--,---,.... -1 .JEr:: - E(k)2 one has a square root singularity when E(k) = Em.

Taking into account the different orbital functions results in the Labbe -Friedel density of states curve shown in Fig. 26.

In this figure,the zero-energy is chosen at the center of the band and the relative shift of the d- sub-band centers is neglected. Using this scheme and locating EF near the band edge has allowed the Orsay Group to account for most of the properties of the A15 compounds. The height of the peaks yield a large N(O) and the narrowness of the peaks produces a large change in N(O) with

changes in temperature. Labbe and van Reuth62 have recently employed this density of states curve to account for the increase in 'Y which accompanies an increase in LRO.

This model was also used to account6le for the structural

C><) 00 ~ ~ [ooTI

3 j!2_ r2 x~ yi! X2._y2

n(E)t d3z2_r2 dxy

dX2_ y2

Fig. 26. Labbe and Friedel d-band density of states curve for A15 compounds.

k TO -1 f- d band

...

E

Fig. 27. Cody et al. proposed d-band density of states for the A15 compounds.

366 ROBERT A. HEIN

transformation which occurs in some A15' s at temperatures beween 20K and 43K. In fact, this transformation from the cubic to the tetragonal phase seems to have motivated much of the theoretical work which has been done since 1965. The RCA6l g ,h group has found that a simple model of the electronic density of states is capable of accounting for the unusual properties of the high T c superconductors.

This assumed density of states curve is shown in Fig. 22. The d band consists of the sum of three independent sub -bands associated with the A chains of the A15 structure. They adjust the parameters of their model by fitting the experimental data obtained for the cubic A15 structure. Having done this, they then go on and predict the properties of the tetragonal phase. An interesting result is that A ~ 0.7 is the critical value for the transformation to occur (i. e. , for A < 0.7 the A15 phase is stable).

The reason for all this interest in the transformation is that it is felt6l h that an unde r standing of the mechani sm which produces the crystallographic distortion will help us understand the mystery of high T c superconductors. The a,bove limitation on 'Y suggest we will not obtain much higher Tc's in the A15 compounds than those already reported, a suggestion in keeping with Muller's empirical deductions.

Recently a group at Bell Laborator ie s14, 15 has proposed that,in the A15's,one can account for the different Tc's as being due to the dependence of T c upon the lattice parameter a o . This dependence is shown in Fig. 28. These workers also concluded that the cubic to tetragonal distortion only reduces the T c of V 3 Si by about O. 3K (T c of V 3Si = l7.lK), the reduction in T c being proportional to the square of the deformation 0 = (c/a-l)~ 2.5 x 10-3 .

T. F. Smith14 immediately set out to check on the predicted dependence of T c upon ao. He used high pressures to change a o and measured T c' He interprets his results (solid lines in Fig. 28) as evidence against Testardi et aL's prediction B. T. Matthias et al. 63 set out to check on the t.T c between the cubic and tetragonal phases and concludes that the difference is orders of magnitude larger than predicted by Testardi et al. 15

Thus the situation among the A15's is, I believe, best summed up by quoting the first and last stanza of a poem, written by J. G. Saxe who lived in England in the 19th Century. It concernS six blind men and an elephant.

It was six men of Indostan To learning much inclined


Who went to see the elephant though all of thezn we re blind That each by observation znight satisfy his znind.

367

As you znay recall, each znan exaznined only one aspect of the elephant ' s anatozny and becazne convinced about the physical appearance of the entire aniznal. The author then goes on to state:

And so tho se znen of Indostan Disputed loud and long Each in his own opinion Exceeding stiff and strong Though each were partly in the right And all were in the wrong.

We experiznentalists are siznilar to the blind znen in that we obtain data only about certain aspects of the Al5 1 s and all the various theories explain sozne of these isolated observations. For exaznple, critical znagnetic field data and,znore convincingly, specific heat data, show that N(O) increases with increase in S. Labbe and Van Reuth ' s treatznent explain this observation as being due to an increase in N(O) which results frozn an increase in the integrity of the A-chains. However, this treatznent says that all Al5 1 s should show this ordering effect and we know Cr based Al5 1 s do not. Well, this zneans we have to argue about the location of the Ferzni level with respect to the peaks in the d-band, etc. Matthias argues about the iznportance of the e/a values. Data exist which shows this is not the case. Testardi et al. argue for the iznportance of a o ' data exist which disputes this. To explain the high T c l sand znagnetic susceptibility data,znodels are postulated which place EF at the edge of a d-band of high N(O) but sozne fairly high T c Al5 coznpounds do not show any teznperature dependent susceptibility, etc. , etc. , and etc.

I cannot help but feel that we old tizne rs are too set in our ways, and that it will take a young znan with a fresh outlook to really write the final chapter of the Al5 saga. So go to it!

368

18 -

16-

~ 14-:.c:

..2....

f-u 12-

10 -

8-

6-

4-

• V3Ge'_XSi X • V3Si,_XV

• V3Si'_XAI

... V3 Si '-XGo X

~.V3Si

."" \. ! ~\ : tf. : -~ . . . . : .. ! . ~ : ~

t\

II. V3Ge'_X AI X o V3Go,-XVX

o V3Go'-XAI X

D V3 Go '-X GeX

I I I , I I I I

4.70 4.72 4.74 4.76 4.78 4.80 4.82 4.84 ROOM TEMPERATURE LATTICE PARAMETER [A]

ROBERT A. HEIN

Fig. 28. Transition temperature of various A3B compounds of the A15 type structure versus the lattice constant. Solid line s depict Smith's results. [Ref. T. F. Smith, S. S. Comm. :t,., 903 (1971) ]

1.

2.

3. 4. 5.

6. 7.

8. 9.

10. 11.

12. 13. 14. 15.

16. 17. 18.

19.

20. 21. 22.

REFERENCES

G.F. HardyandJ.K. Hu1m, Phys. Rev. (L)~, 884 (1953); and Phys. Rev. ,22., 1004 (1954).

B. W. Roberts, NBS Technical Note No. 482 May 1969, u. S. Gov't. Printing Office, Wash. D. C. 20402.

H. Frtlhlich, Phys. Letters 35A, 325 (197l). B. T. Matthias, Phys. Rev. Letters~, 781 (1967). E.F. Burton, H. Grayson-Smith, J.O. Wilhelm, "Phe

nomena at the Temperature of Liquid Helium", Reinhold Publishing Corp. , New York, 1940, page 93.

B. B. Goodman, Nature 167, 111 (1951). R.A. Hein, J. W. Gibson, B.T. Matthias, T.H. Geballe and

E. Corenzwit, Phys. Rev. Letters~, 408 (1962). B. T. Matthias, Phys. Rev. g, 874 (1953). J. G. Daunt and J. W. Cobble, Phys. Rev. (L) 92, 507 (1953). B. T. Matthias, Phys. Rev. :r!..., 74 (1955). A. S. Cooper, E. Corenzwit, L. D. Longinotti, B. T. Matthias,

and W. Zachariasen, Proc. N. A. S. ~, 313 (1970). K. C1ausis, f. E1ectrochem. ~, 312 (1932). R. L. Fa1ge, Jr. , Phys. Letters 24A, 579 (1967) L. R. Testardi, Phys. Rev. B3, 95 (197l). L.R. Testardi, J.E. Kunzler, H.J. Levinstein, J.P. Maita,

and J. H. Wernick, Phys. Rev. B3, 107 (197l). T. F. Smith, J. Low Temperature Phys. ,§" 17l (1972). J. deLaunay and R. D. Do1ecek, Phys. Rev. 7l:.., 141 (1947). R.D. Fowler, B.T. Matthias, L.B. Asprey, H.H. Hill,

J. D. G. Lindsay, and R. W. White, Phys. Rev. Letters..!2., 860 (1965).

McMillan's formula T = aD -(1 + A) c 1. 45

exp A- - /1+ (1+ O. 62 i\.)

in the case where A- > /1+ becomes T c ~ e D exp -1 + AX

If one now replaces aD by < w > and uses the McMillan identity A- = const/M < w2 > where M is the atomic mass and < w2 > is some average phonon energy squared one has Tc :;;; <w > exp -M < w 2 > . Thus any mechan-ism which decreases < w2 > should increase Tc'

J. G. Daunt and T. S. Smith, Phys. Rev. 88, 309 (1952). 1. M. Chapnik, Soviet Physics, Dok1ady,§,:-988 (1962). B. T. Matthias, T. H. Geballe, S. Geller, and E. Corenzwit,

Phys. Rev. 95, 1435 (1954).

370 ROBERT A. HEIN

23. S. Geller, B. T. Matthias, and R. Goldstein, J. Amer. Chern. Soc. 72, 1502 (1955).

24. B.T. Matthias, J. Phys. & Chern. Solids~, 188 (1953); Prog. in Low Temperature Physics, C. J. Gortor, Ed. Vol. II (1957).

25. P. Greenfeld and P. A. Beck, Trans. AIME (J. of Metals) 265 (1956).

26. E. Raub and W. Mahler, Z. Metta1 46, 210 (1955). 27. E.A. Wood, V.B. Compton, B.T. Matthias,andE.

Corenzwit, Acta Cryst . .!!.J 604 (1958) 28. J.H. Wernick, F.J. Monin, F.S.L. Huo, D. Darsi, J.P.

Maita and J. E. Kunzler, Proc. of the Internat. Con£. High Mag. Fields, H. Kohn et al. , Ed. , J. Wiley and Sons, p. 609 (1962).

29. W. E. Blumberg, J. Eisinger, V. Jaccarino, and B. T. Matthias, Phys. Rev. Letters ~, 149 (1960).

3 O. J. H. N. van Vucht, H. A. C. M. Bruning, H. C. Conkers1oot and A. H. Gomes de Mesquita, Phillips Research Reports ~, 407 (1967)

31. E. C. van Reuth and R. M. Waterstratt, Acta Cryst. B24, 186 (1968).

32. J. J. Hanak, G. D. Cody, P. R. Aron, and H. C. Hitchcock, High Magnetic Fields, MIT, Wiley 1961, p. 592.

33. J. J. Hanak, G. D. Cody, J. L. Cooper, and M. Ray1, Proc. of the VIII Internat. Con£. Low Temp. Phys. , R. O. Davis, Ed. Butterwirth, 1963; (b) J.J. Hanak, Metallurgy of Advanced Electronic Materials, G. E. Brock, Ed. , Interscience Publ. 1963, p. 16l.

34. H. G. Jansen and E. J. Saur, Proc. VIIth Internat. Con£. on Low Temperature Physics, Univ. of Tronto Press 1960, p. 184.

35. T.B. Reed, H.G. Gatos, W.J. LaFleur, and T.J. Roddy, "Metallurgy of Advanced Electronic Materials", G. E Brock, Ed. , Interscience Publishers 1963, p. 71.

36. H. G. Jansen, Z. Phys. 162, 275 (1961). 37. L.J. Vie1and, RCA Review 25, Sept. 1964. 38. T.H. Courtney, G.W. Pearsall, andJ. Wulff, Trans.AIME

233,212 (1965); J. Appl. Phys. 36,3256 (1965). 39. J. F. Bachner and H. C. Gatos, Trans. AIME 236,1261 (1966). 40. R. M. Waterstrat - private communication. 41. B.T. Matthias, T.H. Geballe, L.D. Longinotti, E. Corenzwit,

G.W. Hull, R.H. Willens, and J.P. Maita, Science, 156, 645 (lg67).


42. R. A. Hein, J. E. Cox, R. D. B1augher, R. M. Waterstrat, Con£. on the Science and Technology of Superconductors, Aug. 1969. To be publi shed in Physica.

43. E. C. van Reuth, R. M. Waterstrat, R. D. B1augher, J. E. Cox, and R. A. Hein, Proc. of the Xth Internat. Con£. on Low Temperature Physics Moscow (1966).

44. F. Heiniger, R. F1ukiger, A. Junod, J. Muller, P. Spitzli, and J. L. Standenmann, Proc. of the Twelfth Internat. Conf. on Low Temperature Physics, E. Kanda, Ed. , Academic Press Japan, p. 33 (1971).

45. J. Muller, Proc. Summer School for Superconductivity, Oct. 12-16, 1970, Pegnitz/Oberfranken, see also Ref. 44.

46. T. G. Berlincourt, Superconductivity in Science and Technology, M.H. Cohen, Ed. Univ. Chicago Press, p.31 (1968).

47. R.A. Hein, J. E. Cox, R. D. B1augher, and R. M. Waterstrat, Solid State Comm. 7..., 381 (1969).

49. B.T. Matthias, T.H. Geballe, R.H. Willens, E. Corenzwit, and G. W. Hull, Jr. , Phys. Rev. 139, A1501 (1965).

50. A. Muller, Z. Naturforsch. 24a, 11346 (1969). 51. N. Pessa1 and J. K. Hu1m, Physics ~, 311 (1966). 52. T. Asada, T. Horiuchi, and M. Uchida, J. App1. Phys.

(Japan) !!.' 958 (1968). 53. H. L. Luo, E. Vie1haber, and E. Corenzwit, Z. Physik 230,

443 (1970). 54. L. D. Hartsough and R. H. Hammond, Stanford APS Meeting

(1971) and private communication. 55. R. Hagner and E. Saur, Proc. Eighth Internat. Conf. on

Low Temperature Physics, R.O. Davies, Ed. Butterworths, p. 358 (1963).

56. N. E. Alekseevskii,N. V. Ageev, and V. J. Shamrai, Investiya Akademii Nauk SSSR Neorganicheskie Materia1y ~, 2156 (1966) English Trans.

57. R. F1ukiger, P. Spitzli, F. Heiniger, and J. Muller, Phys. Letters A29, 407 (1961).

58. G. Arrhenius, E. Corenzwit, R. Fitzgerald, G. W. Hull, H. L. Luo, B. T. Matthias, and W. H. Zachariasen, Proc. N. A. S. ~, 621 (1968).

59. A. Muller, Z. Naturforsch 25A, 1659 (1970); Paper presented at Tief Temperaturen, Mar. 22-26, 1971.

60. R. M. Waterstrat and E. C. van Reuth, NBS Report 10,061, Oct. 20, 1969.

61. (a) A. M. Clogston and V. Jaccarino, Phys. Rev. 121, 1357 (1961)

372 ROBERT A. HEIN

61. (b) A. M. Clogston, Phys. Rev. 136, A8 (1961). (c) M. Weger, Rev. Mod. Phys. 36, 175 (1964). (d) L. F. Mattheiss, Phys. Rev. 138, A1l2 (1965). (e) J. Labbe, Phys. Rev. 158, 647 (1967); 158, 655 (1967);

172 , 451 (1968). (f) A. P. Levanyuk and R. A. Suris, Soviet Phys. USPEKHI

~, 40 (1967). (g) R.W. Cohen, C.D. Cody, andJ.J. Halloran, Phys.

Rev. Letters!,2., 840 (1967). (h) C. D. Cody and L. J. Vie1and, Electronic Density of

Solids, Nov. 3-6,1937 - NBS, Washington, D. C. (i) M. Weger, J. Phys. Chern. Solids~, 1671 (1970).

62. J. Labbe and E. C. van Reuth. Phys. Rev. Letters 24, 1232 (1970).

63. B. T. Matthias, E. Corenzwit, A. S. Cooper and L. D. Longinotti, J. Proc. N.A.S. 68,56 (1971).

THEORY OF SUPERCONDUCTING SEMICONDUCTORS *

C. S. Koonce

National Bureau of Standards


INTRODUCTION

Superconducting semiconductors are different in many ways from metals, alloys and compounds in which superconductivity is usually found. One of the outstanding differences is the way in which they are constructed. One usually begins with a semiconductor, that is, a material having a concentration of electrons in the conduction band proportional to e-EG/kT where EG, the semiconducting gap,is typically of the order of tenths of electron volts (thousands of Kelvin). Since the semiconducting energy gap is much larger than the superconducting energy gap, pure semiconductors are not superconducting. In order for superconductivity to occur, we must have unpaired electrons in the normal state, that is, a partly filled energy band. Doping is used to achieve this condition. Doping occurs when an element in the semiconductor is replaced by a small percentage of an element in another column of the periodic table, or when a partially ionic semiconductor is prepared with a deficiency of one of the elements in the compound. If the concentration of impurities, which are called donors or acceptors depending on whether they donate or accept electrons, is low, these extra electrons or holes will remain bound to the impurities. Fig. 1 shows an idealized semiconductor band structure with one donor and one acceptor level. In fact, there are a whole series of such levels. If the concentration of impurities is large, however, the overlap energy of the electrons will broaden the donor or acceptor states and a band of the electronic energy states will be formed. If the binding energy of the donors or acceptors is small, this band will be very much like the conduction band in the undoped material. In such a highly doped semiconductor the electrons remain in the conduction band as the temperature is reduced even to zero Kelvin and

373

374 c. S. KOONCE

the conductivity approaches a constant value (in the non-superconducting state) at low temperatures. Such a material has a Fermi energy inside the conduction band at zero Kelvin, and is called degenerate. Figure 2 shows the pockets of electrons in highly doped germanium. We see that in Ge the electron pockets lie at the edges of the Brillouin zone at positions related by the cubic symmetry of the crystal.

k Fig. 1. The energy band structure of a direct gap semiconductor showing one donor and one acceptor level. [From F. J. Blatt, Solid State Phys. ~, 209 (1953)J.

Probably the most impressive difference between degenerate semiconductors and metals is the electron (or hole) concentration. Degenerate semiconductors can be prepared having electron concentrations ranging from metallic densities of 1022 electrons/cm3 down to the order of 1017 electrons/cm3• Strontium titanate, a super conducting semiconductor, has been found to be superconductin~ with electron concentrations over the entire range between 9 x 101

and 3 x 1020 electrons/cm3• (1-3) When about 7 % of Barium is

THEORY OF SUPERCONDUCTING SEMICONDUCTORS 375

added, the superconducting range is extended and superconductivity has been observed at about 5 x 1017 electrons/cm3. (4) Germanium telluride, the first superconducting semiconductor (5) has been found to be superconducting between 8 x 1020 and 9 x 1021 holes/cm3

and tin telluride is superconducting between 5 x 1020 and 9 x 1021 holes/cm3 ,(6) lnTe and La3Se4 are also superconducting at high carrier concentrations. (7,8) More recently, TlBiTe2 has been found to be superconducting at a carrier density of about 6 x 1020 holes/cm3. (9)

Ge Fig. 2. Constant energy spheriods in the conduction band of germanium. The Fermi energy of highly doped germanium could lie on such spheriods. [From F. J. Blatt, Physics of Electronic Conduction in Solids, McGraw-Hill, N.Y. (1968)J.

376 C. S. KOONCE

~INTRAVALLEY

~ k'

-k'

I -k

-k ,

k- K= q - - -

Fig. 3. An electron pair scattering via intervalley and intravalley electron-electron interactions. [From J. F. Schooley, unpublished.]

