120790464 superconductor

The New Superconductors

SELECTED TOPICS IN SUPERCONDUCTIVITY

Series Editor: Stuart Wolf

Naval Research LaboratoryWashington, D.C.

CASE STUDIES IN SUPERCONDUCTING MAGNETSDesign and Operational IssuesYukikazu Iwasa

INTRODUCTION TO HIGH-TEMPERATURE SUPERCONDUCTIVITYThomas P. Sheahen

THE NEW SUPERCONDUCTORSFrank J. Owens and Charles P. Poole, Jr.

QUANTUM STATISTICAL THEORY OF SUPERCONDUCTIVITYShigeji Fujita and Salvador Godoy

STABILITY OF SUPERCONDUCTORSLawrence Dresner

A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment.For further information please contact the publisher.

The New Superconductors

Frank J. OwensArmy Armament Research Engineering and Development Center Picatinny, New Jerseyand Hunter College of the City University of New York New York, New York

and

Charles P. Poole, Jr.Institute of SuperconductivityUniversity of South CarolinaColumbia, South Carolina

Kluwer Academic PublishersNew York / Boston / Dordrecht / London / Moscow

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To our respective wives,

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for encouraging and supporting us

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Preface

In late 1986 and early 1987, a major scientific breakthrough occurred in materials science. A new class of materials was discovered that displayed superconductivity at unusually high temperatures. Superconductors are metals capable of conducting electricity with no losses of any kind. Prior to this the phenomenon was observedonly at very low temperatures, namely, temperatures far, far below the coldest everrecorded in the aretic or antarctic regions of the earth. Prior to 1986 this was the research preoccupation of only a handful of scientists. In 1987 all of this changed due to extensive news coverage of the breakthrough and the recognition of itsenormous economic potential.

Because of the technological possibilities of the discovery, ranging from very fast computers to levitated trains, it is important for national development to havenonspecialists in superconductivity become familiar with the field and its possibleapplications. Managers ofcorporate laboratories, government officials, and univer-sity administrators, most of whom are not specialists in this field and many of whom are not scientists, will be involved in allocating funding for research in this fieldand its applications; these same people will affect future directions oftechnologicaldevelopment. Even now numerous patents are being evaluated by personnel withlittle understanding of the subject. Of course general public interest has also been piqued by extensive news coverage of dramatic superconductivity applications,such as the magnetic resonance imaging (MRI) diagnostic technique in medicineand levitated trains.

Because of these factors, there is a need for a clear explanation of supercon-ductivity at a level suitable for the nonspecialist. Further, nine years worldwideresearch devoted to the new materials have provided a much clearer picture of their nature and potential applications, making the time ripe for a discussion of these developments. For example the past year has witnessed major progress in over-coming obstacles hindering the development of commercial higher field super-conducting magnets, and a prototype levitated train is now operating in Japan.

vii

viii PREFACE

This book assumes some prior exposure to elementary physics and chemistry at the secondary school level; many sections are comparable in style to articles in Scientific American. The object is to provide a descriptive nonmathematical under-standing of superconductors and their properties and a comprehensive grasp of how we are progressing in the marketplace applications. More succinctly the aim is tobridge the gap between the research specialist and the interested nonscientist. Thisbook also introduces the subject to those interested in pursuing further studies inthe field; it can provide supplementary reading for secondary school and university courses in general physics.

The authors wish to thank their friends John Clements, Sue Cluxton, DonDashnaw, Timir Datta, Rev. Gary Dilley, Billy Ellis, Bob Henry, Zafar Iqbal, Bob Mahaffey, Frank Petrusak, Charles Poole III, Mike Schuette, and Jan Owens for helpful comments and suggestions. One of us (CPP) wishes to thank his son,Michael, for drawing a dozen of the more difficult figures.

Frank J. Owens Picatinny, New Jersey

Charles P. Poole, Jr. Columbia,SouthCarolina

Contents

Chapter 1. Discovery of High- Temperature Superconductivity . . . . . . . . 1

Chapter 2 . Conductivity and Magnetism

2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2. Hydrogen Atom Energies . . . . . . . . . . . . . . . . . . . . . . . . . 82.3. Periodic Tableof Elements . . . . . . . . . . . . . . . . . . . . . . . . 92.4. Copper, Silver, and Gold . . . . . . . . . . . . . . . . . . . . . . . . . 112.5. Copper as a Good Conductor . . . . . . . . . . . . . . . . . . . . . . . 122.6. Conductors and Insulators . . . . . . . . . . . . . . . . . . . . . . . . 132.7. Electric Current Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.8. Atomic Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.9. Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.10. Electric and Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . .2.11. Atomic Magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.12. Temperatureand Its Measurement . . . . . . . . . . . . . . . . . . . .2.13. Cryogenic Liquids and Containers . . . . . . . . . . . . . . . . . . . .

17

2222

Chapter 3 . Superconducting State

3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2. Liquefying Helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3. Zero Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.4. Meissner Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.5. Levitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.6. Critical Magnetic Field and Critical Current Density . . . . . . . . . .3.7. Type I and Type II Superconductors . . . . . . . . . . . . . . . . . . . 333.8. Trapped Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.9. Flux Lattice and Pinning . . . . . . . . . . . . . . . . . . . . . . . . . 363.10. Current Flow and Flux Motion . . . . . . . . . . . . . . . . . . . . . .

31

39

ix

CONTENTS

3.11. Critical Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.12. Measuring the Critical Field . . . . . . . . . . . . . . . . . . . . . . . 41

Chapter 4. Superfluidity

4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3. Two-Fluid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.4. Second Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.5. Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.6. Bose–Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2. Superfluid Helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Chapter 5 . Explanations of Superconductivity

5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.2. Isotope Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.3. Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.4. Cooper pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.5. Superconducting Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.6. Correlated Electron Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.7. BCS Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.8. Pairing Mechanisms and Limits to Tc . . . . . . . . . . . . . . . . . . . . 655.9. Josephson Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

66

Chapter 6. Classical Superconductors

6.1, Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.2. Periodic Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.3. Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.4. Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.5. A-15 Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.6. NaCl-Type. Laves, and Chevrel Compounds . . . . . . . . . . . . . . . 806.7. Heavy-Electron Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.8. Charge Transfer Organics . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.10. Perovskites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.11. Borocarbides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.9. Chalcogenides and Especially Oxides . . . . . . . . . . . . . . . . . . 83

Chapter 7. Fullerenes

7.1. Interstellar Dust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857.2. Forms of Carbon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887.3. Soccer Ball Fullerene . . . . . . . . . . . . . . . . . . . . . . . . . . 897.4. Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

71

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CONTENTS xi

7.5. Fullerene Class of Molecules . . . . . . . . . . . . . . . . . . . . . . . 927.6. Tublenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Chapter 8. New High-Temperature Superconductors

8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 978.2. Transition Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 978.3. Layered Structure of the Cuprates . . . . . . . . . . . . . . . . . . . . . 988.4. Hole-Type Cooper Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . 1048.5. Direction-DependentProperties . . . . . . . . . . . . . . . . . . . . . 1078.6. Electron Superconductors . . . . . . . . . . . . . . . . . . . . . . . . 108

Ceramics. Perovskites. and Structures . . . . . . . . . . . . . . . . . . 1088.8. Yttrium Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118.9. Bismuth and Thallium Compounds . . . . . . . . . . . . . . . . . . . 1138.10. Mercury Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 1158.11. Infinite-Layer Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . 1168.12. Summary of Cuprate Properties . . . . . . . . . . . . . . . . . . . . . . . . 119

8.7.

Chapter 9 . Magnets and Their Uses

9.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1219.2. Conventional Magnets . . . . . . . . . . . . . . . . . . . . . . . . . . 1219.3. SuperconductingMagnets . . . . . . . . . . . . . . . . . . . . . . . . 124

High-Field Superconducting Magnets . . . . . . . . . . . . . . . . . . 1259.5. Technological Problems with SuperconductingMagnets . . . . . . . . 1279.6. Uses for SuperconductingMagnets . . . . . . . . . . . . . . . . . . . 1299.7. Nuclear Magnetic Resonance . . . . . . . . . . . . . . . . . . . . . . 1299.8. Magnetic Resonance Imaging . . . . . . . . . . . . . . . . . . . . . . 1309.9. MRI Fluoroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1349.10. SuperconductingMagnetic Energy Storage9.11. Magnetometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

9.4.

. . . . . . . . . . . . . . . 134

Chapter 10. Wires and Films

10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13910.2. Methods of FabricatingTapes and Wires . . . . . . . . . . . . . . . . . 14010.3. Grain Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14110.4. Elimination of Flux Movement . . . . . . . . . . . . . . . . . . . . . . 14310.5. Films and Their Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . 14410.6. Microwave Interactionswith Superconductors . . . . . . . . . . . . . 148

Chapter 11 . Further Applications

11.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

x i i CONTENTS

11.2. Computers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15111.3. Building a SuperconductingComputer . . . . . . . . . . . . . . . . . . 15511.4. Frictionless Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . 15611.5. Levitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15611.6. Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15911.7. ElectromagneticPropulsion . . . . . . . . . . . . . . . . . . . . . . . . 15911.8. Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Chapter 12. Future Prospects

12.1. Possibility of Room-Temperature Superconductivity . . . . . . . . . . 16512.2. Unidentified SuperconductingObjects . . . . . . . . . . . . . . . . . . 16812.3. Role of Fluorine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16912.4. Metastability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17012.5. Ladder Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17112.6. Dimensionality and Fluctuations . . . . . . . . . . . . . . . . . . . . . 17312.7. Room-Temperature Superconductivity . . . . . . . . . . . . . . . . . . 175

Appendixes

A. Units and Conversion Factors . . . . . . . . . . . . . . . . . . . . . . . 177B. Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . 179C. Chemical Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181D. Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

1

Discovery of High-TemperatureSuperconductivity

In March 1987 news media of the world were filled with accounts of a major discovery in material science. A solid had been synthesized that was a supercon-ductor 13°C above the temperature at which gaseous nitrogen liquifies! A super-conductor is a metal that conducts electricity without a loss in energy, hence without any cost to the user. Previous materials could perform in this manner only at muchlower temperatures—temperatures difficult to reach and expensive to maintain.

The importance of the new discovery was the temperature of operation. This new solid functions as a superconductor while in a container cooled with therefrigerant liquid nitrogen. Nitrogen constitutes about 80% of the gas in theatmosphere; condensed as a liquid, it is not only inexpensive, it is easy to handle.Liquid nitrogen can be held in a Styrofoam cup for almost 15 minutes before boiling away. All other known superconductors require much more expensive lower tem-perature coolant liquid helium. Convenient, inexpensive superconductivity wasnow at hand!

Much was said about the future technological possibilities of the discovery. Dr. Chaudhari, director of physical sciences at IBM’s Watson research laboratory, was quoted in the March 10, 1987, New York Times as saying, “If you wanted to dream further, you could see the day when instead of cars with wheels you have levitating cars.” The possibility of room-temperature superconductors and their enormous technological potential was envisioned. Even Business Week devoted a cover story to the discovery and its possibilities. As all this publicity was occurring, many solid-state physicists, chemists, material scientists, and others at universities, and government and industrial laboratories were abruptly postponing whateverresearch projects they were engaged in to work on new superconducting materials. For scientists it was an opportunity of a lifetime, a chance to be part of a major technological breakthrough.

1

2 CHAPTER 1

The story we have just recounted is from the perspective of the news media. From a scientific vantage point, the announcement of the new breakthrough came a year earlier than the flurry of excitement in March 1987, but the background research that went into the discovery began several years before that. During the mid-1980s all known superconductors operated at temperatures far below the boiling point of liquid nitrogen, which is 77 K (i.e., +77 degrees on the Kelvin temperature scale used by scientists). This is equivalent to -199°C or -326°F. Each material superconducts below a characteristic transition temperature denoted by the symbol Tc , and it becomes a normal metal at higher temperatures. Niobium (Nb),with Tc = 9.3 K, has the highest Tc of any element; the compound niobium-germa-nium, with the formulaNb3Geand Tc = 23.2 K, was then the highest of all materials. The fact that this transition temperature is less than a third of the liquid nitrogen boiling point 77 K caused most specialists in the field to believe that the possibility was indeed very remote of ever finding a material to superconduct above, or even close to, the magic value of 77 K.

Another cause of skepticism resulted from mathematical calculations made by some theorists during the late 1960s and 1970s that indicated that the Bardeen-Coopers-Schrieffer (BCS) theory (see Chap. 5) sets an upper limit of 30 K on superconducting transition temperatures. Many experimental physicists believedthese theoretical calculations, and as a result quite a few researchers had given upworking in the field of superconductivity.

The Swiss researchers Bednorz and Müller at IBM’s Zurich research labora-tories were not convinced by these arguments. They believed it was possible to raise the transition temperature by an appreciable amount if the right kind of material could be found. Müller had spent many years working on crystal structure changes, in particular, oxide materials known as perovskites, the prototype of which is the compound strontium titanate. Interestingly this material had been shown to be asuperconductor at 0.35 K, a mere 0.35 degrees above the lowest achievabletemperature, which is absolute zero with the value 0 K. It was perhaps his deepunderstanding of the behavior of this class of materials that led him to believe that crystals of this type would display superconductivity at much higher temperatures. Few others shared this belief, and those few who did so were not sufficiently committed in their view to expend a great deal of time and money on the effort. Bednorz and Müller persisted, making the necessary measurements on numerousmaterials for over 2.5 years before their success.

In late 1985 Claude Michel and coworkers at the University of Caen in France synthesized a copper oxide compound, and in their article they noted that itexhibited an unusual metallic like electrical behavior. When Bednorz and Müllersaw the paper, they were intrigued. They tried the measurements on similar materials, and when they tested a lanthanum, barium copper oxide compound, they observed to their astonishment the phenomenon of superconductivity in the tem-perature range of 30-35 K, over 10 degrees above the niobium-germanium value,

DISCOVERY OF HIGH-TEMPERATURE SUPERCONDUCTIVITY 3

and almost halfway to the liquid nitrogen temperature. After their discovery Müllerand Bednorz did not disclose their results to anyone for almost 3 months during which time they carefully repeated their measurements many times. After convinc-ing themselves of the validity of their results, on April 17, 1986, they submitted a paper to Zeirschrift für Physik, a German physics journal, and the article appeared in the September 1986 issue.

Over the years there had been many reports in the scientific literature of higher superconducting transition temperatures, and all of these had been shown to befalse. As a result such a report usually generated much skepticism in the scientific community. The few readers who did see this Zeitschrift article discounted it asanother false alarm. The overall scientific community did not become aware of the paper for some months. From the beginning it was clear that until results werecorroborated by other independent researchers, no one would pay any attention tothem.

The Bednorz-Müller article was noticed by C. W. Chu, popularly known asPaul Chu, a professor ofphysics at the University ofHouston who had spent muchof his professional research life in search of a higher temperature superconductor.He immediately directed his graduate students to synthesize the material andperform the necessary measurements to verify the original results. By December1986 the Houston group had been able to duplicate the result in the lanthanum, barium copper oxide compound. Having repeated the Bednorz and Müller work,Chu set himself the task of trying to raise the superconducting transition tempera-ture Tc in this material. He knew that some known superconductors had exhibited higher transition temperatures under pressure, so his approach was to apply pressure to the material. By doing this he was able to raise Tc by another 10 degrees. At thesame time a group in Japan headed by Dr. Koichi Kitazawa, an MIT-educatedscientist, also repeated the result with the lanthanum compound. Now two inde-pendent research groups had verified the Zurich work, a necessary requirement for its acceptance by the scientific community. Other laboratories began to join theeffort, and numerous publications appeared in early 1987 by researchers from anumber of institutions.

The Houston laboratory soon realized that it could not raise Tc by more than10 degrees using pressure. Chu’s group now joined forces with Professor M. K. Wu’s research group at the University of Alabama at Huntsville, and they directedtheir joint effort to finding variants of the initial compounds by substituting other atoms for lanthanum. After many tries they obtained a startling result. A material incorporating the element yttrium gave evidence of superconductivity at an almost unbelievable 90 K, 13 degrees above the magic boiling point of liquid nitrogen and three times the previously accepted theoretical limit of 30 K. This was an enormous breakthrough because now the coolant could be liquid nitrogen; this eliminated a need for the much more expensive refrigeration system involving liquid helium at the considerably lower temperature of 4.2 K and thereby opened the door for

4 CHAPTER 1

everyday applications. Announcement of this discovery led to extensive worldwide publicity.

The dramatic event that announced to the world the onset of a new era in superconductivity was the Special Panel Discussion on Novel High-TemperatureSuperconductivity that was held March 18,1987, at the American Physical Society (APS) meeting in New York. A last-minute addition to the program, an all-nightsession that lasted from 7:30 PM to about 3:15 AM the next morning, was attendedby about three thousand people. Most of them were unable to enter the 1200-seatmeeting hall, so they watched the proceedings on television monitors hastilydispersed throughout promenades and corridors in the vicinity of the meeting room. This meeting, filled with such excitement, became popularly known as the Wood-stock of physics. It began with reports from Müller and Chu, Zhao Zhongxian from Beijing, Shoji Tanaka from Tokyo, and Bertram Batlogg of AT&T. There were presentations by panels of theorists and experimentalists and a seemingly endless sequence of 5-minute talks interspersed with discussions and questions from the audience.

A year after the initial Woodstock, there was a follow-up special session at theNew Orleans March 1988 meeting of the APS to discuss the status of recently discovered bismuth and thallium compounds. It was chaired by Timir Datta of the University of South Carolina, and there was a leadoff talk by Allan Hermann of the University of Arkansas, the codiscoverer with Z. Z. Sheng of thallium supercon-ductors. One of the thallium compounds is superconducting at 125 K and below, which at that time was the highest known Tc. This session was attended by 750 scientists, and it lasted from 7:30 PM until midnight.

Since 1988 much work has been done to synthesize materials that becomesuperconducting at higher temperatures. In late spring 1994 superconductivity wasobserved in a new class of copper oxide materials containing mercury. Thesematerials became superconducting at 133 K, but subjecting one of these mercury compounds to high pressure raised this value to 147 K.

In October 1987 slightly more than a year after publication of results on the new materials, the Nobel prize in physics was awarded to Müller and Bednorz ofthe IBM Zurich Laboratory for their discovery of the new class of superconducting materials. Chu, who was involved in high-pressure and 90 K material work, was not included in the prize. Although the discovery of this latter much higher temperature yttrium compound provided the potential for many applications and initiated extensive publicity, it was nevertheless the original discovery of the 35 K lanthanum material that started the search for operation above 77 K and soon ledto the subsequent synthesis of the 90 K yttrium compound.

Starting in early 1987 there was phenomenal growth in scientific literature devoted to the subject of superconductivity. The number of articles increased sorapidly that new journals devoted exclusively to the subject were formed to publish them. This literature growth is shown in Fig. 1.1.

DISCOVERY OF HIGH-TEMPERATURE SUPERCONDUCTIVITY 5

Figure 1.1. Number of superconductivity articles published each year from 1974 to 1994 expressed as a percentage of the total physics literature.

With this discovery hope emerged that many of the useful devices incorporat-ing superconductivity that had been conceived over the years would now becomeeconomically viable. However a high-transition temperature is not all that isrequired to make this possible. The materials also have to possess certain other properties, such as ductility or the capability to draw them into wire; these and other properties are discussed in subsequent chapters.

This book tells the overall story of superconductivity, beginning with the initial discovery by H. Kamerlingh Onnes in 1911 of zero resistance in the elementmercury. Properties of the old or classical materials as well as newer ones discov-ered since 1987 are covered.

2

Conductivity and Magnetism

2.1. INTRODUCTION

To comprehend the meaning of superconductivity and fathom its unusual nature, it is necessary to understand normal conductivity. A material is a conductorof electricity if electrons, carriers of negative charge, move through it when they experience a force. The force is provided by an electric field arising from a voltage applied across the conductor. Not all solids are good conductors of electricity.Copper, which is commonly used in electrical wires, is an excellent conductor ofelectricity, whereas solid sodium chloride, everyday table salt, is not. The ability of a material to conduct electricity is measured by a quantity ρ called the resistivity.The symbol ρ is the lower case Greek letter rho corresponding to r in English. Good conductors have low resistivities, and poor conductors have large values.

Table 2.1 gives typical resistivities of various materials. These values can vary over the very wide range of more than 24 factors of 10 or 24 orders of magnitude, from a good conductor value of about 1 to a typical poor conductor or insulator value of about

Table 2. 1. Electrical Resistivities ρ of Several Materials at Room Temperature

Transition Resistivity(ρ, µΩ-cm)Material Type Superconductivity Temperature (K)a

Copper (Cu) Good metal No — 1.7Aluminum (AI) Good metal Yes 1.2 2.7Niobium (Nb) Poor metal Yes 9.3 12.5Lead (Pb) Poor metal Yes 7.2 22Mercury (Hg) Poor metal Yes 4.1 96

Very poor metal Yes 92. 2000Diamond (C) Semiconductor No — 3 x 108

Typical insulator Insulator No — 1024

YBa2Cu3O6.9

Copper oxide (CuO) Semiconductor No — 6 x 1011

aTransition temperatures Tc for materials that superconduct

7

8 CHAPTER 2

1 x 1024 measured in microohm-centimeters, abbreviated To handle such aµΩ-cm.large range of numbers, the factor of 10 notation is used, where for example:

0.003 is written 3 x 10–3

0.71 is written 7.1 x 10–1

128 is written 1.28 x 102

1,700,000 is written 1.7x 106

(2.1)

This factor of 10 notation is used occasionally throughout the text. To understand why some materials are good conductors and others are insula-

tors (nonconductors), it is necessary to have some understanding of the electronic structure of solids, which arises from the properties of their atoms and theirelectrons. We provide a brief introduction to these topics.

2.2. HYDROGEN ATOM ENERGIES

Some of the properties of atoms are most easily explained in terms of thesimplest atom, namely, hydrogen. The hydrogen atom consists of a negativelycharged particle, called an electron, that circulates about a positively chargedparticle, called a proton. The proton weighs almost 2000 times as much as theelectron, and it constitutes the nucleus. The path of the electron around the nucleus is a circle, called the orbit. The atom has energy due to the rotation of the electron about the nucleus and its electric attraction to the nucleus. Quantum theory wasdeveloped to describe the measured behavior of microscopic particles having sizes on the order of 0.00000001 centimeters [called an angstrom unit (Å)].

0.00000001 cm = 10–8cm = 1 Å (2.2)

This theory tells us that the atom can have only certain distinct values of energy. Since the energy of the electron depends on its distance from the nucleus, it follows that there can be only certain allowed orbits about the nucleus. The allowed energies that the electron-proton system can have are referred to as energy levels of the system. The lowest energy of the hydrogen atom, called the ground state, is 13.6 eV. An electron volt (eV) is the amount of energy an electron acquires when it is accelerated through a potential difference of 1V. When we say the ground state of hydrogen is 13.6 eV, we mean it would take that much energy to move the electron to a position very far from the atom. For atoms with more than one electron, the energy required to remove an outer electron is called the ionization energy. If energy is added to the hydrogen atom, there are only certain higher allowed energies that it can have, and these are called excited states. Figure 2.1 shows the hydrogen atom and its allowed energies. Technically speaking these energies are considered negative in sign because energy must be added to remove an electron from one of these states.

CONDUCTIVlTY AND MAGNETlSM 9

Figure 2.1. An illustration of the hydrogen atom (left) and its allowed energy levels (right).

2.3. PERIODIC TABLE OF ELEMENTS

Atoms larger than hydrogen contain more protons in the nucleus and more electrons orbiting around the nucleus. They are classified according to their elec-tronic structure into groups with similar chemical properties. Each type of atom, called an element, has filled inner shells of electrons that have no effect on the chemical properties and outer electrons, called valence electrons, that determine these properties. The atomic number gives the number of protons in the nucleus, which is also equal to the number of electrons. Figure 2.2 shows the periodic table classification of elements listed in the order of their atomic numbers, which are indicated in the upper left-hand comer of each element’s box. For example the atomic numbers of the first five elements hydrogen (H), helium (He), lithium (Li), beryllium (Be), and boron (B) are 1, 2, 3, 4, and 5, respectively. There are about two dozen radioactive elements with atomic numbers beyond 86 that are not included in the periodic table shown in Fig. 2.2.

We see from Fig. 2.2 that there are three broad classes of elements in the periodic table: ordinary elements that form the chemical compounds with which we are most familiar, such as water (H2O) and sodium chloride or ordinary table salt (NaCl); transition elements, which are such metals as niobium (Nb) and mercury (Hg) that appear prominently in superconducting compounds; and the 14 rare earth metals. Table 2.2 lists the first ten elements, which are all ordinary elements, and some of their characteristics.

Alkali AlkalineEarth

Transition Series Elements HalideChalcogen Rare

GasOrdinaryElements

Rare Earth Elements

[Fits between elements Barium Ba (56)and Lutecium Lu(71)]

Figure 2.2. PeriodicTableof the elements.The types ofelements and their valences are indicated. Some of the high atomic number elements,including the actinide series of radioactive ones, are omitted from this shortened form of the Table.

10C

HA

PT

ER

2

CONDUCTIVITY AND MAGNETISM 11

Table 2.2. Characteristics of the First Ten Elements in the Periodic Tablea

Name Symbol Type Atomic Number Valence

Hydrogen H 1 +1 Helium He Rare gas 2 0 Lithium Li Alkali 3 +1Beryllium Be Alkalineearth 4 +2Boron B 5 +3 Carbon C Amphoteric 6 ± 4 Nitrogen N 7 - 3Oxygen O Chalcogen 8 – 2 Fluorine F Halide 9 – 1 Argon Ar Rare gas 10 0a

Consult a general chemistry book for an explanation of the names of the various types of elements.

The elements are grouped into columns in the periodic table; all elements in a column have the same valence, which is indicated at the top of each column of ordinary elements. An element with a positive valence gives up electrons, and an element with a negative valence accepts electrons in forming a compound. For example sodium (Na), which has a +1 valence, gives an electron to chlorine (Cl), which has a – 1 valence, to form the compound NaCl. In like manner magnesium with a +2 valence gives two electrons to oxygen with a -2 valence to form the compound MgO. Some of the groups of elements with particular valence states have special names, like the alkali metals Li, Na, K, Rb, Cs in the first column with a +1 valence, and the halides F, CI, Br, I in the next to the last column with a -1valence. Appendix C lists elements and their symbols.

We refer to the periodic table from time to time in later chapters, since the number of valence electrons of elements often has a strong influence on the properties of superconductors.

2.4. COPPER, SILVER, AND GOLD

Three elements that are good conductors of electricity are copper (Cu), silver (Ag), and gold (Au). The chemical symbols come from their respective names in late Latin (cuprum) and classical Latin (argentum and aurum). They are transition elements with similar properties because they are in the same column of the periodictable, and they have a +1 valence. They each have relatively large numbers ofelectrons—29 for Cu, 47 for Ag, and 79 for Au. All of their electrons except theouter one are in compact configurations around the nucleus, which constitute what are called closed shells of electrons. For example the copper atom has a positive charge of +29 arising from the 29 protons in its nucleus. The nucleus is surrounded by the closed electron shells with a charge of -28 arising from their 28 electrons,

12 CHAPTER 2

Figure 2.3. Copperatom showing the positively chargednucleus, the closed shells of 28 shielding electrons, andthe outer valence electron.

as shown in Fig. 2.3. Electron shells shield the nucleus so that the outer electron, called the valence electron, circulates in an orbit around a net charge of +1. Given its capability to form chemical bonds, it acts rather like a hydrogen atom. The ionization potential of an isolated copper atom (i.e., the energy needed to remove its outer electron) is 3.49 eV, considerably less than that of hydrogen (1 3.6 eV). A larger amount of energy, namely 4.6 eV, is required to remove an electron from solid copper. Silver and gold have properties similar to those of copper; we are however, mainly interested in solid copper, silver, and gold rather than the isolated atoms, so we proceed to examine the electronic properties of solid copper in more detail.

2.5. COPPER AS A GOOD CONDUCTOR

A solid has a regular arrangement of atoms, ions, or molecules that are relatively close to each other. An ion is an electrically charged atom or molecule, such as the positively charged sodium ion (Na+) and the negatively charged chlorine ion (Cl-). A typical separation between adjacent atoms or ions may be 2–3 Å. In asolid discrete energy levels of individual atoms are influenced by the presence of other nearby atoms. If two copper atoms are brought together, the presence of one atom near the other causes splitting of the energy levels of each atom. When the number of surrounding atoms becomes very large, interaction between them broadens the outer levels into bands. The highest occupied level of atomic copper is capable of holding two electrons, but only one is present, since copper has a valence of +1; therefore it contributes one electron to this band. As a result the energy band is half-full, as shown in Fig. 2.4.

The 28 electrons of copper that are in closed shells remain on the atom in solid copper. In contrast to this the outer electron leaves its copper host and wanders around the material; i.e., it becomes delocalized and capable of carrying electric current. Its energy state is that of the half-full band. The electron can jump to empty levels in the same band but very close to it in energy. As electrons move around,


Figure 2.4. Illustration of how theenergy bandsofan insulatordiffer from those of aconductor. Copperis a conductor because its upper band is half filled with electrons, and sodium chloride is an insulatorbecause its top occupied band is completely filled with electrons.

they are free to carry current because there are many empty levels comparable inenergy that they can occupy during their wanderings.

CONDUCTORS AND INSULATORS

The difference between a conductor and an insulator is determined by whether or not the top-occupied energy band is partly or completely filled, If it is only partly filled, as in the case of copper, then the material is a conductor; if it is completely filled, the material is an insulator. Figure 2.4 shows energy bands for these two cases. We examine the band occupancy for the case of a conductor and an insulator, but before doing so, let us say a few words about how atoms of materials are arranged in space.

Atoms of a solid are arranged in space on a regular array of lattice points extending throughout the crystal. The overall lattice can be considered as generated by a few atoms in a small box, called a unit cell, that repeats itself throughout space. The resulting arrangement of atoms is called the crystal structure; it is defined bythe arrangement of atoms in the unit cell.

The unit cell of a typical insulator, NaCl, or common table salt, is shown in Fig. 2.5. It is a cube with sodium atoms at the apices and centers of the faces, as shown. Chlorine atoms are in spaces between sodium; they are arranged so that each sodium atom has six chlorine atoms as nearest neighbor, and each chlorine atom has six sodium atoms as nearest neighbors. The sodium to chlorine closest approach distance is 2.82 Å. The crystal structure of copper is also cubic. Each copper atom occupies the same position in its unit cell that chlorine occupies in the NaCl unit cell, as is easily seen by comparing the two unit cells in Fig. 2.5. The copper to copper nearest neighbor distance is 2.56 Å.

2.6.

14 CHAPTER 2

COPPER SODIUM CHLORIDE

Figure 2.5. Crystal lattices of metallic copper (left) and sodium chloride (right) showing the positions of the atoms in a unit cell.

Each sodium atom in NaCl transfers its single valence electron to achlorine atomto give chlorine the electronic configuration of the rare gas argon, whose outer shell isfilled with eight electrons. Both atoms have closed shells of electrons, so all energybands are full, and NaCl is an insulator. In metallic copper each copper atom releasesits valence electron to wander around the lattice. Since the energy band associatedwith this peripatetic electron is half-full, copper is a conductor.

We can explain conductivity in another way. A metal is made up of atomswhose outer energy bands are only partly filled and whose outer electrons are notbound to individual atoms but rather are delocalized throughout the lattice. Whenan electric field is applied, these delocalized electrons move through the latticecarrying current. In an insulator on the other hand, the outermost levels are filledand the outer electrons are localized on individual atoms, so they are not able tomove throughout the lattice carrying electric current.

2.7. ELECTRIC CURRENT FLOW

Electrical current can be measured by counting the number of electrons persecond moving through a unit area perpendicular to the direction of electron flow. This quantity is called the current density J. It can be written as:

J = env (2.3)

where v is the velocity of the electrons, n is the number of electrons per unit volume, and e is the charge on the electron. If the wire has cross-sectional area A , then thecurrent I is given by:


I =JA (2.4)

Newton’s second law of motion proposed some 300 years ago tells us that if a force is applied to a body, it accelerates; that is, its velocity increases. Unless something resists motion, velocity should increase continuously while the force is applied. Thus if an electric force is applied to a metal, the velocity of the electrons should continuously increase, and the current should continuously grow. However we know that when voltage is applied to a wire, a constant current results. This is due to resistance, which impedes current build up. As electrons move through a solid, some collide with atoms of the lattice, with foreign atoms, or with defects in the lattice. This causes electrons to stray from the electron flow path. Electrons rebounding from the atoms of solid prevents unimpeded growth of the current. These continual collisions produce an average constant velocity of electron flow. There is a simple relationship between the applied voltage Vand the current I, called Ohm’s law:

V= lR (2.5)

The constant R is known as the resistance of the wire. The units are volts (V ),amperes (A ) for current, and Ohms (Ω) for resistance; the symbol Ω is the upper case Greek letter omega corresponding to a long O in English.