IMPORTANT NORMAL STATE PROPERTIES

For the lower concentrations especially, only a small part of the conduction band is filled and the electrons occupy only small portions of the Brillouin zone. The precise nature of the band structure is therefore important and the electron-electron interactions can be classified as interactions which scatter electrons within a given pocket of electrons in momentum space and interactions which scatter electrons between different pockets of electrons. The scattering of an electron pair is shown in Fig. 3. The electronelectron interactions shown represent not only the direct electronelectron Coulomb repulsion, but also the interaction through phonons, which is attractive below the phonon frequency. The first suggestion that degenerate semiconductors might be superconducting under certain condictions,such as high electron concentration and high static dielectric functions,was made by Gurevich, Larkin, and Firsov. (10) The importance of intervalley interactions was pointed out by Marvin Cohen (11,12),who listed various properties of degenerate semiconductors,such as many-valley band structure, large effective mass

and large static dielectric function,which could lead to a transition temperature high enough to be measureable. Several of the semiconductors which he proposed as likely candidates for superconductivity, GeTe, SrTi03, and SnTe,have been found to be superconducting and all are thought to have a many-valley band structure. (13,14,15,16)

Intervalley interactions are important for superconductivity for several reasons. First of all are the phase space considerations. For a given number of electrons, the density of states at the Fermi energy is maximized by dividing the electrons into many small pockets. Also, the direct Coulomb repulsion which inhibits superconductivity is reduced for intervalley interactions because intervalley interactions involve large momentum transfer, of the order of magnitude of a recriprocal lattice vector. Intervalley interactions usually take place in a relatively narrow range in momentum about some large momentum which connects symmetry points in the Brillouin zone. Hence, selection rules playa role in selecting which phonons contribute to intervalley scattering.

- 2 2 2 (wp+W/-Wtl

/ /

/

/ /

/-2 / wp

/ 2 - - - - - - - -/- - - - - w / R

/ /

/ /

/ /

/ w~ -7---- ---- -- --/ w~ :><w~

/

n

~=4 COUPLED PHONON-PLASMON MODES

Fig. 4. Coupled longitudinal optic phonon-plasmon modes as a function of electron concentration. Here, the plasma frequency wp 2 = 4'JTne 2 /m* and W 2 = Wp 2 /1{ co' Wt is the "bare" longitudinal optic phonon frequengy and wt is the transverse optic phonon frequency, I{ 0 is the static dielectric function, and K is the high frequency dielectric function. 00

378 C. S. KOONCE

Another interaction which plays a larger role in superconducting semiconductors than in metallic superconductors is the interaction through polar phonon modes. This is because of the much lower electron concentration and hence much lower plasma frequency in superconducting semiconductors. The difference between the coupling to polar phonons at low and high electron concentration can be seen by considering the situation at small wave vector. Here, the plasmons are long-lived longitudinal excitations and they couple to the polar longitudinal modes of the crystal through a macroscopic electric field. The frequencies of these coupled longitudinal waves are shown in Fig. 4 plotted as a function of electron concentration. The splitting between the longitudinal and transverse phonon modes at low concentration gives a measure of the electron-polar phonon coupling. We see that at very low carrier concentrations the modes are essentially the bare longitudinal optic (L.a.) phonon frequency and the plasma mode screened by the static dielectric function, I( o' For very high electron concentrations,such as those found in metals,the modes are essentially the plasma frequency screened by the high frequency dielectric function and a triply degenerate phonon mode. At high electronic concentrations the interaction through the macroscopic electric field is relatively small and the transverse and longitudinal modes have almost the same frequency.

While the situation at zero wave vector is two cqupled modes, at large wave vector the plasma mode is not a long-lived excitation and a more appropriate description is a lowering of the phonon frequency arising from virtual electron-hole pair screening. For intermediate wave vector the electron-hole pairs can conserve momentum and energy; that is, the possibility of creating real electron-hole pairs exists. These regions are shown in Fig. 5 which is a plot of F(q,w) = -(l/~)Im(1( II( (q,w» in the wave ~ectorfrequency plan~where I(T(q,w) is the t~taI dielectric function of the system. Since the zeros of the dielectric function correspond to longitudinal modes of the coupled electron-phonon system, the peaks in F(q,w) are the longitudinal excitations of the system. The lower of the two coupled modes at q = 0 is barely visible in the figure. The frequencies of the coupled modes in the randomphase approximation are given by the solution to the transendental equation (17)

where K (q,w) is the electronic dielectric function. e

(1)

w

Fig. 5. Spectral weight function F(q,w) for a degenerate polar material having one optically-active phonon mode. Parameters used were K' = 20, K' 00 = 5, mb_= 2.5 me' \! = 3, w).. = 0.1 eV, electron concen~ration n = 1020 cm 3, transverse optic phonon frequency wt = 0.05 eV, damp,ing ).. = 0.03 eV. The plasma frequency ~ = (4TIne2/m*KooYz = 0.105 eV. Only intravalley interactions are

shgwn. Note that using these parameters for small wave vector q, the high-frequency branch of the coupled phonon plasmon modes is important,while for large q the phonon mode screened by singleparticle excitations is important. All modes are damped in the. region (q2 - 2qkF) < 2m9w < (q2 + 2qkF). The maximum of q shown is 5.127 kF = 0.996 x 10 cm- I • The maximum value of w shown is 0.4149 eV. The viewing angles are 60 0 from the perpendicular to the plane and 255 0 from the q axis.

SUPERCONDUCTING ENERGY GAP EQUATIONS

Since we have seen that some of the differences between a superconducting semiconductor and a metal consist of a Fermi energy and a plasma frequency much lower in the semiconductor than they are in a metal, we may begin to question the application of the theory of superconductivity as it has developed for metals to

380 c. S. KOONCE

superconducting semiconductor5. The current "state of the art" for calculating superconducting transition temperatures and the energy gap as a function of temperature are the renormalized Eliashberg equations. (18) These equations are believed to be more accurate than previous equations because they properly account for the retardation of the interaction through the phonons and the renormalization of the electron mass through self energy effects. The price paid for inclusion of retardation effects is a neglect, Gr approximate inclusion, of variations in the density of electronic states and electron-phonon coupling over energy ranges of the order of phonon energies. For metalS,and even for very highly doped superconducting semiconductors, this is a good approximation,because in these materials the phonon energies are much less than the electron band width and much less than the Fermi energy.

Even in metals, however, difficulty is encountered in calculating the contribution from the direct electron-electron interaction since the range of this interaction is of the order of the plasma energy, which is almost never small compared to the Fermi energy. The use of an approximate method in this case - the Coulomb pseudopotential - can be justified because the direct Coulomb interaction is often smaller than the interactions through phonons.

In metals, the separation of the total interaction into two parts, one part an interaction via phonons (19), and another part, the direct Coulomb interaction, is quite natural, especially for the simple metals, because phonon energies are much less than plasma energiep. The phonon interaction can then be treated to an accuracy (VS/VF)~ where Vs is the velocity of sound and vF is the Fermi velocity. (19) The Coulomb interaction is much less retarded and must be treated more approximately. If, on the other hand,the phonon energies are of the order of the plasma energy, the Fermi energy, or an electronic energy band width, the approximations often made in separating out a "phonon" part become more serious. For example, when such a separation is made, one is left with an electron-phonon coupling which is screened by the electronic dielectric function whose frequency dependence cannot be neglected. In such cases it is often easier to consider the total interaction than to attempt to separate it into "phonon" and "Coulomb" contributions.

Let us consider the zero temperature gap equation for a superconductor, and see what modifications will be necessary, if any, to calculate the energy gap of a superconducting semiconductor. (17) The coupled equations for the energy gan~, the renormalization,Z, and the exchange contribution to the Hartree-Fock energy,x,are given by


L;(P) --x (2)

where cp(£.I) '" /::.(2') Z(J?,') and V is the total interaction between electrons, and Tl and T3 are two dimensional Pauli matrices. L;(P)

• ".. "'" #IIItttI#",;;;J is gJ.ven by

so that Eq. (2) and Eq. (3) form three coupled equations for /::., Z and x; one equation for each of the three independent matrices 1, T1 , and T3. IV _

The integral over~1 may be transformed into an integral over aximu thaI angle e, over wave nurnbe r q '" I.E-.E,I 1/ fl. and over e(,12.) '" p2 /2m-l~. For a spherical Fermi surface the integral over e can be accomplished to give

dp' qdq, (4) -where fI~ is the electron conduction band width, ql '" (l/fI) IIEI -lEI 11, q2 '" (l/fI) lip I + 1£III.and tllt is the "band mass", that is, the result of a band structure calculation that considered an electron moving in the potentials of the fixed ions screened by the electrons in the valence band only. While the gap equation in this form is quite elegant, it is quite difficult to solve for a real material, involving coupled four dimensional integral equations. These integral equations are usually reduced to one dimensional integral equations using the procedure of Eliashberg. First, the total interaction. V(]2.-E; ). is separated into a "phonon" part and a "Coulomb" part. The "Coulomb" part is treated approrimately and the "phonon" part is then treated to a higher degree of accuracy. Since the phonon frequency is taken to be much less than the Fermi energy, band width. and plasma energy, the density of states is replaced by its value at the Fermi energy and the screening of the electron-phonon interaction by its static value. Also p is

382 c. S. KOONCE

replaced by its value at the Fermi surface, PF, q1 ~ 0, q2 ~ 2kF, €F ~ 00 and h llb - €F ~ 00.

We also reason that while the functions Z, 6, and X really depend on momentum as well as energy, p , the momentum dependence will be small compared to the energy de~endence if all phonon processes take electrons from one point near the Fermi surface to another point near the Fermi surface. We then take Z, 6, and X to be functions of Po only, with Ipi evaluated at the Fermi momentum, PF' The integration procedure of Eliashberg (IS) can then be used to evaluate the integral over € and we are left with the equation

~h(p) 00 t P 'I + ~'T ~ -N(O) S dp , o - _1 ph I

= Re Ok K+ (po' po) , ... 0 [(ZIp ')2 _ cp,2]2 -0 0

(5)

where 2kF 00

dw B, (q,w) {tgq~~ , KPh -I: ~ ~ J ~ Ie 2k 2

0 F 0

x [ -1 ± 1 p -p'-w+ i T1 p +p'+w- i T1

o 0 0 0

( 6)

where the sum is over phonon modes, Ie, and the phonon propagator is written in terms of its spectral weight function,BIe(q,w). The plus sign is to be used with the T1 component, and the minus sign with the 1 component. The T3 component gives X which, for low phonon ener~es, can be taken constant and relatively small. The functions K~n(po'po') can then be evaluated and the one dimensional set of coupled equations solved.

For a superconducting semiconductor, the limits q1 - 0, q2 ~ 2kF, E:F - 00, and h~ - €F ~ 00 are not necessarily valid and we can take advantage of the fact that the electrons are located in small pockets in the Brillouin zone to evaluate the integral over wave-vector difference.q. To be more specific, let us consider an intervalley interaction having wave-vector transfer qI' If there are \) valleys there will be \)-1 possible intervalley interactions, and the spectral weight function,~(q,w),will be almost independent of q in a small range of q near~. In addition, the screenedphonon coupling will be almost independent of q in a small range about qI' We can then evaluate the integral over q even though q1 and q2 are not approximated. We will use the approximation of a


and we have used the convention that a positive square root has a positive imaginary part. The approximate position of these square roots in the complex plane is shown in Fig. (6). We see that the real parts of the square roots afd, and if we expand the square roots for small [(Zl pO I)2 - WI2]~/(€~ + X), we can see that the correction terms are of order [(Zl pO )2 - ~12]/(€F + X). If (1' + Xl> hiXlA' the important p I IS in our integral are of the order of pnonon ener~ies,so that corre8tlon terms are roughly of the order of (h w/ E:p) •

Fig. 6. Qualitative ~v~ment of and (~+X+[(ZpO)2_W2]~)~ in the direct~on of increasing p •

o

1dz the functions (€F+x - [( ZPo) 2 -w2] 2) complex plane. Arrows indicate

384 c.s. KOONCE

.3

.1

.r

.01 .1 10 100

Fig. 7. Superconducting transition temperature,Tc,as a function of fI~/ €F for SnTe, GeTe and SrTi03 • The phonon frequency used in each case was that of the highest phonon to which the electrons are strongly coupled.

The transition temperature is plotted as a function of the ratio fI~/€F for the strongest interacting phonon for several superconducting semiconductors in Fig. 7. We see that for GeTe and SnTe these effects are probably of little importance and the Eliashberg equations used for metals should be appropriate. In fact these equations and the McMillan phenomenological equations derived from them have had good success in calculating the superconducting properties of these materials. (20) We have omitted TlBiTe2 from the Fig. (7) because its normal state properties are not well known. We note that other materials which are not strictly semiconductors could be placed on this figure. For example, CuRh~S4 and CuRh2Se4 have ratios of enl€F of about 0.17 and superconducting transition temperatures of 4.37 K and 3.48 K respectively. (21)

For SrTi03 , however, it is clear that these modifications will be important. Unfortuna tely, they are so important that some of our other approximations, such as the evaluation of the functions ~, Z,and X at the Fermi momentum,Pf,and the neglect of the frequency

THEORY OF SUPERCONDUCTING SEMICONDUCTORS 3BS

dependence of the dielectric screening by electron~ are doubtful. It is probably a better approximation to evaluate the functions zC::',Po'), 6(e:',po'), and x(e',po') at the value of Po' which is large in the integral £f Eq. (2), that is at p , = ± E' = [ « e:' + X') Iz' ) 2 + 62 ]'2. The gap equation obt~ined in this way is

where

qdq ~

x V(q p - p ,) , 0 0'

( 8)

where V(q,p -po') is the total electron - electron interaction, that is, the bar~ electron-electron interact~n screene~by the tota~ dielectric function~ and A(e,\e') = (2n:ttYl)(e + q')'2 - (e' + eF)'21h, and B(e,e') = (2~)21(e + ep-)2+ (e' + eF JI 11. If we define

K F( q, w) -(lin) Im[ t )]

KT q,w , (9)

we can write Eq. (8) as

~n r( e, e') B KCI = dq qV (q) x

(2n ) 311 [2~ (e + e ) J!z c F A(e,e')

- 1 ~ w + i 11 Po + E( e') + w - i'l1 •

10)

The similarity between this equation for K~ and Eq. (6) is instructive. We see that we have the bare Coulomb term plus an integral

386 c.s. KOONCE

over a spectral weight function times a resonant denominator of the Eliashberg form. The spectral weight function is the imaginary part of the recriprocal total dielectric function. For intravalley coupling to one polar mode, this function is ~ust that given in Fig. S. We can see that for small wave vectors the coupling is to the coupled plasmon - L.O. phonon modes, while for large wave vectors the coupling is to a screened phonon mode. The parameters used in Fig. S are similar to those measured in SrTi03 • From this figure and Eq. (10) I think it is clear that the frequency dependence of the electronic screening in SrTi03 is important in calculating its superconducting properties.

The study of superconductivity in semiconductors is, I think, important because band structure effects, which are often mentioned in regard to metallic superconductors, especially transition metals, playa critical part in determining the superconducting properties. For example, selection rules on the symmetry of electronic states which can be scattered by a phonon of given symmetry are important in calculations involving superconducting semiconductors. (3) Interactions through polar phonon modes are present to a much larger degree in superconducting semiconductors than in metals as previously discussed, (18,22) and in a sense represent a t1differenttl interaction to study. And, probably most importantly of all, these interactions can be studied as a function of the number of electrons in the conduction band from very low densities up to metallic densities.

REFERENCES

~e Contribution of the National Bureau of Standards, not subject to copyright.

1. J. F. Schooley, W. R. Hosler, and M. L. Cohen, Phys. Rev. Letters ~ 474 (1964).

2. J. F. Schooley, W. R. Hosler, E. Ambler, J. G. Becker, M. L. Cohen, and C. S. Koonce, Phys. Rev. Letters 14, 30S (196S).

3. C. S. Koonce, Marvin L. Cohen, J. F. Schooley, W. R. Hosler, and E. R. Pfeiffer, Phys. Rev. 163, 380 (1967).

4. J. F. Schooley, H. P. R. Frederikse, W. R. Hosler , and E. R. Pfeiffer, Phys. Rev. lS9, 301 (1967)

S. R. A. Rein, J. W. Gibson, R. Mazelsky, R. C. Miller,and J. K. Hulm, Phys. Rev. Letters ~ 320 (1964).

6. R. A. Hein, J. W. Gibson, R. S. Allgaier, B. B. Houston, Jr., R. Mazelsky,and R. C. Miller in Proceedings of the Ninth International Conference on Low Temperatures Physics, edited by J. G. Daunt, D. V. Edwards, F. J. Milford,and M. Yaqub (Plenum Press, Inc., New York, 1965),p 604.


7. S. Geller, A. JayaraITlan, and G. W. Hull, App1. Phys. Letters.!t. 35 (1964).

8. F. Ho1tzberg, p. E. Seiden, and S. von Molnar, Phys. Rev. 168, 408 (1968).

9. R. A. Hein and E. M. SWiggard, Phys. Rev. Letters 24, 53 (1970).

10. V. L. Gurevich, A. 1. Larkin, and Yu. A. Firsov, Fiz. Tverd. Tela 4, 185 (1962) [ English trans1: Soviet Phys. - Solid State ±, 131 (1962) J.

ll. Marvin L. Cohen, Phys. Rev. 134, A5ll (1964).

12. Marvin L. Cohen in Superconductivity, edited by R. D. Parks (Marcel Dekker, Inc., New York, 1969), p. 615.

13. A. H. Kahn and A. J. Leyendecker, Phys. Rev. 135, A1321 (1964).

14. H. p. R. Frederikse, W. R. Hosler, and W.R. Thurber, Phys. Rev. 143, 648 (1966).

15. H. P. R. Frederikse, W. R. Thurber, W. R. Hosler, J. Babiskin, and p. SiebenITlann, Phys. Rev. 158, 775 (1969).

16. Y. W. Tung and Marvin L. Cohen, 180, 823 (1969).

17. C. S. Koonce and MarvinL. Cohen, Phys. Rev. 177,707 (1969) •

18. G. M. Eliashberg, Zh. EksperiITl. i Teor Fiz. 38, 966 (1960) [English trans 1. : Soviet Phys. - JETP ~ 696 (1960) J.

19. A. B. Migdal, Zh. EksperiITl. i Teor. Fiz. 34, 1438 (1958) [English transl.: Soviet Phys. - JETP l...!. 996 (1958) ~

20. Phillip B. Allen and Marvin L. Cohen, Phys. Rev. 177, 704 (1969).

21. G. M. Schaeffer and M. H. Van Maaren, Proceedings of the 11th International Conference on Low TeITlperature Physics, St. Andrews, edited by J. F. Allen, D. M. Finlayson, and D. M. McCall (to be published).

22. M. L. Cohen and C. S. Koonce, J. Phys. Soc. Japan Supp1.~, 633 (1966).

ENHANCEMENT EFFECTS: THEDRY"'~

C. S. Koonce



INTRODUCTION

The obvious desirability of obtaining superconductors with high transition temperatures has led to a great many theoretical attempts to find high transition temperature superconductors. There have been many suggestions concerning one and two dimensional superconductivity, interactions which take place outside the superconducting material, ultrasonic-induced superconductivity, laserinduced superconductivity, high-magnetic-field-induced superconductivity, magnon-interaction-induced superconductivit~ and more. I will restrict myself here, however, to discussing interactions which occur inside three dimensional inorganic solids under the influence of no outside fields. This will make possible a more accurate estimation of transition temperature, although transition temperatures are very difficult to calculate accurately in even the simplest actual cases. The problem of theoretically predicting effects which enhance the superconducting transition temperature is, I believe, intimately related to the problem of calculating transition temperatures of real materials, and the lack of success of many theoretical predictions of enhancement effects arises from the fact that they are often not incorporated in realistic calculations of the transition temperature. Before we can know how to change material properties to get higher transition temperatures, we must first know just how the superconducting transition temperature depends on these properties.

The BCS theory of superconductivity uses an attractive interaction between zero total momentum pairs of electrons. The interaction which is usually most important is through the phonons, especially short wavelength phonons. In general, however, the total

389

390 c. S. KOONCE

interaction between electrons must be used. The attractive interaction leading to superconductivity is closely related to the phenomenon of a driven oscillator. If an oscillator is driven at a frequency lower than its resonant frequency the displacement will be in phase with the force applied, and for driving frequencies higher than the resonance frequency the force and displacement are out of phase. If an electron is driving the oscillator the motion of the electron will be correlated in a positive way with ,the motion of the oscillator for frequencies below the natural frequency of the oscillator. When two electrons are interacting with an oscillator the interaction with the oscillator can lead to an attractive interaction between the electrons when the difference in energy between the electrons is less than the resonant energy of the oscillator.

In solids, all interactions of importance for superconductivity are electromagnetic in origin. We may therefore think of the total interaction between electrons as being the bare Coulomb repulsion screened by everything polarizable in the solid. One thing which can screen the bare electron-electron interaction is a lattice deformation (or phonons); another is the creation of electron-hole pairs (or plasmons, at long wavelength). For details, see Appendix.

For simple metals,the density of electronic states near the Fermi energy is slowly varying over energies of the order of phonon energies, and plasma frequencies are much larger than phonon energies. In this case, the total interaction between electrons can be easily separated into a resonant interaction through the phonons and a roughly constant repulsive direct Coulomb interaction. The electron-phonon interaction is then screened by the square of the static dielectric function.

On the other hand, when the density of states or the electronphonon coupling is changing rapidly on a scale of phonon energies, the separation into phonon and Coulomb interactions is not possible in such a simple manner. This fact is of interest because Itrigid lattice lt resonances, those not involving phonons, usually require a rapid variation in the density of states to be significant. We note that we may make a classification of Itrigid lattice lt resonances. These have been ca!led dielectric resonances, excitonic interactions, and interactions via d or f bands. Essentially they consist of an attractive region of the electron-electron interaction at frequencies lower than a characteristic energy for electronhole formation,such as the energy difference between the bottom of a band and the Fermi energy. We note that when the dielectric function for an electron gas is calculated, we consider intermediate states consisting of electron-hole pairs, that is, virtual excitons.

ENHANCEMENT EFFECTS: THEORY 391

Fig. 1. Dielectric function for a free electron gas (ml~ = ~e K~ = 1) calculated in the random phase approximation and plotted in the energy, wave-vector plane. ~ne dielectric function has an imaginary part in the region labeled "Landau damping". In the region bounded by the plasmon energy outside the Landau damping region and the dashed curve inside the Landau damping region, the real part of the dielectric function is negative, leading to an attractive interaction. since Vc(q,w) = 4TIe 2/[q2K(q,w)].

In Fig (1) we show the plasma frequency and region of Landau damping~he region in which real electron-hole pairs can be formed) obtained from the electronic dielectric function calculated in the random phase approximation for free electrons in the frequency, wave-vector plane. For a parabolic band the electron-electron Coulomb interaction is attractive at zero wave-vector and frequencies below the plasma frequency. In Fig (1) the attractive region is bounded by the plasma frequency (solid line) and a damped transverse mode (dashed line). However, when this interaction is used

392 C. S. KOONCE

in the Eliashberg equations and appropriate integrals over wavevector are performe~the interaction is repulsive at all frequencies. This is a consequence of the relative importance of large wave-vector components. If the band structure is not parabolic, in particular if there is another band separated from the band under consideration by not too large an energy, we have the possibility of a large increase in the available density of states for pair formation and a stronger resonance in the dielectric function. The intermediate state is then an exci tion and we can call the process polarization of a d or f band, or, if it is unfilled, interaction via a virtual d or f band. - -

We see then that several of the so-called "other interactions" are in principle included in the calculation of the transition temperature when the proper function is used to screen the electronphonon interaction and the Coulomb interaction. If one does not make the approximations that both the Fermi energy and band width are much larger than all phonon frequencies, and that the electronphonon coupling varies slowly with energy, these "other interactions" are treated on an equal basis with the phonon interactions. These equations are somewhat more complicated than the usual Eliashberg equations and will be treated in the talk on superconducting semiconductors. The interactions are known to be present, however, and the question to ask is not "Should these interactions be included in our theory?" but rather, "Were they included to a sufficiently high degree of accuracy in a given calculation?" That is, ''Is a Coulomb pseudopotential, screened by a static dielectric function, and static screening of the electron-phonon interaction, a good enough approximation?" The sharp spikes in the density of states of transition elements would lead one to believe that in this case a more detailed calculation might be appropriate for elements or alloys whose Fermi energy is near a peak in the density of states,since the density of states can change by 10-20% over an energy of a few hundred degrees. (See Fig. 2~ However, the Coulomb pseudopotential calculation seems to be correct for Ta, on which tunneling has been done by Shena (1) Shen finds that the tunneling results used with the Eliashberg theory give the phonon density of states measured by neutron work,using a Coulomb pseudopotential of iJ,~~ ~ 0.11, which is close to those usually assumed in the transition metals. If "other interactions" had been important a much smaller value of iJ,~~ would have been required. Figure 3 shows the density of phonon states, F(w), obtained from neutron measurements (2) ,compared with the product of the square of the electron-phonon coupling times the density of phonon states, a?(w)F(w), obtained by Shen from his tunneling results and inverting the Eliashberg equations.(3) From this tunneling work one can conclude that phonons are almost entirely responsible for superconducti vi ty in Ta.


t.5

t.4

1.3

1.2

t.t

:J ~ t.O o( , > .. 0.8 It w .. z 0.8 iL

'" ~ 0.7 0

~ O.ts III

'" ~ 0.5 ... III

0.4

0.3

0.2

O.t

o

20

t8

t8

~t4 t-0( , >-: i2

'" .. z ~ iO w z 0 ... 0 e

'" '" t-o( t-III 8

4

2

,----------------------.,..---------,to .: ......

9

e

..".,.., 4

....

3

2

We could look to more unusual systems in which dielectric resonances (or at least the band structure) are more important,such as in superconducting semiconductors. However, for the semiconductors found to be superconducting so far,the Coulomb interaction has not played a beneficial role. Polar interactions and large wavevector intervalley interactions do enhance the transition temperature for these materials, however.

THEORETICAL IJMITATIONS ON ENHANCEMENT OF TC

Some progress has been made in recasting the Eliashberg equationsin an approximate form in order to see what material properties lead to high transition temperatures. This approach has led McMillan (4) to predict an approximate maximum attainable transition temperature within a given class of materials.

McMillan noted that if one solves the Eliashberg equations using a two square well approximation with the cut off between the

394 c. S. KOONCE

wells being the maximum phonon frequency, Ub' one obtains an equation for the transition temperature

T = w exp [ -(1 + ,..) c 0 A-j..L4~ - « w >/w hj..L* o

,

where

(1)

(2)

and A corresponds roughly to N(O)V of the BCS model. Here F(w ) is the phonon density of states and a(w ) is the electron-phonSn coupling. j..L7~ is the Coulomb pseudopoteHtial,

j..L7~ = N(O)Vc 1 + N(O)V £neE Jw ) c B' 0

,

where ~ is of the order of the band width or plasma energy, whichever is smal]ar. Vc is the Coulomb interaction and N(O) the density of states at the Fermi energy. Also

SWo 2 n dw a (wq)F( wq) w q .....Jl

o Wq

n < w > =

(4)

One would expect Eq. (1) to give an approximate dependence of Tc on A and w ,but not necessarily be very good quantitatively_ McMillan f8und, using the complete Eliashberg equations and the phonon structure of Nb,that the equation,

T = e exp \ - 1.04(1 + A) ] c ~ A-j..L7~(1 + 0.621..) •

(5)

gave a good approximation to his calculations over a wide range of A and j..L7~.

The parameter A can also be expressed as

A ~ N(O) 2

M < w > (6)

where < d2 > is the average over the Fermi surface of the square of the electron-phonon matrix element.


3 " I , ,

(

en ~

z ::J

>- 2 II:

"" II: ~

CD II:

"" 0.6 Q

~ N 1L ....

0.4 0

~ PI Z !!? 0

0.2 z r PI en (I)

0

ENERGY (meV)

Fig. 3. Comparison between oP(w)F(w) (solid line) determined from tunneling and F(w) determined from neutron diffraction [Ref 2J (dashed line) for tantalum. [From L. Y. L. Shen, Ref. lJ.

When McMillan used his equation, (Eq. 5),to calculate ~ for transition metals and transition metal alloys from the known superconducting transition temperatures,Tc' and Debye frequencies,e, he found that

(7)

and that C was fixed for the series of transition metals and their alloys. In the simple metals McMillan found that the same relation held, but that C was different.

This was a rather surprlslng result, because it implied that the product of density of states and the average of the square of

396 C. S. KOONCE

the matrix element was constant for a given class of materials. That is, within a given class of materials. such as the bcc transition elements.or the hcp transition elements (5).0~ the simple metals, the density of states has little importance. By using the result that A, which can be associated with the N(O)V parameter of BCS, decreases as the phonon frequency is increased.it is clear that the superconducting transition temperature in Eq.(~will approach zero exponentially for very high phonon frequencies and also approach zero linearly at very low phonon frequencies (although the equation was derived assuming reasonably small A). One can then find the maximum transition temperature for a given class of materials by allowing only A (or < 0)2 » to change.

We note that the maximum transition temperature for one class of materials can be very different from the maximum transition temperature of another class of materials. In particular, the transition materials should have a higher transition temperature than the simple metals. We see that while we have predicted the maximum Tc within a class, we have not predicted the best class. Also.the apparent unimportance of the density of states within a class does not mean that the density of states is unimportant for selecting a promising class. For example, the density of states is generally higher in the A-15 compounds than in the transition metals and higher in the transition metals than in the simple metals, so that the transition temperature may go like the variation in density of states between classes.

A more general limit to the transition temperature is given by the stability of the crystal itself. The electron-phonon interaction which leads to pairing of the electrons at the Fermi energy also lowers the phonon energy, and the condition that phonon energies be positive is a condition for the stability of the lattice. Unfortunately, at present this restriction has been stated only qualitatively.

SOME ENHANCEMENT POSSIllLITIES

The relation that McMillan noticed between the electronphonon coupling and the phonon frequency (Eq.(7)invited experimentalists not only to produce alloys of transition elements having lower < 0)2 >.but also to try to get lower < 0)2 > and therefore larger coupling by producing very thin amorphous films. Garland and Allen (6) have generally been successful in predicting maximum transition temperatures of thin disordered or amorphous simple metal films using a modification of McMillan's equation. The experimental results on disordered and amorphous films are generally what one would expect using the McMillan-Garland-Allen approach; that is, the transition temperatures of non-transition metals are enhanced by introducing disorder or increasing film surface area


(and thereby lowering the average phonon frequency) (7), and this enhancement is greatest for the materials with the highest phonon frequencies. Lead, having low phonon frequencies and large electronphonon coupling,is not strongly affected by further reduction of its phonon frequency and increase of its electron-phonon coupling, since its value of electron-phonon coupling is near the optimum value for its class of material. Beryllium, on the other hand has a high phonon frequency and the possible enhancement is large. (6)

Changes in transition temperature when transition metals are made into disordered films are not so easily interpreted, however. Tnese difficulties are often attributed to changes in the band structure. Because of the relatively narrow d bands, the band structure of the transition metals is more important in calculating a superconducting transition temperature than is the band structure of simple metals. Tnis is reflected in a larger value of the Coulomb pseudopotential, u~t, for transition metals. Experiments on transition metal films indicate that the high Tc elements, Nb and Ta, have transition temperatures which decrease when they are made into films, while Ti, Mo, and W have transition temperatures which are increased. This may indicate that an averaging procedure, either in density of states or phonon structure or both,is favoring the low Tc materials at the expense of high Tc materials, although I understand that it is difficult to prepare pure transition metal films. Such an averaging process in density of states has been used to explain the increase of the transition temperature of Re when impurities are added. (5)

The application of pressure has been suggested as a means of enhancing the transition temperature. While the application of pressure to the simple metals in general decreases the transition temperature,principally because the average phonon frequency is increased, the application of pressure to the transition elements can either increase or decrease Tc. (8) For very high pressures, the possibility of producing crystal structures which are not stable at pressures near atmospheric pressure exists. These new crystal structures may have energy band structures entirely different from those of the low pressure phase. For example, at high pressures germanium and silicon become metallic (and superconducting). We then have, added to the large number of alloys and compounds that can be produced at atmospheric pressure,a large number of phases of elements and compounds which are stable only at very high pressures, and which can be investigated for superconductivity.

The constancy of the product of the density of states and the average squared matrix element that McMillan has found for the bcc transition metals and the simple metals, and which has more recently been observed in the hcp transition metals, still lacks a satisfactory theoretical basis. McMillan advanced a qualitative explanation for the simple metals. The argument for the simple metals

398 c. S. KOONCE

is that the phonon frequencies observed are much smaller than the "bare" frequencies which would be observed if there were no electrons in the metal. Hence, the phonon frequencies are primarily determined by the electron-phonon coupling. Hopfield (9) has suggested that for the transition metals the explanation lies in the inability of d electrons to scatter other d electrons via phonons which tra~sform like ~, Z, ~; that is,-of £ symmetry. The electrons must then scatter into £ states and the relative constancy of the density of electronic £ states leads to an observed independence of the product of the effective density of states and the electron-phonon matrix element. The angular momentum selection rule proposed by Hopfield is successful in qualitatively explaining the independence of A on the total density of states, but a quantitative calculation of the importance of this selection rule to the actual matrix elements in transition elements (10,11) is still lacking.

The highest transition temperatures so far obtained have been obtained in the A-IS or B-tungsten structure,A3B,where A stands for a transition metal ion and B for a non-transition metal ion. Compounds involving transition metals in the NaCI cubic structure also have high Tc' The density of states at the Fermi energy is large and is made up primarily of d electrons from the transition metal ion, so the problems of unde~standing the A-IS compounds are very similar to those of understanding the transition metals.


Appendix

We would like to calculate the total electron-electron interaction considering the polarizability of the lattice and the polarizability of the other electrons which screen the interaction. A diagrammatic expansion of the interaction in the random phase approximation is shown in Fig. 4. Here, the double dashed line represents the total interaction, the single dashed line represents the bare Coulomb interaction,Vc , and a wavy line represents interaction via a phonon, V. The bubbles represent the polarization of the electron gas (the formation of electron hole pairs),-P.

Fig. 4

=--- + --0-- + --0--0-- +'"

~+~-+~---o---+ •••

-~+-~-+ •••

~+~-+ •••

+~+ •••

+ •••••

An expansion of the total interaction within the random phase approximation

If the lattice is rigid and no phonons are allowed, only the first terms contribute to the screening and we have the result that the total interaction, in the absence of phonons, is

(AI

Since this is an infinite series, the terms may be regrouped to give

VNP Vc + V {P VNP ) (A2

or V V

VNP c c

I + PV K (A3

C e

where K is the dielectric function of the electrons only. When e phonons are included, all terms may be regrouped as in Fig. 5,to

400

give for the total interaction

v

or

v

v c

v + V c P

1 + PVc + PVp

===~+~+

---0- ------- + ---- = --- ----

C. S. KOONCE

(A4

(A5

Fig. 5 A more condensed expansion of the total interaction within the random phase approximation

If there is no screening by electrons, P=O and the interaction is

V + V c P

V c

K P

where K is the dielectric We may ¥ewrite Eq.(A5)as

V V /K

c p 1 + PV /K

C P

function of the phonons only.

V c

K + K - 1 P e

If we define the total dielectric function by

V

(A6

(A7

(AS

we have KT = K + K - I, and if K = 1 + P V and K = 1 + PV , p e p_ pc e . c

we have the result that KT = 1 + PV + P Vc; the total d~elec-tric function in this approximation i~ obtRined by adding the electronic and phonon polarizabilities. The frequency dependence of the polarizabilities is determined by the natural frequencies


of resonance; for the phonon polarizability this is the phonon frequency and for the electronic polarizability at short wavelengths it is the plasma frequency. Als~ resonances in the electronic polarizability can be caused by a large density of states available for pair formation. When the plasma frequency and the phonon frequency are the same order of magnitude, Eq. (A~ is useful for calculating the total interaction. If the plasma frequency is large compared to the phonon frequencies, it is useful to separate the total interaction into a "Coulomb" and a "phonon" part. Eq. A5 may be written as

v v

c 1 + P(V + V ) c p

+

and separating the first term

V c

v p

1 + P(V + V ) c p

V P V c E V E V 1 + PV - (1 + PV )[1 + P(V + V )] + 1 + P(V c c c p

or V V

V c + E

1 + PV (1 + PV )[1 + P(V + V )] c c c p

(A9

(AIO + V ) ,

c p

(All

We may use the relation for the electronic contribution to the dielectric function ,Eq. (A3),to obtain

V V /K 2 V c + E e

K 1 + PV /K e p e (A12

or V V /K 2

V c + E e K

1 + Ke- l ~ e (-) K Vc e

(Al3

and we now have separated the interaction into two terms. The first term is the Coulomb interaction screened by electron interactions only. To interpret the last term we note that the interaction via phonons consists of the first electron interacting with ions in the solid,with an electron-phonon interaction g, followed by a propagation of the phonon to another location,having the resonant form n = 2wA/(W2-wA2), where wA is the phonon frequency, followed by a second electron interacting with other ions, again with an interaction g. Then V = g2n. When electronic screening is introduced it is reasonRble that each electron-ion interaction be screened by the electronic dielectric function.