The resistivity ρ mentioned in the Introduction to Chap. 2 is defined by:

ρ = AR(2.6)L

Equation 2.6 provides the expression:

(2.7)

for a wire of length L and cross-sectional area A. Resistivity is a unique property of a material, whereas resistance is a property of a particular wire. Some materials have greater resistivitity than others; for example conductors have very low resis-tivitity compared to insulators.

When electrons collide with atoms of the material, their velocity and therefore kinetic energy can decrease. The lost energy is dissipated in the form of heat: hot wires in a toaster are bright red because of electrons colliding with atoms of the wire. In electrical terminology we say that the toaster wires have a high resistance and they carry large currents so they become hot due to their energy dissipation.

The following table summarizes electrical quantities, and their units and symbols used in this section.

16 CHAPTER 2

Electrical Quantity Unit

Current (I) Ampere (A )Currentdensity (J=I/A )Voltage (V) Voltage (V )Electric field (E = V/L)Resistance (R) Ohm (Ω )Resistivity (ρ =RA/L )

Ampere persquare meter(A/m2)

Volt per meter (V/m)

Ohm meter (Ω m)

The electric field E is included here for completeness; it is defined and explained in Section 2.8 in terms of the expression E= Jρ,which is a variant of the usual formof Ohm’s law V = IR.

2.8. ATOMIC VIBRATIONS

In a crystal the atoms or ions that make up the lattice do not sit perfectly stillin designated positions (for example see Fig. 2.5), but instead they vibrate back and forth about their equilibrium positions very much like a weight on the end of a spring. The vibration of an atom in a crystal is not independent of vibrations ofneighboring atoms because nearby atoms tend to coordinate their vibrations. The energy of these vibrations is given by the expression kBT, where kB is Boltzmann’s constant; hence this energy determines the temperature T of the crystal.

Figure 2.6. Plot of the resistance vs the temperature of metallic sodium.


When atoms are vibrating rapidly, the lattice is hot; in other words the temperature of a solid is a measureof the average vibrational kinetic energy of theatoms or ions of the material. Kinetic energy is the energy a body has due to its motion. As the temperature is raised, the atoms of the solid vibrate more energeti-cally; i.e., they move faster and make larger excursions from their lattice sites. This increase in the amount of vibration increases the probability that an electron moving through the lattice will collide with a vibrating atom; and as a result the resistance of a metal increases with temperature.

Figure 2.6 shows the change in the resistance of metallic sodium with tem-perature. NOTE: As temperature decreases, resistance decreases and eventuallyreaches a constant value (below 10 K in this case). Resistance decreases at low temperatures because the amplitudes of vibrations of the atoms become smaller, thereby reducing vibrational energy. In a normal conductor resistance never reaches zero, not even at the lowest attainable temperatures.

2.9. MAGNETISM

In addition to the electric force that arises from applied voltage, there is a magnetic force that plays an important role in superconductivity. Most of us have experienced this force when we use a compass or place reminders on the refrigerator door with small magnets. The word magnetism comes from the name of a region in Asia minor, called Magnesia, where stones that attract small pieces of iron are found. The phenomenon hasbeen known formanycenturies; themagnetic compassis referred to in twelfth-century European literature.

The first major step in understanding the origin of magnetism occurred in 1820 when a Danish physicist from Copenhagen, Hans Christian Ørsted, noticed that a compass placed near a wire carrying electric current was deflected. Scientists now know that a moving electrical charge produces a magnetic field and an electric charge moving nearamagnetexperiences aforce. NOTE: This force is different fromthat between two charged particles that are not moving: This magnetic forcerequires one of the charges to be in motion. There is indeed an intimate relationship between electricity and magnetism. If an electrically charged particle moves through a region in space where there is no electric field and it is deflected in its path, then a magnetic force must be present (assuming gravitational forces can be neglected).

2.10. ELECTRIC AND MAGNETIC FIELDS

Scientists use the concept of a field to describe a region where an object experiences a force. The strength and direction of the force is described by the magnitude and direction of the field. An example is the gravitational field of the

18 CHAPTER 2

earth, which causes a mass dropped from a height to fall toward the center of the earth. We are interested in a charged particle, stationary or moving, that experience a force in an electric field and a moving charged particle that experiences a force in a magnetic field.

A voltage V impressed across a wire of length L produces the electric field Egiven by:

E = V (2.8)L

An electron of charge -e in such a field experiences the force:

F = –eE (2.9)

NOTE: The negative sign is inserted because the electron’s charge is negative. Thusan electric field has the units force per unit charge, or newton (N) per coulomb (C). An electric field of 1 volt per meter exerts a force of 1 newton on a charge of 1 coulomb. CombiningEqs. 2.4(I=JA), 2.7 (V= ILρ/A ) and 2.8 gives the expression:

E=Jρ (2.10)

which is the reduced units form of Ohm’s law.

field B experiences the force: An electron moving at a velocity v in a direction perpendicular to a magnetic

F = –evB (2.11)

The unit of magnetic field B is tesla (T); it is measured in millitesla (mT) andmicrotesla (µT).

1 tesla = 1,000 millitesla = 1,000,000 microtesla (2.12)

which is abbreviated

1 T = 103 mT = 106µT (2.12a)

A magnetic field of 1 T causes a force of 1 N to be exerted on a charge of 1 C moving at a speed of 1 meter per second perpendicular to the direction of the field. The magnetic field at the surface of the earth is approximately 0.00005 T, and the magnetic field of the small magnets that hold notes to a refrigerator has a strength of about 0.01 T, corresponding to the following values:

Earth’s magnetic field B = 50 µT


Refrigerator magnet field B = 10 mT (2.13)

Another quantity of interest is the amount of magnetic field B that passesthrough an area A ; and this is called magnetic flux Φ:

(2.14)

with the units tesla meter squared (Tm2). There is a minimum value orquantum ofmagnetic flux Φo called the fluxoid, which is given by:

Φο =2.07 x 10 –15 Tm2 (2.15)

Quantum of flux plays an important role in superconductivity. The magnetic field B in a region can be considered as the flux per unit area Φ/A that is present just asthe current density J in a region is the current per unit area I/A that is present.

2.11. ATOMIC MAGNETS

An electron behaves like a tiny bar magnet, and it orients itself in the presence of a magnetic field just like a compass needle. We say that an electron has a spin that makes it magnetic. Two electrons that are near each other can interact so that their spins are paired, i.e., parallel to each other; and they form a stronger bar magnet. If their spins are oppositely directed, antiparallel, in scientific terminology, then their magnetic properties cancel one another. Two bar magnets placed side by side would like best to have the south pole of one next to the north pole of the other because this is the state of lowest energy.

Ordinary atoms, such as sodium, oxygen, sulphur, and chlorine, which are not in transition series in the periodic table (see Fig. 2.2), form chemical compounds in which all of their electrons are paired; hence the compounds are nonmagnetic. Circulating electrons however act like tiny electric currents in the presence of anexternal magnetic field, so they exhibit a weak negative magnetic interaction called diamagnetism.

Atoms in the various transition series shown in the periodic table in Fig. 2.2ordinarily have unfilled inner energy levels in which all of the electrons are not paired; such atoms act like tiny bar magnets, which we call atomic magnets. The iron atom for example has 26 electrons circulating the nucleus, most of which are paired. The third energy level of iron, denoted by D, is only partially filled, so there is a net spin and associated with it, an atomic magnet. Other transition ions act the same way.

Many materials containing transition ions with atomic magnets are not them-selves strongly magnetic in their properties. The reason for this can be understoodby referring to Fig. 2.7, where atomic magnets are represented as arrows. The point

Φ = B/A

20 CHAPTER 2

of each arrow is its north pole. If all the little atomic magnets are randomly oriented in the material, as in Fig. 2.7a, the net magnetic field outside the material is on the average zero. In an applied magnetic field, these randomly oriented atomic magnets interact with the field; the result is a weak, disordered magnetic state called paramagnetism.

There are also magnetic states in which atomic magnets are ordered relative to each other. If they all point in the same direction, as shown in Fig. 2.7b, then their magnetic properties accumulate, so the material acts like a strong bar magnet producing a net magnetic field outside the material. This arrangement is calledferromagnetism; we say that the material is ferromagnetic.

Another type of magnetic ordering that occurs frequently in compounds containing transition ions is one in which atomic magnets alternate in direction, as shown in Figs. 2.7c and 2.7d. When all transition ions are identical, we obtain theantiferromagnetic state shown in Fig. 2.7d. Figure 2.8 shows positions of magnetic manganese ions in the manganese oxide lattice and the spacial alignment of their

(a) PARAMAGNETIC

(b) FERROMAGNETIC

(c) FERRIMAGNETIC

(d) ANTIFERROMAGNETICFigure 2.7.Possible arrangements of atomic magnets in materials that are (a) paramagnetic, (b) fer-romagnetic, (c) ferrimagnetic, and (d) antiferromagnetic. In the paramagnetic state the atomic magnets are randomly oriented in zero field and slightly aligned in the presence of an applied magnetic field.


Figure 2.8.An illustration ofthe arrangement ofmagnetic man-ganese ions in the lattice of manganeseoxide showing the actualordering of the atomic magnets in the antiferromagnetic phase.

atomic magnets. This kind of an arrangement of atomic magnets may play a role in determining superconducting properties of the new high-temperature supercon-ductors. The new high-temperature materials contain a paramagnetic copper ion that can form an ordered magnetic state.

When half of the transition atoms are of one type and the other half are of another type, such as iron and chromium, and each is antiparallel relative to the other, the result is the ferrimagnetic state shown in Fig. 2.7c. NOTE: In this figure the length of the arrow is an index of the strength of the atomic magnet. Such materials are called ferrites. They have interesting magnetic properties that make them useful for storing information: many computers use ferrite chips for their memory storage.

All ferromagnetic, ferrimagnetic, and antiferromagnetic materials can be transformed into a paramagnetic state if the temperature is raised high enough. To form one of the three ordered magnetic states, a paramagnetic material must be cooled below a characteristic transition temperature. However all paramagnetic materials do not form a magnetically ordered state; this results from the strength of the magnetic interaction between neighboring atomic magnets. This interaction tries to align them, but the thermal agitation interaction kBTarising from the effect of the temperature T continually shakes the atomic magnets, and hinders them from aligning. When the temperature is lowered sufficiently so that magnetic interactions are stronger than the temperature-shaking effect kBT, then atomic magnets align to form an ordered magnetic state, either ferromagnetic, antiferromagnetic, or ferrimagnetic, depending on the circumstances of the magnetic interactions. Raising the temperature causes atomic magnets to vibrate and perhaps destroy the alignment.

To visualize how this occurs, consider a number of pins arranged on a table such that they are all parallel to each other, then gently vibrate the table. We see that the pins gradually become disordered, and after a while, they are no longer parallel. If we continue shaking the table long enough, their orientations become

22 CHAPTER2

random. In a like manner we can demagnetize a material by heating it to a sufficiently high temperature where the thermal-shaking process dominates. Every magnetic material has a characteristic temperature at which it become unmagnet-ized and is transformed into the paramagnetic state. This is called the Curie temperature for the ferromagnetic case and the Néel temperature for the antiferro-magnetic one.

2.12. TEMPERATURE AND ITS MEASUREMENT

We spoke at length about temperatures of materials; now we must discuss the various scales used to designate temperature. Liquid mercury expands and contracts with changes in temperature more than most materials, so a thin capillary tube filled with mercury is often used as a thermometer, i.e., a temperature-indicating device. When it is placed in boiling water, a mark is made on the tube at the top of the mercury level; on the Centigrade scale, this is designated as 100 degrees. If the tube is placed in a well-stirred mixture of ice and water, the mercury column drops; a mark placed on the outside of the tube at the top of the mercury level is defined as zero degrees. Thus on the centigrade scale, the distance between the two marks is divided into 100 spaces. If the upper mark is labeled 212 degrees and the lower one 32 degrees, the scale is called Fahrenheit.

Scientists like to use a scale called the absolute or Kelvin scale. On this scale the boiling point of water is 373.2 K, and its freezing point is 273.2 K, as indicated in Table 2.3. This scale is chosen because 0 K, which is -273.2 centigrade, is accepted as the lowest possible temperature that can be achieved. It is the tempera-ture at which all atomic motion ceases and atomic particles have no kinetic energy. Thus the absolute temperature scale is somewhat less arbitrary than the others because zero corresponds to a definite physical state of matter. A temperature in centigrade can easily be converted into degrees Kelvin by adding 273.2 degrees.

2.13. CRYOGENIC LIQUIDS AND CONTAINERS

Table 2.3 shows some temperatures at which various materials change phase by freezing or melting. Phase refers to whether a material is a gas, liquid, or solid; a phase change is the transition from one state of matter to another, such as water freezing. Of particular importance is the temperature at which gaseous helium becomes a liquid, 4.2 K or –268.9°C; the temperature at which nitrogen gas liquifies, 77.4 K or –195.8°C, is also important. These cooled liquids are used to cool materials to very low temperatures; thus they have played an important role in research on the effects of temperature on materials. Such cooled liquids are often referred to as cryogenic fluids.

CONDUCTIVITYAND MAGNETISM 23

Table 2.3. Comparison of Fahrenheit, Celsius (Centigrade), and KelvinTemperature Scales. Phase transformation temperatures are given for several

cryogenic fluids.

Fahrenheit Celsius Kelvin (°F) (°C) (K) Material Change

32 0 273.2 Water freezes (H2O) (i.e., ice melts) 212 100 373.2 Water boils (H2O)

-148 -78.5 194.7 Carbon dioxide sublimates (CO2, usually called

-320.4 -195.8 77.4 Liquid nitrogen boils (N2)-345.8 -209.9 63.3 Liquid nitrogen freezes (N2)-452 -268.9 4.2 Liquid helium boils (He) -459.8 -273.15 0.0 Absolute zero

dry ice)a

aDry ice sublimes, i.e., passes directly from the solid to the gaseous state. It is often placed in ace-tone to produce a liquid at 194.7 K.

These very cold liquids must be kept in special containers to insulate them from outside heat, so that they evaporate very slowly. Sir James Dewar invented a flask, not unlike the common home thermos bottle, to hold very cold liquids; scientistscall suchcontainersDewars.AtypicalDewarismade ofadoubled-walledglass that is painted inside with silver reflecting paint; the region between the two walls is evacuated. The silvered walls and the vacuum severely limit the amount of heat that enters from the outside. Such a double-walled container can hold liquidnitrogen for many hours, but it is not a good enough thermal insulator for liquid helium.

Figure 2.9 shows a four-wall Dewar that is used to study the properties of materials at liquid helium temperature, namely 4.2 K. This Dewar consists of a double-walled glass vessel inside a larger doubled-walled glass container. The vacuum between the walls of the inner container must be especially high. The region between the two containers is filled with liquid nitrogen, and the inner Dewar is filled with liquid helium. The sample to be studied is immersed in the liquid helium of the inner Dewar.

In studying the superconducting properties of a sample, the resistance of the material is measured while it is in liquid helium. This is done by connecting wires to the sample that are attached to a voltage source, a voltmeter, and an ammeter. The latter two instruments measure voltage and current, respectively. The Dewar in Fig. 2.9 does not allow the sample temperature to be raised in a controlled way above the boiling point of helium. More complex equipment exists that allows the temperature to be varied, thereby permitting the study of various properties as a function of temperature. As we see later, when investigating superconductivity, it is often necessary to measure changes in the resistance as temperature varies.

24 CHAPTER 2

Figure 2.9. A four walled Dewar used to cool a sample to 4.2 K by immersing it in a liquid helium bath surrounded by a chamber containing liquid nitrogen.

In recent years technology has advanced to the state where an alternative type of refrigeration unit, called a cryocooler, is now available to operate various devices efficiently and economically at temperatures near 30 K.

3

Superconducting State

3.1. INTRODUCTION

In Chap. 2 we discussed the properties of conductors, i.e., metals that conduct electricity well; from the viewpoint of superconductivity we would say that Chap. 2 examined the properties of a superconductor in its normal state at temperatures above its transition temperature Tc. Chapter 3 presents the properties of these materials in their superconducting state below Tc.

Superconductivity is a phenomenon that occurs at very low temperatures. Every superconductor has a transition temperature Tc below which it superconducts and above which it is a normal metal. In the superconducting state, the material has no electrical resistance, so it conducts electricity without losses; in the normal state the material does have resistance, and the flow of electric current accompanies the development of heat and the dissipation of energy.

Prior to the discovery of the new superconductors in 1986, the highest known transition temperature was 23.2 K for a metallic compound made of niobium and germanium, with the chemical formula Nb3Ge. To observe the phenomenon, the compound is cooled below its transition temperature; the easiest way to do this is to immerse the compound in a cold liquid (a cryogenic fluid). Before 1986 the fluid used was usually liquid helium (He).

3.2. LIQUEFYING HELIUM

In 1908 H. Kamerlingh Onnes of the University of Leyden in Holland developed a procedure to liquify helium, a process that occurs at 4.2 K. To bring about this liquefaction, helium gas is compressed to a smaller volume, which causes its temperature to rise. The warmed compressed gas is then passed through a pipe surrounded by a colder fluid; this causes heat to be removed from the gas and transferred to the fluid. The still compressed but cooler gas is now forced through a small hole beyond which it expands, and further cooling occurs. These steps are

25

26 CHAPTER 3

repeated over and over again until the temperature of the compressed gas reaches 4.2 K where it liquifies. This process that Onnes used to liquify helium is still used today in modem refrigerators.

3.3. ZERO R ESlSTA NCE

Having successfully liquified helium, Onnes now had the ability to cool othermaterials to 4.2 K by simply lowering them into a helium bath. Onnes then began to carry out experiments on materials immersed in liquid helium—experiments that had never before been possible. One of the first of these, carried out in 1911, was to measure the electrical resistance of solid mercury. In the course of carrying out this measurement, Onnes discovered the remarkable behavior that is now calledsuperconductivity. The measurement consisted in applying a voltage V across themercury, recording the current I that flowed, and calculating the resistance R bydividing the voltageby thecurrent, (R = V/I ), as explained in Chap. 2 (seeEq. 2.5).

While the mercury was at 4.3 K, Onnes noticed that if he turned off the voltage, the current stopped flowing, as expected. He also noticed that if he turned off the voltage below 4.2 K, the current continued to flow! In fact the current continues to flow without loss for months and even years: The resistance of the material had become zero. The state of matter in which resistance is zero is known as the superconducting state. More precisely zero resistance is the first characteristicproperty of a superconductor; the second such property is magnetic in nature, which we describe in Section 3.4.

Figure 3.1 shows a plot of Onnes’s original resistance measurements, the first observation of superconductivity. We see that resistance drops sharply over a very narrow temperature range, from 0.11 ohm at 4.22 K to 0.00001 ohm (10-5 ohm) at4.19 K. Other metallic elements, metals that contain only one type of atom, such as aluminum and zinc, were found to superconduct at temperatures below that of mercury. Two years later the element lead was found to be superconducting at 7.2 K, and 17 years later in 1930, niobium was found to superconduct at 9.2 K. Table 3.1 lists transition temperatures of some elemental superconductors.

The unusual nature of this phenomenon can be realized by recalling the origin of resistance. In Chap. 1 we saw that materials have electrical resistance because their conduction electrons collide with intrinsic defects and impurity atoms of thesolid, which scatter them from the direction of flow. The phenomenon of superconductivity implies that electrons move through the lattice without undergoing these collisions. It took almost 40 years before physicists were able to provide an acceptable explanation for this unusual behavior.

To measure the resistance of a sample, typically in the shape of a disk the size of a dime, we attach two wires to the sample using a conducting glue, such as silver paint. A known voltage V is applied between the two wires, the current I that flows

SUPERCONDUCTING STATE 27

Figure 3.1. Temperature dependence of the resistance of mercury showing its disappearance as it becomes superconducting. This was the first observation of superconductivity made by Kamerlingh Onnes in Leiden in 1911. [From H. Kamerlingh Onnes, Leiden Commun. 120b, 122b, 124c (191 1).]

Table 3.1. Transition Temperatures Tc and Critical Magnetic Fields Bc at T= 0 K of Some Superconducting Elements

Element Symbol Tc (K) Bc (T)

Cadmium Cd 0.52 0.0028 Zinc Zn 0.85 0.0054Gallium Ga 1.1 0.0058 Aluminum AL 1.2 0.01 1 Indium In 3.4 0.028 Tin Sn 3.1 0.031

Tantalum Ta 4.5 0.083 Vanadium V 5.4 0.141 Lead Pb 7.2 0.08 1 Niobium Nb 9.2 0.206

Mercury Hg 4.2 0.042

28 CHAPTER 3

is measured, and the resistanceR is obtained by dividing thevoltageby thecurrent,i.e.,R= V/I. Forsampleshavinglargeresistances, thismethodisadequate,butwhenresistance becomes very small, which occurs when the material starts to become superconducting, then this method encounters problems. Difficulty arises because at the contact point where the wires are glued to the sample, there is a hindrance to theflowofcurrentbetweenthesampleandthewire; this iscalled contact resistance. When the intrinsic resistance of the sample becomes comparable to the contact resistance, our method no longer measures the resistance of the sample but rather the sum of the sample and contact resistances. To avoid this difficulty, four wires are attached to the sample; current is sent through two of them, and the voltage is measured across the other two. Superconductivity is attained when the voltage becomes zero.

3.4. MEISSNER EFFECT

It took 22 more years (until 1933) for the German professor Alexander Meissner and his graduate student R. Ochsenfeld to discover the second charac-teristic property of the superconducting state, a property that became known as the Meissnereffect. When an ordinary metal is placed in a magnetic field Bapp, the fieldpasses through it as shown in Fig. 3.2a. In this figure horizontal, sometimes curved, lines are drawn closer together where the magnetic field is stronger. A magneticmaterial, such as iron, augments the field by causing nearby lines to go through it, as shown in Fig. 3.2b. Meissner and Ochsenfeld discovered that when a supercon-ducting metal is placed in a magnetic field and then cooled below the transition temperature, the magnetic field is expelled. It turns out that no magnetic field is allowed inside a metal when it is in the superconducting state, a phenomenon known as the Meissner effect. Figure 3.2c illustrates the bending of magnetic field linesaway from a superconductor.

In 1938 the London brothers, Fritz and Heinz, provided an explanation for the Meissner effect. They applied the well-known theory of electromagnetism based on what are known as Maxwell’s equations to the case of a superconductor. They showed that an applied magnetic field Bapp induces a surface current; this surfacecurrent in turn produces an internal magnetic field that exactly cancels the applied field within the superconductor, so that no magnetic field is present in its interior. In other words the induced internal field is equal in magnitude and opposite in direction to the applied field Bapp, so the two cancel each other.

A more detailed analysis of this situation shows that some magnetic field does indeed penetrate surface layers of the superconductor, and it is present where the surface current flows. The statement that the superconductor has no magnetic field inside refers to the bulk of the material, not to surface regions; Fig. 3.2d shows this penetration at the surface. The surface layer where penetration occurs has a


Figure 3.2. Behavior of magnetic fieldlines (a) as they pass through a normal metal, (b) as they pass through a magnetic material, (c) in the presence of a superconductor when surface effects are neglected, and (d) in thepresence of a superconductor showing penetration of the field into the surface layer.

a

b

c

d

thickness called the London penetration depth, denoted by the symbol lL. This penetration depth is a parameter in the so-called London equations that were derived by the brothers to explain magnetic field penetration into superconductor surface layers. The value of the penetration depth λL is a characteristic of each individualsuperconducting material; typical values range from 0.2–0.8 µm. NOTE: A micrometer (µm) is a millionth of a meter (m):

1 meter = 1,000,000 µm = 106 µm (3.1)

The penetration depth increases with increasing temperature as the transition temperature is approached from below.

There are a numberofways using both DC and AC magnetic fields to measure theMeissner effect. In the former approach the probe of a DC magnetometer, a device that measures magnetic field strength, is placed close to the surface of the sample located in a magnetic field. As this sample is cooled below its superconducting transition temperature, the field inside of it is expelled, and the resulting increase in the magnetic field strength outside the surface of the superconductor is detected by the magnetometer. A very sensitive superconducting quantum interference device (SQUID) magnetometer

30 CHAPTER 3

whose operation depends on its superconducting properties, which is explained inChap. 9, is often employed for this measurement.

The Meissner effect can also be determined by an AC method. To accomplishthis two small coils are wound around the superconducting sample, an innerexciting coil and an outer probe coil; the latter is attached to a voltmeter. When anAC voltage is applied to the exciting coil, the resulting current flow causes an AC

magnetic field to sweep across the sample. This induces an AC voltage in theprobecoil that is detected by the voltmeter. When the sample is cooled below thesuperconducting transition temperature, theAC magnetic field is excluded from it, causing the AC field outside the sample to increase and thereby increase the voltage induced in the probe coil. The increase in probe voltage is proportional to thestrength of the magnetic field ejected from the sample; therefore it measures themagnitude of the Meissner effect.

3.5. LEVITATION

A particularly fascinating and intriguing manifestation of theMeissner effectis called levitation. The fact that a superconductor expels a magnetic field causes it to be repelled by a magnetic field in its vicinity and to move away from nearby magnetic fields. A magnet is an object that produces its own magnetic field around it, so it is natural to expect a superconductor and a magnet to repel each other. Thismutual repulsion is responsible for the phenomenon of levitation.

Using a high-temperature superconductor, which requires cooling only to liquid nitrogen temperature, makes this a relatively easy Occurrence to demonstrate with the aid of the apparatus in Fig. 3.3. A thin slab of the superconductor is cooled

Figure 3.3. Illustration of the levitation of a small magnet over a piece of material in the supercon-ducting state.


below its transition temperature by placing it in a pool of liquid nitrogen, then amagnet with dimensions smaller than those of the slab is placed above the slab. Themutual repulsion between the slab and the magnet causes the magnet to be levitated, i.e., held suspended in space above the superconductor,

Dr. Chaudhari of IBM referred to this levitation effect when he talked aboutcars not needing wheels: Superconductors would be on the bottom of cars, whilethe roads would be made of magnets. In fact the Japanese built a train that ismagnetically levitated above the tracks using the old low-temperature supercon-ductors. In the not too distant future, it should be possible and probably feasible todesign such a train with the new materials.

3.6. CRITICAL MAGNETIC FIELD AND CRITICAL CURRENT DENSITY

There is an upper limit to the strength of the magnetic field Bapp that can beapplied to a superconductor without destroying its superconducting properties. If a metal is in the superconducting state and Bapp is slowly increased, the fieldeventually reaches a value that removes the material from the superconducting state.The magnitude of the magnetic field that does this is called the critical field Bc. Thevalue of Bc depends on the material; values for several elemental superconductorsare listed in Table 3.1. For a particular superconductor, the magnitude of this critical field Bc(T) increases as the temperature is lowered below the transition temperature.The Bc(T) curve of Fig. 3.4 shows this temperature dependence for the supercon-ducting element lead.

Figure 3.4. Regions of superconductingand normal state behavior sketched as a function of the applied magnetic fieldBappin tesla and temperature T. These regionsare separated by the critical field curveBc(T). This temperature dependent critical field has the zero temperature value Bc(0) atT=0 which isordinarily designated by thesymbol Bc. The plot is for the element lead.

32 CHAPTER 3

The existence of a critical magnetic field that removes the superconducting state implies that there is also a maximum current density, called the critical current density Jc, that can be sustained by the superconductor. In other words there is amaximum amount of current that can flow before the superconducting state is removed. This is a direct consequence of the existence of the critical magnetic field, because the current produces a magnetic field, and a magnetic field produces a current. A critical current produces a magnetic field at the surface of the material that quenches the superconducting state.

The critical field Bc and critical current Jc are related to each other through thesimple expression:

Bc= µoλLJc (3.2)

where µo, called the permeability of free space, is a universal physical constant with the value:

µo = (4p)10-7 N/A2 (3.3)

where N is the unit of force; we use tesla for Bc2, meter for λL, and ampere persquare meterforJc. The critical currentdensityJc(T) has a temperature dependencesimilar to that of the critical field, as indicated by the Jc(T) curve in Fig. 3.5. Thepenetration depthλL(T) also depends on the temperature; its smallest value is atabsolute zero, and it becomes very large as the critical temperature is approached.

Figure 3.5. Regions of superconducting andnormal state behavior of the alloy 52% nio-bium, 48% titanium sketched as a function ofthe applied current density J and temperatureT. These regions are separated by the criticalcurrent density curve Jc(T). This temperaturedependent critical current density has the zerotemperature value Jc(0) at T = 0 which isordinarily designated by the symbol Jc. Thescale on the left covers the range of Jc valuesfrom 104 amp/cm2 to 106 amp/cm2.


The value of the critical current as we see later is a very important factor in thedevelopment of superconducting magnets.

In Table 3.1 Bc = 0.206 T is the critical field of the element niobium (Nb); it isthe highest critical field of any elemental superconductor. If niobium wire were used in a superconducting electromagnet, then the highest magnetic field that could be generated would be 0.206 T. Much higher fields, considerably beyond 2 T, caneasily be produced by a conventional electromagnet, so niobium is not a suitablematerial for magnet wire.

We see in Chap. 4 that a further breakthrough was necessary to find materialswith highenough critical fields so that superconductingmagnets couldoutperformconventionalmagnets.

3.7. TYPE I AND TYPE II SUPERCONDUCTORS

Superconductors can be classified by the way they behave in an externally applied magnetic field Bapp; Fig. 3.6 shows how the magnetic field B in inside the material depends on this external field. We see that as the external magnetic fieldof a Type I superconductor is increased, it remains excluded from the material until the critical field Bc is reached. At this point the superconducting state is removed,andtheappliedfieldBappcompletelypermeatesthematerial.TypeIsuperconductors

TYPE I SUPERCONDUCTOR

TYPE II SUPERCONDUCTOR

Figure 3.6. Plot of the magnetic field B ininside a Type I (upper sketch) and Type II (lower sketch) superconductoras a functionof the external applied field Bapp. Notice that the internal field has different values when the applied field is increasing than it does for a decreasing applied field.

34 CHAPTER 3

are the variety that we have been discussing until now, and they are usually very pure elements. Their transitions are sharp, with resistance dropping to zero over a narrow temperature range, as shown for mercury in Fig. 3.1.

A Type II superconductor behaves much differently. For this variety the magnetic fieldbegins to penetrate at a lowercritical fieldBcl andcontinues to growwith increases in the applied field until Bapp reaches the value of the upper criticalfield Bc2; this removes the superconducting state of the entire material. For applied fields below Bcl, where no flux penetrates, the material is in what is referred to as the Meissner state. For applied fields between the two critical fields Bcl and Bc2,there is a mixed state with regions of nonsuperconducting (i.e., normal) material embedded in a perfectly superconducting matrix. All chemical compounds and alloys are Type 11; only elements are Type I superconductors.

3.8. TRAPPED FLUX

Figure 3.2 shows how magnetic field lines penetrate different kinds of mate-rials in different ways; Fig. 3.7 shows the situation in Fig. 3.2b for the case of amagnetic cylinder that undergoes changes in radius. Since the strength of the magnetic field B is proportional to the closeness of the magnetic field lines, the field is weakest for the large radius on the left and strongest for the small radius on the right. On the other hand the magnetic flux Φ, which is proportional to the totalnumberoffield lines, is the same in all three regions. In more quantitative language, we say that the flux present in a region is equal to the field B times the area A throughwhich the field lines pass

(3.4)

B SMALL B MEDIUM B LARGE

Figure 3.7. Confinement of magnetic field in a magnetic cylinder of changing radius. The magneticfield varies in the different regions, as indicated, but they all have the same amount of flux.

Φ = BA


If Eq. 3.4 is rearranged to read B = Φ/A, we see that the magnetic field B can beviewed as the magnetic flux per unit area. This concept is analogous to therelationship J = I/A in Eq. 2.4 between the electric current I and the current densityJ, where J is the current per unit area.

A Type II superconductorbehaves differently in an appliedmagnetic field than itsType I counterpart. If a magnetic field Bapp in the range of Bcl -Bc2 is applied while thematerial is in the superconducting state, then some of this field penetrates the interior;ifthe applied field is subsequently turned off, i.e., returnedto zero, some magnetic fieldremains inside the superconductor. This is illustrated by the dashed line in Fig. 3.6; ineffect the superconductor remembers that it was exposed to an applied magnetic field.We see from Fig. 3.8 that the retained flux, called trapped flux, is less than the flux that entered the material when the external field was applied. The presence of the trappedflux causes the superconductor to behave like a magnet.

Experiments show that in the mixed state, normal regions are tubes, calledvortices, arranged parallel to the applied field direction. Figure 3.9 shows howvortices begin at one surface and terminate at the opposite surface of the supercon-ductor. Applied magnetic field lines enter each vortex at one end and exit at the other, as illustrated in Fig. 3.10 for the case of a superconducting film.

INITIAL STATE

B=0

B APPLIED

Figure 3.8. Superconducting sphere in the absence of an applied magnetic field Bapp (top), inthe presence ofan applied field with a magnitudein the mixed state range between BcI and Bc2

(center), and after removing the applied field (bot-tom). Note that some ofthe flux remains trappedinside the sphere after the external field is re-moved.

B REMOVED

36 CHAPTER 3

Figure 3.9. An illustration of the arrange-ment of vortices, tubes of normal materialsurrounded by superconducting regions in aType II superconductor. The magnetic fieldinside the material is confined to the vortices.

One of the more interesting properties of the mixed state is the fact that eachvortex contains exactly the same amount of flux Φ designated by the symbolΦo.In technical terms we say that the flux penetrating the sample is quantized and the quantum of flux Φo has a value equal to one-half the ratio of Planck's constant h to the charge e of the electron:

==Φo (3.5)h 2.067910–15 Tm2

2e

Some scientists use the term weber to designate a tesla meter2. Each vortex shown in Figs. 3.9 and 3.10 contains an amount of flux equal to the quantum value ΦoinEq. 3.5.

3.9. FLUX LATTICE AND PINNING

Figures 3.9 and 3.10 show vortices viewed from the side. Viewed from the top when the temperature is very low, vortices form an ordered triangular array (see Fig. 3.1la), which is called an Abrikosov lattice ; it is named after Alexei Abrikosov, who


Figure 3.10. Channeling of magnetic field linesthrough the vortices in a superconducting thin filmoriented perpendicular to an applied fieldBapp [adaptedfrom Poole et al., Superconductivity, Academic Press,Boston, 1995, p. 355].

contributed significantly to our understanding of the behavior of these vortices. This triangular array corresponds to how drinking glasses are arranged on a shelf to accommodate the largest number in the available space; in technical terms this is the close packing arrangement of identical circular objects on a plane surface. Close to the temperature of absolute zero, vortices have very little heat energy, so they stay in place on this triangular lattice, forming what can be called a two-dimensionaltriangular solid.

When the temperature is raised, vortices acquire heat energy and tend to vibrate or move about their equilibrium positions on the Abrikosov lattice. If the tempera-ture is raised high enough, vortex motion is so pronounced that the arrangement becomes disordered, as shown in Fig. 3.11b. This corresponds to the vortex latticemelting, and we have the two-dimensional equivalent of a liquid. This is expected behavior if the only forces acting are mutual repulsions between vortices.

In practice the arrangement of the atoms in a superconductor is not perfect; there can be lattice imperfections or misalignments of atoms that trap or pin flux lines (i.e., hold them in place), thereby preventing them from moving. When an individual vortex is held at a pinning site, the motion of nearby vortices is also restricted. A higher temperature is then required to provide vortices with enough energy to free themselves from pinning sites and bring about the vortex lattice melting.

When a magnetic field is removed from this flux liquid region, vortices are free to leave the material, so no flux remains trapped inside. The magnetic behavior of the sample is said to be reversible, since flux can be introduced or withdrawn simply by increasing or decreasing the applied field. In the flux solid region, many vortices are held in place at pinning sites so they are not free to leave the material; as a result some flux remains trapped inside when the applied field is decreased in

38 CHAPTER 3

a

b

Figure 3.11.A group of vortices viewed from thetoprepresentedby small circles. (a) Arrangement ofvortices in the regular triangular array, called anAbrikosov lattice, that forms in the flux solid region of the magnetic phase diagram of Fig. 3.12. (b)Irregular arrangement of vortices in the flux liquid phase of Fig. 3.12.

value or removed. This means that the magnitude of the field inside the materialdepends on the previous history ofhow the applied field was applied or changed.

The phenomenon of flux melting can be observed by an experiment calledmagnetic decoration. Very small magnetic particles are evaporated, then passedover the surface ofa Type II superconductor in the mixed state. Magnetic particlesare attracted to the surface of the crystal where vortex filaments begin (or end) asflux enters (or leaves) the bulk; this is observed by viewing the surface with anelectronmicroscope. Whentheexperiment isperformedwith thehigh-temperaturesuperconductor YBa2Cu3O7 at 4 K, the observed pattern is an ordered triangulararray like that shown in Fig. 3.1 la. However when the measurement is repeated fora crystal held at a higher temperature of 77 K, then magnetic particles trapped on the surface reveal a disordered arrangement of vortices, as shown in Fig. 3.11 b; this indicates that the flux lattice melted.

The flux-trapping properties of a Type II superconductor that we are describingcan be summarized in what is called a magnetic phase diagram; an example is shown in Fig. 3.12. The diagram plots applied magnetic field strength versus the temperature, showing the Meissner state at low fields, below the Bcl (T )line, whereno flux can enter the material, and regions between Bcl(T) and Bc2 (T) lines, wherethe flux solid state exists for temperatures below the melting line; the flux liquid state is present for temperatures above the line.


Figure 3.12.Magnetic phase diagram of a Type II superconductor showing the Meissner region ofexcluded flux below the lower critical field curve Bcl(T), the flux solid and flux liquid phases separated by the flux melting curve, and the normal region outside the upper critical field curve BC2(T).

3.10. CURRENT FLOW AND FLUX MOTION

If a current I is sent through a Type II superconductor in a directionperpendicular to an applied field, then each vortex experiences a force from the current, called a Lorentz force. Figure 3.13 shows that this force is perpendicularto both the applied magnetic field and the direction of the current flow; its strength is proportional to the magnitude of the current I that flows through thematerial. In the flux flow region, the Lorentz force easily overcomes the pinning forces, so vortices are set into motion. The presence of moving flux is undesir-

Figure 3.13. Illustration of the force F on avortex when a current J flows perpendicular to it. The magnetic field B inside the vortex isindicated.

CHAPTER 340

Figure 3.14. Plot of the resistance versus temperature for the high temperature superconductorYBaCuO in zero field (O) and in an applied field of 5.7 tesla (•). This figure shows how the resistance increases with increases in the applied magnetic field as a result of the movement of flux. [Adapted fromM. K. Wu et al., Phys. Rev. Letters 58, 908 (1987).]

able because the energy that keeps the flux moving is drawn from the current. The result is the dissipation of energy, an effect equivalent to the presence of electrical resistance. Therefore the superconductor is no longer a zero resistance material!Strong pinning in the flux solid region can prevent the movement of vortices, so the supercurrent flows without resistance. Moderately strong pinning results in very slow flux motion called flux creep. When a magnetic field is applied, the presence of flux motion lowers the transition temperature and broadens the resistance drop at Tc, as shown in Fig. 3.14.

3.11. CRITICAL SURFACE

We saw in Fig. 3.4 that aType I material superconducts for low applied fieldsand reverts to the normal state above the critical magnetic field Bc(T). A similar situation occurs with an applied current density J; Fig. 3.5 shows the conversionfrom the superconducting to the normal state at the critical current density Jc(T).When both a magnetic field and a current are simultaneously applied to a super-conductor, the state of the material is determined by the critical surface shown in Fig. 3.15. For combinations of values for B, J, T below the critical surface, thematerial superconducts, while field, current density, temperature combinationscorresponding to points above the surface convert the material into the normal state.


Figure 3.15. Critical surface in the B,J , T space. The material is superconducting below the surface and normal above it.

3.12. MEASURING THE CRITICAL FlELD

Figure 3.16 shows an experimental arrangement for measuring the critical magnetic field Bc2. The sample is immersed in a cryogenic fluid, such as liquid nitrogen, contained in aDewar at a temperature below the critical temperature Tc,and voltage- and current-measuring instruments (voltmeter and ammeter, respec-

Figure 3.16. An experimental arrangement for measuring the critical magnetic field Bc(T) at 4.2 K.

42 CHAPTER 3

tively) are wired across the sample. The battery causes a current to flow through the superconductor, and the whole arrangement is placed within the coils of anelectromagnet. At the beginning of the experiment, the magnet is off. Since thesample is in the superconducting state, the ammeter reads a current, but thevoltmeter reads zero. The magnet is now turned on, and the applied magnetic field is gradually increased. The voltmeter continues to read zero until the applied fieldreaches the critical field. At this point the voltmeter begins to register a voltage, and the current decreases due to the resistance of the superconducting material in the normal state.

Superfluidity

4

4.1. INTRODUCTION

In Chap. 2 we learned that normal electrical conductors exhibit a resistance tothe flow of electrical current and when current does flow, it dissipates energy, which makes the conductor hot and the electricity bill higher. In Eq. 2.6 we noted a quantity r, called resistivity, that ischaracteristicofaconductoranddetermines itsresistanceto electric current flow. In Chap. 3 we found that some electrical conductors calledsuperconductors can be cooled below a characteristic temperature Tc, where resistivity drops to zero, thus they carry electric current without the generation ofheat.A superconductor also has a number of other remarkable properties, such as theconfining internal magnetic flux in vortices with a supercurrent circulating aroundeachvortex.

In Chap. 4 we discuss a related super-type phenomenon that is characteristicof fluids. There are two kinds of normal fluids, a gas and a liquid. Both types offluids can flow, as in the case of theair in a wind, or water moving along in a river.When water moves through a pipe, it experiences resistance to the flow. Theproperty of thewater that is responsible for this resistance is viscosity, designatedby η. This symbol is called eta—lower case long e in the Greek alphabet. When afluid has large viscosity, it flows with difficulty like molasses or heavy oil; when it has low viscosity, it flows easily like water or acetone. Viscosity causes waterflowing in apipe to generate heat; that is, fluid current flow is a dissipative processas is electric current flow.

Liquid helium is a normal fluid between its boiling temperature TB = 4.2 Kand its so-called lambda temperature Tλ = 2.2K, below which it is a superfluid. Inthe superfluid region its viscosity drops virtually to zero, so it flows in a pipe without resistance and without generating heat. In a rotating superfluid the overall rotational motion can break up into local regions of circulating currents of helium atoms called vortices; these vortices are quantized, as in the superconductor case.

43

44 CHAPTER 4

Thus there is a great deal in common between the superconductivity of pairedelectrons and the superfluidity of helium atoms. Both phenomena exhibit flowwithout resistance as well and form vortices that are quantized. In addition theydisplay otherremarkable properties that are inmany ways analogous to each other;Chap. 4 introduces the reader to some of these.

The helium atom has two protons in its nucleus, and there are two stableisotopes, namely, 3He and 4He, with one and two neutrons in the nucleus, respec-tively. The superfluididty of 4He is the main topic of Chap 4. The lighter isotope3He also exhibits some superfluid type properties, but these are not discussed here.

4.2. SUPERFLUID HELIUM

Superfluidity was discovered by Pyotr L. Kapitsaof the Institute for PhysicalProblems in Moscow in 1937,26 years after the discovery of superconductivity by Onnes in Leiden. When Kapitsa cooled liquid helium below its lambda temperature, it immediately became clear, suggesting that something dramatic had happened. Above Tλliquid helium forms many bubbles that move rapidly in the manner

Figure 4.1. Helium phase diagram for the heavy 4He isotope showing the lambda (λ) line that separatesthe normal liquid from the superfluid.

SUPERFLUlDlTY 45

characteristic of any boiling liquid; as soon as the temperature drops below Tλ,bubbling ceases, and the surface of the liquid becomes very still. It is now asuperfluid! It is customary to call the normal state helium I and the superfluid statehelium II. Helium remains in the superfluid state to 0 K; in fact it is the onlysubstance that remains liquid under ordinary pressure at 0 K—all other substancessolidify before approaching that temperature. Figure 4.1 shows the phase diagramof helium; we see that helium condenses to a solid at pressures in excess of 25atmospheres. We also see that the lambda (λ)line where normal helium is trans-formed into the superfluid state depends somewhat on pressure.

When a liquid flows through a small tube, i.e., a tube with a small diameter,such as a capillary, it does so slowly because of friction with the walls. A measureof this friction is the viscosity of the liquid, as previously mentioned. If pressureisapplied to the end of the tube, it acts to counteract the effect of viscosity and causes the liquid to flow faster. When superfluid helium flows through a narrow channel, such as one 10–4 mm in diameter, the speed of flow is essentially independent ofthe applied pressure; this indicates that the viscosity is virtually zero.

Another strange property of superfluid helium is the formation of a film covering the exposed surface outside any container holding the liquid. Consider a

Figure 4.2. Illustration of how superfluid motion in the surface film acts to equalize the helium level inside and outside the beaker. [Prepared by Michael A. Poole.]

46 CHAPTER4

beakercontaining thesuperfluid in aheliumbath where theheliumlevel is the sameinside and outside the beaker. If the beaker is lowered, helium flows along thesurface film into the beaker; if the beaker is raised, its contents flow out along thisfilm, as shown in Fig. 4.2. The speed of theliquid flow in the film is typically about20 cm/sec. Other liquids, such as water, wet such surfaces as glass and form surface films, but their films move slowly because of viscosity. Helium is the only liquid that forms a rapidly moving film. This creeping surface film has a thickness of about 100 layers of helium atoms (300 Å). Its presence makes it difficult to confine superfluid helium in a vessel. Superfluid helium is also a very efficient conveyer of heat, capable of conducting heat 200 times faster than copper.

A property of historic importance is the specific heat of helium. Heating a material raises its temperature, and specific heat is the amount of heat energy needed to raise the temperature of 1 kilogram of a material by 1 K. When the specific heat of helium is measured, we obtain the discontinuity shown in Fig. 4.3 at the transition temperature. This anomaly in the plot of specific heat versus temperature resembles the Greekletter lambda (λ) in appearance (λ corresponds to 1 in our alphabet), which led early workers to call the transition temperature the lambda point (Tλ).

Figure 4.3.Discontinuity in the specific heat of helium at the lambda point Tλ.

SUPERFLUlDlTY 47

4.3. TWO-FLUID MODEL

Many of the properties of superfluid helium can be explained in terms of thetwo-fluid model. This model assumes that helium in the superfluid state is a mixture of two liquids—one normal and the other a superfluid—and these two liquids have helium atom concentrations nn and ns, respectively, given by:

nn + ns = n (4.1)

where n is the total helium concentration. It is assumed that the two liquids can move past each other and still remain intimately mixed. The superfluid component has no viscosity, and it is incapable of transporting heat; the normal component moves like a viscous fluid and transports heat. Above the lambda point 2.2 K, allthe helium is of course normal; below about 0.9 Kit is all superfluid. Between these limits both components are present, with the superfluid fraction gradually increas-ing as the temperature is lowered below Tλ, as shown in Fig. 4.4, which plots thetemperature dependence of the quantities nn/n and ns/n as percentages.

Figure 4.4. Percentage of normal and superfluid helium as a function of the temperature below thelambda point Tλ.

48 CHAPTER 4

We know that in fluid flow experiments as through pipes, superfluid helium acts as though it has zero viscosity. In these cases the flow of the superfluid component is detected. There are other experiments involving oscillating disks and vibrating wires in which fluid is dragged along by the disk or wire above Tl andcontinues to be so dragged alongjust below Tλ. A particularly dramatic example is the pile of equally spaced disks suspended in a helium bath by a fiber, as shown in Fig. 4.5. When these disks vibrate in the normal state, they drag fluid along with them, and their vibration frequency is lowered. Below Tl the vibrational frequency gradually decreases as temperature is lowered, indicating that less and less fluid isbeing dragged along by disks. The explanation of course is that only the normal component is pulled along by the moving disks. The frequency of vibration is a measure of the percentage of helium in the superfluid state; an experiment of this type provided data for Fig. 4.4.

The two-fluid model also explains why helium stops boiling below Tλ. Abubble forms in a normal liquid when the local temperature is higher than that ofthe surrounding medium; when a normal liquid leaves the surface, it carries away the extra heat. In the case of superfluid helium, the heat transport is so efficient that heat is carried away very rapidly, so a bubble has no time to form.

If a channel is sufficiently small, then superfluid can flow through it readily, but the normal component cannot get through; such a channel is called a superleak.If a superleak connects two containers of helium at different temperatures but both below Tλ, then the superfluid component flows from the low-temperature to thehigh-temperature end of the superleak in an attempt to equalize temperatures at the

Figure 4.5. Rotating pile of disks used tomeasure the viscosity and determine the percentage ofsuperfluidfraction in the sur-roundinghelium. [FromE. M.Lifshitz,Sci-entific American , June (1958).]

SUPERFLUlDlTY 49

Figure 4.6. Fountain effect showing how radiation heats the superleak formed by emery powder, andthe influx of superfluid helium rises in the narrow tube and forms the fountain at the top [After Allen and Jones (1938), Wilkes (1967).]

two ends. An example of this phenomenon is the fountain effect observed by Allen and Jones the year after the discovery of superfluidity. The closely packed emery powder shown in Fig. 4.6 constitutes a superleak; when it is heated by radiation, superfluid flows into the region so fast that it is forced out the top of the capillary tube in a jet (as shown) that can rise to a height of perhaps 40 cm (16 in.).

4.4. SECOND SOUND

Sound waves moving through air consist of variations in density, or variationsin the closeness of molecules to each other, that move forward in the air. Suchvariations in density or closeness are heard as variations in the pressure of the air that strikes the ear drum. This ordinary type of sound passes through water in the form ofsonar waves. In supeffluid helium ordinary sound, called first sound, ariseswhen superfluid and normal components vibrate in unison in the same direction to produce density variations that move forward in the medium, as in the case of sound waves in air. This first-sound wave is relatively easy to detect by pressure changes

50 CHAPTER 4

that it produces in a detector. Second sound, a property peculiar to superfluidhelium, arises when the superfluid and normal components vibrate in opposite directions so that one of them becomes denser when the other becomes less dense. Since there is very little density change, there is very little pressure variation to measure; thus second sound is very difficult to detect. Its speed is much less than that of first sound.

Second sound was predicted to exist, but several years passed without observ-ing it. Eventually it was realized, as we can deduce from Fig. 4.4, that changes in the ratio of normal to superfluid helium correspond to temperature changes, sosecond sound involves variations in temperature. In other words second sound is a thermal or heat wave. Once this was realized, detectors sensitive to temperature variations were employed successfully to detect the phenomenon.

4.5. VORTICES

We saw in Chap. 3 that the magnetic field in a Type II superconductor is present in the form of vortices. These vortices form because it is energetically more favorable for the magnetic field to be condensed into small regions of quantized flux rather than to spread uniformly throughout the superconductor. The amount of magnetic flux in each vortex is given by the expression:

Quantum of flux = —h (4.2)

where h is Planck’s constant and 2e is the charge of the Cooper pairs carrying thecurrent. We saw that supercurrents circulate around the vortex, and the total circulating current is closely related to the quantized amount of flux.

When superfluid helium is caused to rotate, it acquires rotational energy. While it is rotating cylindrical holes along the direction of rotation can spontaneously appear with nearby surrounding helium flowing in circles around these holes, as shown in Fig. 4.7; this is analogous to air flow in a tornado or a hurricane. A typicalhurricane has a central low-pressure region called the eye, with wind blowing incircles around the eye at perhaps 75 miles perhour (120 km/hr), while the eye itselfmoves forward at a mere 5-10 miles/hr (8–16 km/hr).

Consider a cylinder of superfluid helium being rotated on its axis at thefrequency f. Each helium atom experiences a centrifugal force given by:

2e

Centrifugal force = 4π2mHerf 2(4.3)

acting to move it in the outward direction, where r is the distance of the atom fromthe axis and mHe is the mass of a helium atom. This means that the force is greaterfor helium atoms on the outside of the container (large r) than it is for those near

SUPERFLUlDlTY 51

Figure 4.7. Superfluid helium circulating around a vortex.

the center (small r). As a result the fluid level rises on the outside and drops at thecenter, with the surface assuming the shape of a parabola, as shown in Fig. 4.8. Figure 4.8 also shows six vortices that form during rotation, whose axes are parallel to the main axis of rotation. The superfluid helium circulation around each vortex is indicated.

A measure of the strength of a vortex is a quantity called the circulation, which we designate by the symbol C. If the flow is circular at a constant velocity, then

Figure 4.8. Array of vortices in rotating superfluid helium. The rotation causes the helium surface toassume the shape ofa parabola [Adapted from Tilley and Tilley, Superfluidity and Superconductivity,2nd Ed., Adam Hilger, Bristol, (1986).]

52 CHAPTER 4

circulation is the product of the velocity vs times the path length L = 2πr around the hole, where r is the radius of the circular path:

C = 2πrvs (4.4)

The remarkable thing about a vortex is that circulation is only permitted to have certain values that are multiples ofa fundamental circulation designated by Co. Inphysics we say that the circulation is quantized, and the quantum ofcirculation Co

is given by:

h (4.5)Co =

where again h is Planck’s constant. Some vortices have more than one quantum of circulation, so more generally we have

=Cnh

(4.6)

where the integern = 1, 2, 3 is the number of quantum units, or quanta, in the vortex. Thus superfluidity, which is characterized by the resistanceless flow of mass

(helium atoms) and vortices quantized in terms of this mass (h/mHe), is analogous to superconductivity with its resistanceless flow of charge (paired electrons) and vortices quantized in terms of this charge (h/2e).

4.6. BOSE–EINSTEIN CONDENSATION

mHe

mHe

Physicists have known for almost a century that material in the world is made oftwo kinds ofparticles, distinguishedfromeachotherby their spin. Wementionedin Chap. 2 that electrons and atoms have a property called spin that is responsible for their magnetic properties; in Chap. 9 we explain how the spin of the hydrogenatom nucleus (i.e., a proton) and spins of the nuclei of other atoms can be detected by nuclear magnetic resonance. We also describe how the proton spins of hydrogen atoms can provide magnetic resonance images of the brain and other organs.

On a more basic level, particles that are building blocks for matter in the world are of two types: particles whose spins have half-integer values—1/2, 3/2, 5/2; they obey Fermi–Dirac statistical laws, and they are called fermions; and particles whose spins are whole numbers or integers—0, 1, 2; they obey Bose–Einstein statistical laws, and they are referred to as bosons. These statistical laws are named in honor of scientists responsible for their discovery, namely, Enrico Fermi and Albert Einstein of the United States, Paul A. M. Dirac of England, and the Indian theorist Satyendra Nath Bose. Examples of fermions with half-spin are electrons, protons,

SUPERFLUlDlTY 53

and neutrons. A particle of light, called a photon, has a spin = 1, so it is a boson.The mathematical treatment of these two types of quantum statistics is rathercomplicated, so we provide only some general indications of their properties.

One main difference between fermions and bosons is how they distribute themselves in their available energy levels as temperature varies. We describe this for the simple case in which there are ten particles occupying 13 equally spaced energy levels. At high temperatures both types of particles behave in a similar manner by distributing themselves in their energy levels somewhat haphazardly but with more of them toward lower energies, as shown in Fig. 4.9. At the other temperature extreme, namely, at absolute zero, the two types of particles rearrange themselves in their lowest energy configuration in a totally different way. Fermions obey the Pauli exclusion principle, so that no two fermions are in the same energy state. Therefore at absolute zero, each level from the bottom up to the Fermi energy EF is occupied by two half-spin fermions, one with spin up and the other with spin down, as shown in Fig. 4.9. All energy levels above the Fermi level are empty at absolute zero, as shown. In contrast to this, bosons have no exclusion principle, so at absolute zero they all consolidate in their lowest energy state, as shown on the right-hand side of Fig. 4.9, where collectively they exhibit special properties. Noninteracting fermions and bosons in their lowest energy states may be viewed

Figure 4.9. Occupation ofenergy levels for, from left to right, fermions at high temperature, fermionsat absolute zero, bosons at high temperature and bosons at absolute zero. Spin zero is assumed for the boson case so no spin direction is indicated.

54 CHAPTER 4

as constituting quantum liquids, since their properties are explained by quantum mechanics based on their statistics.

Do these special low-temperature configurations occur in nature? To reply we give three examples of half-spin fermions. Conduction electrons in an ordinary metal at temperatures in the millikelvin range, such as at 0.01 K, behave in many ways like fermions at absolute zero, and they are said to constitute a Fermi liquid. Neutrons in a neutron star are also believed to be a good approximation of a Fermi liquid at its lowest energy state. The lighter helium isotope 3He with atomic weight3 is only 0.0001% abundant. It has two paired protons and one neutron and spin = 0.5. At the extremely low temperature of 0.0026 K, this isotope is transformed into a state somewhat resembling the superfluid state, but manyof the propertiesof 3Hebelow this transition temperature are quite different from those described in this chapter for the dominant 4He isotope that is a boson. The disparity between the properties of these two helium isotopes in their low-temperature states is a rather dramatic demonstration of the important role played by statistics.

This book is primarily about systems that have undergone Bose–Einsteincondensation. The two electrons of a Cooper pair each have half-spin, but in the pair state their spins point in opposite directions and cancel each other; therefore the spin of a Cooper pair is zero, making it a boson. The superconducting state can be considered a state of Cooper pairs that are in their Bose–Einstein condensed state. This helps explain why Cooper pairs exhibit a great deal of coherence and tend to act in unison. Ordinary 4He helium atoms with an atomic weight of 4havetwo protons with oppositely directed spins and two neutrons with oppositely directed spins. Since these protons and neutrons cancel, the zero-spin 4He helium atoms are bosons. Many of the properties of superfluid helium (4He) can be explained in terms of the properties of an ideal Bose gas, i.e., a gas of noninteracting bosons. Thus in both the superconducting and the superfluid cases, there is boson condensation to a ground energy state that exhibits very unusual super-type properties.

For many decades physicists dreamed of cooling a sufficiently large numberof ordinary atoms to low enough temperatures to undergo Bose–Einstein conden-sation spontaneously. In mid-1995 this was accomplished by two groups acting independently. As reported by spokesman Eric A. Cornell, the research team involving members of the National Institute of Standards and Technology and the University of Colorado successfully formed a Bose –Einstein condensate of rubidium atoms on June 5,1995; 2 months later this accomplishment was independently corroborated by the research team of Randall G. Hullet of Rice University in Houston using lithium atoms. Both groups used magnetic fields to confine the atoms and lasers to cool them evaporatively. A temperature of 2 x 10–9 K wasreached in making the condensate—the lowest temperature achieved to date. As many as ten institutions working on this problem were getting close to their goal when the results were announced.

5

Explanations of Superconductivity

5.1. INTRODUCTION

In Chap. 3 we learned about theproperties ofmaterials in the superconductingstate, but we did not say much about why they have these properties. We begin Chap. 5 by describing an additional property of many superconductors, namely, the isotope effect. The remainder of Chap. 5 is devoted to explaining why variousmaterials superconduct.

Onnes discovered superconductivity in the element mercury in 1911, and anumber of other superconducting elements and alloys were found soon thereafter. However it took a while for an understanding of the phenomenon to develop. In 1935 Cornelius Gorter and H. B. G. Casimir of Leiden proposed a two-fluid model of superconductivity in which the material contains both normal electrons and superelectrons that exist side by side as interpenetrating liquids. This is analogous to the two-fluid model of superfluid helium described in Chap. 4.

As the material is cooled below the transition temperature Tc, normal electrons begin to convert into superelectrons; the lower the temperature, the greater the percentage of superelectrons and the stronger the superconductivity. In that same year 1935, the London brothers published a simple theory based on electromagnet-ism; the equations they derived, called the London equations, are still widely used today. The London model explained the Meissner effect and provided an expression for the first characteristic length of superconductivity, namely, what became known as the London penetration depth λL, or simply the penetration depth.

In 1950 the Russian physicistsV. L. Ginzburg and L. Landau developed a moresophisticated theory (henceforth the GL theory) based on what they called an order parameter φ whose square φ2 is proportional to the density of superelectrons ns that carry current in a superconductor. The GL theory provided the same expression forthe penetration depth as the London model and also an expression for the second characteristic length parameter ξ, called the coherence length. The symbol ξ is the lowercase Greek letter xi, which does not have a counterpart in our alphabet. The

55

56 CHAPTER 5

length parameter is the shortest distance over which the density of superelectrons nscan undergoan appreciablechange in value. In addition the GL theory furnisheda formula for the quantum unit of magnetic flux, called the fluxon . Thus the GLtheory provided a much more comprehensive explanation of superconductivity than the London approach, and the London equations followed as a natural consequence of the GL theory. Unfortunately this theory was not widely accepted outside the Soviet Union until the Russian physicist L. P. Gor’kov showed in 1959 that the GL theory is derivable from the more fundamental BCS microscopic theory which we will describe below.

The two-fluid model, the London approach, and the GL theory constitute what are called phenomenological theories, since they provide correct descriptions of phenomena or observable characteristics of superconductors without an underlying basis. They suffer from the liability that they do not take into account the micro-scopic structure of materials that superconduct. As a result they did not satisfy physicists who always want their theories to be of a fundamental, not a superficial,

type.It took over 45 years after the discovery in 191 1 for the emergence of a really

fundamental understanding of superconductivity based on the microscopic struc-ture of matter. In the year 1957 J. Bardeen, L. N. Cooper, and J. R. Schrieffer published an article explaining the phenomenon. In 1972 these three researchers were awarded the Nobel prize in physics for the theory in this paper, now commonly referred to as the BCS theory. As in the GL case, the BCS theory also introduces an order parameter φ,but the approach followed in expounding the theory differscompletely from that of Ginzburg and Landau. A working understanding of this theory cannot be acquired without some familiarity with quantum theory and its applications to solids. Because this book is written for a readership that does not have this background, the discussion that follows is qualitative and therefore to some extent limited. However, we do provide some background material on aspects of quantum theory.

5.2. ISOTOPE EFFECT

Before proceeding to the next theoretical development, which is the prediction that superconductivity is due to the electron-phonon interaction between conduc-tion electrons and lattice vibrations, we explain phonons, then provide somebackground on the nature of isotopes.

Lattice vibrations are a wavelike phenomenon associated with atoms of a solid oscillating in unison at characteristic frequencies. These vibrations can also exhibit particlelike properties with localized regions of vibration moving about the lattice and perhaps being scattered by obstacles that are encountered. For example an earthquake or seismic wave forms a localized region of earth vibrations that acts

EXPLANATIONS OF SUPERCONDUCTIVITY 57

like a giant particle as it moves along at the speed of sound. A localized lattice vibration exhibiting particlelike properties is called a phonon.

Atoms of the same element with different weights are called isotopes; thequestion arises as to how isotopes differ in their contribution to superconducting properties. Before examining this question, we explain a little about isotopes. The nucleus of an atom consists of two types of particles, namely, neutrons and protons. Protons are positively charged, and in Chap. 2 we mentioned that the number of protons in the nucleus equals the element’s atomic number. The number of electrons orbiting the nucleus, which determines chemical and bonding characteristics, equals the number of protons (also explained in Chap. 2). The neutron, as its name implies, is neutral (no charge); the weight of an atom is proportional to the number of protons plus the number of neutrons. The element Mercury has 80 protons in its nucleus and seven stable isotopes with 116, 118, 119, 120, 121, 122, and 124 neutrons, respectively. The atomic weights of these isotopes accordingly range from 196-204 atomic mass units, where:

80 + 116 = 196

80 + 124 = 204 (5.1)

In 1950 the theorist Herbert Frohlich concluded that vibrating atoms of a material must play an important role causing it to superconduct. Frohlich proposed that an electron-phonon interaction between electrons carrying the supercurrent and the lattice that vibrates bring about superconductivity. He knew that no one would accept his conjecture unless it were supported by an experiment, and he also knew from infrared spectroscopy that the frequency of vibration of an atom in a solid is proportional to the reciprocal of the square root of its weight. The isotope effect, or dependence of infrared vibrational frequencies on atomic weight, was a well-known phenomenon among spectroscopists. Accordingly Frohlich proposed that searching for an isotope effect in superconductors would establish whether or not lattice vibrations play an important role in the interaction responsible for the onset of superconductivity.