402 c. S. KOONCE

If we define the screened electron-ion interaction g = g/K , we e

have

v

or

v

v c

K e

v c

K e

+

+

(A14

K -1 (A15 2 2 + (_e_)

W -w;x, K

e

If the plasma frequency is large, compared to phonon frequencies, and if the density of states is approximately constant over an energy range of phonon energies, we may neglect the frequency dependence of the electronic screening in Eq.(A15). The only resonant term is then the last term of Eq.{A15). We see that this term represents an interaction between two electrons through the phonons. The electron-phonon interaction is screened by the other electrons in the solid and the phonon frequency is lowered. The phonon frequency in the presence of screening by the electrons is given by

W 2 ;x, (A16

If the crystal is to be stable ~ 2 must be positive. This imposes an upper limit to the electron-p~onon coupling for a given material.


REFERENCES

*Contribution of the National Bureau of Standards, not subject to copyright.

1. L. Y. L. Shen, Phys • Rev. Letters 24, 1104 (1970).

2. A.D.B. Woods, Phys. Rev. 136, A781 (1964).

3. G. M. Eliashberg, Zh. Eksperizn. i Teor. Fiz. 38,.966 (1960) [English Transl.: Soviet Phys. - JETP ~ 696 (191)0)] • J. R. Schrieffer, Theory of Superconductivity (W. A. Benjaznin, Inc., New York, 1964).

4. W. L. McMillan, Phys. Rev. 167, 331 (1968).

5. C. W. Chu, W.L. McMillan, and H.L. Luo, Phys. Rev. B3, 3757 (1971).

6. J. W. Garland and P. B. Allen, Proceedings of the International Conference on the Science of Superconductivity, Stanford, California, 1969 (to be published).

7. J. W. Garland, K. H. Benneznann, and F. M. Mueller, Phys. Rev. Letters ~ 1315 (1968).

8. G. Gladstone, M.A. Jensen, and J.R. Schrieffer, Superconductivity, edited by R. D. Parks (Marcel Dekker, Inc., New York, 1969), p. 665.

9. J. J. Hopfield, Phys. Rev. 186, 443 (1969).

10. D. C. Golibersuch, Phys. Rev. 157, 532 (1967).

11. S. K. Sinha, Phys. Rev. 169. 477 U968).

12. L.F. Matthies, Phys. Rev. 139, A1893 (1966).

ENHANCEMENT EFFECTS

J. F. Schooley



Superconductivity research has had as one of its continuing aims the production of high-transition-temperature materials. This situation arises from the realization that superconductivity can be applied to transportation, communication, power transmission, and instrumentation on a wider and more efficient basis, the higher the transition temperature. Many of these applications will be discussed during this course by several of the lecturers.

In this lecture, I will discuss various ways in which experimenters have attempted to generate high transition temperatures.

EFFECTS OF ALLOYING AND IMPURITIES

~ne major interest in alloying and impurity enhancement of Tc lies in the transition metals (TM), since, as we shall see, only in the TM is this technique very effective. Here the rigid band model of the electronic density of states, due principally to Mott (1), provides a plausible frameworK on which to build a discussion of the behavior of Tc. The chapter by Gladstone, Jensen and Schrieffer (2) in "Superconductivity" provides an exhaustive treatment of the TM elements and alloys from this point of view.

Figure 1 shows a section of the periodic table which includes the superconductive transition elements. Columns V and VII contain all the TM elements with TclS greater than 1 K; they also show maxima in density of electronic states.

Contribution of the National Bureau of Standards, not subject to copyright.

405

406 J. F. SCHOOLEY

The situation certainly is not a simple correspondence between density of states and transition temperature, however. Indeed, elements in columns III and VIII possess greater densities of states than column VII, but only La is Known to become superconductive.

3d

4d

5d

r WI------.

fcc 1 hep bee bee hep hep fcc

rn-l-e-r\Il-t- Cr ~ L.:J

Co Ni Mn Fe

I '\. / 1.5 /0 t

I '\. /

rZrl_,, _rNb'l_02-rM;l_'4_rTcl rRu"l-t - Rh ~ L.!..J L!.J L.!!..J L:J

/" " .f ! ,/ Pd

/ ,0 , '\. ~ " / ",. I /

rHf'l-7-rTa'l 1w1- 7-rRe'l-2-rOs"l-.9-f""i7l-. - Pt L!.J ~ ~ l..2.!J L!J ~

Fig. 1. Yne superconductive transition metal symbols are enclosed in boxes, along with the corresponding Tc's. Maximum or minimum Tc values for alloys between neighbors are indicated by intervening numbers, and an arrow indicates a continuous decrease of Tc with alloying.

The simplest and most effective Tc-predictor for transition metal alloys still is the 5-7 rule proposed by Matthias in 1955 (3). According to his formula, a value of electrons per atom can be computed for TIM alloys simply by averaging the column number of the constituent elements. [This type of formula was used quite successfully by Hume-Rothery (4) to correlate phase equilibria in binary alloysoJ If the column numbers are weighted according to the atomic fractions of the corresponding elements in the alloy, the transition temperature of the alloy is liKely to follow reasonably well a common curve with peaks at 5 and 7 valence electrons per atom. The numbers in Fig. 1 support this statement for neighboring elements. Here the Tc's are listed directly beneath the symbol, and maximum or minimum Tc's resulting from alloys of neighboring elements are indicated by intervening numbers. Alloys for which Tc simply goes to zero are denoted by a downwards-pointing arrow.

The utility of the Matthias rule does not end with neighboringelements alloys, either. The alloys (which actually may be true compounds) Nb3 Sn, Nb3 Ge, and V3 Si, in which the TIM is alloyed with a non-TIM, also follow the rule, as do (approximately) Y - Rh and Y - Ir, which are alloys of non-neighboring TIM elements. The Y - Rh

ENHANCEMENT EFFECTS 407

system which is shown in Fig. 2, is particularly illustrative, since much of the system is superconductive above 0.5 K in spite of being composed of non-superconductive elements. One might note that the Tc peaks occur at 5.4 and 7.8 electrons per atom.

z 1.5

~ 11.1 :.: 1/1 1&.1 1&.1 It: C)

~ 1.0

1&.1 It: ~

~ It: 1&.1 a.. 2 ~0.5 z o ~ iii z « a: I-

\ r \ I \ I ...

, t I , I , , I , , , , , •

I , , , , l

• •

o

O~--~----~--~----~----~--~----~--~----~--~ Y 10 20 30 40 50 60 70 80 90 Rh

COMPOSITION. ATOMIC PER CENT

Fig. 2. Superconductivity in the Y-Rh system. From Matthias, et al, Revs. Mod. Phys., :2£., 155 (1964).

The Matthias rule does not WOrK unfailingly. There are,for example, low maxima in T beyond Re and Os, and one can make Matthias rule alloys which ~e neither superconducting nor magnetic. However, like the Hume-Rothery rules, the Matthias rule is a simple relation which correlates a great deal of experimental data and has predictive power. The success of a substantial amount of superconductivity research is due to its recognition.

Thus we see that the first rule for enhancing Tc in the TM might be stated, ''Make an alloy with 4.8 - 5.0 or 6.8 - 7.0 electrons per atom." .M we see from Fig. 1, the Tc' s of V, Nb, and Ta can be raised to 8 K, 11 K, and 7 K simply by alloying with their Col. IV neighbors; respectively 25% Ti, 25% Zr, and 25% Hf. Similarly, 25% Mo-Tc and 25% W-Re alloys provide maxima in Tc of 14 K and 5-7 K. It should be noted, of course, that most of these alloys show broad, rather than sharp, maxima and that in the Col VII cases a change in crystal structure with composition intervenes near the 75% Col.

408 J. F. SCHOOLEY

VII point, where the hexagonal structures of Tc and Re begin to yield to the cubic structures of Mo and W, respectively. These data can be found in Refs. 5-7, as well as in Ref. 2.

A second rule for enhancing Tc by alloying in the 'I'M might be, "Start with Rule #l and then look for a maximum by varying the composition a littJ..e bit". Among the alloy systems studied for highest Tc work, the maximum Tc is seen at Z values ranging from 4.0 to 5.1 and 6.4 and 6.75, as shown in Table I. A glance at Table I reveals the multitude of possible alloys to be considered for Tc - enhancement experiments. Considering the number of possible ternary and quaternary (why stop?) combinations, one is quite happy to have a guideline such as Rule #l in looking for high Tc materials.

Among the "p-band" superconductive elements alloying is nearly ineffective in enhancing Tc (excepting, of course, those alloys with 'I'M elements). A few bismuth alloys (with In (B) and Pb (9), for example) and a Ga 05Sn.95 (10) alloy have higher Tc IS than the component elements, b~t the enhancement is not large.

In certain alloy systems (11), a striking enhancement of Tc results. Notable exampl~s are Bel-~Rex' which reaches 9.75 - 9.5 K for 1-7% Re~ Agl_xGax , W1th Tc.= 6.j-BK ~or x = .2-.7; AU.725-.4 ~275-.6 , W1th Tc = 1-1.6 K; Bil-xClix, W1th Tc up to 2.2 K; the metallic sodium, barium, and rubidium tungsten bronzes, whose TclS range up to 2 K; and PdTe.L.o2 _ 1. 08 , with Tc ranging from 2.5 -1.9 K. Many of these systems contain no superconductive elements, and in this sense, there is, indeed, a strong "enhancement" of Tc1

There are other interesting systems in which alloying yields strong enhancements of Tco One of these is Th4~5' a material unusually rich in hydrogen for a superconductor, with Tc = B K (vs. 1.5 for Th) (12). A second is the semiconductive system S~_x-yCaxBayTi03' in which a few percent Ca or Ba increases Tc (see Fig 5) by factors ranging from 1.5 to five or more (13). Another is the ternary system Nb3Snl_A, where M is Ga (Tc = IB.3 for x =~5-.20), In (Tc = IB.3 for x ~ :10-.20), Tl (Tc = ~B.2 for x = .02-.2), Pb (Tc = IB.2 for x = .05-.25), As (Tc = IB.2 for x = .05), and Bi (Tc = IB.3 for x = .1-.2) (14).

Before leaving the discussion of Tc enhancement by alloying, we should mention alloying between members of a particular column in the Periodic Table. Not so much WOrK has been done in this area, but both maximum (Ti - Zr) and minimum (V-Nb, Ru-Os) TclS can be found; these results must be explained without invoking valency arguments, of course.


TABLE I. MAnMUM - TC ALLOYS

Alloy Tc Zeff Structure

(Nb3Al)4Nb3Ge 20.05 4.55 Al5

Nb Sn 3 18 4.75 Al5

Nb3Ge 18 4.75 Al5

NbN C .72 .28 17.9 4.86 NaCl (fcc)

Nb3Al 17.5 4.5 AI5

V3Si 17.1 4.75 Al5

NbN 16.1 5.0 (NaCl)

V3 Ga 14.6 4.5 A15

Mo Hi' C 14.2 5.1 NaCl .95 .05 .75

Mo Re .57 .43 14.0 6.4

Mo C .56 .44 13.0 5.1 cubic + hex

Nb3Au 11.5 4.0 Al5

Nb Au Rh 3 .95 .05 11.0 4.1 Al5

ZrN 9.8 4.5 (NaCl)

Mo Ru 7.2 6.75 Tet • • 61 .39

410 J. F. SCHOOLEY

TC ENHANCEMENT BY COMPOUND FORMATION

The differences between alloys and compounds are not always distillct, and one could well argue that the two should be discussed together. MY own preference is to treat them separately, emphasizillg the differences that exist. The limi tillg case of an alloy is a materialasuch as Nb-Ta or Nb-STi, which forms solid solutions over the whole concentration range (15). The elemental lattices are nearly the same size and have the same structure and the atoms have similar valences, so that the ions can replace each other without difficulty. The archetype of compound formation, on the other hand, might be found ill the Cu-S or Ca-Pb systems. Here differences ill ionic sizes, valences, and elemental structures, preclude any substantial alloy formation, and such systems are limited ill composition to a few well-defilled compounds with the usual properties of uniform, simple atomic ratios and crystal structure distillct from that of the component elements.

Between these limi tillg cases lie a great number of systems which form both solid solutions and compounds. Often the "compounds", as for example NbsSn, do not form readily with the simple atomic ratio which could be expected from the chemical law of Defillite Proportions. These problems notwi thstandillg, one can discuss many cases of enhancement of Tc by compound formation.

Certainly the most impressive "enhancements" of Tc occur ill compounds whose elements are not superconductorsl Two of these, CuS and Au2 Bi, were among the first superconductive materials discovered. The group of superconductive compounds composed of non-superconductive elements is a large one and it illcludes most of the remailling elements exceptillg the halogens and the rare gases. In Table II are listed representative compounds, arranged accordillg to their groups ill the Periodic Table.

No ready rule of thumb serves as a Tc-predictor for compounds wi th anything like the accuracy of the Matthias rule for alloys. The length of the catalog in Table II illdicates that it is not necessary for one constituent of a superconductive compound to be a superconductor. On the other hand, the superconductive compounds containillg at least one superconductive element are much more numerous than the group composed entirely of normal elements, as can readily be seen ill Ref. li.

Fo~ the most part, the highest Tc materials are compounds; it has been noted already that these mostly occur in the TM alloy systems with Nb and V. It is perhaps significant that ill many of the billary systems of Table I, superconductivity occurs over a substantial range of composition, but that the peak Tc occurs for the compounds.

TABLE II

Superconductive Compounds from Non-Superconductive Elements

Group A Elements

I II III IV

® LiBe B Sc C K

® 12 8

NaBi CoSi

KBi CaBi GeTe 3 RbBi SrBi 3 CsBi BaBi 3

Group B Elements

I II III IV

GuS B12SC

Ag2 F

Au Bi 2

Rare Earths

V VI

® Ag7 BF 0 4 8

2 P3Pd7 CuS

As3Pd5 GuSSe

PdSb GeTe

LiBi

V VI VII

@ 9 @

VII

Ag F 2

@ ® <D

VIII

CoSi 2 , Bi3Ni YRh3, BiPd2

Y3Pt 2

LuRh 5

Non-superconductive elements wnlcn are not known to combine with other non-superconductive elements to form superconductive compounds are circled.

412 J. F. SCHOOLEY

The two crystal structures most favored for superconductivity are the Al5 or ~-W and the NaCl structures. Dr. Hein will discuss the Al5 materials tomorrow.

Besides the group of non-superconductive-element compounds, there are many other classes into which the superconductive compounds may be divided. One class, notable for its character rather than for its high Tc's, is that of the doped semiconductors. Semiconductive compounds and elements were examined for superconductivity over the years, and they were never found to be superconductive. For intrinsic semiconductors such as Si and Ge this is not surprising, since these materials are insulators at low temperatures. After }L L. Cohen worked out the elements of a theory for doped manyvalley semiconductors (16), however, superconductivity was discovered in several of these materials, including GeTe, SrTi03 , and SnTe (17). The Tc values for these three are all less than 1 K, but they provide a very interesting display of superconductive properties. C. S. Koonce and Cohen have given a fairly complete discussion of superconductivity in semiconductors (18), and from this it appears that SrTi03 is currently the limiting case, since its Fermi energy is less than its phonon energy. For GeTe, SnTe and other materials like InTe and L~Se4 the reverse is true, and they bridge fairly smoothly the region between semiconductive and metallic superconductors. It was noted in the section on alloys that a few percent of Ca or Ba in SrTi03 increases Tc markedly, and a similar effect occurs for Ag in GeTe and SnTe.

Besides the Al5, and NaCl structure compounds and the semiconductors, particularly interesting series of superconductive compounds are the tetragonal alkali metal bismuthides (19), the cubic alkaline earth bismuthides (19), the alkali tungsten bronzes (20), the alkali layered-graphites (21), and the chalcogenides (22).

ENHANCEMENT BY VARIATION IN ATOMIC ORDER

Quite large enhancements of Tc can accompany changes in crystal structure or atomic order of a given material. In the former case, a great deal of data exists showing the relatively high Tc values associated with the ~-W and NaCl structures. Rather than simply repeat much of what has been said already, we might summarize the influence of crystal structure by examining the data of Bucher, Heiniger, Muheim, and Muller on the system Cr-Re (23), shown in Fig. 3. The system is antiferromagnetic for low Re concentrations but from 20% - 50% Re, Tc increases from 1-5 K. As the sigma phase appears, however, Tc drops below 1 K. It is not unusual, of course, to find that only one structure in a system is superconducting at all.

ENHANCEMENT EFFECTS

60

50

..... lI::

0 -I-z 40 w a:: ~ .-« 0: IAJ

300 0-:E IU .-

GJ \GI 200 z

100

Cr

ANTIFERROMAGNETIC AND SUPERCONDUCTING

lRANSITION TEMPERATURES OF Cr- Re

SOLID SOLUTIONS

1"·"\ ( . •

p~

~l~ TC

Y rf~ ?~

! ~r ~~

b.c. c. .. - ,,---+

20 40 60 CONCENTRATION (at.% Re)

Fig. 3. Influence of crystal structure on Tc.

413

6

5

4

..... lI::

• -.,.u

2

414 J. F. SCHOOLEY

A very conclusive set of data on the effects of crystalline order on Tc was given by Hein, Cox, Blaugher, Waterstrat and Van Reuth on the material V3Au (24). The Bragg-Williams order parameter S was measured along with Tc as the samples were annealed, and Tc rose from less than .015 K to 3.2 K as S rose from 0.75 - 0.99.

The effect on T of vapor deposition of a material can result from film thickness (or thinness) as well as from structure changes, but in some cases,- such as disordered Bi and Ga films, as examined by Buc'Kel and Hilsch, the increase of Tc to 6.2 and 8.4 K, respectively, is very probably due to disorder introduced in the deposition (25).

Stritzker and Wuhl presented evidence that the denser, metallic state of Ge seen in the liquid form can be stabilized by the IB group of elements to produce superconductivity over a broad range of composition (26).

One can treat non-magnetic impurities in Group A superconductors as affecting lattice order only, according to WOrK done by Serin, Lynton, and Zucker over a period of years (27). They found that changes in Tc , as shown in Figure 4, accompanying small additions of Bi, Sb, Pb, Cd, Zn to Al and Sn depended only on the relative electronegativity of the solute to that of the solvent. By plotting 6Tc/Tc vs So/~, they were able to represent virtually all the data by two curves, one for a relatively electropositive solute and another for a relatively electrogegative one. Both enhancement and depression of Tc were seen, but only to the extent of a few percent.

ENHANCEMENT BY VARIATION IN ELECTRON DENSITY

On the simple free-electron picture of metals, t~/~ensity of states at the Fermi surface should increase slowly (~ ) as the electron concentration n rises. Since the BCS expression for Tc contains an exponental dependence upon the electronic density of states at the Fermi surface, there has been a corresponding interest in the effect on Tc of variation in electron concentration.