In this same year the experiment was carried out at Rutgers University by C. A. Reynolds, B. Serin, W. H. Wright, and Z. B. Nesbit, who investigated the effect of using different isotopes of the element mercury on its superconducting transition temperature. They found that the transition temperature decreased when the mer-cury atom became heavier and the change was proportional to the reciprocal of thesquare root of the weight of the atom, in striking agreement with infrared spectro-scopic results. This may be expressed mathematically as follows:

= (5.2)Tc (M')1/2

T 1

c (M)1/2

58 CHAPTER 5

Table 5.1. Isotope-effect Parameter α from Eq. 5.3 forSeveral Elemental Superconductors

Material Symbol Tc αa

Cadmium Cd 0.5 0.5a

Carbon (fullerene) 30 0.25 Lead Pb 7.2 0.48 Mercury Hg 4.1 0.5a

Molybdenum Mo 0.9 0.37 Osmium Os 0.7 0.20 Rhenium Re 1.7 0.23Ruthenium Ru 0.5 0a

Thallium TI 2.4 0.50a

Tin Sn 3.7 0.41 Zinc Zn 0.9 0.45Zirconium Zr 0.6 0a

aα= 0.5 ifthe BCS phonon prediction is satisfied; α = 0 if there is no isotope effect.

or more generally in the form:

(5.3)

where the isotope effect exponent α = 0.5 if the square root relation in Eq. 5.2 exists and α= 0 if there is no isotope effect. Subsequent work showed that a number ofother elemental superconductors also exhibit an isotope effect, but some did so to a lesser extent than mercury, as indicated by data in Table 5.1. We have been discussing elemental superconductors but some superconducting compounds dis-play an isotope effect, while others do not.

These experimental results convinced physicists that a microscopic theory of superconductivity must involve the electron-phonon interaction to explain the isotope effect; however the phenomenological theories just described did not accomplish this. Knowing the basic mechanism provided guidance and motivation for theorists looking for a fundamental theory. In 1956 Leon Cooper showed that two conduction electrons in the presence of very weak electron-phonon interaction are capable of forming the stable paired state referred to as a Cooper pair; this provided the final fact needed to formulate a microscopic theory of supercon-ductivity. Accordingly Cooper teamed up with Bardeen and Schrieffer, and within a year the theory was spawned. This however is getting a little ahead of our story.


5.3. WAVES

Supercurrent flow can be viewed as the forward movement of electron waves, so it is helpful to say a few words about the nature of waves. In quantum theory microscopic atomic-sized particles are sometimes represented as waves. A wave is some occurrence that repeats itself in time and space. What is waving in quantum mechanical terminology is the probability of finding a particle at a given place and time; a simple example is concentric waves of water, which move radially outward from the point where a small rock is dropped. Figure 5.1 shows this wave in waterviewed from both the top and the side. The wave is characterized by a wavelength λ,which corresponds to the distance between successive crests where the waterrises to a peak height (see Fig. 5.1). There is clearly more water at the crest than at

(a)

(b)

Figure5.1. Top and side views of a surface wave generated by dropping a rock in water, showing (a) The distance between the crests which is defined as the wavelength λ, and (b) two waves that are out of phase since their crests do not occur at the same place or time.

60 CHAPTER 5

the dip. In a sense the height of this wave is a measure of the probability of finding a given amount of water at a given place and time.

If these waves of water encounter an obstacle in the water much larger in dimension than the distance between crests, then they are reflected from the obstacle; in the terminology ofquantum mechanics, we say they are scattered. Onthe other hand, if the obstacle is very small with respect to the distance between crests, then the wave passes over the obstruction without scattering. Technically speaking the criterion for scattering can be stated as follows: Ifthe wavelength λ islarge compared to obstacle dimensions, the wave does not see the obstacle but rather passes over it undeflected. If the obstacle is comparable or larger in its dimensions than the wavelength, the wave is scattered.

Normal metals have resistance because some electrons in the flowing current are scattered by defects and impurity atoms in the lattice. In the superconducting state, something must happen to electron waves so that they can move through the lattice without being scattered by its atoms. The basic task of explaining superconductivity boils down to figuring out how conduction electron movement takes place without scattering.

5.4. COOPER PAIRS

There were indications that supercurrent flow involved pairs of electrons, so physicists were curious about how such pairing occurred. The BCS theory proposes an attractive interaction between electrons in the solid that overcomes their normal electrostatic repulsion, thereby causing themto formboundpairs knownas Copperpairs. Electrostatic repulsion refers to the repulsive force between two electriccharges of the same sign that pushes them apart from each other. Two nearbyelectrons with negative charges experience this force. Before development of the BCS theory, there was experimental evidence that the charge of particles carrying supercurrent is twice the charge of the electron, The question then is what consti-tutes the glue that causes electrons tobind into pairs or topose this question anotherway, what is the nature of the force of attraction that forms electron pairs by overcoming the strong electrostatic repulsion between the electrons. Without this new attractive force, two moving conduction electrons try not to be near each other.

The BCS theory explained how vibrating atoms in a lattice caused electrons to form bound pairs. In the phonon mechanism of a Cooper pair formation, a phononis a lattice vibration wave that moves through the lattice. A classical (nonwavelikedescription) can be used to give some insight into how such pairing occurs. Atomsof the metal lattice are positively charged because their outer valence electrons aredetached and delocalized, thereby becoming part of the conduction process. Outer electrons form the current that moves through the lattice. When electrons move past


Figure 5.2. Illustration of how a conduction electron(indicatedby•)movingthroughalat-tice causes a distortion of the lattice atoms along its path. The four atoms in the center are shown displaced slightly toward the electron. This distortion produces a region near the path which is slightly more positive in charge than the remainder of the lattice, and a distant electron is at-tractedtoit.

positively charged atoms, atoms are attracted to the electrons, and there is a slight distortion in the lattice around each speeding conduction electron, as illustrated inFig. 5.2. This region ofdistortion becomes more positively charged, and the forceof interaction of these distorted atoms with their neighbors is altered. The passingelectron slightly distorts the lattice, so vibrational frequencies are changed. Theregion ofdistortion follows the conduction electron as it speeds through the lattice.This more positive cluster is seen by a distant electron, which is then attracted to itand follows the motion ofthe electron that initially caused the distortion. The resultis the formation of a bound pair ofelectrons, called a Copper pair, held together bya binding energy that is typically about 0.0001 eV in magnitude, a very small value. The separation of electrons in the pair often ranges from 10-1000Å. If we recallthat the typical separation of an atom in a lattice is about 3 Å. two bound electronscan be as far apart as 300 lattice spaces. Put another way theremay perhaps be 100or more atoms in between the two electrons ofeach bound pair.

5.5. SUPERCONDUCTING GAP

The existence of these bound electron pairs alters somewhat the picture of the lattice band structure discussed in Chap. 2. We recall that the top band in a metal ispartly filled with electrons, and the highest energy of an electron in this band is called the Fermi energy. The presence of bound Cooper pairs in the lattice intro-duces a small gap in the band at the Fermi level, known as the superconducting gap. The energy of this gap corresponds to the binding energy that holds the pairs together. The difference in the band structure of the superconducting state and that of the nonsuperconducting state of a metal is illustrated in Fig. 5.3.

62 CHAPTER 5

Figure 5.3. The electron band structure of a nor-mal metal (a) and a metal in the superconducting state (b). In the superconducting state there is a small gap Eg at the top of the filled energy level.

Experimental evidence of the existence of the superconducting band gap comes from measurements using infrared light or microwave radiation. An infrared light beam, whose wavelength can be changed by a device called a monochrometer, is incident on the metal in the superconducting state. The amount of light reflected from the surface of the metal is measured as the wavelength (and hence energy) of the light is changed. At a certain wavelength corresponding to the energy gap, some of the infrared light is absorbed by inducing excitations across the gap; this causes the amount of light reflected from the surface to decrease. Figure 5.4 plots the percentage of incident infrared light reflected from a superconductor versus the

Figure 5.4. A plot of the percentage of infrared light that is reflected from a superconductor as a function of the wavelength of the incident light. The wavelength where the amount of reflected light begins to decrease provides a value for the superconducting energy gap.


wavelength of the incident light. The wavelength at which a reduction of reflected light is first observed measures the energy of the superconducting gap. When the superconductor begins to absorb infrared light, the absorbed light energy inducesthe Copper pairs to break up into individual electrons.

5.6. CORRELATED ELECTRON PAIRS

The existence of isolated bound pairs of electrons moving through the metal is not enough to explain superconductivity, since the pairs could also be scattered just as single electrons. Something more is needed to account for the lack of scattering. The waves of the Cooper pair exhibit most unusual behavior: They all have the same wavelength, but even more remarkable, the wave of every Copper pair is in phase with the wave ofevery other. Another way of saying this is that the motion of all electrons is correlated in space and time. In quantum mechanics wavelength and momentum (mass times velocity) are related. All pairs have the same momentum and thus the same velocity. The concept of phase is illustrated in Fig. 5.5. In phase means that the wave of every pair reaches its maximumand minimum amplitudes at exactly the same time and the same point in space. In effect superconductivity results from the fact that waves of paired electrons are all in phase, and they have the same wavelength. This unusual condition means that there is a massive tidal wave of paired electrons moving through the lattice.

Because the wave of each pair is connected to that of every other, an individual wave cannot be scattered without many other waves scattering in unison. The analogy of a large wave of water moving unobstructed over a small obstacle in its path is a simple picture of what happens in superconductivity. Robert Schrieffer, whose name provides the S in BCS theory, used the analogy of men and womensliding down a hill rippled with bumps to explain superconductivity. Suppose each man and woman loop arms to form a pair before starting down a hill. Some couples who encounter bumps will certainly fall, although fewer will do so than if they were unlinked. Couples represent Cooper pairs. On the way down the hill, some couples may go faster than others, so their motion is not correlated. When a couple falls, this is analogous to scattering, with a corresponding reduction in current flow. Since fewer people are going down the hill, the formation of pairs is not sufficient to explain why none of them fall. If all pairs linked arms, they would move down the hill at the same speed; in other words, their motion would be correlated. If a pair hit a bump, the pair would be prevented from falling because its neighbors would hold it up. It would require a massive bump to knock down all the pairs all at the same time. Superconductivity occurs because electron pairs are linked together and move through the lattice as an entity.

64 CHAPTER 5

(a) IN PHASE

(b) INTERMEDIATE

(4OUT OF PHASE

Figure 5.5. (a) Illustration of how two waves of the same wavelength which are in phase can combineto yield a larger amplitude wave, (b) how an intermediate amplitude is produced by two waves out of phase by a quarter ofa wavelength, and (c) how two waves cancel each other when they are phase shifted by half a wavelength with respect to each other.

5.7. BCS THEORY

There is of course a more fundamental question about the motion of the electron pairs described in the previous section: What causes all the pair waves to have the same wavelength and same phase? The answer to that question is the essence of BCS theory. The theory shows that for certain metals at low temperature, the presence of correlated bound pairs produces a state that is lower in energy than the normal state in which isolated electron waves move through the lattice in an uncorrelated fashion. It is a general law of nature that all physical systems seek to reach the lowest state of energy that they can attain. Thus BCS theory predicts the stability of a unique state in which probability waves of electron pairs move through a lattice without being scattered; this comesponds to a state of zero resistance. The formulation of BCS theory in terms of Cooper pairs held together by the lattice vibration mechanism is known as phonon BCS; materials conforming to thisformulation are known as conventional superconductors.


The theory can also account for other properties of the superconducting state, for example the existence of a critical field. This is the value of an applied magnetic field that supplies enough energy to separate electrons of Copper pairs. We see later that the electron has a magnetic field associated with it, and it behaves not unlike a small bar magnet. The interaction of tiny magnets of paired electrons with the applied field breaks up a Copper pair. The superconducting gap and thus the bindingenergy of the pairs increase as the temperature is lowered below Tc ; therefore thecritical field increases as temperature is lowered, as shown in Fig. 3.4.

The theory also explains the origin of the quantization of the magnetic field inside the superconductor. In a Type II superconductor, there are thin threadlike filaments of normal conductivity called vortices, as shown in Fig. 3.9. Super current flows around the cores of these vortices as illustrated in Fig. 3.9. If the waves of all the electrons are to be in phase at all times and at every point around the surface, the length of the waves must fit exactly around the circumference of the vortex to ensure that they connect smoothly; otherwise the phase relationship would be lost. Since the circumference must be an exact multiple of the wavelength for this to happen, it follows that only certain wavelengths are allowed, and thus only certain energies. Since the energy of the electrons is quantized, so is the magnetic field produced by their motion.

5.8. PAIRING MECHANISMS AND LIMITS TO Tc

We have discussed what is sometimes called phonon BCS, meaning the usual version of BCS theory in which bound electron pairs are held together by the attractive force arising from localized disturbances in the vibrating lattice from its equilibrium configuration. The overall formalism of BCS theory however is also valid if two electrons of Cooper pairs are held together by a mechanism that differs from the phonon-based one, and a number of alternative mechanisms have been proposed. We say a few words about limitations arising from the phonon case, and then comment on alternative mechanisms.

The A-1 5 compound Nb,Sn held the record for the highest transition tempera-ture, Tc = 18 K, from 1954 to 1966; in 1966 this value was surpassed by 3 degrees by another niobium-based A-1 5 compound Nb3(Al0.75Ge0.25), which continued to hold the record for 5 more years. As a result of seeing Tc rise by a mere 3 degrees in over 14 years, in 1968 the scientific community believed that superconductivity might well have reached its limits. At this time W. L. McMillan wrote a landmark article in which he analyzed phonon BCS in various metals and alloys. His calculations for A- 15 compounds suggested a maximum possible value of Tc = 40 K; since this was for a compound that was too unstable to form, he recalculated and found Tc = 28 K as a maximum possible transition temperature for a feasible compound. This maximum possible value was rounded off to Tc = 30 K, which became the fairly universally accepted ultimate limiting value. This did not

66 CHAPTER 5

deter experimentalists who continued to work hard to reach 30 K, but the closest they came was Tc = 23.2 K for Nb3Ge in 1973. In 1975 Allen and Dynes reexamined McMillan’s treatment, but did not propose a higher possibility for Tc; thus the 30 K limit remained fixed in ceramic, so to speak, for another decade.

The belief in the electron-phonon 30 K limit encouraged physicists to look for alternative mechanisms to bring about Cooper pairing, and a number of them were proposed. In some of the alternative coupling schemes, the role of phonons is taken over by for example the following localized or particlelike quantities: excitons, plasmons, polaritons, and polarons. These quantities are described in solid-state physics texts, so we do not define them but rather give an example of how an exciton mechanism brings about Cooper pairing.

Ordinarily electrons on the atoms of a solid are in what is called their ground states; this means that they have their lowest possible energy. We are referring to both the outer electrons that bring about the bonding of atoms to each other and the inner-shell electrons that we discussed in Chap. 2. If one of the atoms acquires some extra energy, it can be raised to an excited energy level where it remains for a short time, then returns to its ground state. Another possibility is for the atom to transfer its excitation state to a nearby atom, which could cause the excitation to move around the lattice; such a moving excitation is called exciton. If the movingexcitation involves shifting electronic charge distributions in a region of the lattice, it is called polarization. A conduction electron can interact with such a polarization, and if two conduction electrons interact with the same region ofpolarization, theymay form a Cooper pair. This is not a new idea, since W. A. Little first proposed an exciton mechanism for Cooper pair formation in 1964, four years before the McMillan article appeared.

Currently there is a great deal of research on other possible mechanisms for bringing about pairing, and non-BCS approaches are being adopted to explain superconductivity in the cuprates. Some of this research involves exotic-soundingquantities called anyons, spin bags, slave bosons, Fermi surface nesting, etc.Theoretical physicists sometimes make up strange names for the mathematical constructs they propound and the observables they postulate as existing.

5.9. JOSEPHSON EFFECT

In 1962 a young graduate student in physics at Cambridge University, Brian Josephson, theoretically predicted an unusual effect involving superconductors that was subsequently observed experimentally. Josephson was awarded the Nobel prize in 1973 for his contribution. His prediction is illustrated in Fig. 5.6; the device depicted has become known as the Josephson junction. It consists of a thininsulating material, about 10-20 Å thick, sandwiched between two supercon-ducting metals. In many instances the insulating layer is a thin oxide coating on an


Figure 5.6. Josephson junction consisting of a sandwich formed by an insulator no more than 20angstroms thick located between two superconducting metals. The switch connects or disconnects thebattery from applying a voltage across the superconductor. The ammeter measures the current flowingthrough the junction. The current I versus voltage V characteristic is shown at the bottom.

evaporated metal film. A voltage is applied to the junction, which is then cooledbelow the transition temperature of the superconductor. When the voltage is disconnected by opening the switch, a DC current continues to flow as though thesandwich were one continuous slab of superconductor: Copper pairs move through the insulating nonsuperconducting layer without breaking up.

The phenomenon cannot be explained in purely classical terms but requires the quantum mechanical representation of pairs as waves. The explanation of how the current crosses the insulating layer involves the purely quantum mechanical phenomenon of tunneling. Normally for an electron to pass from a conductor to an insulator, it must overcome an energy barrier at the interface. Treating the electron as a classical particle, the only way the electron can enter the insulator is by possessing an amount of energy greater than the energy associated with the barrier height. For example, to throw a ball over a building, sufficient kinetic energy must be applied to get the ball over the top; the barrier in this case is the gravitational force pulling the ball down before it goes over the building. If the electron is represented as a probability wave, quantum mechanics predicts that there is some probability that it can penetrate the insulator even if its energy is less than the barrier

68 CHAPTER 5

height. This phenomenon is referred to as tunneling, and tunneling explains how the Copper pairs get through the insulator. The effect is very pronounced in the superconducting sandwich because of large tidal waves of Copper pairs in the superconducting metal layers on each side of the insulating barrier.

This tidal wave results from the fact that all waves in each metal are in phase, andtheyallhavethesamewavelength.NOTE: The phase of waves in each supercon-ducting part of the junction may not necessarily be the same. The flowing current depends on the relative phases ofwaves in the two superconductors. These relative phases determine how waves combine to give a net current. Figure 5.5 shows three cases of the phase difference between waves, and the net wave produced by adding the phase differences. When two waves are in phase, they combine to produce the largest current, as in Fig. 5.5a. On the other hand, when two waves are out of phase, i.e., shifted by half a wave length with respect to each other, as in Fig. 5.5c, the combined wave produces no current. Phase shifts between these two extremes produce an intermediate amount of current.

The relative phase ofthe waves in the two superconductors can be changed byapplying a magnetic field. Figure 5.7 shows how the current through the junction is changed as the strength of an applied magnetic field perpendicular to the plane of the sandwich is increased. The points where there is no current are points where waves from each superconductor are a half-wavelength out of phase. The current in a Josephson junction is very sensitive to a small magnetic field, and magnetic-

Figure 5.7. Dependence of the current in a Josephson junction on the applied magnetic field.


field measuring devices, i.e., magnetometers, are based on this effect; they are called SQUID magnetometers. SQUID is an acronym for superconducting quantum interference device. We learn more about how they work later.

The current versus magnetic field plot from a SQUID very much resembles what physicists call an interference pattern produced when a single wavelength beam of light is sent through a very thin slit, then projected on a photographic plate. Both phenomena are a result of the addition of waves with different relative phases;this is called interference.

6

ClassicaI Superconductors

6.1. INTRODUCTION

In Chap. 3 we examined the characteristics of superconductors, and in Chap. 5 we described various theories explaining these characteristics. Chapter 6 surveys the properties of various individual elements and compounds that superconduct at temperatures below about 25 K (see Table 6.1 for the transition temperatures of some of them). We begin by introducing the periodic table classification of theelements, then we discuss elements that superconduct, followed by various com-pounds. Of particular interest are the A-15 compounds, such as Nb3Sn, which provided the highest transition temperatures for classical superconductors; the heavy-electron systems, which were widely studied prior to the discovery of the cuprates; and the organics, which have experienced a rapid rise in transition temperature in recent years. Then we say a few words about compounds related to high-temperature superconductors discussed in Chap. 8. The more recently discov-ered K3C60-type fullerene compounds, sometimes called buckyballs, are the topicof Chap. 7.

6.2. PERIODIC TABLE

In Chap. 1 we mentioned that superconductivity was first observed in 1911 inthe element mercury with Tc = 4.2 K, as shown in Fig. 3.1. We know from general chemistry and general physics that elements, which are the basic building blocks ofmatter in the universe, can be classified into groups according to their chemical properties; this classification is given by the periodic table in Fig. 6.1. This table indicates elements by their chemical symbols (see Appendix C for the name of each element). Fourteen additional elements, called rare earths, and over 20 radioactiveelements are not included in Fig. 6.1.

Chemical properties of an element are determined by the number of electrons in its outer shell; these are called valence electrons (Ne). Elements in a particular

71

72 CHAPTER 6

Table 6.1. Transition Temperatures Tc of SomeSuperconducting Elements and Compounds

Compound Type Tc (K)La Element 6.3 Pb Element 7.2 Nb Element 9.3TiZr Alloy 1.5 NbTa Alloy 6.0 NbTi Alloy 9.5Zr3Bi A-15 3.4 Nb3Sn A-15 18.Nb3Ge A-15 23.2ZrN NaCl 10.7MoC NaCl 14.3NbN NaCl 17.CaIr2 Laves 6.2ZrV2 Laves 9.6

CU2Mo6S8 Chevrel 10.7PbMo6S8 Chevrel 12.6Sn1,2.Mo6S8 Chevrel 14.2UPt3 Heavy electron 0.43UBe13 Heavy electron 0.85

Zr1/2Hf1/2V2 Laves 10.1

UNi2Al3 Heavy electron 1 .0 (TMTSF)2FSO3 Organic 3.(ET)2I3, Organic 8.(ET)2CU(NCS)2 Organic 10.CuRh2Se4 Chalcogenidespinel 3.5 cuv2s4 Chalcogenide spinel 4.5 LiTi2O4 Oxide spinel 13.7 SrTiO3 Perovskite 0.3BaPb1–xBixO3 Perovskite 13.B0.4K0.6BiO3 Cubic perovskite 30.ThPd2B2C Borocarbide 14.YNi2B2C Borocarbide 16.

K3C60 Buckminsterfullerene 19.Rb2KC60 Buckminsterfullerene 25.

La3Ni2B2N3Boronitride 12.

Cs2Rb60 Buckminsterfullerene 33.

Transition Series Elements HalideChalcogen Rare

GasOrdinary Elements

Alkali Alkaline Earth

Rare Earth Elements

[Fits between elements Barium Ba (56) and Lutecium Lu (71)]

Figure 6.1. Periodic table showing the elements which are superconductors. Those elements which superconduct only as thin films or underpressure are so designated with anasterisk. The number of valence electrons Ne of the atoms is indicated at the top of each column.

CLA

SS

ICA

L SU

PE

RC

ON

DU

CT

OR

S 73

74 CHAPTER 6

column of the periodic table in Fig. 6.1 all have the same number of valence electrons; this number is indicated at the top of each column. For example the alkali elementslithium(Li),sodium(Na),potassium(K),rubidium(Rb),andcesium(Cs),which are all metals that conduct electricity well, are in the first column, and Ne =1. At the other side of the periodic table is the column with the rare gas elements helium (He), neon (Ne), argon (Ar), krypton (Kr), xenon (Xe), and radon (Rn), which have eight valence electrons; these elements are chemically inert. The number of valence electrons can have a strong influence on the superconducting properties ofelements and compounds.

6.3. ELEMENTS

Figure 6.1 lists 14 other elements that cannot be made to superconduct bysimply cooling them but do so only when irradiated, subjected to high pressure, or made into thin films. These are indicated by an asterisk. Thus over half of the elements in the table in Fig. 6.1 can achieve the superconducting state. The great majority ofthe superconducting elements are Type I, as explained in Chap. 3. These elements are not very suitable for applications because of their low transition temperatures and low critical fields.

When an element or a chemical compound becomes a solid, its atoms arrange themselves in space in a regular manner; the majority of the elements adopt one of the three structures shown in Fig. 6.2. The face-centered cubic (fcc) structure in the center of Fig. 6.2 has its atoms in the same positions as the sodium atoms in the NaCl structure in Fig. 2.5. The body-centered cubic (bcc) structure on the left has an atom in each vertex or comer of the cube and another atom in the center. In the hexagonal close-packed (hcp) structure on the right, atoms form hexagonal layers.Structure does not seem to be a dominant factor in determining superconducting properties, since elements with the highest transition temperatures belong to all three types:

Element Symbol Structure Tc (K)Niobium Nb bcc 9.3Technetium Tc hcp 7.8Lead Pb fcc 7.2

Among the elements niobium not only has the highest Tc, but it is also a constituentof many higher Tc compounds, like Nb3Ge. Niobium has not appeared prominently in the newer oxide superconductors.

Many elements that superconduct are classified chemically as transition ele-ments, and their location in the periodic table is indicated in Fig. 6.1. The highest

CLASSICAL SUPERCONDUCTORS 75

BCC FCC HCP

Figure 6.2. Sketch of the body centered cubic (left), face centered cubic (center) and hexagonal close packed (right) unit cells of three structures found commonly in the elements.

values of Tc occur for transition elements with five or seven valence electrons, asdata in Fig. 6.3 demonstrate. We see from this figure that niobium with Ne = 5 and technetium with Ne = 7 are the best elemental superconductors.

The transition temperature Tc of some elements is raised dramatically by preparing them in thin films. For example the Tc of tungsten (W) was increased from its bulk value of 0.015 K to 5.5 K in a film; molybdenum (Mo) exhibited an

Figure 6.3. Dependence of the transition temperature Tc of transition elements on the number of valence electrons Ne. The other two transition elements with 5 valence electrons, namely vanadium V(Tc = 5.4 K) and tantalum Ta (Tc = 4.4 K), also have high transition temperatures, and the remainingelements with 4, 6, 8, and 9 valence electrons all have Tc below 0.5 K. Two alloys Zr1/3Nb2/3 andMo1/3Tc2/3 with high transition temperaturesandN eequal to 4.7 and 6.7, respectively, are also shown.

76 CHAPTER 6

increase from 0.92 K to 7.2 K and titanium (Ti) from 0.42 to 2.52 K. Chromium(Cr) and lithium (Li) superconduct only in the thin-film state; other nonsupercon-ductors, such as bismuth (Bi), caesium (Cs), germanium (Ge), and silicon (Si) can be converted into superconductors by either applying pressure or preparing them as thin films. Figure 6.1 summarizes this information.

Of the elements most commonly found in the newer ceramic-type supercon-ducting compounds lanthanum (La) is a superconductor; barium (Ba) supercon-ducts under pressure; and lead, which is added in low concentrations to stabilize the bismuth, thallium, and mercury high-Tc compounds, is also a superconductor.The elements copper (Cu), oxygen (O), and strontium (Sr), prominent in these high-Tc compounds, do not themselves superconduct. Thus superconducting prop-erties of elements are not indicative of the properties of their compounds, althoughniobium seems to be an exception.

6.4. ALLOYS

An alloy is a solid solution or atomic mixture of two or more different kindsof atoms whose constituents are randomly distributed on lattice sites. When two kinds of atoms are present, the alloy is called binary. Some alloys have their atoms arranged in an ordered manner on lattice sites for particular ratios of the constitu-ents. Both random and ordered alloys are known to superconduct. As an example consider a random binary alloy in which two elements are mixed in various proportions. The transition temperature Tc of such an alloy can be higher than thatof both elements, between the two values, or lower than either one alone. Figure 6.4 plots Tc versus the composition for two binary alloys involving the transitionelements niobium, vanadium (V), and zirconium (Zr). We see from the figure thatalloying Nb with Vbrings aboutadecrease in Tc, while combining Nbwith Zrraisesthe transition temperature; the maximum Tc occurs at a composition of about 33%Zr and 67% Nb. Thus we now have a method for raising the transition temperatureby selectively alloying one transition metal with another. Since Tc can eitherincrease or decrease with alloying, what is needed is a way ofpredicting how it willchange.

During the early 1950s B. Matthias found out that alloys with an averagenumber of valence electrons given by Ne = 4.7 and Ne = 6.7 are most likely to have high transition temperatures; 20 years later A. R. Miedema proposed empirical rules for estimating the transition temperature and other physical properties of superconducting alloys. We illustrate his approach by noting that in the periodic table in Fig. 6.1 niobium has five and zirconium has four valence electrons; we calculate theaverage number of valence electrons Ne of the highest Tc Nb-Zr alloy as follows:

(4.1)Ne = – (4)+ – (5)=1.33 + 3.33=4.71 23 3


Figure 6.4. Dependence of the transition temperature Tc on the concentration for the binary alloys niobium-zirconium(top sketch) and niobium-vanadium (bottom sketch). The abscissae are in percentage of V and Zr atoms, respectively. [Adapted from S. V. Vonsovskii, Yu. A. Izumov and E. Z. Kurmaev, “Superconductivity in Transition Metals,” Springer, New York (1982). p. 235.]

This gives the Matthias value Ne = 4.7. In like manner adding the element molybdenum (Mo) (Tc = 0.9 K) to technetium (Tc) (Tc = 7.8 K) also increases thetransition temperature: the maximum value Tc = 13 K is achieved at a composition of about 33% Mo and 67% Tc. Repeating the preceding calculation for this case:

1 2 (4.2)Ne = –(6) + –(7) = 2.00 + 4.67 = 6.7

we obtain the average valence electron numberNe = 6.7. These two alloys are at the highest Tc points of the plot in Fig. 6.3.

3 3

78 CHAPTER 6

Perhaps the most important of the classic superconductors is the niobium titaniumalloy (NbTi). Its maximum Tc of 10.1 Koccurs fora composition ofabout33% Ti and 67% Nb, which corresponds to an average number of valence electrons Ne equal to 4.7, as in NbZr. In practical applications it is more important to select the NbTi ratio to give the highest upper critical field and the best ductility for wire fabrication. Standard magnet wire has a critical temperature Tc of 9.5 K and anupper critical field Bc2 of 13 T at 4.2 K. The ability to draw NbTi so easily into fine wire filaments is largely responsible for it being the wire of choice for use in superconducting magnets. In a 6-T magnetic field, the critical temperature is 6.5 K; this makes the wire convenient for use in winding magnet coils for operation at a temperature of 4.2 K. This is discussed further in Chap. 9.

We have referred to percentages of atoms in such alloys as NbTi and TcMo, but magnet design engineers use percentages of weight. These percentages are not the same because each element has a characteristic weight or mass that differs from that of other elements; this must be taken into account when converting atom percentages into weight percentages. For example the niobium atom weighs almost twice as much as titanium (the exact ratio of their weights is 1.939); as a result anNbTi alloy that is 67% niobium by atom count is about 75% niobium by weight.

6.5. A-15 COMPOUNDS

The highest transition temperatures of the older superconductors were ob-tained with the so-called A-15 compounds A3B, andextensive data are available onthem. These compounds received their name from an old system of classifying crystal structures, called theStrukturberichtnotationinGerman (StructureReportsin English). This notation uses the letter A for elements, B for AB compounds, Cfor AB2 compounds, D for AmBn compounds, with additional letters used forcompounds containing three or more elements. Superconductors of the class Nb3Snwere originally assigned to what is called the beta tungsten or β-W structure of theelement tungsten (also called wolfram). This structure has two types of tungsten atoms that are in the lattice positions oftheA and B atoms, respectively, ofthe A3Bcompound; its Structurbericht symbol is A-15. The designation A-15 has endured over the years despite the fact that Nb3Sn is really a D-type compound AmBn withm = 3 and n = 1; hence A-15 is not really an appropriate classification.

The A3B compounds have a cubic structure with the B atoms in the body-centered cubic arrangement in Fig. 6.2; the A atoms are paired on the faces, as indicated in the left-hand side of Fig. 6.5. The A atoms form chains along the three crystal-lographic directions, as clearly seen in the cluster of eight unit cells on the right-handside of Fig. 6.5. The A atom is one of the eight transition elements Cr, Mo, Nb, Ta, Ti, V, W, and Zr; the B atom can be any one of 20 different elements.