One might argue that light doping of one element with another from a different group in the Periodic Table, as noted in the previous Section, could vary the electron density. The Rutgers group did find that light doping of Al and Sn affects Tc. However, both higher valence and lower valence impurities produced a lower Tc ' and the 6Tc correlated quite well with 6~, the change in mean free path, indicating that a more likely explanation for their results lay in Anderson1s theory of dirty superconductors (28).

Glover and Ruhl (29) constructed an experiment in which the electron concentration appeared to be varied independently of most

ENHANCEMENT EFFECTS

+.0.1

1-0

'" ~ <J -.01

o Sn • AI

(electfopositlve)

o

o

(electronegative)

•

Fig. 4. The dependence of Tc upon ~lectron mean free path.

415

416 J. F. SCHOOLEY

o

other properties. They deposited ~ 100 A films on mica and found that reversible effects on Tc resulted from electrostatic charging of the mica capacitor. About 1/104 increase in n, the electron concentration, produced a liKe increase in Tc of tin and unannealed indium. Reverse polarity of charging reversed L'lTc. An interesting peculiarity was the opposite effect found for Tl and annealed In films from that on unannealed In and Sn films; since Tl is p-type, negative charging serves to decrease n and thus Tc ' whereas for n-type Sn, negative charging increases both n and Tc' The effect of depositing oxygen on these films was much larger, and corresponded to removal of electrons.

The equivalence of Tc shifts in the capacitor-plate experiments with those found on oxidation of the film surfaces indicated to Glover and Ruhl that the principal effect of surface oxidation of films might be to reduce n in the films. Yney estimated from the size of L'lTc ' which was ~ 0.1 K, that oxidation of their Sn film produced about a 6% change in n, or one electron per surface Sn atom.

When the phenomenon of superconductivity in semiconductors was discovered (17), it became possible to vary the charge carrier concentration by a factor of ten or more rather than the few percent variation accomplished by oxidation or the one-hundredth percent change through electrostatic charging. The corresponding changes in Tc are similarly larger, as shown in Fig. 5.

10r---------------------------------------------------------~--_.

II)

A'La So 3"X • B· Sn(Ag) To

C -GeT. and Ge(Agl1i

D· S',...fa. 0751iO •.•

E=S'92,B:b7,Ti 03"X F - S, TiO._x Ii- Stli(Nb)O.

-8

QI~--~----------~~------------~----~~----~~------------~ 10'·

Fig. 5. Superconductivity in low-carrier-density systems. Tg, in kelvins, is plotted logarithmically against n, in carriers/cm •

In GeTe and SnTe, n can be varied by about a factor two by self-doping. With silver doping SnTe can be given a carrier concentration of nearly 1022 , at which point Tc > 1 K. SrTiOs is superconductive for n = 1-30 x 10~9, with carriers generated either by reduction or Nb-doping, and the carrier concentration range is extended to n < 1019 for Sr(Ba)TiOs and Sr(Ca)Ti03 • These materials have been of considerable interest because of their unique properties, not the least of which is the variability of n within a single structure. The problem of changing mean free path experienced with lightly doped metal superconductors is absent in this class because these semiconductors are short-mean-free-path materials and are "dirty" superconductors in the manner discussed by Anderson (28).

Metallic systems in which n appears to be widely variable within a given structure exist as well. Some of these are also shown in Fig. 5.

The system Inl_xTe, with n = 0.8-1.7 x 1022 and Tc = 1-3.5 K was examined by Geller and Hull (30). It is one of several NaClstructure materials discussed briefly in their paper.

Holtzberg, Seiden, and von Molnar (31) have prepared a very interesting series of superconductors with n in the range 1-6 x 1021 • They accomplished this by preparing solid solutions of L~Se4' a metallic superconductor with Tc = 10 K, in La_Ses ' an isostructural insulator. They have pointed out that even tKe lattice parameter is invari&!t within this system, and Seiden has successfully discussed the Tc(n) curve in terms of free-electron superconductivity.

A number of metallic tungsten bronzes have been observed to be superconducting, and Shanks and Danielson (32) have prepared a series of RbxWOs with x = 0.26 - 0.33 for which Tc = 1.5 - 2.1 K. While these authors did not measure n, it is commonly in the range 1022 for these materials and approximately linear with x.

Finally there is a series of superconducting spinels, CuRhl-x(Sn)xSe4' and pyrites, Rhl_x(Ru)xSea' discovered by the Philips Research Laboratories group at Eindhoven (33). For the spinels, Tc varies from 3.3 K for CuRhSe4 to less than 0.1 K for Cu~Sn.5Se4' and within this range the material changes from an n- to a p-type metal. The Tc of RhSea is 4.2, and Tc becomes less than .05 K at Rh.55Ru.45Sea. The value of n appears to be nearly linear with (I-x).

As you can see from the foregoing, in almost all cases of variable-charge-density superconductors increasing the charge carrier density increases Tc. Unfortunately (and perhaps not COincidentally) the charge carrier densities are all lower than for ordinary metals, and no Tc is very high. Clearly the experiment to do is to increase the carrier densities beyond 10G4 carriers per cm3 l

418 J. F. SCHOOLEY

ENHANCEMENT BY PROXIMITY EFFECTS

Yne effect on T of proximity to various sub- or superstrates is a little complica%ed. In the first place, the experiments are commonly performed by depositing films of various thicknesses on various substrates which are held at various temperatures. The films themselves, as will be discussed in the next section, may show a thickn~ss-dependent Tc; most commonly, for example; Al films of about 100 A thickness show Tc = 2 K. The electrical properties of the substrate may interact with the film to produce a change in Tc as, for example, Glover and Sherrill and Ruhl have found for oxidized layers. These latter two effects may have either sign of 6Tc. Particularly for high-temperature deposition~the thermal contraction of the heavy substrate may stress the film, producing a 6Tc due to strain. Thus there are several problems to worry about before one can really begin constructing a proximity effect experiment.

In the second place, it is difficult to deposit films of uniform thickness, as is required to avoid different Tc values in different parts of the film. This is particularly true in multiplelayer films.

Finally, there is the ever present problem of contamination, that is to say the accidental deposition of some material which the experimenter did not intend, or the accidental presence of oxygen or some other gas in the deposition apparatus.

All this is said not to destroy your faith in film worK, but merely to point out the difficulties under which such experiments are performed.

These difficulties notwithstanding, there appear to be two separate effects discernible in proximity studies. If the superconductor is contigious with a non-magnetic metal, a value of Tc is found which is between those of the two materials taKen independently. Experiments by Hauser and Theurer on Pb-Al films illustrate this effect (34); lead was deposited over a heavy aluminum layer (4400 A thiCK), and the Tc of the combination was found to increase from ~ 1.2 K to 7 K as the lead film thickness increased from a few tens to a thousand angstroms. Apparently Tc is not going to rise above that of the higher Tc constituent in those experiments, so that there is no reason to discuss metal-proximity effects in great detail. An intriguing facet of these experiments, however, is the possibility, discussed by Von Minnigerode (35) and by Hauser, Theurer, and Werthamer (36), of predicting the (low) Tc value of a normal metal such as copper.

More variety, and some enhancement of Tc , is found when dielectric materials are placed in proximity with a superconductor.


TABLE III - OXIDATION EFFECTS

Film T(dep) ATe (oxid) Sign of H( Hall)

All 3 K

Heat to 40 K for Oxidation

Al 40 K +

In 40 K +

TI 40 K + +

Pb 40 K +

Sn 40 K

Ga (unannealed) 40 K

In 40 K

Tl nLown + +

In (annealed) nLown +

In (unarmealed) nLown

Sn nLown

420 J. F. SCHOOLEY

Ruhl (37) has discussed some of the results of depositing oxygffi1 on and of actually oxidizing (at 40 K) Al, In, Tl, Pb, and Sn films. These results are shown in the upper part of Table III. In some cases Tc rises slightly and in others it is depressed; the results agree with those of Glover and Sherrill mentioned under charge variation (Section D) and also summarized in the lower part of Table III, but only a few of them have been satisfactorily explained. For example oxidation raises the Tc of Tl and lowers that of Sn, and these effects can be interpreted as resulting from decreases in electron density in p-type Tl and n-type Sn; however, in AI, In, and Pb, oxidation produces 6Tc's of the wrong sign based on the changing-electron-density idea. Hilsch (38) discusses some of the ideas which have been proposed to explain these experiments, but some of the results may be due to faulty techniques.

Small enhancements of Tc in V films have been seen as a result of depositing organic materials by McConnell, et ale These results are summarized in a paper by Gamble (39).

Fig. 6. Alternating Al and (Al + 02) layers. Curves C, E, G, I and J show resistive superconductive transitions after Al depositions at pressures below 8 x 10- 7 torr. Curves D, F, and H were made after Al depositions in 1-2 x 10-4 torr O2 gas. The substrate temperature was 4-10 K.


A very substantial enhancement of the Tc of Al was achieved by Strongin, Kammerer, Douglass and Cohen by alternating aluminum and aluminum oxide films (40). These results are shown in Fig. 6. Depositing the films on a 4 K substrate, these WOrKers found Tc's as high as 5 K. In a similar experiment, (tin)-(tin oxide)-(tin) sandwiches gave Tc's up to 6 K. One should note here, however, that the Tc of Al can be raised b,y other means as well; deposition in granular form and deposition with organic inclusions produce Tc = 5-6 K.

ENHANCEMENT BY VARYING SAMPLE DIMENSIONS

Two common techniques are used to produce small samples. One is to deposit a thin layer, essentially a two-dimensional sample, and the other is to deposit the sample in an oxygen atmosphere, producing 200-1000 AOislands of metal separated from each other by oxide layers. Yne latter materials approach zero-dimensionality, since they are smaller than the coherence distance in all directions, and are called "granular superconductors".

There is not much in the way of a "general rule" in these small superconductors; perhaps one can say that most transition metal Tc's go down, and most non-TM Tc's go up, but that is not a very strong rule. Yne biggest enhancements are found in unannealed Be films deposited at 4.2 K, where Glover, Baumann and Moser (41) observed Tc of 9.6 K, Al films and grains, which yield Tc's from 2 K to above 3 K (42) and unannealed Mo films, which show Tc's up to 6.8 K (43).

Besides Ga, Be, and Al, the non-TM's which show enhancements are Sn (to 4.2) (44) and In (to 4.2) (45). These,of course, are not large changes in Tc.

Apart from Mo, which has been studied by the Brookhaven and the RCA groups, no other transition metal appears to show enhanced Tc in small samples. HanaK and Gittlemen find grains of Mo, with Tc = 5.7 - 8 K, that show evidence of remaining in the bec structure characteristic of bulk Mo. (43).

Yne same caveat concerning vacuum deposition problems which was mentioned in Section E must be renewed here. In preparing zero -, one- or two-dimensional samples, care must be taken to achieve the desired dimensions, metallurgical state, and composition. It must be said, however, that very c~nsiderable care has generally been taken to show the very large enhancements of Tc which often accompany a reduction in one or more dimensions of superconducting materials.

422 J. F. SCHOOLEY

TABLE IV

Superconductivity Induced by Presaure

Elerrent Pressure T c

Group I

Cs IV 75 Kbar 1.6

Group II

BaIl 55 Kbar 1.3 BallI 140 Kbar 5

Group III y 150 Kbar 2.5

Rare Earths

Ce II 50 Kbar 1.7

Group IV

Si II 120 Kbar 5 GeII 115 Kbar 7

Group V

BiII 25 Kbar 3.9 Bi III 28 Kbar 7.2 p 170 Kbar 5.8 Sb II 85 Kbar 2.6 - 3.6 As

Group VI

Se II 130 Kbar 6.8 Te II 56 Kbar 3.3

ENHANCEMENT EFFECTS

Probably the largest enhancement is not primarily due to reduction in size at all. Buckel (46) condensed Bi at 4.2Kand

423

found T = 6 K, which is near the usual value found for the metallic, high-prgssure phase of that element. Buckel also found Ga films superconducting at 8.4 K, which is somewhat above the S and y phase Tc 's.

ENHANCEMENT BY PRESSURE

A very recent review article by Boughton, Olsen, and Palmy has been written on pressure effects in superconductors (47), and it should serve as a background for problems in theory, methods and experimental results.

The most striking enhancements of Tc are of course the cases where a superconductive state is induced for the first time under pressure. The case of Bi has already been mentioned; two superconductive phases exist with Tc's at 4 K and 7 K. A listing of elements which become superconductive under pressure is given in Table IV (48). Besides the elements, compounds such as InSb and InTe become superconductive under pressure. In many of the cases noted above, the effect of pressure is simply to transform the material into a metal. The fact that many of the metals so generated are superconductive is perhaps not surprising in view of the ubiquity of superconductivity throughout the Periodic Table; every major grouping of metals excepting the Groups IA and IB contains one or more superconductor.

Besides the "manufactured" superconductors, the Tc of many superconductive elements and compounds is enhanced by pressure. Among the elements Ir, Ti, V, Tl, Zr, and La all have positive values of dTc/dp.

In some systems, pressure both enhances and depresses Tc ' depending upon the circumstances. In certain superconductive SrTi03 samples, as shown in Fig. 7, uniaxial stress along the (100) axis was found to enhance T , while the general effect of uniaxial or hydrostatic stress in that material is depressive (49). Smith has found that the La-Ce alloys change sign of dTc/dp between 0.5 and 1% Ce (50). In the case of Tl, pressure less than 2 kbar enhances Tc ' while above 2 kbar Tc is depressed by pressure (51).

In other systems, particularly in the high-Tc materials and especially for Nb, there is disagreement among the various sets of data. Among the several experiments involving pressure-tension on Nb appear dTc/dp results which are positive, zero, and negative. Kohnlein eventually decided that the real Nb shows an initial value of dTc/dp of + .01 KrKbar, see Fig. 8, and that this derivative

424 J. F. SCHOOLEY

gradually falls off with increasing pressure (52). He warns that deformed specimens often show an initially higher Tc which then decreases with pressure until it reaches the equilibrium line. Luders (53) however, finds 6Tc = + .4 K accompanying a 12 kbar tension on a Nb wire. The effects of cold-wOrK and impurities are probably quite important in experiments on such materials. Besides Nb, the compounds VsSi, VsGe and VsGa show positive values of dTc/dp ranging from 1-8 x 10-2 K/Kbar of hydrostatic pressure (54).

·02

o

-·02

~

- -·04 ~ <I

-·06

-·08

-·10

o

"-

~*' 1 I

~ -:k-.

"- I

"-

H:.t+

·1

o

-·1 ...... ~ c

-·2 .::::; <I

OIl] -·3

~ --4

2·0

Fig. 7. Stress effects in Nb-doped SrTiO • Compression in the (100) direction increases Tc , while (110) and tIll) compression and hydrostatic pressure decrease Tc.

ENHANCEMENT EFFECTS

T. oK c, r-----------------------------------------~~ 10.0

\ \ , \

9.8 \ \ \

\

\ ... , , 9.6 ', .....

HEAVILY DEFORMED NIOBIUM

UNDEFORMED NIOBIUM

I I

I I

I

I I

I

425

o 10 20 30 40 r. kbor

Fig. 8. The pressure dependence of Tc for strained (dashed curve) and annealed (solid curve) Nb.

426 J. F. SCHOOLEY

REFERENCES

1. N. F. Mott, Proc. Phys. Soc. (London) 47, 571 (1935)

2. G. Gladstone, M.A. Jensen, & J.R. Schrieffer, Ch. 13 of "Superconductivity" ed. by R.D. Parks, Ma,rcel Dekker Inc., NY, 1969.

3. B. T. Matthias, Phys. Rev. 97 74 (1955)

4. W. !fume-Rothery, "Structure of Metals and Alloys" Institute of Metals, London 1947

5. V.B. Compton, E. Corenzwit, J.P. Maita, B.T. Matthias, & F. J. MOrin, Phys. Rev. 123 1567 (1961)

6. J.K. Hulm & R.D. Blaugher, Phys. Rev. 123 1569 (1961)

7. C.W. Chu, W.L. McMillan, & H.L. Luo, Phys. Rev. B3 3757 (1971)

8. J.V. Hutcherson, R.L. Guay, & J.S. Herold, J. Less Common Metals 11 296 (1966)

9. H.W. King, C.M. Russell, & J.A. Hulbert, Phys. Lett. 20 600 (1966)

10. G. Knapp & M.F. Merriam, Phys. Rev. 140A 528 (1965)

11. Transition temperatures of most of the superconductive materials are listed in the compilations of B.W. Roberts. These are National Bureau of Standards Technical Note 482, May 1969, and Progress in Cryogenics, IV, 160 (1964). The compila,tions contain extensive references to the original Ii tera,ture.

12. C.B. Satterthwaite and 1.L. Toepke, p.365, Proc. 12th Int. Conf. on Low Temp. Phys., ed. by E. Kanda, Academic Press of Japan, 1971.

13. J.F. Schooley, H.P.R. Frederikse, W.R. Hos.Ler, & E.R. Pfeiffer, Phys. Rev. 159 301 (1967)

14. R. Hagner & E. Saur, p.358, Proc. 8th Int. Conf. on Low Temp Phys. , ed. by R.O. Davies, Butterworths, Washington, 1963.

15. A very useful source for phase equilibria in binary systems is "Consititution of Binary Alloys", by M. Hansen, McGraw-Hill, New York, 1958.

16. M. L. Cohen, Phys Rev 134, A511 (1964).

17. Discussions of both the theoretical and the experimental aspects of the superconductive semiconductors have been given by M. L. Cohen in Chapter 12 of "Superconductivity", ed by R. D. Parks, Marcel Dekker, Inc., N.Y. 1969, and by J. K. HulriJ., M. Asbkin, D. W. Deis, and C. K. Jones in Ch V of Prog. Low Temp. Phys. VI, 1970, C. J. Gorter ed.


18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

C. S. Koonce and M. L. Cohen, Phys. Rev., 177 , 707 (1969).

For a listing of literature references for these superconductors, see D. Shoenberg, "Superconductivity", Cambridge Univ. Press 1962.

II. R. Shanks and G. C. Danielson, p 359, Proc. 12th Int. Conf. on Low Temp. Phys., ed by E. Kanda Academic Press of Japan, 1971.

N. B. Harmay, T. II. Geballe, Be T. Matthias, K. Andres~ P. Schmidt, and D. McNair, Phys. Rev. Lett. ~ 225 (1965).

F .. R. Gamble, Science 168 568 (1970), F. J. D. Salve, Jr., Bull APS ~ 376 (1971).