Figure 6.5. Cubic unit cell of the A-15 compound (left) showing the locations of the A and B atoms.There is a second B atom in the center of the cell that is not shown. Group of eight unit cells (right)showing the chains of A atoms along the three perpendicular crystallographic directions x, y and

Typical A-15 compounds A3B only form for the 3: 1 ratio of A atoms to B atoms. Attaining this precise 3 to 1 ratio is important, and paying attention to it hasproduced higher transition temperatures. For example when samples of the com-pound NbxGe were prepared with x moving closer and closer to the exact value of3, the transition temperature increased from 6 K to 17 K and finally to the previous record value 23.2 K of Nb3Ge. Other A-15 compounds underwent a similar evolution, such as Nb3Ga(Tc increased from 3.8 to 17.9 K) andV3Sn (Tc increasedfrom 6.0 to 11.2 K) as the A/B ratio of atoms approached 3. In contrast to this, in some compounds the highest Tc does not occur at the ideal composition; for example Tc = 0.16 K for the exact-ratio case Cr3Ir, but it increases to Tc =0.75 K in the inexact case Cr3.0Ir0.66.

The B atom and each of the three A atoms of an A-15 compound contribute individual valence electrons to the compound. If we add three-fourths of the A atom electrons to one-fourth of the B atom electrons, we obtain the quantity Ne, which is the average number of valence electrons per atom. For example in the compound Nb3Ga, niobium has a valence of 5 and Ga a valence of 3, for a sum of 18 valence

(4.3)

electrons and an average of 4.5:

Ne = –(5) + –(3) = 3.75 + 0.75 = 4.53 1

For the A-15 compounds, as for the binary alloys just discussed, there is a close correlation between transition temperature and the average number of valence

4 4

80 CHAPTER 6

electronsNe . Four of the A- 15 compounds with the highest values of Tc have valence electron numbers Ne as follows:

A-15 Compound Ne Tc (K)Nb3Ga 4.5 20.3 Nb3Ge 4.15 23.2Mo3Tc 6.25 15 Ta3Au 6.5 16

These favorable Ne values are close to the optimum ones, 4.7 and 6.7, of the binaryalloys.

Several physical properties of A-15 compounds show a dependence on the average number of valence electrons Ne that is similar to the Tc dependence. Two examples of this are specific heat, which measures how effectively the material absorbs heat energy, and magnetic susceptibility, which measures the strength of magnetism in the material.

The A-15 compounds exhibit the highest transition temperatures of the classic superconductors, but they are not widely used in applications because they are too brittle and therefore not flexible enough to be drawn into wires. Despite its lower transition temperature (Tc = 9.5 K), the alloy niobium-titanium is easily drawn intowires; hence it is much more useful for applications.

6.6. NaCI-TYPE, LAVES, AND CHEVREL COMPOUNDS

There are a number of superconducting binary compounds AB with the sodium chloride NaCl structure shown in Fig. 2.5 in which A is a metallic element and B is a nonmetallic element. Examples are niobium nitride (NbN) with Tc = 17 K and molybdenum carbide (MoC) with Tc = 14.3 K. Sometimes the ratio of atoms is not a ratio of whole numbers or integers, as in the cases of vanadium carbide (V1.0C0.84),vanadium nitride (V1.0N0.75), and vanadium oxide (V0.76O1.0). This means that there are vacant sites in the lattice.

There are several dozen metallic AB, compounds called Laves phases that aresuperconducting. Some of them have critical temperatures above 10 K and highcritical fields; for example Zr1/2Hf1/2V2 has Tc = 10.1 K, Bc2 = 24 T; a compound with a different Zr:Hf ratio has similar Tc and Bc2 values, with the critical currentdensity Jc 4 x10 5 A/cm2. These materials also have the advantage of not being sohard and brittle as some other compounds and alloys with comparable transition temperatures.

≈

The Chevrel-phase compounds AxMo6X8 are mostly ternary (i.e., three-atom)transition metal compounds, where A can be almost any element and the element Xis one of the chalcogens sulphur (S), selenium (Se), or tellurium (Te). A chalcogen is an element of the Ne = 6 column on the right-hand side of the periodic table; two

CLASSlCA L SUPERCONDUCTORS 81

Figure 6.6. Structure of the Chevrel phase compound AxMo6Xu8. [Adapted from S. V. Vonsovskii, Yu. A. Izumov and E. Z. Kurmaev, “Superconductivity in Transition Metals,” Springer, New York (1982), p. 431.]

other chalcogens are the common element oxygen and polonium (Po), which is highly radioactive. The parameter x in the formula AxMo6X8 assumes variousvalues, such as x = 1 (e.g., YMo6S8) andx = 2 (e.g., Cu2Mo6Se8). These compoundshave relatively high transition temperatures and critical magnetic fields Bc2 ofseveral tesla, but their critical currents are rather low.

The crystal structure shown in Fig. 6.6 is quite interesting: White cubes in the figure represent Mo6X8-group building blocks (enlarged in the upper right-handcorner). The A atoms are in the center of the black cubes, one of which is shownenlarged in the lower right-hand corner of the figure. Various cubes in the structure are actually somewhat distorted, but these distortions are not shown in Fig. 6.6. The electronic and superconducting properties of these compounds depend mainly on the Mo6X8 groups, with the A ion having very little effect.

6.7. HEAVY-ELECTRON SYSTEMS

For several years prior to 1987, there was a great deal of interest in the studyof heavy-electron superconductors in which conduction electrons act as though they areveryheavy.Theseelectrons are said tohavelargeeffectivemassesm*, typicallyabout 200 times the free electron mass mo. Physicists have given these materials the rather pretentious name heavy Fermion superconductors, but we use the moredescriptive term heavy electron. The first such superconductor, CeCu2Si2, wasdiscovered in 1979, and some time passed before the heavy-electron phenomenon was confirmed by the discovery of UBe13 and then UPt3; since then quite a fewadditional cases have been found. Some of the researchers now active in the field of oxide superconductivity obtained their experience with the heavy-electron types.

82 CHAPTER 6

Superconducting charge carriers for these materials are Cooper pairs formed fromheavy electrons, so they have also large effective masses. The mass m* entersthe formulae for a number of normal and superconducting state properties, so these properties are influenced by the large mass. In fact much of the evidence for high effective mass comes from experimental observations of the normal state. Oneimportant superconducting property so affected is the penetration depth l, which is proportional to the square root of the effective mass m*. Therefore an effectivemass of 225 increases the penetration depth 15-fold, since 15 is the square root of 225.

Heavy-electron systems often exhibit two ordering transitions, a supercon-ducting transition at Tc and an antiferromagnetic ordering transition at the Néeltemperature TN, which is typically several degrees higher than Tc. Heavy-electronsuperconductors have anisotropic properties; i.e., they vary with direction in the crystal. These anisotropic properties include for example critical fields and electri-cal resistivity.

Some researchers believe that heavy-electron materials involve an unconven-tional type of superconductivity, one not invoking the usual electron-phononinteraction; a similar claim has been made about the newer oxide superconductors. There have also been reports of high effective masses in oxide superconductors; for example m

*/mo 12 in LaSrCuO, and m*/mo ranges from 5–100 in YBaCuO.

6.8.

≈

CHARGE TRANSFER ORGANICS

Organic compounds and polymers are ordinarily insulators, but it is now known that some of them form good electrical conductors; these conducting organics were widely studied during the 1970s. A typical organic compound (ED)2X that superconducts consists of an electron donor organic molecule (ED)and an inorganic atom group (X). An example of an electron donor is the organic molecule bis(ethylenedithio)tetrathiafulvalene, called BEDT-TTF for short, whichhas the chemical formula C6S8(CH2)4; examples of counterions X– areClO

–4, FSO

–3, and PF

–6. The electron donor ED transfers an electron to X:

(ED) + X (ED)+ + X–

to form a negatively charged counter ion, such as PF–6 . The separation of charge

creates electrons and holes that can become delocalized to render the compound conducting and at low temperatures, superconducting.

The electrical properties of organic conductors are often highly anisotropic: TCNQ salts behave as quasi-one-dimensional conductors; salts of other organics, such as BEDT-TTF, exhibit low dimensional behavior. Some of these organics show interesting similarities with the cuprates because of their layered structures. At the present time transition temperatures Tc of the organics are in the range of typical


classic superconductors; although they have been rising rapidly during the past decade, the best of them is still far below those of the cuprates.

6.9. CHALCOGENIDES AND ESPECIALLY OXIDES

Many of the classic superconductors contain one of the chalcogens—oxygen,sulphur (S), selenium (Se), or tellurium, with oxygen by far the least represented among the group. These elements have six valence electrons, as previously men-tioned. The Chevrel phases discussed earlier are examples of chalcogenides, i.e., compounds containing a chalcogen; in contrast to this newer superconductors are oxides. Thus the presence of Ne = 6 elements is a commonality that links older andnewer superconductors.

There are two oxide compounds listed in Table 4.1; one is the well-knownperovskite strontium titanate, (SrTiO3), which has the low transition temperature Tc = 0.3 K; the other is the spinel LiTi2O4 with a moderately high Tc = 13.7 K. The system LixTi3–xO4 is superconducting in the range of 0.8 x 1.33, with Tc in therange of 7-14 K.

6.10. PEROVSKITES

Perovskite is the name of the mineral calcium titanate with the chemical formula CaTiO3; it has the cubic structure shown in Fig. 6.7 with calcium (Ca) on the vertices, titanium (Ti) in the center, and oxygen at the center of the faces. More generally any compound with the formula ABC and the structure in Fig. 6.7 is referred to as a perovskite.

In their pioneering article, Bednorz and Muller’ called attention to the discov-ery over a decade earlier of superconductivity in the mixed valence compound BaPb1–xBixO3 by Sleight et al.† They pointed out that the bismuth (Bi) atoms ofthiscompound have two different valence states Bi3+ and Bi5+ and that structurally it is a distorted perovskite. The Bi5+:Bi3+ ratio plays an important role in determiningcritical temperature and other superconducting properties. The highest Tc obtainedfor this system was 13 K. In their original article, Bednorz and Muller reasoned that “Within the BCS system, one may find still higher Tc’s in the perovskite type orrelated metallic oxides, if (some parameters) can be enhanced further.” It was their determination to prove the validity of this conjecture that led to the greatest breakthrough in physics in the present decade. Their choice of materials to examine was influenced by the 1984 article of Michel and Raveau‡ on mixed valent

*J. G. Bednorz andK. A. Muller,Z. Phys., B64, 189(1986).†A. W. Sleight, J. L. Gillson, and P. E. Bierstedt, Solid State Commun. 17, 27 (1975). ‡C. Michel and B. Raveau, Rev. Chim. Miner 21, 407 (1984).

84 CHAPTER 6

Figure 6.7. The cubic perovskite unit cell of thecompound Ba1–xKxBiO3–y.

Cu2+—Cu3+ lanthanum-copperoxides containing alkalineearths. NOTE: Controllingthe copper ion charge ratio Cu2+:Cu3+ has always been an important factor in thequest for higher temperature cuprate materials.

≈≈

The compound Ba1–xKxBiO3–y with Tc 30 K for x 0.4, is structurally relatedto the Ba-Pb-Bi system; it was not discovered until after the advent of high Tc, andit is of particular significance for several reasons. This compound is the first oxide superconductor without copper with a transition temperature above that of all the A-15 compounds; the high Tc occurs without the presence of a two-dimensionalmetal-oxygen lattice; and the compound has the regular cubic perovskite structure sketched in Fig. 6.7, whereas BaPb1–xBixO3 is not a cubic but rather a distorted type of perovskite. The cubic potassium perovskite has attracted a great deal of attention, and the hope that it may elucidate the mechanism of high-temperature supercon-ductivity has prompted many experimental measurements.

6.11. BOROCARBIDES

During 1994 there was considerable interest in the borocarbide class of superconductor with the general formula RM2B2C, where M is usually the element nickel (Ni) and R a rare earth element. Transition temperatures in the low 20s have been reported for less well-characterized members of the series. Some of these compounds are simultaneously superconducting and antiferromagnetic. Related compounds, such as the boronitride La3Ni2B2N3, are also found to superconduct.

7

Fullerenes

7.1. INTERSTELLAR DUST

The course of scientific discovery often takes twisting paths. The discovery of a soccer-ball-shaped molecule containing 60 carbon atoms resulted from studying the transmission of light through interstellar dust—the small particles of matterfound in the regions of outer space between the stars. When light from a distant star passes through the cosmos to reach Earth, its intensity is greatly reduced; this is referred to as optical extinction. It occurs because of absorption and scattering from interstellar dust lying in the path of light on its way to Earth. Scientists study this extinction by measuring the intensity of light coming from the stars at different wavelengths, i.e., with different colors. When these studies were made, an increased extinction in the ultraviolet region at a wavelength of 2200 Å, (5.6 eV) was noted. It was attributed to light scattered from small particles of graphite throughoutregions between stars. Figure 7.1 plots extinction versus photon energy. Thisexplanation for the optical extinction in the 2200-Å, region was widely accepted by astronomers.

Donald Huffman of the University of Arizona and Wolfgang Kratschmer ofthe Max Planck Institute of Nuclear Physics in Heidelberg were not convinced by this explanation, so they decided to study the question further. Their approach wasto simulate graphite dust in the laboratory to investigate light transmission through it. They made smokelike particles by striking an arc between two graphite elec-trodes in a helium gas environment, then condensing the smoke on quartz plates.Various spectroscopic methods, such as infrared and Raman spectroscopy, whichcan measure vibrational frequencies of molecules, were used to investigate the condensed graphite. The researchers did indeed obtain spectral lines known to arise from graphite, but they also observed four additional infrared absorption bands that did not originate from graphite.

Although a soccer-ball-shaped molecule consisting of 60 carbon atoms withthe chemical formula C60 was envisioned by theoretical chemists for a number of

85

86 CHAPTER 7

Figure 7.1. Plot of the extinction (reduction) in intensity of light coming from stars versus the energy of the photons of the light. The bump at 5.6 eV (wavelength of 2200 angstroms) arises from theabsorption by interstellar dust.

years, no evidence of its existence had ever been found. Many detailed properties of the molecule were however calculated by theorists, including a prediction of how the infrared absorption spectrum of the molecule would look. To the amazement of Huffman and Kratschmer, the four bands observed in the condensed graphitematerial corresponded very well to those predicted for a C60 molecule. Could the extinction of ultraviolet light coming from stars be due to the existence of C60

molecules? To further verify this scientists studied the infrared absorption spectrum using carbon arcs made of the 1% abundant 13C isotope, then compared it to their original spectrum that arose from the usual 12C isotope. It was well-known that thischange in isotope would shift the infrared spectrum by the square root of the ratio of the masses, which in this case has the value:

131/2

121/21.041

corresponding to a shift of 4.1%. This is exactly what was observed when theexperiment was performed. The two scientists now had firm evidence of theexistence of an intriguing new molecule consisting of 60 carbon atoms bonded in

-= (7.1)

FULLERENES 87

the shape of a sphere. Other experimental methods, such as mass spectroscopy, were used to verify this conclusion, and results were published in Nature in 1990. NOTE:From the discussion in Chap. 5, we know that the isotope effect involving a shift in the superconducting transition temperature Tc by the square root of the isotope mass ratio played a crucial role in establishing the electron-phonon interaction as the dominant mechanism for Cooper pairing in classic superconductors.

Other research groups were also verifying the existence of the C60 moleculeby different methods, although ironically cosmological issues were also drivingtheir research. Harlod Kroto, a chemist from the University of Sussex in England, was part of a team that found evidence of the presence of long linear carbon chain molecules in outer space. He was interested in how these chains came to be, and he speculated that such molecules could be created in the outer atmosphere of a type of star called a red giant. To test his hypothesis, Kroto wanted to recreate conditions of the star’s outer atmosphere in a laboratory setting, then see if linear carbon chains formed. Kroto knew that high-powered pulsed lasers would simulate the hot carbon vapor conditions that might exist in the outer surface of red giants. He contacted Professor Richard Smalley of Rice University in Houston, who built the apparatus shown in Fig. 7.2 to make small clusters of atoms using high-powered pulsed lasers. In this experiment a graphite disk is heated by a high-intensity laser beam that

Figure 7.2. Apparatus used to make clusters ofcarbon atoms andeventually C60 molecules. The carbonevaporated from the graphite disk by the heating from the laser beam condenses into atomic clusters which include C60 molecules to form the cluster beam. The cluster beam proceeds to the massspectrometer where the masses of the various sized carbon aggragates are measured and the presence of C60 is confirmed.

88 CHAPTER 7

produces a hot vapor of carbon. A burst of helium gas then sweeps the vapor through an opening, and the beamexpands. The expansion cools the atoms so they condenseinto clusters. This cooled cluster beam is narrowed by a skimmer, then fed into amass spectrometer—a device to measure the mass of molecules in the clusters. When the experiment was done using agraphite disk, the mass spectrometeryieldedan unexpected result: A mass number of 720, which would consist of 60 carbon atoms, each of mass 12. Evidence for a C60molecule had been found! Althoughdata from this 1990 experiment did not give information about the structure of thecarbon cluster, the scientists suggested that the molecule mightbe spherical, so theybuilt a geodesic-like dome model of it.

7.2. FORMS OF CARBON

Carbon is one of the most common substances on earth. It is an essentialconstituent in the fundamental organic molecules like amino acids that make up such living things. Until the discovery of C60 only two forms of carbon were known, namely, graphite and diamond; Fig. 7.3 shows the structures of these two forms.

Graphite, which is the lead in lead pencils, has a layered structure consistingof a series of perfectly flat layers stacked on top of each other. Within a particular

Figure 7.3. Structures of the two forms of carbon, namely graphite (top) and diamond (bottom).

FULLERENES 89

layer, each carbon atom is covalently bonded to three other carbon atoms in the same layer; each bond forms 120-degree angles with the two adjacent bonds.

Carbon also forms the diamond structure; this material is the hardest naturally occurring substance known to man. In this structure each carbon atom is covalently bonded to four carbon atoms that form the vertices of a tetrahedron; all of the bond angles are very close to 109 degrees.

From our modern viewpoint, we can say that there is a third form of carboncalled fullerene, with the formula C60. The bond angles of the carbons in the various fullerenes are close to but slightly less than 120 degrees.

7.3. SOCCER BALL FULLERENE

The C60 molecule was named fullerene after the architect and inventor R.Buckminister Fuller, who designed the geodesic dome that resembles the structureof C60. Originally the molecule was called Buckministerfullerene, but that was

Figure 7.4. Structure of the C60 fullerene molecule.

90 CHAPTER 7

Figure 7.5. Crystallographic unit cell of the C60

structure showing one molecule in the center and parts of eight others at the vertices, and showing the location of smaller alkali atoms added bydoping. A tetrahedral site in which the alkali has four surrounding C60 spheres and an octahedral site with six such C60 neighbors are indicated.

unwieldly, so the name was shortened to fullerene; a picture of the molecule isshown in Fig. 7.4. It has 12 pentagonal (5-sided) and 20 hexagonal (6-sided) faces symmetrically arrayed to form a molecular ball; in fact a soccer ball has the samegeometric configuration as fullerene. These ball-like molecules bind with each other in the solid state to form a crystal lattice with a face-centered cubic structure shown

Figure 7.6. Plot of the superconducting transition temperature of alkali doped C60

crystals versus thelength of the side of the unit cell.

FULLERENES 91

in Fig. 7.5. In the lattice each C60 molecule is separated from its nearest neighborby 10 Å (the distance between their centers is 10 Å); they are held together by weak forces called van der Waals forces. Because C60 is soluble in benzene, single crystals of it can be grown by slow evaporation from benzene solutions.

In the face-centered cubic fullerene structure (see Fig. 7.5),26% of the volume of the unit cell is empty, so alkali atoms easily fit into empty spaces between molecular balls of the material. When C60 crystals and potassium metal are placedin evacuated tubes, then heated to 400°C, an atmosphere of potassium vapordiffuses into these empty spaces to form the compound K3C60. This compound isno longer an insulator but becomes conducting. Figure 7.5 shows where the alkaliatoms go when they enter the lattice to occupy the three vacant sites per C60

molecule.

7.4. SUPERCONDUCTIVITY

As we mentioned crystals of graphite consist of parallel planes of the graphitic sheets (see Fig. 7.3). It is possible to put other atoms between the planes of thesesheets; this procedure is called intercalation. When intercalated with potassiumatoms, crystalline graphite becomes superconducting at the extremely low tempera-ture of a few tenths of a Kelvin. Perhaps motivated by this observation, in 1991 A. F. Hebard and coworkers at Bell Telephone Laboratories doped C60 crystals withpotassium by the previously described methods, then tested the crystals for super-conductivity. To the surprise of all, there was evidence of a superconductingtransition at 18 K. A new class of superconducting materials had been found that had a simple cubic structure and containing only two elements. Not long after theinitial report, it was found that many alkali atoms could be doped into the lattice and the transition temperature increased to as high as 33 K in Cs2RbC60. As theradius of the dopant alkali atom increases, the cubic C60 lattice expands, so thesuperconducting transition temperature rises. Figure 7.6 plots the transition tem-perature versus the lattice parameter.

A major question is, What is the mechanism of superconductivity in thealkali-doped fullerenes? Is it a phonon-mediated BCS mechanism? The criticalexperiment for testing this was thought to be the isotope experiment. Carbon has an isotope whose atomic mass is 13, compared to a more abundant carbon of mass 12. Thus a C60 molecule made from 13C is 8.3% heavier than a C60 made from 12C.As discussed in Chap. 5, the BCS theory predicts a downward shift in the transitiontemperature in a superconductor made from a heavier molecule. Figure 7.7 showsa measurement of magnetization as a function of temperature in K

3

13C60

andK3

12C60. The isotopically labeled material becomes superconducting at 0.4 K below the nonlabeled material. However the phonon-mediated BCS theory predicts adownward shift of 0.8 K, twice that obtained in the measurement. The measure-

92 CHAPTER 7

Figure 7.7. A measurement of the magnetic susceptibility versus the temperature of K3C60 made fromthe heavy isotope carbon-I3 (O) compared with the susceptibility of the same compound made from theordinary isotope carbon- 12 There is an evident 0.4 K downward shift of the transition temperature in the heavier C60 made from 13C. [Adapted from A. F. Hebard, Physics Today 29, November (1992).]

ment therefore offers supportbut does not unequivocally prove that the mechanismis a phonon-mediated BCS process. As a matter of fact an electronic process hasbeen demonstrated to predict a 0.4 K downward shift. On the other hand, we seefrom data in Table 5.1 that several elementary superconductors believed to be BCSphonon types have an isotope-effect exponent α that is only half of the predictedvalue. Thus the establishment of the mechanism remains a matter for futureresearch.

7.5. FULLERENE CLASS OF MOLECULES

We have been using the name fullerene to designate the C60 molecule, but C60

is only one of a class of molecules, and all members of this class are calledfullerenes. It is instructive to examine the characteristics of this class of molecules.

A fullerene molecule contains only carbon atoms; each carbon is bonded tothree other carbons to form a closed surface. In geometrical terms a fullerene formsa polyhedron or many-sided figure that has three edges coming together at eachvertex. Figure 7.8 shows the five regular polyhedra, called Platonic solids after theGreek philosopher Plato who lived 2400 years ago. Three of the solids, namely, the tetrahedron, the cube, and the 12-sided dodecahedron, have three edges at each vertex. If we place carbon atoms at the vertices of these figures, we obtain fullerene molecules. In contrast to this, the 8-sided octahedron has four edges at each vertex, and the 20-sided icosahedron has five edges at each vertex, so they would not form fullerenes. None of the carbon molecules associated with these five Platonic solids has ever been synthesized.

FULLERENES 93

Figure 7.8. The five Platonic solids arranged from left to right in the order of increasing number offaces: tetrahedron (4 faces), cube (6 faces), octahedron (8 faces), dodecahedron (12 faces) andicosahedron (20 faces). [Adapted from F. Durell and E. E. Arnold, “New Solid Geometry,” Merrill Co., NewYork(1932).]

The reason for the three bonds per vertex rule is that it leads to a molecule of particularly high stability. Consider the case of the molecules benzene and naph-thalene, which have the structures shown in Fig. 7.9. These molecules havealternating single and double bonds between the carbon atoms; stability arises from the possibility of having different arrangements of single and double bonds. We see from Fig. 7.9 that benzene has two and naphthalene has three such arrangements,which are called resonant structures.

Larger molecules of this series with three, four, or more fused benzene rings have many more than three resonant structures; hence they are very stable. Exten-sive sheets of graphite have virtually uncountable numbers of resonant structures. The C60 molecule has an incredibly large number of resonant structures, hence its great stability.

The actual structure of any conjugated molecule, meaning one with a multi-plicity of resonant structures, can be viewed as a combination of all possiblestructures existing simultaneously. This is because extra electrons of the doublebonds are not in fixed positions like single-bond electrons; instead they wander around the molecule and are said to be delocalized. When C60 is doped with alkalimetals to form for example the potassium-doped compound K3C60, potassiumatoms become ionized to form the positive ion K+ plus an electron:

(7.2)K K+ +e-

These pass on their electrons e– to the C60 molecules, making them negativelycharged:

3e- + C60 C3–60 (7.3)

Thus each fullerenehas three extra delocalized electrons. These extraelectrons notonly wander around their respective C60 molecules, but they can jump from one C60

to another and thereby carry electrical current. This explains why alkali-dopedcrystals of fullerene are good conductors of electricity.

We discussed C60 and smaller fullerenes, but besides C60 almost all fullerenes studied are larger molecules Cn such as C76 with 76 carbon atoms. A fullerene with

Figure 7.9. Resonant structures of the benzene (C6h6 top) and naphthalene (C10H8 bottom) molecules.

94 C

HA

PT

ER

7

FULLERENES 95

n carbon atoms has the following number of bonds between carbons and numberof faces in its polyhedron:

Number of atoms: nNumber of bonds: 1.5nNumber of faces: 1/2n + 2

Therefore C60 with n = 60 has 60 carbons, 90 bonds, and 32 faces. Each largefullerene has 12 pentagonal (5-sided) faces, and all of the remaining faces arehexagons (six-sided).

Figure 7.10. An illustration of the structure of a tubular carbon sheet called tublene.

96 CHAPTER 7

7.6. TUBLENES

Many of the scientists involved in synthesizing C60 are now making yet another new form of carbon, which they call tublene; an illustration of a part of the structure of tublene is shown in Fig. 7.10. We can think of it as being formed by rolling asheet of graphite into a tube. Scientists have been able to make these tubes of carbon, but generally they are nested; this means they consist of a number of concentrictubes of smaller radii inside each other. In other words they resemble rolled up graphite containing a small number of sheets. The Japanese scientist Sumio Iijima*reported an electron micrograph picture of single-sheet carbon tubes, sometimes called nanotubes, since their diameters range from 0.7– 1.4 nm. The interest in thesenanotubes is very high because they are expected to have a number of propertiesthat could lead to important applications. Theoretical calculations have shown that how the sheets are connected when rolled up to form tubes strongly influences their electronic properties; thus it is possible to engineer them from semiconductors toinsulators. In other words the band gap depends on a parameter called helicity thatmeasures the degree of twist in the tubes. Furthermore long fibers of these tubes wound around each other are expected to be the bases for materials that may bestronger than steel. Perhaps alkali metals inside small-diameter tubes will transferelectrons to the tubes to make them conducting and hopefully even supercon-ducting.

*S. Iijima, Nature 354, 56 (1991).

8

New High-TemperatureSuperconductors

8.1. INTRODUCTION

We have discussed properties of superconductors in general, and our examples were primarily classic types with transition temperatures Tc below 24 K thatoperated at liquid helium temperature (4.2 K). The discovery of materials that superconduct above 77 K raised the possibility of commercial applications using liquid nitrogen as a cooling fluid. Thisconsiderably reduces the cost of operation, since liquid nitrogen sells for $0.25 a liter unlike liquid helium, which costs $4 a liter.

These superconductors, often called cuprates because they contain copper atoms bonded to oxygen, constitute different classes of compounds than the old ones, raising the intriguing possibility of an entirely new mechanism of supercon-ductivity. Chapter 8 describes characteristics of high-temperature materials and examines possible explanations for their novel features. Later chapters discuss progress in developing applications based on these materials.

8.2. TRANSITION TEMPERATURE

Progress in developing materials with higher transition temperatures was very slow prior to 1986. After the first observation of the phenomenon in 191 1 in mercury at 4.2 K 19 years passed before reaching 9.2 K with niobium. It took another 24 years to reach 18.1 K with niobium-tin and 19 additional years to go up a mere 5more degrees to attain 23.2 K with niobium-germanium. We see from Table 8.1 that in only 10 years since 1986, the transition temperature was raised progressively from 23 K to 35, 52, 95, 110, 125, and 133; applying a pressure of 30 GPa raised the Tc of the mercury superconductor to 147 K. There appears to be no end in sight, andperhaps someday there willbe aroom-temperaturesuperconductor. Wediscussthis possibility in Chap. 12.

97

98 CHAPTER 8

Table 8.1. Progress in Raising the SuperconductingTransition Temperature TcSince the Discovery of Cuprates in 1986

Material Tc (K) Year

BaxLa5–xCu5O9 30-35 1986(La0.9Ba0.1)2Cu4O4–x (at 1-GPa pressure)a 52 1986YBa2Cu3O7–x 95 1987Bi2Sr2Ca2Cu3O10 110 1988Tl2Ba2Ca2Cu3O10 125 1988

HgBa2Ca2Cu3O8+x 133 1993HgBa2Ca2Cu3O10 (at 30-GPa pressure) 147 1994aA pressure of 1 GPa is about 10,000 atm.

Tl2Ba2Ca2Cu3O10 (at 7-GPa pressure) 131 1993

While this increase in Tc itselfis an amazing result, ahigh-transition tempera-ture is not the only property required to make new compounds useful for applica-tions. For example if materials are to be used as wires in magnets, they must be malleable and ductile rather than brittle; in addition they must have high critical currents in large magnetic fields. Critical currents as high as those in niobium-tinhave not yet been achieved in forms of the new materials that can easily be made into wires, although there are reports of comparable values in thin films on various substrates.

The Holy Grail that is being sought is a transition temperature much above room temperature. We say much above because devices must operate significantly below the transition Tc so that the critical current Jc and critical magnetic field Bc

are sufficiently high. Very close to the transition temperature, the critical magnetic field is usually quite small, but we see from Figs. 3.4 and 3.5 that Bc and Jc

continuously increase as the temperature is lowered below Tc. We need an operating temperature far below the critical surface in Fig. 3.15 so that both Bc and Jc aresufficiently large for the desired application.

8.3. LAYERED STRUCTURE OF THE CUPRATES

All cuprate superconductors have the layered structure shown in Fig. 8.1 : The flow of supercurrent takes place in conduction layers, and binding layers support and hold together the conduction layers. Conduction layers contain copper-oxide(CuO2) planes of the type shown in Fig. 8.2; each copper ion (Cu2+) is surrounded by four oxygen ions (O2–). These planes are held together in the structure by calcium (Ca2+) ions located between them, as indicated in Fig. 8.3. An exception to this is the yttrium compound in which the intervening ions are the element yttrium (Y3+)instead of calcium. These CuO2 planes are very close to being flat. In the normal state above Tc, conduction electrons released by copper atoms move about on these

NEW HIGH-TEMPERATURE SUPERCONDUCTORS 99

Figure 8. 1. Layering scheme of the cuprate superconductors. Figure 8.3 shows details of the conduction layers for different sequences of copper oxide planes, and Fig. 8.4 presents details of the binding layersfor several cuprates.

Figure 8.2. Arrangement of copper and oxygen atomsin a CuO2 plane of the conduction layer.

100 CHAPTER 8

Conduction layer with one copper oxide plane

Conduction layer with two copper oxide planes

Conduction layer of yttrium compound with two copper oxide planes

Conduction layer with three copper oxide planes

Figure 8.3. Conduction layers of the various cuprate superconductors showing sequences of CuO2 andCa (or Y) planes in the conduction layers of Fig. 8.1.

CuO2 planes carrying electric current. In the superconducting state below Tc, thesesame electrons form the Cooper pairs that carry the supercurrent in the planes.

Each particular cuprate compoundhas its own specific binding layerconsistingmainly of sublayers of metal oxides MO, where M is a metal atom; Fig. 8.4 gives the sequences of these sublayers for the principal cuprate compounds. These binding layers are sometimes called charge reservoir layers because they contain

Lanthanum Superconductor La2CuO4

Neodymium (electron) Superconductor Nd2CuO4

Yttrium Superconductor YBa2CU3O7

Bismuth Superconductor Bi2Sr2Can–1CunO2n+4

Thallium Superconductor T12Ba2Can–1CunO2n+4

Mercury Superconductor HgBa2Can–1CunO2n+2

Figure 8.4. Sequences of MO sublayers in the binding layers of Fig. 8.1, where M stands for various metal ions. The parentheses around the oxygen atom O in the lowest panel indicates partial occupancy.