E. Bucher, F. Heiniger, J. Muheim, and J. Muller, Rev: Mod. Phys. ~ 146 (1964).

R. A. Hein, J. E. Cox, R. D. maugher, R. M. Waterstrat, E. C. van Reuth, paper 5.1, Int. Conf. of Superconductivity, Stanford Univ. 26-29 Aug. 1969.

W. Buckel and R. Hilsch, Z. Physik. 138, 109 (1954).

B. Stritzker and H. Wuhl, P 339, Proc. 12th Int. Conf. on Low Temp. Phys. ed by E. Kanda, Academic Press of Japan, 1971.

A short summary of this work appears in the paper by B. Serin, p 391, Proc. 7th Int. Conf. on Low Temp. Phys., ed by G. M. Graham and A. C. Hollis Hallett, University of Toronto Press, 1961.

P. W. Anderson, J. Phys. Chem. Solids !b 26 (1959).

R. E. Glover, III, and W. Ruhl, paper sl44, lOth Int. Conf. on Low Temp. Physics, Moscow, 1966. See also R. E. Glover, III and M. D. Sherrill, Phys. Rev. Lett 2. 248 (1960). '

S. Geller and G. W. Hull, Jr., Phys. Rev. Lett. 13 127 (1964).

F. Ho1tzberg, P. E. Seiden, and S. von Molnar, Phys. Rev., 168, 408 (1968).

II. R. Shanks and G. C. Danielson, p 359 in Proc. 12th Int. Conf. on Low Temp. Phys., ed by E. Kanda, Academic Press of Japan, 1971.

Me II. van Maaren, II. B. Harland, and E. E. Havinga, p 357 in Proc. 12th Int. Conf. on Low Temp. Phys., ed by E. Kanda, Academic Press of Japan, 1971.

J. J. Hauser and II. C. Theurer, Phys. Lett ~ 270 (1965).

B. von Minnigerode, Z. Physik 192, 379 (1966).

J. J. Hauser, H. C. Theurer, N. Werthamer, Phys. Rev. 136, A637 (1964).

428 J. F. SCHOOLEY

37. W. Ruhl, Z. Physik, 196, 464 (1966).

38. P. Hilsch, p 979 in Proc. 11th Int. Conf. on Low Temp. Phys., ed by J. F. Allen, D. M. Finlayson, and D. M. McCall, Univ. of St. Andrews, 1968.

39. F. R. Gamble, Proc. of 1969 Superconducting Symposia at NRL, NRL Report 6986, 16 May, 1969.

40. M. Strongin, o. F. Kammerer, D. H. Douglass, Jr., and M. H. Cohen, Phys. Rev. Lett 19 121 (1967).

41. R. E. Glover, III, F. Baumann, and S. Moser, p 337 in Proc. 12th Int. Conf. on Low Temp. Phys., ed by E. Kanda, Acad. Press of Japan, 1971.

42. R. Cohen and B. Abeles, Phys. Rev. 168, 444 (1968).

43. J. E. Crow, M. Strongin, R. S. Thompson, and O. F. Kammerer, paper 9-8, and J. J. Hanak and J. 1. Gittleman, paper 9-1, Int. Conf. of Superconductivity, Stanford Univ., 26-29 Aug. 1969.

44. I. Giaver and G. Zeller, Phys. Rev. Lett., 20, 1504 (1968).

45. J. H. P. Watson, p 977, Proc. 11th Int. Conf. on Low Temp. Phys. ed by J. F. Allen, D. M. Finlayson, and D. M. McCall, Univ. of St. Andrews, 1968.

46. W. Buckel, paper 14-3 in Proc. 5th. Conf. of Low Temp. Phys., ed by J. R. Dillinger, Univ. of Wisconsin Press, 1958.

47. R. I. Boughton, J. L. Olsen, and C. Palmy, Chapter 4 in Prog. of Low Temp,Phys VI, North-Holland, 1970, ed by C. J. Gorter.

48. An up-to-the-minute note on superconductivity under pressure is included in an article by B. T. Matthias in Physics Today, Aug. 1971, p 23.

49. E. R. Pfeiffer and J. F. Schooley, J. Low Temp. Phys. ~, 333 (1970) •

50. T. F. Smi~h, Phys. Rev. Lett. ~ 386 (1966).

51. L. D. Jennings and C. A. Swenson, Phys. Rev. 112, 31 (1958).

52. D. Kohnlein, Z. Physik. 208, 142 (1968).

53. K. Luders, Z. Physik. 193, 73 (1966).

54. T. F. Smith, Phys. Rev. Lett ~ 1483 (1970).

AUTHOR INDEX

Abeles, B. Abrahams, E. Abrikosov, A. A. Adkins, C. J. Ageev, N. V. Alekseevskii, N. E. Allen, J. F. Allen, P. B. Allgaier, R. S. Alphonse, G. A. Ambegaokar, V. Ambler, E. Amendas, w. Anacker, W. Anderson, P. W.

Andres, K. Appleton, A. D. Aron, P. R. Arrhenius, G. Asada, T. Ashkin, M. Aslamazov, L. G. Asprey, L. B. Atherton, D. Averill, R. F. Axe, J. D.

Babiskin, J. Bachner, J. F. Baker, D. Bakker, J. W. Baldus, W. Baratoff, A. Bardeen, J.

Barisic, s. Barnes, L. J. Barrett, C. s. Barsa, F.

A

247,261,428 164,167,183,748,753 20,71,93,99,128,146,289,710 329,332

xi

360,371 282,284,285,288,360,361,362,371 428 384,387,396,400,403 375,386 616,622 98,684,748,753 374,386 466,470,472,481 621,623 64,70,114,147,157,161,414,417, 427,547,563,631,632,633,640,650, 654,676,687,690,746,749,752,753 282,288,427 486,495 346,370 291 358,371 426 643,647,652,684,695 338,369 437,457 57,69,143,148,212,253,260 746,752

B

377,387 348,370 604 628,630 190 548,563 7,18,25,29,42,57,68,69,71,74, 77,78,88,90,91,94,95,96,122, 125,128,130,135,140,141,143, 144,147,148,149,152,161,287, 292,325,327,330,331,563,626, 629,685,687,692,737,743,744 751 291 277,288 281,288

Pages 1-428 will be found in Volume 1, pages 429-778 in Volume 2.

xii

Bass, I. Batterman, B. W. Baumann, F. Baym, G. Beall, W. T. Bean, C. P. Beasley, M. R. Beck, P. A. Becker, J. G. Benda, R. Bennemann, K. H. Benz, M. G. Berk, N. F. Berkl, E. Berlincourt, T. G. Berruyer, A. Berstein, J. T. Bierstedt, P. E. Bishop, J. H. Black, w. c. Blackburn, J. A. Blackford, B. L. Blaisse, B. S. Blank, C. Blatt, F. J. Blatt, J. M. Blaugher, R. D. Bloch, F.

Blount, E. J. Blumberg, W. E. Boesenberg, E. H. Bogoliubov, N. N. Bohm, D. Bol, M. Bonera, G. Bosio, L. Bostock, J. Boughn, s. Boughton, R. I. Bozorth, R. M. Bragg Bremer, J. W. Brennemaun, A. E. Brewster, P. M. Briscoe, C. v. Britton, R. B. Brown, R. E. Bruning, H. A. C. M. Bucher, E.

AUTHOR INDEX

603 277,288 421,428 71,93,98,99,101,117,139,146,147 439,442,447,457 317,325,331 668,671,675,677 343,370 374,386 746,751 396,403,751 509,536 745,752 474,481 297,330,353,354,371 516,537 457 291 268,288 268,288 548,563 255,257,261 445,457 445,457 375 62,70,374,375 291,293,330,353,371,414,426,427 38,66,149,155,160,161,171,183, 684,739,751 746,752 343,370 433,434,456 115,147 77,146 593,605 281,288 226,260 57,69,142,148,682,685,716 604 423,428 32,69 414 611,622 613,614,622 268,288 248,261 498,535 594,605 343,370 291,412,427

AUTHOR INDEX

Buchler, R. Buck, D. A. Buckel, w. Buckner, S. A. Buehler, E. Buhrman, R. A. Buravov, L. I. Burger, J. P. Burns, L. L. Burton, E. F. Buttner, H. Byers, N.

Cabrera, B. Cadieu, F. J. Cairns, D. N. H. Caroli, C. Carroll, P. J. Carslaw, H. S. Carter, J. T. Casimir, H. B. G. Cave, E. C. Chandrasekhar, B. S. Chanin, G. Chapnick, I. M. Chester, P. F. Chu, C. w. Clarke, J.

Clausis, K. Clem, J. R. Cline, H. E. Clinton, w. L. Clogston, A. M. Cobble, J. W. Cochran, J. F.

Cody, G. D. Coffey, H. T. Cohen, B. M. Cohen, D. Cohen, M. H. Cohen, M. L.

Cohen, R. W.

Coles, W. D.

313,325 607,622 414,423,427,428 647,652 313,325,331 667,668,675,676 282,288 299,301,330 616,622 334,338,369 144,148,747,753 149,152,160

C

587 291 449,458 327,751 212,260 233,260 157,161 13,54,69,625,629,722 449,450,458 38,69 242,260 340,369 503,535 396,403,426 552,561,563,564,568,581,584, 585,631,636,651,671,732 336,369 218,249,251,252,261 318,319,321,331 144,148 298,330,364,365,371,372 336,341,369 211,217,218,219,231,236,237, 244,245,248,259,260,261 346,347,370,372 682,713,714 203 603,605,675,677,699,753 428,720,721,733 285,374,377,378,380,384,386, 387,412,421,426,427 247,261,366,372,428,746,747, 749,750,752 516,536

xiii

xiv

Colling, D. A. Collins, s. C. Collver, M. Colwell, J. H. Compton, V. B. Conkers loot, H. C. Cooper, A. s. Cooper, J. L. Cooper, L. N.

Corak, w. s. Corenzwit, E. C.

Cosentina, L. s. Courtney, T. H. Cox, J. E. Crane, L. T. Crippa, M. L. Critchlow, P. R. Crow, J. E. Crowe, J. W. Cullen, A. L. Cupp, J. D.

Daniels, A. Danielson, G. C. Darnell, F. J. Darsi, D. Daunt, J. G.

Davies, R. o. Dayem, A. H. Deaver, B. S.

Defrain, A. deGennes, P. G.

Deis, D. W. DeLaunay, J. Delile, G. Denestein, A.

DeSorbo, W. Deutscher, G. Devlin, G. E.

AUTHOR INDEX

291 8,190 272,288 209,627,630 292,341,370,426 343,370 336,369,372 347,370 7,18,20,68,71,74,78,87,88,90, 91,94,95,96,112,125,126,128, 130,135,141,143,147,149,152, 161,292,330,629,692,737,744,752 625,629 291,292,316,331,336,341,346, 369,370,371,372,426 614,622 311,313,348,370 209,360,371,414,427,736,750 545,563 281,288 321 428 614,622 435,457 571,572,584

D

192,203 417,427 291 343,370 193,217,260,291,336,338,339, 341,369,375,386 426 640,651,668,670 7,15,69,539,563,589,591,592, 593,602,605,682,687,738 226,260 59,70,82,144,147,299,317,325, 327,330,331,548,563 426 337,338,369 437,457 152,567,568,569,573,578,579, 584,682 142 751 56,69

AUTHOR INDEX

DeWaele, A. T. A. M. Diepers, H. Dietrich, W. Dillinger, J. R. DiNardo, A. J. Dolecek, R. D. Doll, R. Douglass, D. H. Dowman , J. E. Drangeid, K. E. Dudley, J. C. duPre, F. K. Dy, K. S. Dynes, R. C. Dziuba, R. F. Dzyaloshinski, I. E.

Eagar, T. Eck, R. E. Edelsack, E. A.

Edwards, D. R. Edwards, D. V. Ehrenfest, P. E. Ehrenfests, T. Eigenbrod, L. K. Eilenberger, G. Einstein, A. Einstein, T. H. Eisinger, J. Eliashberg, G. M.

Erdboin, I. Essmann, u. Evans, D. M. Evenson, K. M.

Fack, H. Fairbank, W. M. Fairbanks, D. F. Falge, R. L.

595,605,654,676 328,332 746 428 631,637,650 337,338,369 7,15,38,69,589 291,421,428,628,630 254,255,256,257,261,689 608,622 489,490,496 192,203 143,148 690,742,751 577,578,584 71,93,99,146

E

316,331 67 5,25,603,605,675,677,681,682 684,685,686,687,688,690,692, 694,695,696,697,698,699,701 702,703,704,705,706,707,708, 709,711,712,713,714,715,716, 736,737,750 447,458 375,386 47,69 69 439,442,447,457 144,148,686 719,732 490,496 343,370 167,183,380,381,382,386,387, 392,396,402,403,748,753 226,260 310,331 247,261 571,572,584

F

xv

569,573,580,584,647,652 7,15,38,69,589,602,603,604,605 509,536 224,260,336,369

xvi

Falicov, L. M. Falkoff, D. Fast, R. F. Fermi, E. Ferrell, R. A. Fetter, A. L.

Feynman, R. P. Field, B. F. Filippovich, E. I. Finlayson, D. M. Finnegan, T. F.

Firsov, Yu. A. Fishlock, D. Fitzgerald, R. W. Fiukiger, R. Foner, A. Foner, s. Forgacs, R. L. Forshey, R. K. Fowler, R. D. Fox, G. R. Freake, S. M. Frederick, N. V. Frederikse, H. P. R. Freeman, D. C. Frenkel, R. B. Friedel Frohlich, H.

Fuchs, K. Fulde, P. Furdyna, J. K.

Gallop, J. C. Gambino, R. J. Gamble, F. R. Gange, R. A. Garland, J. W. Garwin, R. L. Gatos, H. C. Gauster, W. F. Gavaler, J. R. Geballe, T. H.

Geiger, A. L.

AUTHOR INDEX

628,630 93,147 516,536 168,169 298,330,695 71,93,99,135,136,139,142,146, 317 70 579,585 284,288 428 565,567,568,569,573,578,579, 584,585,647,652,682,699,700 376,387 750 268,288,291,371 353,354,371 316,717 460,481 595,605,667,676 291 338,369,691 435,457 329,332 749,753 375,377,386,387,426 437,439,457 569,580,584 365,372 7,16,77,146,334,369,387,625, 629,747,753 245,260 749,753 301,330

G

569,579,584 751 420,427,428 616,622 396,403,691,743,751 437,457 291,347,348,370 501,535 313,331 266,282,287,288,316,331,341, 346,369,370,371,427 143,148,247,261

AUTHOR INDEX

Geist, J. M. Geller, s. Gerlach, E. Giaever, I.

Gibson, J. W. Gierke, G. V. Gifford, W. E. Giftleaen Ginsberg, D. M. Ginzburg, V. L.

Giorgi, A. L. Gittleman, J. I. Gladstone, G. Glover, R. E. III

Goldschuartz, J. M. Goldstein, R. Golibersuch, D. Gollub, J. P. Golovashkin, A. I. Gomes deMesquita, A. H. Goodkind, J. Goodman, B. B.

Goodman, W. L. Goodstein, J. Goree, W. s. Gor'kov, L. P.

Gorter, C. J.

Graham, G. M. Graneau, P. Green, D. L. Greenfield, P. Gregory, E. Gregory, W. D.

Greytak, T. J. Grieger, G. Griffin, A. Grigsby, R. Grimes, c. c.

188 341,369,370,375,387,417,427 747,753 7,23,209,247,261,428,562,564, 627,630,724 336,369,375,386 468,481 191 421

xvii

242,260 16,58,68,69,70,149,152,153,161, 163,165,167,187,285,289,294, 297,330,676,748,751 273,288 325,326,331,421,428 397,403,405,426 209,247,261,414,416,418,420, 421,427,428 445,457 341,370 399,403 675,677 747,753 343,370 660,668,675,676,677 7,294,297,330,333,334,369,625, 629 563 732 193,591,592,593,594,721,733 71,112,146,147,167,168,169,183, 294,297,330,748,753 13,54,69,290,426,428,625,629, 722 427 435,457 490,496 343,370 321 25,57,69,143,148,185,209,211, 212,218,219,242,244,245,246,247, 249,251,253,254,255,259,260, 261,321,627,629,681,682,689, 703,737,738,750 291 468,481 749,753 449,450,458 631,635,636,637,640,650

xviii

Gubser, D. U. Gurevich, V. L.

Haden, c. R. Hafstrom, J. W. Hagner, R. Haid, D. A. Hake, R. R. Halas, E. Ha1britter, J. Halloran, J. J. Halperin, W. P. Ham, F. S. Hamer, W. J. Hamilton, C. A. Hamilton, W. o. Hammond, R. H. Hamon, B. V. Hanak, J. J. Hancox, R. Hannay, N. B. Hansen, M. Harding, J. T.

Hardy, G. F. Harland, H. B. Harris, E. P. Hart, H. R. Hartsough, L. D. Hartwig, W. H. Harvey, I. K. Hatch, A. M. Hauser, J. J. Havinga, E. E. Heath, F. G. Hechler, K. Hein, R. A.

Heiniger, F. He1f, E. Hempstead, C. F. Hendricks, J. B. Hende1s, W. H. Henning, C. D.

209,217,246,254,261 376,387

H

327,331 251,261,328,329,332 360,361,371,426 682,707,708,713 291,297,330 486,495 327,329,331,711 366,372 667,675,676 246,261

AUTHOR INDEX

568,584 209,632,636,643,647,651,652 38,587,588,682,701,702 244,272,288,358,359,371 575,576,584 346,347,348,370,421,428,751 465,468,479,481,504,535,536 282,288,427 426 581,585,649,652,660,662,668, 676 333,340,341,369 427 626,629 244,254,503,535 358,359,371 203,327,328,331 568,584 486,495 418,427,747,752 427,743,752 608,622 291 209,224,260,268,270,272,333, 336,369,371,375,386,387,412, 413,414,427,682,689,692,693, 694,700,703,705,706,715,739, 745,746 291,353,354,371,412,427 299,330 306,308,331 501,535 632,651 516,520,536

AUTHOR INDEX

Herold, J. S. Hertel, P. Hill, D. C. Hill, H. H. Hilsch, R. Hitchcock, H. C. Hoffstein, V. Hogan, W. H. Hohenberg, P. C. Holland, L. Hollis Hallett, A. C. Holtzberg, F. Hood, W. Jr. Hopfield, J. J. Hoppe, L. O. Horigome, T. Horinchi, T. Hosler, W. R. Houghton, A. Houston, B. B. Jr. Hsu, F. S. L. Hu, G. R. Huang, K. Hudson, W. Hulbert, J. A. Hull, G. W. Hulm, J. K.

Hume-Rothery, W. Huo, F. S. L. Hurwitz, H. Hutcherson, J. V.

Ihara, S. Inoue, K. Irie, F. Iwase, Y.

Jaccarino, V. Jach, T. Jackson, J. D. Jackson, J. E. Jacobs, A. E.