102 CHAPTER 8

Figure 8.5. Layering schemes of three thallium compound superconductors Tl2Ba2Can–1CunO2n+6,where there are n = 1, 2, 3 CuO2 planes in the conduction layers, from left to right. [Adapted fromTorardi et al., Science 240, 631 (1988).]

103NEW HIGH-TEMPERATURE SUPERCONDUCTORS

ConductionLayer

BindingLayer

ConductionLayer

BindingLayer

ConductionLayer

Figure 8.5. (Continued)

104 CHAPTER 8

Table 8.2. Superconducting Transition Temperatures of theCuprates

NumberofCuO2Planes

Compound Type Formula 1 2 3

Lanthanum (La0.9 Sr0.1)2CuO4 35Nedodymium (Nd 0.9Ce0.1)2CuO4 35Yttrium YBa2Cu3O7 —Bismuth Bi 2Sr 2Can–1CunO2n+4 — 84 110

— — ———92

Thallium T12Ba2Can–1CunO2n+4 90 110 125 Mercury HgBa2Can–1CunO2n+2 95 122 133

the source of charge required to bring about hole doping in the conduction planes through thefollowing interaction:

cu2+ → Cu3+ +e- (8.1)

taking place in a copper-oxide plane. Figure 8.5 provides a three-dimensionalperspective of how conduction and binding layers are arranged in the thalliumcompounds containing one, two, and three copper oxide planes, i.e., having n = 1, 2, and 3 in the formula T12Ba2Can–1CunO2n+4.

We may expect to be able to make 18 different compounds by combining eachof the six binding layers in Fig. 8.4 with conduction layers containing 1, 2, or 3 copper oxide planes, but not all of these compounds can be synthesized. In addition some of them, such as the single-layer bismuth compound Bi2Sr2CuO6, can besynthesized but do not superconduct. Table 8.2 gives the transition temperatures of the 11 compounds that do superconduct; dashes in the table designate compounds that either do not exist or do not superconduct. From this table we see that mercury compounds are the best conductors and thallium compounds are second; bismuth compounds seem to be more easily made into wire, and they may be preferable for applications. However because of its ease of synthesis and interesting properties, the yttrium compound has been the most widely studied of the group. We also see from Table 8.2 that increasing the number (n) of CuO2 planes in the conductionlayers raises Tc , but unfortunately Tc begins to decrease for further increases in n > 3.

8.4. HOLE-TYPE COOPER PAIRS

In classic superconductors electric current is carried by paired electrons called Cooper pairs, with a charge of –2e. In most of the new cuprate materials, Cooperpairs have the positive charge +2e; in other words they are paired positive holesdesignated by h. By holes we mean absent electrons in an electron shell or level


Figure 8.6. Hopping (a) of electrons e, and (b) of holes h, at the top of an almost filled band.

that is otherwise filled with electrons. A plus one hole is formed in an electron shellthat is one electron short of being filled. When hole notation is employed, the charge transfer process in Eq. 8.1 can be written

(8.2)

Figure 8.6 depicts electron motion in an almost filled band when electrons experience a force inducing them to hop to the left to occupy a hole if one exists. Figure 8.6a shows three successive hops by an electron tooccupy a hole on theleft;Fig. 8.6b shows that this is equivalent to hopping to a hole on the right. Figure 8.7 shows a Cu3+ ion moving along a line of Cu2

4+ ions; this trivalent ion Cu3

4+ can be

viewed as Cu2+ plus a hole. Successive electron exchange interactions of the type:

Cu2+ = cu3+ + e– (8.3)

Cu3+ + e– = Cu2+ (8.4)

can cause the hole on Cu3+ to move along the line of divalent copper ions, asindicated.

Cu2+ + h+ → Cu3+

Figure 8.7. Hopping of the hole on a Cu3+ ionalong a line of Cu2+ ions. The hole moves to theright on successfully higher panels.

106 CHAPTER 8

Another way of seeing that this is a hole conduction process is to consider the outermost electronic shell of the copper ions, which is called a 3d level in atomic spectroscopy. This level can hold a maximum of ten electrons; it is filled for the ion CU1+. The ion Cu2+ has only nine electrons in its third level, which corresponds toten electrons plus one hole; Cu3+ has eight electrons or ten electrons plus two holes. Electrical current in the normal state is carried by these holes, which are in the conduction band, via the hopping mechanism depicted in Fig. 8.7. Electric current in the superconducting state is carried by Cooper pairs formed from these holes.

The number of holes in the copper-oxide planes depends on the relative amounts of Cu3+ to Cu2+ valence states. To see how this can be controlled by doping, consider La2CuO4 as a purely ionic compound composed of two La3+ ions, one Cu2+

ion, and four O2– ions. If 10% of the trivalent La3+ is replaced by divalent Sr2+ toform the compound (La0.9,Sr0.1)2CuO4–x, there are two ways of achieving electrical neutrality, i.e., to make the positive charges balance negative charges in the compound. One way is to let all of the copper remain as Cu2+ and reduce the amount of oxygen; if we do this, we obtain the compound (La0.9,Sr0.1)2CuO3.9. Copper-oxideplanes are not changed by this process, since all of the copper is still in the Cuz+

state. Another way of doping the original compound is to keep the oxygen content

Figure 8.8. Plot providing the superconducting transition temperature versus the strontium content xof the compound La2–xSrxCu4–y. The region of antiferromagnetic behavior is also indicated.


corresponding to the formula (La0.9,Sr0.1)2CuO4 and achieve charge neutrality by converting 20% of the divalent Cu2+ ions to trivalent Cu3+. The result is copper-oxideplanes with many holes arising from the presence of so many Cu3+ ions. The charge balance equation for achieving charge neutrality in this manner is

La3+ + Cu2+ → Sr2+ + Cu3+ (8.5)

In practice both processes can occur: There can be some decrease of oxygen and some conversion to Cu3+ ions. The concentration of holes on the copper-oxideplanes can be calculated when the parameter x in the formula (La0.9,Sr0.1)2CuO4–x isknown. This superconductor is often referred to as the lanthanum compound.

The superconducting transition temperature depends on the hole concentration as shown in Fig. 8.8. The graph shows that superconductivity occurs only over a certain range of hole concentrations and there is an optimum concentration for the highest Tc. Figure 8.8 also shows that there is an ordered antiferromagnetic phase at low-strontium contents.

8.5. DIRECTION-DEPENDENT PROPERTIES

One of the consequences of a hole-hopping process involving a two-dimensionalarray ofcopper ions is that the superconducting current is very direction-dependent oranisotropic. Hopping tends to occur between coppers ions that have the smallest separation from each other, namely, those in the plane. The distance between coppers ions in adjacent planes is much larger than within the planes; hence charge hopping between planes is much less efficient. This means that current flows much more easily within a plane than by jumping from one plane to another. In a single crystal, current measured above Tc and critical current density Jc below Tc depend on thedirection in which the determination is made. Many other properties of cuprate superconductors, such as the critical fields Bcl and Bc2, penetration depth (λL), andthe coherence length (ξ) also depend on the direction.

In many applications single crystals provide the best performance, since they can be aligned in a preferred direction for each particular application. Thin-filmsamples, called epitaxial films, have their atomic planes aligned parallel to thesurface of the film so they conduct well along the film; they are important for some applications. When single crystals are not available, a technique called grainalignment may be employed to convert a collection of randomly oriented grainsinto a set of grains with their atomic planes preferentially arranged parallel to each other. Three ways of accomplishing this alignment involve applying pressure to compress grains in a particular direction; embedding grains in a resin, such as epoxy, then applying a strong magnetic field to the composite; and melting a granular powder sample, then reforming it in an oven whose temperature varies in a regular manner across the sample. Grain-aligned samples are not so suitable as

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single crystals, but they are far superior to powders of randomly oriented grains. In a later chapter, we say more about increasing the current flow capability of superconductors.

8.6. ELECTRON SUPERCONDUCTORS

Section 8.5 explains that such cuprates as (La1–x,Srx)2CuO4 are hole-typeconductors. There is a material, neodymium-cerium-copper oxide, (Nd 1–x

Cex)2CuO4, whose charge carriers are electrons rather than holes. The parent compounds Nd2CuO4 and La2CuO4 both have trivalent positive ions:

La ⇒ La3+ + 3e– (8.6)

Nd ⇒ Nd3+ + 3e– (8.7)

but they differ in the valences of the dopants strontium (Sr2+) and cerium (Ce4+),respectively:

Sr ⇒ Sr2+ + 2e– (in La2CuO4) (8.8)

Ce ⇒ Ce4+ + 4e– (in Nd2CuO4) (8.9)

Thus when cerium replaces neodymium, it adds one extra electron to form an electron-like superconductor. In contrast to this strontium subtracts one electron, which is the same as saying it adds a hole, so the superconductor is hole-like. Anytheory of superconductivity must take into account both of these examples of superconductors. These two compounds have similar, but not identical structures; their electrical properties are also quite similar because most experiments are not sensitive to the sign of the charge carrier.

8.7. CERAMICS, PEROVSKITES, AND STRUCTURES

Although new superconductors are widely referred to as ceramics, they are more properly classified as perovskites. The term perovskite refers to the particular arrangement of atoms shown in Fig. 8.9 for the mineral perovskite, calcium titanate (CaTiO3). Atom positions in the upper (and lower) parts of the lanthanum com-pound unit cell in Fig. 8.10 are similar to those in perovskite, with Cu present in the titanium sites of perovskite, La in the calcium (not shown in Fig. 8.9) positions, and oxygen at the same location in both structures. Similarities between these two atom arrangements induced crystallographers to call La4CuO4 a perovskite-typematerial.

NEW HIGH- TEMPERATURE SUPERCONDUCTORS 109

Figure 8.9. Sketch of the cubic unit cell of the mineral Perovskite, CaTiO3, showing titanium at thevertices and oxygen in the middle of the edges. Calcium, not shown, is in the center of the cube.

In contrast the ceramic designation is not based on structural grounds but on the similarity of the cuprate-superconducting compound and ceramic manufactur-ing process. For example La-Sr-Cu-O is made by heating mixtures of lanthanum oxide, strontium carbonate, and copper oxide in air at 900-1000 °C for 20 hours. Proportions of atoms in the initial mixture should be the same as in the end product, and for the compound (La0.9Sr0.1)2CuO4 the ratio La:Sr:Cu is 1.8:0.2: 1. Materials are usually ground to a fine mixture before heating; after heating in air, they are cooled, pressed into pellets, and reheated from 900-1000 °C for several more hours.

We see in Fig. 8.10 that the superconductor (La1–xSrx)2CuO4 has only one copper oxide plane in its conduction layer and each copper ion is surrounded by

Figure 8.10. Atom positions in the tetragonal unit cell of the La2CuO4 compound. When strontium issubstituted for lanthanum in the superconducting compound (La1–xSrx)2CuO4 it replaces lanthanum in some of the La sites.

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six neighboring oxygen ions; these form an 8-sided figure called an octahedron, as shown. The CuO6 complex of one copper and six oxygens is present in all cuprate superconductors that have a single CuO2 plane in their conduction layer. Figure 8.11 shows atom arrangements in the mercury compound HgBa2Ca2Cu2O10, which has three such planes in its conduction layer. In the upper and lower planes, copper ions have five neighboring oxygens forming a CuO5 group with the shape of a pyramid, as shown. The middle copper ions have only four nearby oxygens, forming what is called a square planar group CuO4. If we consider removing the centralcopper oxide plane and one calcium layer from Fig. 8.11, we generate the two-planestructure in which all copper ions form CuO5 pyramids. These structural details may somehow constitute important factors in determining why cuprates are such good superconductors.

Figure 8.11. Atom positions in four unit cells of the superconducting compound HgBa2Ca2Cu3O8+x

which has Tc = 133 K. The copper ions of the upper CuO2 plane are hidden by the pyramids, and some partially occupied oxygen sites in the mercury Hg plane are not shown.


8.8. YTTRIUM COMPOUNDS

The discovery that generated extensive news coverage in early 1987 was not the initial report of superconductivity in the lanthanum compound by Bednorz and Muller (see Fig. 8.12) but rather the observation of superconductivity in the compound YBa2Cu3O7–x at 92-94 K, well above the boiling point (77 K) of liquid nitrogen, as shown in Fig. 8.13. This discovery was the outcome of a collaboration between the research groups of C. W. Chu of the University of Houston and M. K. Wu of the University of Alabama.

Figure 8.12. First reported drop to zero resistance for a high temperature superconductor. [from J. G.Bednorz andK. A. Müller,Z. Phys. B64, 189 (1986).]

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Figure 8.13. A plot of the historic observation of the disappearance of resistance in YBa2Cu3O6+x, thefirst material in which superconductivity was observed above the boiling point of liquid nitrogen (77 K). [Adapted from M. K. Wu et al., Phys. Rev. Lett. 58, 908 (1987).]

Like the lanthanum compound just described, the present material is deficient in oxygen, and the x in the chemical formula denotes the possibility of differentamounts of oxygen in the crystal lattice. The parameter x is zero when all oxygen sites are full, and x is about 0.1 for the highest Tc. Figure 8.14 shows that thetransition temperature depends on the oxygen content and the average copper ion charge. Many of the important electrical, magnetic, and other properties are also sensitive to the amount of oxygen in the crystal. As the oxygen content is varied, relative amounts ofCu3+ to Cu2+ change to keep the crystal electrically neutral. Thestructure is orthorhombic with latticeconstants a b in the superconducting regionand tetragonal with a = b for x < 0.4 where it no longer superconducts. There aretwo copper oxide planes in the conduction layer, with yttrium ions between these two planes. We see from Fig. 8.4 that the middle of the binding layer also has copper ions, and they form chains Cu-O-Cu-O- . . . along the crystallographic b-axis thatmay contribute to superconducting properties.

The substitution of first-transition-series ions, such as zinc (Zn) or nickel for copper in YBa2Cu3Ox generally causes a reduction of the transition temperature;this underlines the important role that copper plays in the superconducting mecha-nism. Compared to the magnetic nickel ion, deterioration of superconducting


Figure 8.14. A plot of the superconducting transition temperature versus the oxygen content (7—x)of the c ompound YBa2Cu3O7–x..

properties is greater when the nonmagnetic zinc ion is substituted for copper; this suggests the importance of the magnetic nature of copper in the superconducting mechanism. Most of the rare earth elements can replace yttrium without a signifi-cant change in superconducting properties.

Like the lanthanum compound, many recipes can be used to synthesize this yttrium compound; in fact it is one of the easiest cuprates to make in the laboratory. A typical process involves grinding and pressing into pellets mixtures of Y2O3,BaCO3, and CuO in the molecular ratio of 1:4:6. The pellet is heated to 900 °C for 16 hours in an oxygen atmosphere, then rapidly cooled to room temperature. Afterward it is reground, heated to 900 ºC for 18 more hours, then allowed to cool slowly to room temperature.

8.9. BISMUTH AND THALLIUM COMPOUNDS

In January 1988 a Japanese scientist, H. Maeda, reported the discovery of a new family of high-temperature superconductors consisting of bismuth-strontium-calcium-copper oxide having a Tc around 110 K. Not long thereafter Alan M. Hermann and Z. Z. Sheng of the University of Arkansas reported an even higher transition temperature of 125 K in a similar series of materials, namely, thallium-

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barium-calcium-copper oxide. In this family there are many compositional vari-ations that give superconductivity; some of them involve for example the addition of lead. Table 8.2 lists transition temperatures for some members of these new series of compounds:

Compound Formula na

Bi2Sr2Can–1CunO2n+4 Bi 22(n–1)n 1-4Tl2Ba2Can–1CunO2n+4 Tl 22(n–1)n 1-4

an is the number ofcopper atoms present; this corresponds to the number ofplanes in the conduction layer.

The binding layers for these two families of compounds are shown in Fig. 8.4. Thetransition temperature rises as n increases from 1 to 3, then decreases for n = 4, as indicated in Fig. 8.15, with Tl 2223 or Tl2Ba2Ca2Cu3O10, having the highest value Tc = 125 K. Initial reports of this increase with n caused some excitement in thescientific community because of the obvious indication that adding a large enough number of copper planes may lead to much higher transition temperatures; however this possibility did not materialize.

The similarity of all structures having different n values has resulted in anotherproblem: It is difficult to synthesize materials that are entirely of a single phase-having only one n value. Most fabrication processes result in a mixed phase product;

Figure 8.15. The superconducting transition temperature of the Bi2Sr2Can–1CunO2n+4, Tl2Ba2Can–1

CunO2n+4 and HgBa2Can–1CunO2n+2 compounds versus the number n of copper oxide planes in the conduction layers.


for example the Bi-Sr-Ca-Cu-O superconductor is made by grinding and mixing Bi2O3, SrCO3, CaO, and CuO in the ratio Bi:Sr:Ca:Cu = 1:2:1:2, then heating thematerial in air at 850 oC for 12 hours. Following this the material is cooled,reground, and reheated for 9 more hours in air. This process gives a mixture of the n = 2 and the n = 3 phases; duration of the second heating determines relativeamounts of the two phases.

8.10. MERCURY COMPOUNDS

In spring 1994 a new superconductor was synthesized that had zero resistance at 133 K; some samples had an onset of superconductivity as high as 140 K. The material is HgBa2Ca2Cu3O8+x; the structure of this copper oxide superconductor is tetragonal; it has the unit cell shown in Fig. 8.11. Its conduction layer has three copper oxide planes, as shown at the bottom of Fig. 8.3, and the binding layer is shown at the bottom of Fig. 8.4; a small additional amount of oxygen in the mercury layer is not shown in Fig. 8.1 1. This oxygen is not strongly bonded in a particular site in this layer, so it can diffuse between them, causing undesirabletime-dependent changes in the superconducting properties. However a lead-substituted version of this material Hg0.7Pb0.3Ba2Ca2Cu3O8+x in which about one-third of the mercury sites has lead was synthesized, and it is more stable. This is because an oxygen at a site between lead and mercury is more strongly bonded than one between two mercury atoms. Table 8.1 and Fig. 8.15 show the dependence of Tc on the number n of copper oxide planes.

The method of synthesizing this compound is more complicated than the other cuprates. First an initial compound Ba2Ca1.75Cu3Ox is made from a mixture of Ba, Ca, and Cu nitrate powders heated in flowing oxygen at 900 oC for 16 hours. ThenHgO and PbO2 are added in a 2:1 ratio, and a pressed pellet of this material is wrapped in gold foil and placed in a quartz tube. The tube is then evacuated, filled with pure oxygen at a high pressure, and heated at 800 ºC for about 2 hours. This process yields a 100% pure stable material having zero resistance at 133 K.

There is much interest in this superconductor for two reasons. Measurements of the effect of pressure on the material indicate that the onset transition temperature increases to 147 K when pressure is raised to 140,000 times atmospheric pressure, as indicated in Fig. 8.16. The result excited many researchers because pressure on a material can be created chemically by replacing some fraction of ions by a similar ion of smaller radius. The obvious choice in this case was to replace the larger barium with smaller strontium; this had worked with the lanthanum com-pound discussed earlier. Unfortunately nature is not always so predictable: Replace-ment of barium by strontium reduced Tc to 127 K instead of increasing it beyond 133 K.

116 CHAPTER 8

Figure 8.16. Effect of applying pressure on the transition temperature of the mercury superconductorHgBa2Ca2Cu3O8+x. The data plotted here are the temperatures at which the resistance drops half way to zero. These data are from L. Gao, Z. J. Huang, R. L. Meng, and C. W. Chu, Nature365, 323 (1993).

The other reason for interest in this material seems much more important:Resistance measurements in DC magnetic fields show that fields up to 10 T do not increase the resistance at 77 K. This means that flux is more strongly pinned in this superconductor than in other cuprates at 77 K. Therefore if the mercury material can be fabricated into wires, it may be possible to have a high-temperaturesuperconducting magnet that operates with liquid nitrogen as the coolant.

8.11. INFINITE-LAYER PHASES

In 1993 superconductivity was discovered in the series of compounds with the general formula Srn+1CunO2n+1+8; these compounds represent perhaps the simplest of the copper oxide superconductors containing only two metallic elements, stron-tium and copper. Like the cuprates these are layered compounds, and the parameter n designates the number of copper oxide layers. The layering scheme is very simple, and it can be visualized from Fig. 8.17. The binding layer Sr2O for all of these


Binding layer of infinite layer phases Srn+1CunO2n+1+8

Conduction layer of Sr2CuO3+d

coduction layer of Sr3Cu2O5+d

Conduction Layer of Sr4Cu3O7+d

Figure 8.17. Binding layer (top) followed by, in succession, conduction layers of the first three infinitelayer phase compounds, namely Sr2CuO3+d Sr3Cu2O5+d and Sr4Cu3O7+d. The figures are drawnassuming δ = 0.

118 CHAPTER 8

compounds consists of successive Sr, O, and Sr planes, as indicated at the top of Fig. 8.17. This binding layer is much thinner than those of the cuprates; conduction layers are the same as the cuprate ones shown in Fig. 8.3 but with strontium atoms between the CuO2 planes instead of calcium or yttrium. Thus these compounds can properly be considered as cuprate types.

The n = 1 compound has a structure similar to that of La2CuO4, discussedearlier in Chap. 8 and shown in Fig. 8.10. This n = 1 compound with the formula Sr2CuO3.1 has a large number of vacancies in the binding layers and a transition temperature of 70 K. These vacancies provide the doping mechanism for holes in the CuO planes. The n = 2 compound Sr3Cu2O5+d has a Tc of 100 K.

The limit of the series for very large n is SrCuO2; it has the infinite layerstructure shown in Fig. 8.18. This material can be made into an electron-dopedsuperconductor with Tc = 43 K by replacing some of the divalent Sr2+ with trivalent La3+; this has the effect of putting electron carriers in the copper oxide planes. The large n material doped with holes occurs only in a small fraction of the samples; it

Figure 8.18. Crystal structure of the infinite layer phase SrCuO2.


has a transition temperature of 110 K. It is not clear what causes hole doping, but it is believed to involve some kind of defect structure. One idea is that an oxygen atom may be trapped between two Sr atoms, forming an Sr-O-Sr defect. Another proposal is that a corrugated Sr-O layer is substituted for one of the copper oxide layers.

8.12. SUMMARY OF CUPRATE PROPERTIES

We discussed characteristics of a number of copper oxide superconductors, a group of materials that has many properties in common. These shared properties may well provide the key to explaining the mechanism of high-temperature super-conductivity. The most important common features are

Presence of magnetic copper ions.

Arrangement of copper and oxygen ions in two-dimensional arrays.

Conduction by hole hopping is largely confined to copper oxide planes.

Presence ofbinding layers (charge reservoir layers), which provides a source

Dependence of transition temperature and superconductivity on concentration

Evidence for short-range antiferromagnetic ordering in copper oxide layers.

Some of these properties are summarized in what is referred to as a universal phase diagram for superconductors; an example is shown in Fig. 8.8 for the LaSrCuO superconductor. This diagram clearly shows the strong influence of the hole concentration on the properties. At low hole concentrations, the material is an antiferromagnetic insulator and does not superconduct. As the hole concentration increases, the transition temperature to the antiferromagnetic phase decreases. At a higher hole concentration, the material is no longer antiferromagnetic, and it begins to superconduct. The phase diagram underlies the intimate relationship between antiferromagnetic order and superconductivity in these materials. In fact experi-mental evidence has been found for the existence of short-range antiferromagnetic order in the range of hole concentration where the material superconducts. Short-range order means that there are regions in the superconductor on the order of a fewhundred cubic angstroms in volume within which copper ion spins are ordered. These observations suggest that magnetic interactions may play a role in the superconductivity of copper oxides, perhaps by helping to bond the holes into Cooper pairs.

for hole doping copper oxide planes.

of holes.

9

Magnets and Their Uses

9.1. INTRODUCTION

In Chap. 9 we discuss magnets and how superconductors have contributed to their advancement. We begin by explaining the principles behind producing mag-netic fields and the construction of ordinary electromagnets. Then we describe the limitations of standard iron-core magnets, and we show how using superconductors has led to improvements in magnet technology. Next we present three applications of magnets constructed from classic low-temperature superconductors, namely, high-field magnets; magnetic field measuring devices, called magnetometers; andnuclear magnetic resonance imaging (MRI). As the level of technology advances, these three devices will eventually be replaced by others fabricated with new high-temperature superconductors; because of the higher transition temperatures of the new materials, the devices should be less costly both to manufacture and operate.

9.2. CONVENTIONAL MAGNETS

A conventional electromagnet is based on the principle that a wire carrying an electric current produces a magnetic field. This is done by winding the wire on a cylindrical coil, then attaching the ends of the wire to a battery so that current flows through it (see Fig. 9. la). The magnetic field produced by the current in each loop of the coil adds to the current from the other loops, thereby causing a relatively intense magnetic field to exist along the central axis of the coil. The strength of the field depends on the number loops, the diameter of the coil, and on the amount of current flowing in the wire.

If an iron bar is inserted in the coil, as shown in Fig. 9. 1b, then the strength of the magnetic field is increased more than a thousand times. This increase isindicated in Fig. 9.1 by the closeness of the magnetic field lines at the end of the coil with the iron bar compared to the coil without the iron bar. The former is called

121

122 CHAPTER 9

AirCore IronCoreElectromagnet Electromagnet

Figure 9. 1. Magnetic field lines produced by an air core magnet (left) and by an iron core magnet (right). Several return paths of magnetic field lines are indicated.

an iron-core magnet and the latter, an air-core magnet. The iron acts in this manner because it is a ferromagnet of the type illustrated in Fig. 2.7b. Tiny bar magnets in the iron atoms are aligned in the same direction by the magnetic field produced by current flowing through the coil; when this happens, the bar magnets produce their own magnetic fields, which add to the field from the current. This enhances the overall magnetic field inside the coil a hundredfold or perhaps a thousandfold.

One of the problems associated with the magnets in Fig. 9.1b is that the field is not uniform at the ends, but rather it becomes weaker as we move away from the end. In addition since magnetic field lines are loops with a return path through space, as shown in Fig. 9.1, the field is not confined. Both of these problems are solved by the design shown in Fig. 9.2 in which an iron yoke is added to confine the field lines of the return path, and magnetic field measurements are carried out in the narrow gap where the magnetic field is uniform.

For low currents the magnetic field produced by the iron-core magnet just described is proportional to current flowing through the coils. For high currents the

MAGNETS AND THEIR USES 123

Figure 9.2. Design for a conventional electromagnet.[Adapted from C. P. Poole, Jr., EIectronSpinResonance,2ndEd.,Wiley,NewYork, 1983,Fig.6.1].

Figure 9.3. Dependence of the magnetic field B of an iron core electromagnet on the current through the coils, showing the saturation that occurs at high currents. The data are for a commercial 1.35 cm(1/2-inch) gap magnet.

124 CHAPTER 9

magnetic field increases less and less with additional increases in current, as indicated in Fig. 9.3. This phenomenon is called saturation(the iron is becomingsaturated); this sets a limit to the highest field that can be reached by an iron-coremagnet.

To overcome this limitation, the highest field magnets were constructed from copper wire coils with no iron core, in accordance with the earlier magnet design just described. To generate extremely large magnetic fields on the order of 10 T, enormous currents and electrical power levels are required. Furthermore because copper wire has resistance, the very large current flowing through it produces heat, thus requiring the coils to be cooled. A conventional magnet of this type at the Francis Bitter Magnet laboratory at MIT capable of producing a field of 10 T required 1.5 million Watts of electrical power and a thousand gallons per minute of cooling water to remove heat generated in the coils. To make this feasible, the Magnet laboratory was built on the banks of the Charles River near Boston. These considerations put practical limits on the highest magnetic field that can be achieved in practice by an electromagnet.

9.3. SUPERCONDUCTING MAGNETS

The advantages of making coils of superconducting wire are evident. Because the wire has no resistance, no heat is generated, thereby eliminating the need forcooling water. Further the only electrical energy consumed is that applied to charge the circuit initially with supercurrent. Once the supercurrent is flowing, coils can be disconnected from the power supply; as long as they remain cooled below the transition temperature, current flows indefinitely.

In principle a superconducting magnet is capable of producing much higher magnetic fields than a conventional electromagnetic. The limit to how high a magnetic field can be produced is the uppercritical field Bc2 of the superconductor used for the wire coils. A superconducting magnet is also much lighter than a conventional one because it does not require heavy iron pole faces.

The idea of constructing a superconducting magnet emerged very soon after the discovery of superconductivity; in fact K. Onnes himself tried to build such a magnet but gave up the effort. A limitation at the time was the very low value of the critical magnetic field of the elemental superconductors, as shown by the data in Table 3.1. We see that niobium has the highest critical magnetic field of 0.206 T. Therefore if the current in a niobium wire produces a field of 0.206 Tat the surface of the wire, then the superconducting state is quenched, i.e., removed. Thus there is a rather low limit to the maximum magnetic field that can be generated by superconducting coils made of elemental metals; this makes conventional iron-core magnets far superior in performance.

MAGNETS AND THElR USES 125

9.4. HIGH-FIELD SUPERCONDUCTINGMAGNETS

The effort to build a superconducting magnet was not undertaken until the discovery in 1961 that alloys can have critical fields much higher than those of elements. An alloy is an atomic mixture of two or more metals in which constituent atoms form an ordered crystal structure. Niobium-tin (Nb3Sn) is such an alloy, with three parts of niobium to one part of tin; it also happens to be an A-15 type superconductor with a transition temperature of 18 K. In 1961 it was discoveredthat this material has the very high critical magnetic field Bc2 of 28 T. Other A-15alloys were subsequently found with high critical magnetic fields, as shown in Fig. 9.4. We see from Fig. 9.4 that at 4.2 K, the temperature at which superconducting magnets ordinarily operate, these A- 15 compounds have critical fields far above that of the standard magnet material NbTi. Figure 9.4 also shows that just below the transition temperature, the critical magnetic field of each compound becomes quite small, so magnetic operation becomes feasible only at temperatures consid-erably below Tc.

This discovery of materials with high critical fields opened the door to manufacturing magnets with superconducting coils reaching field strengths far

Figure 9.4. A plot of the critical magnetic field Bc2 versus temperature for three superconducting compounds. [Adapted from J. E. Kunz and M. Tanenbaum, ScientificAmerican June (1962).]

126 CHAPTER 9

Figure 9.5. A toroidal warm core superconducting magnet.

above those previously achievable. The A-15 compounds are rather brittle, so 95% of present-day magnet wire is made from the more ductile niobium-titanium alloy NbTi; hence is much more easily drawn into wire. These NbTi magnets operate at 4.2 K. Most of the remaining 5% of magnetic wire is made from the A-15 compound Nb3Sn; these A-15 magnets operate at 10 K using refrigeration obtained from cryocoolers. It is much more economical to operate a magnet at 10 K than at 4 K; we expect that eventually magnets constructed from wire of the high-temperaturesuperconductor BiSrCaCuO will operate even more inexpensively at 30 K.

Figure 9.6. Experimental arrangement for establishing a persistent current in a loop of superconducting wire. Switch S2 is closed to send current through the loop, and switch S1 is then closed and S2 openedafter the loop is cooled below Tc to confine the current flow to the loop. [Adapted from C. P. Poole et al., Superconductivity , Academic Press, Boston (1995), Fig. 2.15.]


Figure 9.5 shows a warm-core toroidal superconducting electromagnet of the type used in MRI. Warm core means that the useful magnetic field region is at room temperature. Coils producing the magnetic field are refrigerated by imbedding them in a ceramic in contact with a surrounding liquid helium bath.

To generate a magnetic field by a superconducting coil, a supercurrent mustbe established in the coil; Fig. 9.6 shows an electrical circuit that can accomplish this. It operates as follows:

The superconducting loop is held above the transition temperature. Both switchs are initially open. Switch S2 is closed and current flows around the circuit through the loop. The loop is cooled below the transition temperature, then Switch S1 closed and Switch S2 opened. Because the loop is in the superconducting state, current continues to flow indefinitely around the loop, even though it is not attachedto a power supply.

9.5. TECHNOLOGICAL PROBLEMS WITH SUPERCONDUCTING MAGNETS

Because there are large forces between the adjacent wires in a coil when they carry current, it is necessary for the ceramic to be strong enough to withstand the resulting tensions. Two parallel wires carrying current in the same direction, like the wires in the ceramic, experience a force of attraction that tends to pull them together. This force is a result of the interaction of electrons moving in one wire with the magnetic field produced by current in the other wire. Any movement of the wires brought about by this force could result in frictionally generated heat due to the motion of the wire through the ceramic material. This heat could raise the temperature enough to quench the superconducting state; this would result in a large pulse of normal current that would in turn generate further heat, causing rapid evaporation of the cryogenic fluid. Excessive pressure generated by evaporated helium gas confined in the Dewar could cause an explosion; clearly rapid quenches of superconducting coils can be dangerous.

Flux slippage is another factor that can generate heat and induce a transition from the superconducting to the normal state. Such slippage is caused by the interaction of current flowing in the wire with flux trapped there in the form of flux lines or vortices. Each vortex experiences a force perpendicular to both the direction of the current flow and the direction of the trapped vortex, as shown in Fig. 3.13.If the flux is not pinned, it starts to move. This motion creates a kind of frictional heating that generates localized regions ofheat called hot spots. Such hot spots canraise the surrounding temperature above the transition temperature, thereby induc-ing a transition from the superconducting state to the normal state and perhaps cause a quench.

128 CHAPTER 9

To eliminate the danger of hot spots, it is necessary to devise ways of rapidly dispersing heat if flux slippage occurs and of pinning vortices, i.e., holding them more firmly in place, as explained in Chap. 3 so they do not slip. To accomplish the first task of heat dispersal, many thin filaments of superconducting wire can be imbedded in a good thermal conductor, such as copper; Fig. 9.7a shows a cross section of such a copper cable 1 mm (One-twenty-fifth of an inch) in diameter containing 2035 filaments of superconducting wire arranged in a regular hexagonal array. There are 37 filament bundles in the array; each bundle contains 55 filaments. Figure 9.7b expands the view of the region where three bundles come together and delineates individual filaments. This arrangement allows a rapid transport of heat away from a hot spot in the superconducting cable in the event of flux slippage.

To accomplish the second task of vortex pinning, defects can be intentionally introduced into the wire filaments that serve as pinning centers to trap vortices and impede the movement of flux. This can be achieved by cold working the wire, a process that involves repeated deformation of the material to introduce many kinds of defects, such as disiocations where planes of crystal atoms have slipped over each other so they are no longer in normal alignment. Techniques to accomplish efficiently pinning vortices have been developed to a fine art.

Figure 9.7. Cross section of a 1.0 mm diameter copper cable containing 2035 niobium-titaniumsuperconducting wire filaments arranged in 37 bundles containing 55 filaments each. (a) overall view and (b) enlargement of region showing individual filaments where three bundles meet. [From M. N. Wilson, Superconducting Magnets, Clarendon Press, Oxford (1983). Fig. 12.91.


9.6. USES FOR SUPERCONDUCTING MAGNETS

Superconducting magnets have many uses, and energy storage is one obvious application. Continuously flowing large current and its associated high magnetic field represent an enormous amount of stored electrical energy that can be released at the appropriate time by a controlled quench. In effect a superconducting magnet is an energy storage cell. Applying this method awaits future development, but superconducting magnets have already had a major impact on medical diagnosis and health care by their use in nuclear MRI—taking three-dimensional pictures of the inside of the human body. Before discussing this application, we provide some background on atomic magnetism and the underlying experimental technique.

9.7.

In Chap. 2 we saw that some atoms behave like small bar magnets, and hydrogen is an example. The nucleus of hydrogen is a proton, and the proton bar magnet is detected in nuclear magnetic resonance (NMR). Hydrogen atoms are bound in chemical compounds that make up the material of the human body, such as proteins, DNA, and neurons (nerve cells). In addition a large percentage of thehuman body consists of water within and between cells; since the chemical formula for water is H2O, two-thirds of its atoms are hydrogen.

The NMR is an experimental technique that detects the presence ofhydrogenatoms; since the NMR signal produced by hydrogen protons is slightly different in different chemical environments, different tissues respond to NMR in a slightly different manner. This difference can be used to produce images ofthe human body, a technique called MRI (magnetic resonance imaging).

To understand the principles behind NMR, consider a magnetic field applied to a molecule containing hydrogen. The tiny proton bar magnets of the hydrogen atoms, which in NMR terminology are called spins, tend to align themselves alongthe applied magnetic field direction the way a compass aligns itself along the Earth’s magnetic field—pointing north. A force must be applied and energy expended to rotate the compass needle to the opposite direction, i.e., south. In like manner energy must be expended to turn the spins to the opposite direction. It turns out that the application of radio wave energy at a particular frequency called the Larmorfrequency rotates protons and points them in the direction opposite to the appliedfield. The Larmor frequency is proportional to the strength of the applied field, and it has a characteristic value for every atom whose nucleus has spin. For example the Larmor frequency in a 4-T magnetic field is 170.3 MHz for hydrogen, 12.3MHz for nitrogen, and 68.9 MHz for phosphorus—three atoms commonly found in biological tissues.

A simple NMR spectrometer is shown in Fig. 9.8. The sample to be studied is placed inside a wire coil into which radio wave energy is introduced. The coil is

NUCLEAR MAGNETIC RESONANCE

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Figure 9.8. Sketch ofa simple nuclear magnetic resonance spectrometer. [From G. E. Pake, ScientificAmerican, August (1958).]

placed between the poles of a Iarge electromagnet or in the field of a supercon-ducting magnet. The oscillator varies the frequency at which the coil oscillates. When this frequency reaches the Larmor value, the nuclei reorient themselves in the opposite direction, radio wave energy in the coil is absorbed by the sample, and this absorption is picked up by the detector and displayed on the oscilloscope. Commercial NMR spectrometers and MRI installations involve much more sophis- ticated enhancements of this simple circuit.

9.8. MAGNETIC RESONANCE IMAGING

We saw that the NMR absorption frequency depends on the strength of the magnetic field. If the field were arranged to increase gradually across the dimen-sions of a sample, then the frequency of the proton NMR absorption would correlate with the location of the hydrogen in the sample. The idea of having the magnetic field vary spatially across the sample, thus making measurement of the frequency


in effect a measurement of proton positions in the sample, is the essence of the concept of MRI.

To illustrate this idea, let us consider the case of two water drops trapped inside a sodium chloride crystal (NaCl—common table salt), as shown in Fig. 9.9a. The applied magnetic field Bapp is arranged to increase linearly from a value ofBo at one edge of the sample to a value of B1 at the other side, as shown in Fig. 9.9b, so theincrease in the field is directly proportional to the distance x from the left edge of the sample. Since the Larmor frequency of energy absorption is proportional to the strength of the magnetic field, it is also directly proportional to the position across the sample. Thus a plot of the distribution of trapped water drops across the crystal can be obtained by measuring the intensity Iof energy absorption at each frequency. For this simple case, Fig. 9.9c shows what the spectrum may look like. NOTE: Thesmall drop produces a weak signal and the large drop a strong one. We can obtain three-dimensional pictures by allowing the magnetic field to vary linearly in all three dimensions.

Nuclear magnetic resonance imaging, whose name is shortened by the medical community to MRI to avoid frightening patients with the word nuclear, is a very

a)

b)

C)

Figure 9.9. (a) Two trapped water bubbles in acrystal of NaCl, (b) Linear variation of the magnetic field from one end of the sample to the other, and (c) Proton NMR spectrum consisting of two reso-nant lines originating from the two droplets of water in the crystal.

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powerful medical diagnostic technique for a number of reasons. Living tissue is relatively transparent to radio waves and radio waves do not harm this tissue. In addition living tissue is made of organic molecules that contain hydrogen atoms. The cytoplasm within cells, material between the cells, plasma in the blood vessels, etc., principally consist of water (H2O), which is the main contributor to the proton signal detected by MRI.

Magnetic resonance imaging is able to distinguish between tissue types, such as kidney and liver tissue, because there is something different about the magnetic resonance spectra in different tissues (if this difference did not exist, then NMR spectra

Figure 9.10. Four MRI side view (sagittal slice) images of a human head obtained using differentinstrumental settings which provide various levels of contrast and accentuate different features of the brain. [From K. H. Hausser andH. R. Kalbitzer, NMRinMedicine andBiology [in German], Springer,Berlin, 1989 (English translation 1991)].


of all tissues would look the same). Fortunately the viscosity of body fluids, which measures how easily they flow, differs from environment to environment through-out the body. For example distilled water, which flows easily, has a low viscosity; sea water, as well as water in tissues, contains many salts, and so it has a higher viscosity, while molasses, which flows with considerable difficulty, has a very high viscosity. One of the parameters detected by NMR is called the spin-lattice relaxa-tion time (T1); this is the time it takes for flipped spins to revert to their originalorientations. Anotherparameter, called thespin-spin relaxation time (T2), measuresthe time it takes reversed spins to pass their absorbed energy to nearby spins throughmutual spin flips. Both of these relaxation times depend on viscosity; hence they vary with tissue type.

In an NMR experiment, radio wave energy reverses spin directions, after which they revert to their original orientation, passing absorbed energy to their surround-ings, which is called the lattice. The signal is analyzed to provide the two relaxationtimes and identify characteristics of the tissue being observed. Furthermore since protons of malignant tumors have unusually long relaxation times, their resonance signals can easily be distinguished from those of normal tissue. As a result NMR imaging has become a powerful tool for detecting malignant tumors in for example the brain. Figure 9.10 presents four MRI images of a human head and brain obtained under different NMR instrumental conditions, i.e., different contrast settings. Figure 9.11 shows how well a brain tumor is revealed by MRI.

Figure 9.11. Two MRI frontal cross section images of a human head obtained with different instru-mental settings. The white area at the left of center is a cystic brain tumor [from Hausser and Kalbitzer, NMR in Medicine and Biology [in German], Springer, Berlin, 1989 (English translation 1991)].

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This description of the MRI technique is overly simplified for the purposes of explaining the concept. In the usual MRI technique, the actual measurement is made differently: Instead of applying continuous radio wave radiation at a single fre-quency, a sequence of many short pulses within a broad frequency band is used. The width of the pulse and the separation between pulses depend on the nature of the tissue being examined and the relaxation times of protons in the tissue.

Absorbed energy is measured at a fixed time after each pulse, then stored in a computer. This is done repeatedly, with each successive amount added to the previous one. The resulting stored signal, which has many frequency components, is then subjected to a mathematical analysis called Fourier transformation, which identifies how much absorption occurs at each frequency. The result is a picture of how various normal and diseased tissues are distributed in space, as illustrated in Figs. 9.10 and 9.1 1.

9.9. MRI FLUOROSCOPY

As just discussed MRI provides a diagnosis of tumors and other medical problems without surgery. In the future MRI is expected to evolve into a very important surgical tool called magnetic resonance fluoroscopy that will guide the surgeon’s hand during an operation. As he or she operates the surgeon will watch a computer screen with an MRI picture of the area on the patient’s body where the incision is being made. The image on the screen shows exactly where to cut, in contrast to present surgical techniques that often involve cutting through healthy tissue to locate malignancies.

In another application MR fluoroscopy could be employed to direct inserting and positioning an optic fiber at the precise point where a small cancerous growth is located; a powerful laser pulse could then penetrate the growth without affecting surrounding healthy tissue. If the optic fiber was originally inserted with the aid of a needle, when it is removed, a small bandage would be sufficient to seal the wound.

9.10. SUPERCONDUCTING MAGNETIC ENERGY STORAGE

The demand for electrical power varies with the time of day. It is desirable to be able to store excess electrical energy produced during the night when demand is low, then release it when power demand is high. There is also a need for backup power when electricity supplied to a factory, hospital, or other facility experiences temporary voltage drops or prolonged outages, which can be very inconvenient, costly, or even dangerous.

Energy can be stored in various ways, such as electrically by charging capacitors or batteries; or gravitationally by pumping water uphill for temporary storage in a basin, then inducing it to generate electricity when it flows downhill.


Energy can also be stored in the magnetic field associated with electric current flowing in a wire loop or coil. A conventional coil continuously dissipates energy due to the resistance of the wire, so it is not practical for storing electricity. A superconducting coil on the other hand carries persistent current that continues to flow indefinitely without loss; thus such a coil provides a convenient way of storing energy, a technique called superconducting magnetic energy storage (SMES).

Figure 9.12 shows an SMES system using a coil to retain magnetic energy. The power source may be for example a hydroelectric generator or a nuclear power plant; the loads are devices or machines operated by electricity. A switching unit can directpower from the source to the coil to charge it, and it can also directpowerfrom the coil to loads when extra energy is needed. The invertor-convertortransforms between AC power generated by the source and consumed by the loadand DC current stored in thecoil.

Superconducting magnetic energy storage systems come in various sizes. A typical small one may have a storage capacity of 0.25 kWh, and it may be able to

Figure 9.12. Superconducting magnetic energy storage system. The switching unit can shunt the generated power directly to the loads where it is consumed, direct it to the superconducting coil to charge it, or transfer energy from the coil to the loads. The invertor-convertor transforms power back and forth between the ac type of the source and loads, and the dc type of the superconducting current storage.

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deliver power at the rate of 450 kW for 2 seconds or I kW for 15 minutes. The power of I kW will operate ten television sets. A large superconducting magnetic energy storage system capable of storing 5000 MW has been designed. This system can release energy at the rate of 1000 MW, so all of the energy could be delivered within 5 hours. However from 5-10% of stored energy is wasted in converting from stored DC current to applied AC current and in the energy required for refrigeration.

Experimental SMES systems using niobium-titanium alloy operating at liquid helium temperatures have been built. A storage system capable of operating at 77 K using liquid nitrogen as the coolant increases efficiency by considerably reducing refrigeration costs. The main obstacle to be overcome is developing commercial quantities of superconducting wire that can carry currents on the order of 105–106

A/cm2; in Chap. 8 we discussed progress being made toward this goal.

9.11. MAGNETOMETERS

TheJosephsonjunction, explained inChap. 5, can be used as amagnetometer,which is an instrument for measuring magnetic fields. Figure 9.13 shows an arrangement of two junctions that can be used to measure small magnetic fields. Essentially two thin oxide layers are placed in a ring of superconducting material,thereby forming two Josephson junctions. When the circuit is cooled below the superconducting transition temperature, current flows through the junctions. The phase of the current emerging from each junction may be different. When a magnetic field B is applied through the loop perpendicular to the junctions, therelative phases of the two currents at point A where they combine is altered, and

Figure 9.13. A SQUID magnetometer consisting of two insulators (i.e., thin oxide layers) sandwiched in a superconducting loop circuit. A magnetic field B applied through the loop perpendicular to the figure changes the combined current that emerges at A.


the two current waves combine differently. Thus a measure of how the current changes at point A as a function of the applied magnetic field is a measure of the magnetic field traversing the circuit. This device, called a SQUID, is a very sensitive detector of small magnetic fields.

The SQUID has many important applications; for example it is used to detect magnetic fields generated by neurons in the brain. These fields can be as small as one-millionth of a tesla. Because regions of the brain having different functions emit different amounts of electromagnetic radiation, the SQUID is used to map brain regions and identify areas of disease. In another application, the US Navy developed a SQUID to detect the presence and location of submarines. When a submarine moves through the water, its metal hull slightly disturbs the Earth’s magnetic field; this small distortion can be measured with a SQUID.

10

Wires and Films

10.1. INTRODUCTION

Most devices are based on superconductivity require materials to be in the form of wires or thin films deposited on the surface of an insulator or metal. In Chap. 10 we discuss techniques used to fabricate these forms and progress that has been made toward achieving high quality. To be useful fabricated films and wires must have certain properties, the most important of which is a high critical current. Progress made to date in achieving high critical currents with cuprates, particularly at 77 K,is examined.

The methods of synthesizing new high-temperature superconductors pre-sented in Chap. 8 produce pellets consisting of small randomly oriented microcrys-tals pressed together. These materials are brittle, so they cannot be easily shaped or bent into wires. The critical current of pellets is considerably lower than that of niobium-tin wires at 4 K, and it is especially low at 77 K where superconducting materials are most likely to be operated. This current decrease occurs mainly because spaces between grains act as barriers to current passing from one grain to another. Another reason for the current decrease is that current flow depends on direction in the microcrystal. Some microcrystals of the pellet are properly oriented, and some are not. Another cause of the lower critical current is flux creep, which is discussed in Chap. 3. We saw earlier that superconducting properties of the YBa2Cu3O7–x material are extremely sensitive to oxygen content. Therefore surface degradation, which may change the oxygen fraction and thereby alter the transition temperature, can be a problem.

An intense effort to develop fabrication methods that eliminate some of these difficulties is now under way, particularly in industrial laboratories. Details of new fabrication processes being developed are proprietary and therefore not available to the public because of the intensely competitive nature of the enterprise and the potentially large rewards for companies that succeed. There will be a great deal of profit for the company developing the first high critical current superconducting

139

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wire that functions well at 77 K. In the following sections, we discuss in general some of the methods now being developed to fabricate wires, and we assess theprogress ofvarious approaches.

10.2. METHODS OFFABRICATING TAPES AND WIRES

The mechanical properties of a brittle ceramic can be improved by dispersing it in a binder, such as polyurethane-a bendable organic polymer with mechanical properties not unlike a rubber band, so it can be stretched and bent easily. Thepolymer is first dissolved in acetone, then ceramic particles are introduced into the resulting viscous solution. The slurry is stirred, then the acetone is allowed to evaporate to produce a collection of particles held together by the binder. Other chemicals are often added to the solution, such as a dispersant to distribute particles uniformly and a plasticizer to improve flexibility of the final product.

Superconducting tapes were made by scientists at AT&T using an analogous process: YBaCuO powder synthesized by the usual method, a binder, and a plasticizer were mixed into a solvent that dissolves the binder. Then a controlled thickness of this well-stirred solution was deposited on a material called a substrate, which passes below it. After evaporating the solvent, the tape was separated from the binder, shaped into the desired form, then sintered (i.e., heated in air) near 950 °C to produce a dense, brittle ceramic material. Such tapes showed zeroresistance at 92 K, and they had a critical current density of 103 A/cm2.

Another process developed at Argonne National Laboratory uses an extrusion technique: The viscous solution just described was allowed to dry until it had a consistency somewhat like modeling clay. It was extruded (forced through a small hole), then sintered as in the tape process to form a brittle superconducting wire.This process produced a wire with a critical current density of 104 A/cm2 at 77 Kin zero-magnetic field.

A cold-drawing method is also under development as a way of making wires. The YBaCuO is packed into a silver tube, since silver can be drawn, and it does not chemically interact with the superconductor. The silver tube containing the super-conductor is slowly stretched, reducing its diameter from 10 to 100 times its original size. The tube is then shaped into whatever form is required, such as a coil, and cooked at 900 °C to form a dense superconducting wire. After this the silver sheath is removed. Critical current densities of 104 A/cm2 at 77 K in zero-magnetic fieldhave been obtained in wires made by this method.

Since a YBaCuO pellet can carry currents up to 104 A/cm2 at 77 K inzero-magnetic field, the previously described methods have not improved the current-carrying capability. Their importance lies in demonstrating that materials can be made into wires. As previously discussed three factors limit the critical current in granular superconductors, namely, dependence on the direction of flow

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in an individual grain, spaces between grains acting as barriers to flow, and flux creep. As current is increased, the force that it exerts on the trapped flux increases, so the flux may begin to slip and move. The effect of this is to introduce resistancein the material and perhaps to remove it from the superconducting state.

10.3. GRAIN ALIGNMENT

The tiny microcrystals of wires made by the previously described processes are randomly oriented; the critical current is therefore far below its maximum possible value. If the microcrystals of a pellet were aligned with their direction of maximum current flow parallel to the axis of the wire, then critical current in the wire would be substantially increased. Once the grains are oriented, it is then necessary to remove intergranular barriers to increase current flow further. One approach to this would be to develop methods to fuse or weld the grains together. A welding process would involve melting the surface of the grains, pushing them together, then letting the material cool.

A melt-textured growth method was developed to produce aligned grains. The material is first melted, then cooled in an oven that has a spatial variation in the temperature across the melt on the order of 50 °C/cm. When a material, such as YBaCuO, cools in this temperature gradient, it forms long needlelike crystals, with the axis of each needle along the copper oxide planes and all of the needles lined up parallel to each other. This axis is of course the direction of easy current flow in the crystal, and critical current densities of 2 x 104 Ncm2 have been attained at 77K in zero-magnetic field; this demonstrates that grain alignment does indeed increase critical current.

Another approach to producing materials with preferential microcrystal align-ment is to solidify them in high magnetic fields. Work at the National Institute of Science and Technology (NIST) demonstrated that magnetic alignment of grains is possible. When a cuprate compound cools from the melt in an applied field, tiny atomic magnets of the paramagnetic copper ions interact with the field. This causes microcrystallites to shift in position so that they all have their copper oxide planes perpendicular to the magnetic field direction. Wires of course are drawn along directions parallel to these planes, so larger currents can flow.

A cold-rolling method was developed that seems promising for aligning grains. A mixture of the superconductor powder and about 10% of an organic binder and acetone are sprayed on the surface of a thin silver foil, then dried. Next this mixture is heated in air at 800 °C for 4 hours to burn off the binder. The superconductor-coated film is cold rolled by pressing and rolling a metal cylinder over it, then subjected to further heating. The resulting ribbons have preferentially aligned grains and lower intergranular resistance and pinning centers to control flux flow. This process produces a BiSrCaCuO ribbon with a critical current of 2.3 x 106

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A/cm2 at 4.2 K in an 8-T field and a YBaCuO ribbon with a critical current of 3 x104 A/cm2 at 77 K in a 1-T magnetic field. Unfortunately BiSrCaCuO has seriousflux dissipation problems arising from the extent of the melted flux lattice region of its magnetic phase diagram, as explained in the following sections. These problems severely limit the current-carryingcapability at 77 K, butbelow about 30K, these tapes can sustain current densities as high as 105 A/cm2. A number ofcompanies have developed superconducting magnets from these tapes that operate at 30 K with the aid of closed-cycle refrigerators. These refrigerators are much more economical than liquid helium cooling, but they are not so inexpensive as liquid nitrogen cooling.

Very recently scientists at Los Alamos Scientific Laboratory found a much more effective way of aligning grains of YBa2Cu3O7–x; they developed a process using these aligned grains to make tapes with critical current densities at 77 Kgreater than 106A/cm2 in zero-magnetic field. Such tapes sustain current densities as high as 3 x 105 in applied fields up to 9 T, as shown by data plotted in the upperleft-hand corner of Fig. 10.1. This could result inmarketing high-field supercon-ducting magnets that operate at 77 K.

To produce this alignment, a malleable nickel alloy tape is first coated with a layer of zirconia (ZrO2) by using a laser beam to remove ZrO2 molecules from a zirconia target so that they migrate to the tape. A second laser is focused on thenickel tape at a specific angle to force the ZrO2 to align preferentially as they are deposited. A pulsed laser process then deposits the YBaCuO on top of this zirconia layer, and oriented crystallite grains of zirconia force grains of YBaCuO to align. This latter alignment occurs with the easy current flow directions of the YBaCuO grains arranged parallel to the flat surface of the tape so that current can readily

Figure 10.1. Critical current density versus the applied magnetic field for the new YBaCuO tapes (•)compared to those for previous YBaCuO melt textured samples The measurements were made atthe temperature 77 K. [Adapted from S. Foltin, High TcUpdate 9, No. 10.2 (1995).]

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flow along the tape; Fig. 10.1 shows the improvement over melt-textured growth preparation. Whether or not this process becomes technologically important de-pends on whether it can provide large quantities of tape at competitive costs. In addition to magnets current levels reached with these tapes make otherdevices possible, such as superconducting generators, superconducting en-ergy storage devices, as well as superconducting motors; all of these requirecooling to 77 K.

10.4. ELIMINATION OF FLUX MOVEMENT

Critical current can be further improved by reducing or eliminating flux flow from YBaCuO and BiSrCaCuO materials that are ordinarily used to manufacturewires and films. Flux flows in the flux liquid region above the melting line TM(T)on the magnetic phase diagram in Fig. 3.12; one way of improving a superconductor is to reduce the extent of this liquid region.

We see from Fig. 10.2 that the compound BiSrCaCuO has a large region of flux flow, while this region is much smaller in YBaCuO. This means that the

Figure 10.2. Temperature dependence of the melting line TM(T) in single crystals of the bismuth(BSCO, left) and yttrium (YBCO, right) compounds. For each superconductor the region of flux flowis in the liquid region to the right of the melting line. Figure 3.12 provides a sketch of a more complete magnetic phase diagram.

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BiSrBaCuO superconductor must be cooled much farther below Tc than YBaCuO to reach the flux creep region where current flows without resistance. In fact for low applied magnetic fields, at 77 K YBaCuO is in the flux creep region where large currents can be sustained; for these same operating conditions, BiSrCaCuO is in the flux flow region where such currents transform the material to the normal state. See Chap. 3 for a discussion on how flux flow produces this energy dissipation.

One reason for the superiority of YBaCuO is the fact that it is orthorhombic,with its a- and b-axes having slightly different lengths, namely, a = 3.83 Å and b =3.88 Å. During the growth process, the a and b directions tend to interchangeoccasionally, so the direction in which the atoms are closest together (3.83 Å) becomes the direction in which they are furthest apart (3.88 Å), andviceversa. Theresulting crystals are said to be twinned. Many boundaries between these two types ofgrowthregions act as pinning centers to restrict flux motion andprevent its flow.The higher Tc superconductors BiSrCuO, TlBaCuO, and also HgBaCuO are alltetragonalwith a= b, so they lacktwinningplanes and therefore have fewer pinningcenters. We mentioned earlier that such methods as irradiation and cold working artificially introduce pinning centers to reduce the size of the melted flux lattice region and thereby improve the quality of a superconductor.

Although YBaCuO has the smallest region of the magnetic phase diagram where the flux lattice is melted, it has the disadvantage that its superconducting properties depend more strongly on the oxygen content than do other cuprates; in addition YBaCuO is more easily degraded by exposure to moisture and carbon dioxide. This chemical decomposition alters the surface layer oxygen content; it can be circumvented by a plastic coating that seals the surface from moisture and other gases. Also, grain alignment is more difficult.

10.5. FILMS AND THEIR USES

A thin film is a layer of material only a few lattice parameters thick depositedon another material called a substrate. Much present-day basic research in super-conductivity is being carried out with thin films, which are of great importance in many applications. Films suitable for Josephson junctions are now commercially available, and SQUID magnetometers using such films and operating at 77 K are already on the market with the capability ofdetecting magnetic fields as small as amicroTesla (0.000001 T).

A typical apparatus used to make thin films is shown in Fig. 10.3. Component materials used to make YBa2Cu3O7–x, namely, Y2O3, BaCO3 and CuO, are placed in three heated holders in a vacuum chamber along with the substrate on which the film is to be deposited. The holders are heated to temperatures high enough to allow the materials to evaporate, while the substrate is maintained at a lower temperature, typically 375 ºC. Evaporated materials are deposited on the surface of the substrate

WIRESAND FILMS 145

Figure 10.3. A system used for vacuum deposition of thin films of the new superconductors. Thestarting materials are placed in heated holders, then evaporated by heating them in the vacuum chamber, and subsequently they are deposited as a thin superconducting layer on the cooler crystalline substrate. The oxygen content is controlled by gases introduced into the vacuum chamber at the gas intake valve.

in the proper ratios to form YBa2Cu3O7–x. Since superconducting properties depend on oxygen content, it is necessary to introduce oxygen into the chamber after the material has been deposited. The substrate must have lattice spacings compatible with YBa2Cu3O7–x; an excellent choice is a crystal of strontium titanate, SrTiO3, cut and mounted with a certain crystallographic orientation. After deposition, the film is annealed (heated) in an oxygen atmosphere at 850 °C for a number of hours. Films developed by the preceding process have critical current densities of 106 A/cm2 at77 K andasharp transition tozeroresistance at90K. However these films are too thin and too brittle to be used as wires; their important applications are in electronic devices.

A number of other techniques are being used to make films. One such technique, sputtering, involves accelerating positive argon ions onto a target of the material to be made into a film. Argon ions remove atoms from the target; the atoms then migrate to the cooler substrate where they condense; this process is carried out in a vacuum chamber. The schematic of a single-target sputtering device is shown in Fig. 10.4. Another approach, laser ablation, employs a high-power pulse laser to heat the surface of the target to a high enough temperature to vaporize atoms; these atoms then condense on the cooler substrate to form the film.

An example of a thin-film application is an infrared detector, a device that indicates how much infrared radiation is present. An efficient radiation detector can use the sharp change in the resistance of YBaCuO (see Fig. 10.5) in the neighbor-hood of its transition temperature Tc = 90.5 K. This resistance change occurs over

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Figure 70.4. Schematic of a single target sputtering system. In this method the atoms of the startingmaterial are knocked out of the target by high speed argon ions, and the ejected material then condenses on the cool substrate.

Figure 10.5. Sharp drop in resistance at Tc of a YBaCuO epitaxial film [From Hopfengärtner et al.,Phys. Rev. B44, 741 (1991).]

WIRES AND FILMS 147

the narrow temperature range of about 0.3 K. Suppose the temperature is maintained at precisely 90.7 K so that resistance of the material is half of its above-Tc value. Ifinfrared radiation now strikes the superconductor, it heats the material, raising its temperature. The temperature change is reflected by an increase in resistance, thereby providing a measurement of the amount of radiation present. The drop inresistance at the transition is so sharp that a temperature change of 0.0001 K results in a detectable resistance change.

Figure 10.6 shows a device based on such a principle. The main part of the detector is a superconducting film deposited on a substrate, such as mica; this lies on a copper block in contact with the cryogenic fluid liquid nitrogen. Tiny wires called resistance leads are attached to the superconductor to measure its resistance. The film is enclosed in a vacuum container to isolate it from its environment. A small heateron the blockmaintains the temperature near Tc. When infrared radiationstrikes the film, its resistance increases; this increase is detected by applying a small current, then measuring the voltage across the resistance leads.

It is clear that zero resistance is not essential for all devices. Some flux flow resistance can often be tolerated; the presence of resistance is of course essential for operating the infrared detector. The issue of whether resistance in a particular application must be zero, or how small it must be, is one of engineering design.

Figure 10.6. Sketch of an infrared radiation detector whichemploys asuperconducting film maintainedat a temperature close to Tc by the balance between the energy input from the heater and the coolingeffect of the cryogenic fluid. The slight rise in temperature of the film brought about by the incoming infrared beam produces a large increase in resistance that is measured at the resistance leads attached to the film.

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10.6. MICROWAVE INTERACTIONS WITH SUPERCONDUCTORS

A major application of the new high-temperature superconductors is thecreation of thin films for use in microwave electronics. Microwaves, sometimes referredtoasradarwaves, areelectromagneticwaves thatoscillatewithfrequenciesfin the range of three billion (3 x 109) to 300 billion (3 x 1011) cycles per second;the associated wavelengths (λ ) are in the range of a millimeter to perhaps 10centimeters, The term radar is an acronym for radio detecting and ranging; the technique is used to guide airplanes to airports, then assist them in landing. Air traffic controllers and law enforcement personnel use radar to determine the distance and speed of moving airplanes and automobiles, respectively.

When microwaves impinge on a normal metal, most of the incoming radiation is reflected. When a radar beam scanning the sky strikes a plane, it bounces offandis reflected back to a detector. Microwaves travel at the speed of light, so the distance of a plane can be calculated by measuring the time it takes for the signal to return to the detector. A metal does not however reflect all the radiation incident on it; some of it is absorbed. Microwaves impart energy to the conduction electrons in the metal, so less energy is reflected. An airplane can be made less sensitive to radar by coating it with a material that strongly absorbs microwaves, as was done with the Stealth bomber to help make it invisible to radar.

When a metal becomes superconducting below Tc, it has two kinds of currentcarriers-single electrons and Cooper pairs of electrons. As the temperature is lowered further, more and more pairs are formed, and the number of single electron current carries decreases. Because all Cooper pairs are connected in their motion, they have too much inertia to absorb incident microwave energy; therefore as the

Figure 10.7. Dependence of the micro-wave energy absorbed by aluminum metal on the temperature in the superconductingregion below Tc = 1.2K. The energy is ex-pressed as a percentage of the amount ab-sorbed immediately above Tc.

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temperature is lowered, less and less of this energy is absorbed by the supercon-ductor. At some temperature below Tc, there is no absorption of microwave energy, as indicated in Fig. 10.7. This means a material in the superconducting state attemperatures far below Tc provides an excellent shield for electromagnetic radia-tion.