426 746,751 300,302,303,310,324,331 338,369,691 414,420,427,428 346,370 747,752 192,198 142,299,317,330 247,261 427 375,387,417,427,751 675,677 399,403,742,751 486,495 458 358,359,371 374,375,377,386,387,426 748,753 375,386 313,325,331 299,330 52,53,69,93,147 749,753

xix

426 290,346,370,371,375,387,417,427 291,293,313,330,331,333,340, 341,358,359,369,371,375,386, 391,426,751 406,407,426 343,370 478,481 426

I

458 292 308,331 292,460,481

J

364,365,370,371 603 26,68 248,261 144,148,685

xx

Jaeger, J. C. Jaklevic, R. C. Janocko, M. A. Jansen, H. G. Jayaraman, A. Jennings, L. D. Jensen, M. A. Johnson, R. E. Johnson, R. W. Jones, C. K. Josephson, B. D.

Jostram, P. s. Junod, A.

Kadanoff, L. P.

Kafka, W. Kahn, A. H. Kammerer, O. F. Kamper, R. A.

Kande, E. Kanter, H. Kantrowitz, A. R. Kapitza, P. L. Karimov, Y. s. Kartsev Kazoviskiy Keesom, P. H. Keister, J. C.

Khidekel, M. L. Kim, Y. B. King, H. W. Kinner, H. R. Kirschman, R. K. Kirtley, J. L. Jr. Klaudy, P. A. Klose, W. Knapp, G.

AUTHOR INDEX

233,260 143,148,153,161 313,331 346,347,370 375,387 428 398,403,405,426 613,621,622 190 313,331,426 7,24,62,70,143,149,150,151,152, 153,154,155,157,158,159,160, 161,163,164,169,171,172,175, 177,181,201,209,563,594,617, 622,631,632,633,640,650,657, 658,659,670,676,719,723,748, 753 217,260 353,354,371

K

71,98,99,101,107,121,139,143, 146,147,241,242,244,245,247, 260 478,481 377,387 421,428 628,629,630,632,651,675,677, 721,733,738,750 426,427,428 632,651 713 282,581,585 282 486,495 486,495 217,260 212,244,246,249,251,253,255, 260,261 282,288 143,306,308,330,331,687,744,752 426 486,495 183 486,490,495 437,457 746,751,752 426

AUTHOR INDEX

Kneisel, P. Knight, W. D. Kohn1ein, D. Kohr, J. G. Koonce, C. S.

Korenman, V. Kose, V. Kramer, L. Krikorian, N. H. Krupta, M. C. Kiimme1, R. Kunz, W. Kunzler, J. E. Kuper, C. G. Kusko, A.

Labbe, J. Labusch, R. LaFleur, W. J. Lambe, J. J. Landau, L. D.

Langenberg, D. N.

Langer, J. S. Larkin, A. I.

Lashmet, P. K. Laveriek, C. Lawson, A. C. Lawson, J. D. Leek, G. W. Lee, W. D. Leighton, R. B. Leopold, L. Lerner, E. Leung, M. C. Levanyuk, A. P. Levinstein, H. J. Levy, R. H. Lewin, J. D.

327,331 244 423,428 314,315,316,331 373,378,380,386,387,389,412, 427,682,686,738,739,740,741, 742,745,751

xxi

299,330 556,564,569,573,580,584,647,652 291 273,288 273,288 142,144,148,175,183,685 291 313,325,331,338,343,369,370 625,629 433,456

L

364,365,367,372,746,752 675,677 347,370 143,148,153,161 16,58,70,78,133,146,147,149, 152,161,163,164,165,167,183, 289,297,300,330,676,737,748 67,152,565,567,568,569,573,578, 584,631,633,640,650,682,695, 696,698,702,703,704,714,716, 719,735,739 684 188,376,387,643,645,647,652, 684,695 188 504,536 268 474,481 616,622 489,496 70 57,69 291 144,148,748,753 366,372 338,369 516,536 317,320,321,323,331,536

xxii

Lewis, H. W. Leyendecker, A. J. Lifschitz, E. M. Lindsay, J. D. G. Little, W. A. Livingston, J. D. London, F.

London, H. Long, H. M. Longacre, A.

Longinotti, L. D. Lord, R. Lucas, E. J. Lucas, G. Luck, D. L. Luders, K. Lukens, J. E. Luo, H. L. Lykken, G. I. Lynton, E. A.

Macdonald MacFarlane, J. C. MacNab, R. B. MacNair, D. MacVicar, M. L. A.

Maddock, B. J. Madey, J. Mahler, W. Maita, J. P. Maix, R. Maki, E. Mapother, D. E. March, R. H. Markowitz, D. Marshak, H. Martens, H. Martinelli, A. P. Mason, P. Mathes, K. N. Mathews, W. N. Jr.

Matisoo, J.

AUTHOR INDEX

57,69 377,387 133,147 338,369 9,81,146,285,544,563,732 142 7,13,14,15,16,32,35,69,141, 147,544,563,589,590,605,722 7,13,14,15,16,35,69,722 437,439,442,447,457 631,640,641,642,647,650,651, 652 336,346,369,370,372 514 516,536 748,758 490,496 423,428 672,673,674,675,676,677 291,358,359,371,391,403,426 143,148,247,261 241,242,245,260,414

M

736 569,584 486,495 282,288 247,249,251,254,255,256,257, 271,328,329,330,332,689,704 501,535 603 343,370 338,343,346,369,370,426 516,522,537 142,327,331,748,753 626,629 255,257,261 241,242,244,245,247,260 629 328,332 459,470,472,478,481,682,712 703 445,457 27,71,144,148,185,681,682,685, 686,687,737,739 437,457,607,611,617,619,623, 676,682,696,697

AUTHOR INDEX

Matricon, J. Mattheiss, L. F. Matthias, B. T.

Mattis, D. C. Maxwell, E. Mazelsky, R. McAshen, M. S. McCall, D. M. McCarthy, s. McConnell, McCumber, D. E. McDermott, R. C. McDonald, D. G. McFee, R. McInturff, A. D. McLachlan, D. McMahon, H. o. McMillan, B. McMillan, W. L.

McNichol, J. J. McN iff, E • U. Meats, R. J. Megerle, K. Meissner, W.

Melchert, F. Mellors, G. W. Mendelssohn, K. Mercereau, J. E.

Merriam, M. F. Meservey, R. Meussner, R. A. Meyer, G. Meyerhoff, R. W. Migdal, A. B. Miles, J. L. Milford, F. J. Miller, R. C. Minnich, s. H". Minors, R. H.

327,331 372,399,403 263,266,268,282,291,292,295, 316,333,334,335,336,338,341, 342,353,357,362,366,367,369, 370,372,406,407,410,426,427, 428,682,685,690,691,692,693, 694,706,707,708,709,710,711, 712,714,739,745,751 42,57,69,140,246,261,327,331 77,146,222,241,260 375,386 604,682,709,710,711 428 277,279,288 420 563,643,647,651 631,650 571,572,584,631,647,650 436,437,457 322,323,331 242,245,260 191 286,287,384 338,392,393,395,397,403,426, 683,689,690,691,740,741,751 613,614,622 316,331 445,457

xxiii

627,630 7,12,13,18,27,334,588,589,591, 593 569,573,580,584 439,440,457 32,69 65,67,143,153,161,165,173,176, 177,178,179,180,183,563,591, 594,603,605,668,669,670,732 426 143 209 516,522,537 328,332,433,439,442,447,448,457 380,387 249,261 375,386 375,386 435,457 449,458

xxiv

Mitchell, E. N. Molokhia, F. Monin, F. J. Montgomery, D. B. Moravesik, M. J. Morin, F. J. Morris, K. Morrison, D. D. Morrison, W. A. Morse, P. M. Moser, S. Mota, A. C. Mott, N. F. Mueller, F. M. Meheim, J. Muller, A. Muller, J.

Nahauer, M. Nam, S. B. Nambu, Y. Naugle, D. G. Neighbor, J. E. Nesbitt, L. B. Nethercot, A. H. Neuringer, L. J. Newbower, R. S. Newhouse, V. L. Nicol, J. Nisenoff, M. Norman, J. C. Norris, W. T. Notaro, J. Notarys, H. A. Novikov, Yu. N. Nozieres, P.

Oberhauser, C. J. Ochiai, S. 1. Ochsenfeld, R. 01ien, N. A. Olsen, J. L. Onnes, H. K.

AUTHOR INDEX

148,247,261 675,677 343,370 516,536 69 426 568,569 308,309,310,324,330,331 720,721,733 43,69 421,428 268,288 426 403,396 412,427 353,354,371 291,353,354,357,358,363,363, 366,371,412,427

N

7,15,38,69,589 744,749,752,753 121,147 247,261 231,260 77 242,260 325,331 675,677 611,612,622 249,261 594,603,605 675,677 449,458 439,442,447,457 183 282,288 78,135,136,146,147

o

486,495 329,332 7,12,13,18,27,588,589,591,593 721,733 423,428 7,8,10,11,24,27

AUTHOR INDEX

Opfer, J. Oswald, B. Ouboter, R. deB. OWen, c. S.

Page, C. H. Palmy, c. Papp, E. Parker, W. H. Parks, R. D. Pastuhov, A. Patterson, A. Paul, H. Paul, S. Pearsall, G. W. Pessa11, N. Peterson, R. L. Petley, B. W. Pfeiffer, E. R. Phillips Pierles, R. Pillenger, W. L. Pines, D. Pipes, P. B. Pippard, A. B. Pollard, E. Price, P. J. Purcell, J. R.

Rabinowitz, M. Ralls, K. M. Ramsay, W. Pau, F. Raub, E. Rauch, G. C. Pay1, M. Rayleigh Reed, T. B. Repici, D. Reppy, v. Reuter, G. E. H. Reynolds, C. A. Richards, P. L. Richardson, R. C. Rickayzen, G.

603 474,481 595,605,654,676 564,690

P

576,584 423,428 246,261 152,565,569,584 65,67,69,71,93,98,99,146,426 435,457 313,331 435,457 See Saint Paul 348,370 358,359,371 556,564 569,579,584 374,375,386,426,428 218 32,69 217,260 77,78,135,146 604 39,42,57,69,563,722,732 292 246,261 465,481,516,520,537

R

328,332 291,298,304,330 6 471,472,481 343,370 311,613 347,370 574,584 347,360,370 57,69 667,675,676 41,42,69,239,260 7,77,146 631,633,637,640,650,732 667,675,676 135,136,147,583,585

xxv

xxvi

Ricketts, R. L. Rieger, T. J. Ries, R. P. Rigamonti, A. Roberts, B. W. Roddy, T. J. Rodgers, E. C. Rogers, J. D. Rorschach, H. E. Rose, R. M.

Rosenblum, B. Ross, J. S. H. Rothwarf, A. Rowell, J. M. Ruccia, F. Ruh1, W. Russell, C. M. Russer, P.

Sadagopan, v. Saint-James, D. Saint Paul Salve, F. J. D. Jr. Sands, M. Sard, E. Sarma, G. Sasaki, W. Sass, A. R. Satterthwaite, C. B. Saur, E. J. Sca1apino, D. J.

Schaeffer, G. M. Schaw1ow, A. L. Schmid, A. Schmidt, H. Schmidt, P. Schmitt, R. W. Schneider, R. S.

AUTHOR INDEX

305 165,172,177,183,180,748,753 626,629 281,288 244,254,334,342,369,426 347,370 447,449,450,451,452,453,457,458 516 157,161 247,249,254,261,289,291,298, 300,302,303,307,308,309,310, 316,318,319,324,328,329,330, 331,332,333,339,682,704,705, 709,739,744 325,326,331 486,495 77,81,126,146,147,685,747,753 23,64,70,690 435,457 414,416,418,420,424,427,428 426 35,648,652

S

291 59,70,142,148,299,301,330 341 427 70 631,647,650 142 244,246,255,256,258,261 614,615,622 426,625,629 291,360,361,370,371,426 67,107,143,147,163,165,172, 176,177,178,179,180,183,284, 285,286,564,568,584,626,631, 633,640,650,682,683,684,685, 686,687,688,690,691,694,698, 732,739,748 384,387 56,69 748,753 695 282,288,427 720,721,733 81,147

AUTHOR INDEX

Schooley, J. F.

Schrader, H. J. Schrieffer, J. R.

Schroen, w. Schueler, c. Schuster, H. Schwartz, B. B. Schwenter1y, S. w. Schwidta1, K. Scott, N. R. Seidel, G. Seiden, P. E. Se11maier, A. Senderoff, S. Seraphim, D. P. Seraphim, G. R. S. Serin, B. Shaktarin Sham Shamrai, V. J. Shanks, H. R. Shapira, Y. Shapiro, S.

Shchego1ev, I. F. Sheahen, T. P. Shen, L. Y. L. Shepard, L. A. Shepe1ev, A. G. Sherri1, M. D. Shiffman, c. A. Shirane, G. Shoenberg, D. . Shu1ishova, O. I. Siebenmann, P. Silsbee, F. B. Silver, A. H.

209,213,224,260,269,270,374, 375,376,386,405,426,428,625, 627,630,682,689,692,693,738, 745

xxvii

569,573,584 7,18,20,25,68,71,74,78,88,90, 91,93,94,95,96,99,107,116,121, 122,125,126,128,130,135,141, 143,149,152,161,292,330,398, 403,405,426,629,683,692,737, 743,744,745,752 649,652,702 604 746,752 143,548,563 667,675,676 572,584,700,702 607,622 217,260 375,387,417,427,741,751 190 439,440,457 613,614,622 631,650 77,142,241,242,260,414,427 486,495 683 360,362,371 417,427 325,331 24,66,70,209,249,251,261,631, 635,636,637,638,639,640,641, 647,650,651,652,682,687,688, 694,697,705,707,711,738 282,288 211,218,219,244,245,259 392,395,403 291,307 246,248,261 418,420,427 242 746,752 32,38,69,217,260,427 292 377,387 11 152,156,161,563,595,605,632, 650,654,656,657,658,660,668, 671,676

xxviii

Simon, Y. Sinha, S. K. Slaughter, R. J. Smith, J. L.

Smith, P. F. Smith, P. H. Smith, T. F.

Smith, T. S. Sommerhalder, R. Sondheimer, E. H. Souders, T. M. Soulen, R. J. Jr. Soymar, K. Spitzli, P. Spurway, A. H. Standenmann, J. L. Stans, H. Stekley, Z. J. J.

Stephen, M. J.

Sterling, S. A. Stewart, w. c. Stewart, W. D. Stolan, B. Stolfa, D. L. Stoltz, o. Stoner, E. C. Straus, L. S.

Strauss, B. P. Stritzker, B. Strnad, A. R. Strobridge, T. R. Strong, P. F. Strongin, M. Stuart, R. W. Suhl, H. Sullivan, D. B. Sun, R. K. Superata, M. A. Suris, R. A. Swenson, C. A. Swift, D. A. Swiggard, E. M. Swihart, J. C. Szklarz, E. G.

AUTHOR INDEX

752 399,403 447,458 190,483,486,487,490,495,496, 682,705,706,707 536 249,261,317,320,321,323,331 279,288,338,362,366,368,369, 412,428 338,339,369 608,622 41,42,69,239,260 577,578,584 209,627,630 246,261 371,747,753 317,321,323,331 353,354,371 714 462,463,481,486,495,497,500, 501,511,512,514,516,535,536, 682,702,708,709,713 306,308,325,330,632,651,687, 748,753 631,651 563,614,615,622 643,647,651 743,744 660,668,676 327,331 32,69 143,148,212,244,246,249,251, 253,260,261,738,750 318,319,331 414,427 306,308,331 195 249,261 222,241,260,288,421,428 185,203 287,288 556,564,581,585,643,645,647 328,332 212,242,260 366,372 428 449,458 375,387 569,584,626,629,691 273,288

AUTHOR INDEX

Taber, M. Taber, R. Tachikawa, K. Takken, E. H. Taylor, B. N. Taylor, M. T. Terwordt, L. Testardi, L. R. Theurer, H. C. Thiene, P. Thomas, E. J. Thompson, R. S. Thullen, P. Thurber, W. R. Timmerhaus, K. D. Ting, C. s. Tinkham, M. Tinlin, F. Tiza, L. Toepke, I. L. Tomasch, W. J. Toots, J. Toth, L. W. Trauble, H. Tsebro, V. I. Tsui, D. C. Tsuneto, T. Tung, Y. W. Turneaure, J. P.

Uchida, M. Ulrich, B. Urban, E. W.

Valatin, J. G. van Gelder, A. Van Kempen, H. Van Maaren, M. van Reuth, E.

Vant-Hull, c. van Vucht, J. Vernon, F. L.

P.

H. C.

C. H. N.

xxix

T

603 604 292,460,481 721,733 67,152,565,584 449,458 144,148,685 338,367,369 418,427 581,585,649,652,660,662,668,676 142,168,169 428 486,487,490,495,496 377,387 487,495 287,288 675,676 486,495 57,69 426 212 579,581 476,481 310,331 272,751 738 164,167,183,748,753 377,387 327,328,331

U

358,359,371 631,651 744,752

V

115,147 299,330 628,630 384,387,427,752 346,348,362,364,365,367,371,414, 427 591,605 343,370 632,651

xxx

Victor, J. M. Vieland, L. J. Vielhaber, E. Viet, N. T. Vincent, D. A. Vinen, W. F. Voigt, H. Vol'pin, M. E. Von Gutfeld, R. J. Von Minnigerode, G. von Molnar, s. Von Riedel, E. Vystavkin, A. N.

Wada, Y. Waldram, J. E. Wallecka, J. D. Walters, c. R. Warburton, R. s. Waring, R. K. Warnick, A. Waterstratt, R. M.

Watson, J. H. P. Webb, W. W.

Weber, J. Weger, M. Weisbarth, G. S. Weissman, I. Wells, J. S. Wenner, F. Wernick, J. H. Werthamer, N. R.

Westendorp, W. F. Weston, R. vilexler, A. Wheatley, J. ~. White, R. W. Wilhelm, J. o. Wilkins, J. W. Willens, R. H. Williamson, S. U. Willis, W. D.

327,328,331 348,370,372 358,359,371 327,328,331 563 687 437,457 282,288 242,260 418,427 375,387,417,427 632,647,651 648,652

W

AUTHOR INDEX

626,629 254,255,256,257,261,563,689 71,93,99,135,136,139,146 317,320,321,323,331,536 673,674,675,676,677 291 595,605,667,676 346,348,351,352,362,363,370, 371,384,414,427 428 38,632,651,653,667,671,674, 675,676,677,682,687,694,695, 696,698,699,703,704,711,714, 715,716,752 604,605 364,365,372,747,752 247,261 327,328,331 571,572,584 575,584 291,313,325,331,338,343,369,379 57,144,299,330,418,427,518, 632,638,639,641,647,651 478,481 568 625,629 675 338,369 334,338,369 107 316,331,346,370,371,747,752 301 563

AUTHOR INDEX

Wilman, H. Wilson, M. N. Wipf, s. L. Witt, T. J. Wittgenstein, F. Wittig, J. Wizgall, H. Woo, J. Wood, E. A. Wood, J. H. Woods, A. D. B. Woodson, H. H. Wright, W. H. Wuhl, H. Wulff, J.

Wyatt, A. F. G. Wyder, P.

Yamafuji, K. Yang, C. N. Yangubskii, E. B. Yaqub, M.

Yoshihiro, K.

Zachariasen, W. H. Zanona, A. Zavaritskii, N. V. Zeller, G. Zimmer, H. Zimmerman, J. E.

Zucker, M. Zvarykina, A. V.

xxxi

247,261 317,320,321,323,331,462,481,536 322 579,585 506,509,536 266,288 291 748,753 343,370 256,257,258,261 395,403 486,490,495,496 77 248,261,414,427 291,298,307,311,313,318,319, 330,331,348,370 247,261 628,630

Y

308,331 149,152,160 288,292 217,231,236,237,248,260,261, 375,386 244,246,255,256,258,261

Z

273,291,336,369 433,434,456 244,247,255,256,257,261 428 648,652 156,161,556,563,564,581,585, 595,603,605,632,647,648,651, 654,656,657,658,662,668,671, 673,674,675,676,677,699,749, 750,753 241,242,260,414 282,288

SUBJECT INDEX xxxiii

A

A-15 compounds, crystallographic order in crystal structure of density of states in electron-phonon interaction in general discussion of

353-368 275,291,339,345-347 365 747 273-279

history of 333-368 metallurgy of 292,310-313

353-368 search for high T in Ac losses in supercondUctors; see Electromagnetic

absorption Ac susceptibility technique, 212-241

and isotropic resistivity 226-231 and anisotropic resistivity 231-236

Alternators, superconducting; see Motors, super conducting

Aluminum, critical field curve of enhancement effects in T of a~ a thermometric standard

Anomalous skin effect

Band structure, and enhancement of T c BCC solid solutions, Tc values of, tabulated

BCS theory,

B

effective electron-electron interaction in gap equation in general discussion of and Meissner effect and penetration depth pairing interaction in strong coupling corrections and thermodynamics transition temperature in

Beta-Tungsten compounds; see A-15 Bolometers, superconducting,

noise equivalent power of Boundary scattering in

superconductors

588 415,416,420 294 626 140

391-393

290 71-146

126-130 126 18 135-141 141 87-93

to 394 130-135 141

compounds

724

241-247

Pages 1-428 will be found in Volume 1, pages 429-778 in Volume 2.

xxxiv

C

C-14,C-15 structure C-16 structure Cadmium,

T of Chemicgl potential,

in BCS theory in G-L theory in nonequilibrium superconductors

Coherence length, in BCS theory general definition of in G-L theory Pippard

Computer devices; see also Cryotrons

Constitutive equations, and nonlocal effects for superconductors

Cooper's problem Copper,

density of resistivity of specific heat of and stabilization of superconductors thermal conductivity of

Coulomb pseudopotential, definition of

Critical current, definition of and flux flow and stabilization problems in Type II materials

Critical magnetic field; see also field anomalous in BCS theory defined H , defined HC 1, defined HC2 , defined HC3 , defined iR4 phenomenological theories of Type II compounds, tabulated in Type II materials

SUBJECT INDEX

278,290,313 278

294

96 169-171

171-182

141 18 58,166 40-42

607-622

38-42 26-27,36-38 87-90

500 500 500

317-324,500-504 500

394

11 306-310 461-464 296,306-310

Upper critical

302-305 134-135 10,13 59 59,297 59,299 299-300 48-50

499 297-306

SUBJECT INDEX

Critical temperature; see Transition Cryotrons,

effective noise temperature of and Josephson junctions in logic circuits as memory elements switching time in thin film wire wound

Demagnetizing coefficient, definition of

Dielectric function, for free electron gas

Dielectric resonances in superconductors

Effective electron-electron interaction, in BCS theory calculation of, in RPA in jellium model in semiconductors

Effective mass of electrons, due to phonon dressing

Electrical diffusivity, defined

Electrical resistance,

D

E

in superconductors, upper limit to

Electric dipole moment of 3 He , detection of, using super conducting techniques

Electromagnetic absorption, and nonlocal effects in magnetometer circuits in super conducting motors in a superconducting ring in superconducting transmission lines in Type II materials

Electron-electron interaction, general discussion of

Electron-ion interaction

temperature 611-617 724 617-622 614 614-617 613,621 611-617 607

31

391

390,393

88,126-130 399-402 86 376-387

740

500

8-9

603-604

38-42 661 484 153,157

438-444 324-329

xxxv

77,389-393,741-744 401

xxxvi

Electron-phonon interaction, in A-15 compounds in BCS theory dependence of, on phonon frequency in strong coupling theory upper limit of and virtual phonons

Energy gap, anisotropy of in BCS theory definition of and density of states in semiconductors as a voltage standard

Energy gap equation, in BCS theory

Fermi-Thomas approximation, and jellium model

Fluctuations, of order parameter and persistent currents

F

Flux detectors; see Magnetometers Flux exclusion; see Meissner effect Flux flow,

and Hall effect and pinning effects in type II superconductors

Fluxoid, defined quantization of

Flux quantization, discovery of and the London equations in a superconducting ring

Flux quantum, definition of value of

Flux transformer, use of

Flux vortex; see Vortices

Gapless superconductivity

G

747 18,78-87

SUBJECT INDEX

389-396 394-396,741 402 79

247-255 126 72-74 17 379-386 582-583

583

85

683-684 9

308 306-310 306-310,744

543 542,546,590

15 36-38 151,655-659

38 589

672-673

17,142

SUBJECT INDEX

Gallium, boundary scattering effects on T critIcal field curve of energy gap anisotropy of foils, effect of annealing on T mfiltiple energy gaps in T of t~ansition width in

Ga-In alloys, magnetic susceptibility of

Generators, superconducting; see Motors, superconducting

Germanium, conduction band of

Giaever tunneling; see Tunneling of electrons

Ginzburg-Landau K parameters, K defined Ka, KI, defined and normal state resistivity

Ginzburg-Landau theory, time dependent

Gor'kov equations Gor'kov factorization Gravitational radiation,

detection of using superconducting techniques

Green's functions, and boundary conditions and Nambu matrix formalism and quasiparticles and second quantization time ordering of

Hafnium, superconductivity in T of

H

Hard snperconductors; see Type II superconductors

Hartree-Fock approximation Helium,

density of dielectric strength of film boiling in nucleate boiling in

242-247 218 217,244,252-258

221 252-254 212,217,294 211,217

224

375

58-59,297 297 297 57-60,165-170 167-170 120 112-117

603-604 93-122 99-101 120-122 109-111 93-97 97

270 294

112

500 445-447 462 461

xxxvii

xxxviii

specific heat of thermal conductivity of

HfNb, T of

HfTa, c T of

Hydroggn, possible superconductivity in

Isotope effect, in Uranium

In-Bi alloys, flux pinning in

Indium,

I

and boundary scattering effects on T critrcal field curve of and electron concentration effects on T T of c

I 'd' c rl. l.um, T of c

Jellium model Johnson noise Josephson effect,

ac dc

J

and the determination of e/h and flux quantization Gibbs energy in in a magnetic field in nonequilibrium superconductors in a superconducting ring as a voltage standard

Josephson equations Josephson junctions,

as computer devices and coupling to electromagnetic radiation creation of suitable oxide layers in use in dc voltage measurements equivalent circuit for

500 500

290

290

271

16,76 287

309

242-247 14,588

416 294

294

82-87

SUBJECT INDEX

628 22-24,62-67,149-160 65 64 579-580 151 659 66-67

171-182 149-160,547-550,657 565-582,725,728 64,542,634

617-622,726,728

570-571,635

572 569-573 553-556,643

SUBJECT INDEX

fabrication of noise equivalent power of point contact as a radiation detector as a radio frequency mixer Reidel peak phenomenon in solder blob stabili ty of as a thermometer thin film wave function in

Josephson tunneling; see Tunneling of electrons, Josephson effect

Josephson tunneling current, electromagnetic field dependence

329,662-666 725,728 551,569,632 631-650 637-642 632 551,569,632 572,702,705 726 551,569,632 555

of 557,566,635-642 as a function of junction bias 552 magnetic field dependence of 559,618 Reidel peak phenomenon in 647-648 spatial variation of 558-563 temperature dependence of 628

Kapitza boundary resistance

Landau damping Lanthanum,

T of

K

L

581

391

294 Laves Shases; see C-14, C-15 structures Lead,

critical field curve of phonon modes in T of

Londoncmoment London equations London superconductor,

defined

Magnetic diffusivity, defined

Magnetic field,

M

shielding against, using superconductors

Magnetic pressure, in a solenoid

14,588 572 294 544 34-38

42

317

590-605

532-533

xxxix

xl

Magnetic susceptibility; see also Ac susceptibility technique general discussion of skin depth dependence of

Magnetization, and demagnetizing coefficient general discussion of in mixed state of Type I superconductors of Type II superconductors

Magnetometers, superconducting, as a communication device design of equivalent circuit for practical applications of principles of operation of sensitivity of and thermometry

Magnets, superconducting, applications of in fusion research magnetic pressure in mechanical fabrication of performance of quenching of stabilization of

winding materials for

Matthias' rules Maxwell's equations,

applied to superconductors Meissner effect,

departures from ideal definition of derived in BCS theory

Mendelssohn sponge Mercury,

critical field curve of T of

c Metallurgy of superconductors, general discussion of and manufacturing methods

Molybdenum, alloys, T of T of c cd' Motors, supercon uctlng,

ac losses in advantages of performance of

SUBJECT INDEX

226-241 222,229,237-241

31 28-33 21 19,29,33 20 594-596,653-676 727-728 560-563,667-676 661 675-676 659-667 602,672,727 675 459-480,497-535 520-523,726-727 464-480 516,532-533 514-516,531-535 509-514,516-520 509-514 461-464,500-506, 524-531 316-324,460-461, 498-499,506-509 266,284,406-407

26-28

589,591-594 12 135-141 32

14,588 294

289-332 310-316

409 294 483-494 484 494,729 490-491

SUBJECT INDEX

Nambu Matrix formalism NbsAl,

N

critical current in critical field curve of fabrication techniques for H of u~ed as a magnet material T of

Nb 3 (AI C; 8 Ge. 2 ) ,

critical field curve of H of TC of

(Nb 3AIT Nb Ge, T of

Nb 3AU,c T of

Nb3(AuC;9sRh.os) , T of

NbC, c

TC of Nb 3 Ge,

Tc of NbN,

T NbN-Nbe

Tc Nb 3 Sn,

of alloys, of

critical current in critical field curve of density of fabrication techniques for use in power transmission lines stabilization of T of tHermal conductivity of specific heat of upper limit to T in

Nb-Sn alloys, c T of

NbTi, c density of specific heat of T of tHermal conductivity of

Nb-Ti alloys, critical current in fabrication techniques for

120-122

315 22,314 316 316,460 508 409,499

22 460 460,499

409

409,348

409

290

409

290,409

290,409

499 22 500 310-313 435-436 323-324 409,460,499,500 500 500 286

347

500 500 460,500 500

305,499 310-311

xli

xlii

NbZr,

H of u~e as a magnet material use in power transmission lines T of s~abilization of

density of specific heat of T of tRermal conductivity of

NbZr alloys, fabrication techniques for use in power transmission lines T of

N ' b' c ~o ~um,

critical current in critical field curve of electromagnetic absorption in phonon modes in use in power transmission lines pressure dependence of T in T of c

c

o

Off diagonal long range order Order parameter,

in BCS theory and gapless superconductors general discussion of spatial variation of

Organic superconductors Osmium,

T of c

Protoactinium, T of

P

Pairin§ interaction Paramagnetism in superconductors Penetration depth,

in BCS theory in London theory temperature dependence of

Phenomenological theories of superconductivity

Phosphorous, superconductivity in, under pressure

SUBJECT INDEX

304 506-509,510 435-436 290,293,499 317-323

500 500 290,409,500 500

310 435-436 293

303 588 327-329,438-442 572 436,438-442,452 423-425 294,394

152

132 142 57-60 174-182 269,460

294

338 87-93 298-299

141 14,36,56,571 56,72,75

25-70

266

SUBJECT INDEX

Pippard superconductor, defined

Plasma frequency, in the free electron gas

Propagators; see Green's functions Proximity effect,

and enhancement of T in Josephson junctioRs and magnetometers

Quasiparticles,

Q

recombination time of, in superconductors

Quenching of a superconductor

R

Radiation detectors; see Johsphson Refrigeration techniques,

and Joule-Thomson cycle and Stirling cycle and Vuilleumier cycle and work extraction cycles

Riedel peak phenomenon, in Josephson junctions

Rhenium, T of

Rh-Ir ~hloys, T of

Rhodiuffi, Tc of (extrapolated)

Rh-Os alloys, T of

Ruthenrum, isotope effect in Tc of

Second quantization Semiconductors,

energy gap of many valley phonon modes in

S

and polar phonon interactions superconductivity in

42

391

418-421 177-182 669-670

583 509-514

junctions 185-205 186-188 192-200 192 188-191

632,647,648

294

268

267-269

268

287 294

93-96

379-386 265 376-379

xliii

377-378 373-387,412,416-417

xliv

Sesquicarbide structure, superconductivity in

Shielding of electric and magnetic fields, using superconductors

Skin depth,

SLUG,

anomalous normal

Clarke type defined use in magnetometers

Sommerfeld condition, and superconducting ring

Sodium chloride structure, superconductivity in

Soft superconductors; see Type I superconductors

Splat cooling SQUID,

definition of sensitivity of

Stabilization of superconductors adiabatic cryogenic dynamic effect of twisting on

Superconducting devices, physics of summarized

Superconductivity, discovery of early theories of, reviewed future of unanswered questions in

Supercooling effect in superconductors

Superheating effect in superconductors

Surface sheath, and H

C3

Tantalum,

T

Coulomb pseudopotential in critical field curve of T of c

SUBJECT INDEX

273

587-605

40,237 39,213,224,230

551 671 561

654

273

274

671 725 317-324,500-506 462-463,480 461,465 462-464 323,504-506 688-712,724-732 539-563 540-541

7-8 13-18 682-716 740-749

52-53

52-53 50-54 59,299

392 14,588 294

SUBJECT INDEX

Ta-Hf alloys, T of

TechneEium, T of

Thermal diffusivity, defined

Thermodynamics of superconductors, in BCS theory general discussion of and nonreversible effects

Thermometry, by critical magnetic field by heat capacity techniques using Josephson junctions noise standards using T

Thomas-Fermi screeningClength Thorium,

T of Time dgpendent Ginzburg-Landau

theory Tin,

critical field curve of and effect of electron density on Tc and mean free path effects on T T of

Titanifim, T of

Ti3Ir,c T of

Ti-Mo ~hloys, T of

Thalli8m, critical field curve of and effect of electron density on T c and pressure dependence of T T of c

TomascR effect Transition metals,

and alloying effects on T and bond structure effect~ on T and g/a ratio effects on T and impurity effects on T c Tc of c

xlv

293

294

317,500

130-135 42-54 50-54

217,626 625-626 627 628-629 217,627 168

294

165-170

14,588

416

415 294

294

348

290

14,588

416 423 294 212

293,405-412

271-273 295 405-412 271,406

xlvi

Transition temperature, and alloying effects annealing effects on in BCS theory and bond structure and Curie-Weiss behavior and Debye temperature defined dependence on average phonon frequency and e/a ratio electron density effects on and electronic specific heat and enhancement effects

and ferroelectric behavior and ferromagnetic behavior highest value of and impurity effects isotope effect on mean free path effects on and melting point pressure effects on

and proximity effect of semiconductors size effect in and strain effects in strong coupling theory tabulated values for elements

Transmission lines, superconducting

Tungsten-bronze structure, superconductivity in

Tungsten, density of states in T of

TunnelCjunctions; see also Josephson junctions selection of materials for

Tunneling of electrons, into bulk materials introduction to in multiple gap superconductors selection rules for used as a voltage standard

Two-fluid model, Gorter-Casimir introduction to

SUBJECT INDEX

269,405-413 362 127,740 271-273 282 337-338 9

742 263-288,335-342,355 414-417 338-340 263-288,389-402, 405-425 281-282 282-284 396,460 405-413 16 241-247,414-415 265-277 266-270,280,397, 422-425 418-421 384 421-423 270-271 394-396,740 11,294

433-456

280

393 272,294

329-330

247,259 21-24 251-254,738 255-259 582-583

54-57 13-14

SUBJECT INDEX

Type I superconductors, intermediate state in magnetic properties of

Type II superconductors, critical current in discovery of electromagnetic absorption in flux flow in magnetic properties of mixed state in and Pauli paramagnetism

Upper critical field, defined

U

enhanced by spin paramagnetism limited by Pauli paramagnetism

Uranium, isotope effect in T of c

Vanadium, critical field curve of T of

V3 Al , c Tc of

V3Au, and long range order in T of

V3Ga, c critical field curve of H of

v

p~essure dependence of Tc in T of c V,+.sGa, critical field curve of

V-Ga alloys, H of TC of

V3 Ge , c

142 19,29,33

296,306-310 21 324-329 306-310,744 20,297-305,744 20-21,59,142-143 298

59,297 299 298

287 294

14,588 294

357-358

348-352 348

22 460 424 290,409,460,499

22

344 344

V2 Hf , pressure dependence of Tc in 424

T of 290 Virtual phonons in electron-electron

interaction 79

xlvii

xlviii

Voltage divider, cryogenic

Voltage standard, and Giaever tunneling and Josephson effect

Vortex state, and Type II superconductivity

Vortices, and electromagnetic absorption Lorentz force on and pinning effects

pinning force on repulsion between

V3 Si , critical magnetic pressure T of

field curve of susceptibility of dependence of T in c

u~per limit to T in c V-Ti alloys, T of

V3-X alloys, search for high transition temperature in

VZr, T of c VzZr, T of c Vz (Zr. 5Hf. 5) T of c

W

Weak superconducting link; see Josephson junction

W-Re alloys, Tc of

Y-Rh alloys, Tc of

Zinc, critical field curve of T of . c Zlrconlum, T of c

Y

Z

573-582

582-583 565-582

20-21

324-327

SUBJECT INDEX

306 306-308,324-329, 591,744 325 317

22 278 279,424 290,340,409 286

290,293

356-360

290

290

290

290

407

588 294

294

the science and technology of superconductivity: proceedings of a summer course held august...

Documents