Well below Tc no microwave energy is absorbed provided the microwavefrequency makes the product hf less than the superconducting energy gap, where his Planck’s constant. Figure 10.8 shows the dependence of the absorption of aluminum on the microwave energy hf; the plot shows absorbed energy versus the ratio of the microwave energy hfto the energy of thermal motion kTc at the transitiontemperature. The frequency fg ≈ 2.2kBTc /h where the curve begins its sharp upturnprovides an estimate of the energy Eg of the superconducting energy gap:

Eg hfg 2.2kBTc≈ (10.1)

This method of measuring energy gaps is applicable to superconductors with low

≈

transition temperatures, such as aluminum, with Tc = 1.2 K, because their energygaps (see Eq. 10.1) occur at microwave frequencies. High-temperature supercon-ductors with transition temperatures 100 times as high have gap frequencies fg that are 100 times as high, so infrared radiation is required to measure them.

Figure 10.8. The dependence of the absorbed microwave energy on the microwave frequency f for a superconductor at a temperature far below Tc. The abscissa is the ratio of hf to kBTc. [Adapted from M.A. Bondi and M. P. Garfunkel, Phys. Rev. Lett. 2, 143 (1959).]

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Figure 10.9. A microwave delay line madeof crystalline superconducting thin film strips deposited on the surface of a magnesium oxide (MgO) single crystal substrate.

An example of an application involving microwaves and superconductors is the device called a delay line (see Fig. 10.9), which stores microwave energy for ashort length of time, perhaps 100 microseconds, simply by providing an extra long path for microwaves to traverse. In a good delay line, there is almost no energy loss over the path length. A delay line typically consists of film strips of a low-resistanceconductor deposited on a magnesium oxide (MgO) crystal substrate. Delay lines have been made of strips or films of the high-temperature superconductors operat-ing at 77 K, as shown in Fig. 10.9. Such strips show much less loss than the best normal metal strips operating at 77 K.

11

Further Applications

11.1. INTRODUCTION

A number of commercially available superconductivity applications were discussed in two Chaps. 9 and 10, such as magnets, radiation detectors, and magnetometers. Since the discovery of the phenomenon, innumerable ideas have been proposed for using superconductivity in all types of devices. Some have actually been built for research purposes, and prototypes of others have been constructed. Most of these have not come to commercial fruition because oftechnological or economic factors. The need to cool the devices to liquid helium temperatures has made most of these applications too expensive in practice. One of the exciting aspects of the availability of liquid nitrogen temperature supercon-ductors is the possibility that some of these proposals may now become feasible. In Chap. 11 we discuss a few of the more important ideas advanced over the years that now have the potential for implementation.

11.2. COMPUTERS

One of the most unusual archaeological discoveries in modem times was made on the island of Apraphul off the northwest coast of New Guinea in 1958. It consisted of a large number of boxes through which ropes passed. Many of the boxes had knots on the rope near their entrance and exit holes; three examples of these are shown in Fig. 11.1. By pulling on appropriate ropes, positions of the knots are moved from near to far from the holes on entrances and exits of the boxes. The construction of the device was dated around 850 AD, and it is believed to be anup-down binary calculator, which can be considered as the first known digitalcomputer.

A computer employs a binary or two-digit system for encoding, i.e., storing information. The two digits are designated by zero (0) and one (1). The Apraphullian rope computer used knot positions to store this information. Appropriate arrange-

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152 CHAPTER 11

Figure 11.1. Elements of an ancient digital computer reconstructed from archaeological findings onthe island of Araphul near New Guinea. When the knot on the rope is near an input hole on the left it represents the binary digit zero, and when it is away it represents one. The opposite is the case for the output holes on the right. The three components shown are a) an OR gate, b) an AND gate, and c) an inverter. [From A. R. Dewdney, Scientific American (1958).]

ments of mechanical devices, such as pulleys, levers, and springs within the boxes, produced various combinations of knot positions (see Fig. 11.1). We see in this figure that zero is represented by a knot near an input hole and by a knot far from an output hole; the digit 1 is represented by the opposite knot positions, as shown. In each of these components, the position of the output knot is determined by the input settings. We assume that all of input settings are initially at 1 and examine how changes in these input settings produce changes in the output.

Beginning with Fig. 11.1a: With both inputs initially on 1 the output rope is held to the left so the output is also on 1. We see from the construction that a change in the upper input alone from 1 to 0, a change in the lower input alone from 1 to 0, or a change in both, moves the output knot to the zero position. In modern computer terminology, this component functions as an OR gate.

FURTHER APPLICATIONS 153

Considering the component in Fig. 11.1b: With both inputs initially on 1, the output rope is also on 1. We see from the arrangement of the pulleys that in this case, both input ropes had to be changed to alter the position of the output knot.Changing one input alone has no effect on the output; this component is accordingly referred to as an AND gate.

Considering the component in Fig. 1 1.1c: Initial settings are the one position for the input knot and the zero position for the output knot. Changing the input from 1 to 0 compresses the spring; this changes the output from 0 to 1. These three components provide basic elements required to build an elementary digital com-puter.

In a modern computer a device that can be turned on or off, called a switch, represents the two binary digits. The on position may represent 1 and the off position, 0. Two switches have the following four possibilities: on/on, on/off, off/on, off/off. A modem computer essentially consists of a vast number of switches connected together by wires forming complex circuits through which current flows. The current carries messages to the switches to set them to either 0 or 1. Information is encoded by various combinations of the on and off positions of the many switches. For example an array of three switches can store up to eight numbers, four switches up to 16 numbers, etc.; Table 11.1 shows the three-switch case. To process information rapidly, switches must be changed very rapidly, so speed is essential for computers to carry out large calculations.

There are two basic limitations to computer speed, namely, the intrinsic time it takes for the switch to change once it receives a command and the time it takes for the command to travel between switches. Clearly a computer can operate more rapidly if the switches are very close together, thereby reducing the time current requires to travel between them. However when the current moves through the resistive wires, some heat is generated, so the switches become warm. Since

Table 11.1. Eight Possible Permutations ofThree Switch Settings That Can Be Used to

Encode Integers from 0-7

Switch1 2 3 Number

On On On 0 On On Off 1 On Off On 2 On Off Off 3Off On On 4 Off On Off 5Off Off On 6 Off Off Off 7

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switches are transistors, they may not function properly when heated. Thus com-puter components must be ventilated to remove heat; in fact most personal com-puters (PCs) contain a small fan to exhaust warm air. The necessity of cooling means that components can not be placed together too closely. However if the wires connecting the switches were in the superconducting state, they would have no resistance, no heat would be generated, so computer components could be placed much closer to each other. Such a computer would be much faster, but no faster than the time needed for a switch to change.

Superconductivity can also help decrease switching time. A superconducting device called a Josephson junction (see Chap. 5) can function as a very fast switch. Referring to Fig. 5.6, which shows the current voltage characteristics of the Josephson junction, we see that current suddenly starts to flow at some critical voltage value; in other words at a certain voltage, the device switches to permit current to flow. The range of voltages where no current flows could be the zero position; the voltage at which current flows could be the one position in our binary-encoding system. This superconducting switch has three major advantages over a transistor switch. First it switches from off to on very fast, on the order of 10–9 seconds, almost a thousand times faster than the present-day semiconductor switch. Secondly it requires much lower voltages and currents than a semiconductor switch, so a computer based on a Josephson junction would require much less power to operate. Finally because it is in the superconducting state, current flowing through it does not generate heat.

Besides executing such processes as rapid changes in switches, a computer must store information. Programs providing directions for the computer as well as calculation results must be stored. In a PC this is accomplished with hard or floppydisks, which are magnetic storage systems. Information is stored by applying tiny magnetic fields to orient regions of magnets in magnetic material in the storage disks. A region of the tape with the tiny local magnets oriented in one directionrepresents 1, and a region with the orientation in the other direction is 0.

Superconductors can also be used as storage devices. An externally applied magnetic field penetrates a Type II superconductor in the form of vortices; the magnetic field within the material is parallel to the applied field direction. When the external field is removed, some magnetic field remains trapped inside the superconductor with its orientation unchanged, as explained in Chap. 3. This trapped magnetic flux stores information about the direction of the previously applied external field. Memory cells based on this flux-trapping principle have been built.

One concept of a superconducting storage cell proposed by J. W. Crow of theIBM Corporation in the mid-1950s is illustrated in Fig. 11.2. Two small adjacent D-shaped holes are made in a thin film ofa superconductor, such as lead. The smallD-shaped holes serve as regions to trap flux. Flux is trapped in the holes by meansof a “write” wire running across the top of the film between the two holes. The


Figure 17.2. Superconducting memory device based on the flux trapping properties of type II superconducting films. The small D shaped holes in the film trap flux generated by the current flowing over the hole in the write wire. When this current is turned off flux remains trapped in the holes. A voltage induced in the read wire determines the direction of the flux trapped in the holes, and hence the value of the stored digit.

magnetic field produced by current flowing in this wire is perpendicular to the surface of the page; it is trapped in the D holes and remains there when the currentis turned off. The direction of the magnetic field in the holes depends on the direction of current flow in the wire. For a current flowing in the upward direction in Fig. 11.2, the field is directed outward from the page in the left D and down into the page in the right D. This represents the digit 1. If the current flows down, asshown on the right-hand side of Fig. 11.2, the direction of the field is reversed; it could represent zero. A wire called a sensing or read wire runs below the film, also between the two holes.This wire reads whether a 1 or 0 is stored. Increasing current in the write wire induces voltage in the read wire whose sign depends on the direction of current in the write wire. Thus the read wire tells whether a 1 or 0 is being stored based on the sign of the voltage.

11.3. BUILDING A SUPERCONDUCTING COMPUTER

Advantages that we have mentioned would seem to provide sufficient motiva-tion for building a superconducting computer—at least for special research purposes where great speed is required despite the disadvantage of having to keep computer components cooled to liquid helium temperature. Why then has such a computer not materialized? In 1968 the IBM corporation initiated a program to

156 CHAPTER 11

develop technology to build a computer based on superconducting switches. The challenge was to develop the capability to fabricate millions of switches, i.e., Josephson junctions, all with identical performance characteristics, that could reproducibly switch on and off hundreds of millions of times per second over many years of operation without performance degradation. In 1983 after spending over a hundred million dollars, IBM management decided to discontinue the effort. The speed of superconducting chips was indeed fast, but during this period of time, semiconductor technology had also made advances, so faster but less costly switches were available. In the interim the Cray corporation developed and mar-keted a very rapid computer based on existing technology. Various advantages gained by the superconducting computer were offset by the manufacturing expense and the need to cool the system with liquid helium, so the IBM project was discontinued.

Interestingly the Japanese, funded by a consortium of private industry and government, also embarked on an effort to develop a superconducting computer, but they never terminated the effort, so they are now well-positioned to exploit the new high-temperature materials in this application. It is likely that the first super-conducting computer will be Japanese, and it should be operable at 77 K.

11.4. FRICTIONLESS BEARINGS

Many ideas have been proposed that make use of the repulsion between a magnet and a superconductor; one of these is the frictionless bearing. This is a device to reduce friction between two metal parts moving over each other, such as a wheel rotating on an axle. Figure 1 1.3a shows a ball bearing used to reduce friction between a wheel and axle. The space between the wheel and the axle is packed with metal balls. The frictional force between two materials depends on the area of the surface where the two metals are in contact; the metal balls considerably reduce this contact area, thereby reducing friction. A superconducting bearing is shown in Fig. 1 1.3b; the axle is the superconductor, and the inner surface of the wheel is lined with a coil carrying a current that produces a magnetic field repelled by the superconducting axle. The net effect is a radially repulsive force between the axle and the wheel, which if strong enough, would prevent the wheel from touching the axle, resulting in zero friction.

11.5. LEVITATION

Perhaps the most dramatic manifestation of the repulsion between magnets and superconductors is the phenomenon of levitation (see Chap. 3 and Fig. 3.3).

For many years physicists and engineers dreamed of a superconducting levitated train, and both the Japanese and the Germans have built a prototype of


Figure 11.3. (a) A normal ball bearing arrangement and (b) a frictionless bearing employing superconductivity. In the latter case the coil on the wheel produces a magnetic field which repels the axle made of the superconducting material.

such a train. The idea is that the train is lifted off the tracks by the repulsive forces between magnets and superconductors, thereby eliminating friction between the wheel and the rails. Figure 11.4 shows one type of levitated train called Maglev, magnetic levitation. Superconducting magnets are located on the bottom of the train, and the tracks are a series of closed wire loops aligned down the path of the train. Figure 1 1.4 illustrates the track configuration; there are actually two sets ofloops on the tracks—one set lifts the train and the other propels it. For the moment let us ignore the dashed-line loops, which are part of the propulsion mechanism, and concentrate on the levitation system.

Levitation is brought about as follows. Faraday’s law of induction, which is explained in general physics textbooks, states that when a magnetic field is swept through a closed wire loop, a voltage, and therefore a current, are induced in the wire. As the train starts to move down the track, the magnetic field from supercon-ducting magnets on the bottom sweeps over the coils, inducing a current in the coils that produces a magnetic field. The direction of this magnetic field opposes the field

158 CHAPTER 11

of the magnets on the train. Since the amount of current induced in the coils depends on how fast the field sweeps the coil, the lifting force depends on how fast the train is going. Before picking up speed, the train remains on the tracks, and therefore uses wheels in the usual way for slow speeds. When a sufficient speed is reached, levitated propulsion then takes over.

Electromagnetic interactions can also be used to propel the train. This is achieved by adding a second set of coils to the track—the dashed-line coils in Fig. 1 1.4. A current passes through the coils just ahead of the train. The direction of this current in the coils is chosen so that the magnetic field it produces attracts magnets on the train. This operation is equivalent to a linear synchronous motor. The timing ofthe current flow is very importantbecause it must flow only in the coil just aheadofthe train. This is achieved by applying an AC signal to the coil whose frequencyis chosen so that the train isjust about to cross the coil when the AC current reachesits peak value. The speed of the train is controlled by changing the frequency ofthecurrent in the coil.

Figure 11.4. A superconducting levitated train and the track configuration for the train (at bottom).The train is raised off the tracks by the interaction of the magnetic field of the superconducting magnets on the bottom of the train with the reversed direction magnetic field arising from the current inducedby these magnets in the suspension coils lying between the rails. The forward movement of the train isprovided by synchronously timed currents in the propulsion coils lying just ahead of the train.


11.6. GENERATORS

In its simplest form,aconventionalAC generator consists of a rectangular loop of wire with many turns. Alternating current is generated by rotating the loop in a fixed magnetic field. Changing the magnetic flux through the loop produced by rotating it induces alternating current in the loop. Using superconducting magnets instead of iron-core magnets to produce the fixed field could increase the efficiency and considerably lighten the weight of the device. The US Air Force is seriously investigating the use of superconducting generators, primarily because of their potential to be so much lighter.

Another type of generator, called a homopolar generator, is a relatively simple device capable of producing very large electric currents. It consists of a metal disk made of a good conductor rotated between the poles of a magnet. A current is induced to flow radially, i.e., from the center outward, in the disk. The magnitude of the current depends on the velocity of rotation and the strength of the magnetic field. Since the metal disk is a good conductor, it has low resistance and thus generates large currents and low voltages. The device can generate a large current by introducing electrical contacts at the center and the edge of the rotating disk. In fact this device is used to generate the large currents required in the rail gun described in Section 11.7. Since larger magnetic fields result in larger currents, making magnets from superconducting coils considerably increases the amount of current that can be generated. In addition it reduces the weight of the homopolar motor because a superconducting magnet does not require heavy iron poles.

11.7. ELECTROMAGNETIC PROPULSION

A number of military applications of electromagnetic propulsion are now under investigation, such as a silent submarine engine. One of the ways a submarine is detected is by the sounds it makes, particularly the noise of the engine, and the distribution in frequency of this noise can be used to identify the type of submarine. Obviously an engine with no mechanically moving parts makes much less noise and hence is highly advantageous.

A noiseless electromagnetic propulsion system that employs superconducting magnets is now under development; the concept is illustrated in Fig. 11.5. A superconducting coil capable of generating a large magnetic field is mounted on the stern of the boat so that the magnetic field is perpendicular to the surface of the water. Two plate electrodes attached to the stern with their planes perpendicular to the coil of the magnet are immersed in the water, as shown. A high-DC voltageestablished across the electrodes ionizes the salt water; the sodium (Na+) and chlorine (Cl-) ions that result move through the waterbetween the electrodes. Thisconstitutes a flow of current in the water between the plates. Because the current is moving perpendicular to the magnetic field, the ions experience a force, and the

160 CHAPTER 11

Figure 11.5. An electromagnetic propul-sion system for a submarine using magnetic fields generated by superconducting coils. The high voltage across the electrodes causes a current to flow through the salt water which, because of the magnetic field from the magnet, produces a force that pushes the water backward and hence the submarine forward.

effect of this force pushes the water backward, away from the stem of the submarine. By Newton’s law of action-reaction, reaction to this outward rush of water pushes the submarine forward. The speed of motion can be controlled by either adjusting the high voltage or varying the magnetic field strength. It is desirable to use superconducting magnets to reduce power requirements and magnet weight.

Electromagnetic propulsion techniques are also being explored as a way of shooting projectiles from large guns. Present-day technology propels a shell by igniting a chemical propellant placed behind a shell in the breech of the gun. This results in a rapid (milliseconds) buildup of gas pressure behind the shell, which causes it to accelerate down the barrel of the gun. It is amazing to realize that a chemical propellant can throw a two-ton shell over 20 miles when fired from the 16-inch gun of a Missouri-class battle ship.

At the present time, a number of military and industrial laboratories are working on an electromagnetic gun, and technology based on superconductivity may play an important role in its development. A sketch of a rail gun is shown in Fig. 1 1.6. The metal projectile sits on two parallel rails, and its base is made of an electrical-conducting metal in contact with the rails. To fire the gun, a large amount of current is rapidly transferred onto the rails from the homopolar generator previously described. Current flows down the rails through the base of the projec-tile; this produces a magnetic field in the space between the rails and also through the projectile. The interaction between this magnetic field and current flowing through the base exerts a strong force on the projectile, which propels it down the rails.

A number of experimental electromagnetic guns have been built and tested, and velocities up to 10Km/sec have been achieved using small projectiles. However a number of engineering problems must be overcome before such guns become practical. One is the large size of the device needed to create a charge of about 1000 C by applying an enormouscurrenton theorderof 106 Aflowingfora millisecond;a superconducting device could possibly accomplish this.


Figure 11.6. An electromagnetic rail gun accelerates a projectile down the rails by the interaction of the current through the base of the projectile andthe magnetic field produced by thecurrent in the rails. Additional mag-netic fields from the superconducting coils can enhance the acceleration.

One proposal is to use a device called a cryotron (see Fig. 11.7), which is asuperconducting switch. Loop 1 of the circuit consists of superconducting wire cooled below the transition temperature, so its resistanceRs is zero, and charged tocarry a large amount ofcurrent. Loop 2 contains the load resistor RL, which is nota superconductor. Because the resistance of Loop 2 is large and that of Loop 1 is

Figure 11.7. A superconducting switch called a cryotron consisting of a superconducting Loop 1 anda normal metal Loop 2. The load, a normal metal, has the resistance RL, and the superconductor has theresistance Rs in the normal state and zero resistance in the superconducting state, where Rs is chosento be much larger than RL. At temperatures below Tc the current is confined to Loop 1. Warming thesuperconductor above Tc dumps a large amount of current from Loop 1 into the non superconductingLoop2.

162 CHAPTER 11

zero, all current flows into Loop 1. The resistance Rs of the superconductor in itsnormal state is chosen to be much larger than RL, so quenching the superconductor by heating it above Tc rapidly transmits a large amount of current to Loop 2. In the case of the rail gun, the load resistor is the rails of the gun, and the current transmitted induces projectile acceleration.

Superconductivity could also be used to enhance the velocity of the projectile by placing current-carrying superconducting coils over the rails of the gun, as shown in Fig. 11.6. The coils then produce a magnetic field perpendicular to the plane of the rails, so that current moving through the base of the projectile experiences an increased force due to the presence of this field.

11.8. TRANSMISSION LINES

Electricity is carried to your house from the power source via copper cable.Because cable has resistance, there is some energy loss during the conversion process. It is estimated that about 7% of all electrical energy produced is lost in transmission at an annual cost of about 13 billion dollars. Using superconducting cable, which has no resistance, would reduce this loss, but because of refrigeration cost nothing would be saved by using a low-temperature superconductor like

Figure 11.8. Critical current and magnetic field requirements for various superconducting devices.


niobium-titanium. However operating at 77 K using liquid nitrogen could appre-ciably reduce costs and make the system economically viable. The Brookhaven National Laboratory constructed an experimental transmission line 115 meters long made of niobium cable cooled by liquid helium that demonstrated the technical feasibility of the concept.

The ability of many of the previously described applications to operate at 77 K using liquid nitrogen as the coolant depends on whether copper oxide supercon-ductors can be fabricated to carry sufficient current in the presence of magneticfields. Various applications have their own individual current and magnetic field requirements; Fig. 11.8 summarizes these requirements for a number of the applications that we discussed.

12

Future Prospects

12.1. POSSIBILITY OF ROOM-TEMPERATURESUPERCONDUCTIVITY

One of the most frequently asked questions of researchers in the field of superconductivity is, “Will there ever be a room-temperature superconductor?’ Ironically this question was seriously addressed in 1965, 21 years before the discovery of copper oxide superconductors in a 1965 Scientific American article by W. A. Little entitled, “Room-Temperature Superconductivity.” The article did not predict superconductivity in copper oxide materials but rather the possibility of room-temperature superconductivity in long chainlike molecules.

Little carried out calculations with chain molecules having alternating single and double bonds, called polyconjugate molecules, containing molecular groups attached to certain carbons along the chain, as shown in Fig. 12.1. He used highly polarizable side groups, which means that the electronic charge distribution of the groups can easily be effected by the presence of an electric field or a nearby charged particle. When an electron passes by a side group, its electric field shifts the charge distribution of the side group so that it is positive near the moving electron and negative away from that electron. This causes electrons on the side group to be pushed away from the chain. The electron moves much faster than the charge can be redistributed, so the onset of polarization lags behind the moving electron. The enhanced positive charge near the chain can attract another electron, which then follows the first electron, in effect forming a Cooper pair. When Little carried out detailed BCS theory calculations on this hypothetical molecule, he obtained the incredible value of 2000 K as the transition temperature. Of course it is highly unlikely that the hypothetical molecule exists at this temperature, but the possibility ofroom-temperature superconductivity is certainly predicted in this article.

In Chap. 5 we showed that the usual Cooper pair formation depends on the inverse of the mass of the atoms in elementary superconductors because the passing electron causes the atom to move away from its regular position in the lattice. In

165

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a) hypothetical superconductingchain molecule

R is a very polarizableside group

b) polypropylene

Figure 12.1. (a) Carbon chain molecule with alternating single and double bonds, and (b)polypropylene as an example of a molecular side group R incorporated in molecule (a).

Little’s mechanism the atoms do not move, but the electronic charge is displaced. This mechanism of superconductivity is known as the exiciton mechanism. Because the electron is so much lighter than the atom, the inverse mass rule leads us to expect a very large increase in transition temperature.

In March 1989 three scientists from the Russian Academy of Science published a paper in a Russian scientific journal entitled, “Possible Superconductivity Near 300 K in Oxidized Polypropylene.” The work was apparently confirmed by another Russian research group. Polypropylene films were deposited on metal surfaces; on certain regions on the surface of the oxidized polymer, there were channels where the resistance measured was zero. It was suggested that there were thin filament regions that were superconducting. Polypropylene shown in Fig. 12.1b is not a polyconjugate chain having highly polarizable side groups (the side groups are CH3); thus it does not have the necessary properties of the chain molecules predicted by Little to be superconducting. Most researchers in superconductivity do not accept the validity of these results as proof of superconductivity.

Thus it fair to say that, no polyconjugate chain molecule has been synthesized that displays superconductivity at high temperatures, perhaps supporting the wis-dom of the late Bernd Matthias, the grand old man of superconductivity, who

FUTURE PROSPECTS 167

advised “never listen to theorists.” Matthias was responsible for many of the increases in the transition temperature of classic superconductors prior to thediscovery of the cuprates. In this chapter we ignore the advice of Matthias, perhaps to our peril.

As of this writing, the highest confirmed superconducting transition at atmos-pheric pressure is 133 K in the HgBaCaCuO system. Subjecting this sample to highpressure increases Tc by about 15 degrees. The existence ofa material that becomessuperconducting at a much higher temperature would be easier to affirm if themechanism of superconductivity in copper oxide materials were known. Forexample if, as some researchers suspect, the binding of the holes to form Cooper pairs in copper oxide planes is magnetic in nature, then much higher transitions temperatures may be possible.

To see this let us recall that the electron-phonon interaction responsible for superconductivity in classic materials is based on lattice vibrations. The measure of the strength of the vibrational motion of atoms in a lattice is called the Debyetemperature (θD). The Debye temperature is 102 K and 277 K for the supercon-ducting elements lead and niobium, respectively, and about 250 K for most cuprates. The measure of the strength of magnetic interactions in a material is the temperature of the onset of a magnetically ordered state. Many iron alloys have ferromagnetic-ordering temperatures above 1200 K, and some compounds have antiferromag-netic-ordering temperatures that are high, such as 950 K for iron sesquioxide (Fe2O3) and 670 K for the chromium antimony alloy CrSb. Many materials havemuch lower magnetic-ordering temperatures; temperatures at which materialsbecome antiferromagnetic are generally lower than temperatures at which materials become ferromagnets. The important point is that magnetic interactions can some-times be very strong. In addition there is evidence of short-range antiferromagnetic correlations in copper oxide superconducts, which raises the possibility that mag-netic interactions are involved in superconductivity.

A number of theoretical physicists who have been trying to develop a theory to explain superconductivity in copper oxides have speculated on the possibility of superconductivity at much higher temperatures. For example V. L. Ginzburg, a member of the Russian Academy of Sciences who has made major contributions to the understanding of metallic superconductors, suggested in a 1992 article in the Russian journal Sverkhprovodimost (Superconductivity) that transition tempera-tures from 300-500 K may be possible. This estimate is based on the assumption that the pairing mechanism is magnetic. Interestingly Ginsburg and D. A. Kirzhnits edited a book in 1977 entitled, High-Temperature Superconductivity. Each chapterwas written by a different Russian scientist, and the focus of the book was possible materials and mechanisms for higher transition temperatures. In the book Ginsburg suggests that reduced dimensional materials, such as layered compounds, may be candidates for higher temperature superconductors. It is noteworthy that the eventual discoveryofsuperconductivity above77 Kwas in layeredmaterials—copperoxides.

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T. M. Rice, who holds the chair of theoretical physics at the ETH in Zurich, predicted the possibility of higher temperature superconductivity in copper oxide materials that have a different arrangement of the coppers and oxygens in the copper oxide planes. We discuss these materials in more detail in the following sections.

12.2. UNIDENTIFIED SUPERCONDUCTING OBJECTS

In addition to the speculation of theoretical physicists, a number of papers in reputable journals, such as Science, report evidence of superconductivity at highertemperatures, typically around 250 K. Unfortunately all of these observations have one major characteristic in common—they have not been reproduced by any other laboratory, a necessary requirement for acceptance by the scientific community. Paul Chu, the discoverer of the 90-K superconductor YBaCuO, calls these USOs—unidentified superconducting objects. Chu feels that “all the USO’s are tantalizing enough for us not to ignore them.”

Others however are less certain: T. H. Geballe of Stanford University, who has considered the possibility of higher temperature superconductivity, points out that resistance drops are frequently observed in samples containing many different conducting compounds. Generally the measured conductivity of a multiphase sample reflects the combined conductivity of all phases. If one phase undergoes a metal-to-insulator transition resulting in a sudden increase in the resistance of this phase, the current is then forced to flow in the other conducting phases. If the conductivity of the phases is greater than the conductivity of the material that underwent the metal-to-insulator change, resistance measurements show an appar-ent resistance drop. Further small reductions in magnetic susceptibility may be observed. In the metal phase, free conduction electrons give rise to a weak paramagnetism, called Pauli paramagnetism. When the material becomes insulat-ing, there can be a reduction in paramagnetism, which manifests itself as a decrease in magnetic susceptibility. However measurements indicating diamagnetism tend to be more widely accepted, since they do not require electrical contact with the sample. In these reports the superconducting fraction of the sample is usually quite small; in some instances measurements are on the edge of the sensitivity limitations of the instrumentation. Nevertheless many reports have come out of such respected research laboratories as the CERN, European Center for Nuclear Research in Grenoble, France, and they have been published in peer-reviewed scientific jour-nals.

As recently as May 1995, three Russian scientists, V. D. Shabetnik, S. Y. Butuzov, and V. I. Plaskii published apaper in the Russianjournal TechnicalPhysicsLetters reporting a superconducting transition at an amazing 371 K in YBa2Cu3Se7;presumably the selenium atom (Se) substitutes for oxygen in the YBa2Cu3Ox

material. To date there have been no verifications of this observation. Table 12.1


Table 12.1. Examples of Published but Nonreproduced Observations ofHigh-temperature Superconductivity

Composition Tc (K) Detectiona Fabricationb Instabilityc

YBaSrCu3O7–x 340 R, D Bulk 100 MA BiSrCaCu8O6+x 270 R, D MBE 2 weeks

Bi1.9Pb0.4Sr1.9- 240 D Bulk 2 weeks

Bi2Sr2CuO6+x 300 D Solution gel 1 month HgBa2Ca2Cuu3Ox 235 R, D Bulk Current thermal

recycling

Y2BaSrCu3O8 250 R Bulk —

Ca2Cu3.2Ox

Ca1–xSrxCuO2 180 R, D MBE —

aR means resistance drop measurement; D indicates diamagnetism observed by AC or DC

bMBE means molecular beam epitaxy. cInstability refers to the disappearance of superconductivity intime with increased current or thermal

susceptibility measurement.

recycling.

lists some examples of published, but not reproduced, observations of higher temperature superconductivity. Table 12.1 indicates the assigned transition tem-perature and detection method, as well as the composition of the starting material; the table also includes the synthesis method. Bulk synthesis is the most prevalent method; it is described in Chap. 8. The solution gel method is a refinement of the bulk method in which starting materials are dissolved in a common solution that is then cooked to a lower temperature—typically from 300-500 C—to a gellike substance. This approach results in better mixing in the reactants. The final material is fired at the normal higher temperature of synthesis.

Molecular beam epitaxy (MBE) is a new method of building a structure by depositing one layer of atoms at a time on a substrate. To accomplish this a beam of copper ions is swept over the surface of a substrate, followed by a beam of ions for the next layer, etc. Phases, such as Bi2Sr2Ca7Cu8O20, which are suspected of being superconducting and not possible to make by other methods, can be made by this technique. Generally the amount of the sample claimed to be superconducting is a very small fraction of the total that is synthesized, and its chemical composition may be quite different from that of the starting material.

12.3. ROLE OF FLUORINE

The fluorine atom has also played an intriguing role in many high-temperaturesightings. Fluorine is very electronegative; that is, it strongly attracts electrons.When it is substituted in a copper oxide, it may replace oxygen and induce hole

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Table 72.2. Examples of the Effect of Fluorine on Superconducting Transition Temperatures

Composition Tc(K) Fabrication Method Comments

Nd2CuO4–xFx 27 CuO, NdO3, NdF3 900 °C, Nd2CuO4 not superconducting14 hr

La2CuO4–xFx 35 Expose LaCuO 15 min to La2CuO4 not superconducting90%N2/10%F2 gas at250 °C

Sr2CuO2F2+x 40 Expose Sr2CuO3 to F2/N2 Sr2CuO3 not superconductinggas at 210 °C when made at atmospheric

pressureYBa2Cu3OyFx 155 Bulk synthesis at 950 °C

YBa2Cu3OyFx 148 YBaCuO irradiated with

Bi1.6Pb0.4Sr2Ca2– 121 Bulk synthesis 750 °C Without F, Tc = 113 K,

for 48 hr

180-KeV F ions

Cu3O9.2F0.8Hint of Tc = 172 K

doping in copper oxide planes. For example when La2CuO4+x, which is not asuperconductor, is subjected to fluorine gas at 350 °C, the material becomes superconducting at 35 K. Presumably the presence of fluorine in the lattice produces holes in copper oxide planes. In late 1987 there was a report of superconductivity at 155 K in a bulk-synthesized sample of YBaCuO in which some oxygen was replaced by fluorine. A number of efforts were made to repeat the experiment in other laboratories but to no avail. Oddly enough at that same time, a Chinese research group from Beijing, apparently unaware of this work, reported zero resistance at 148 K in a YBaCuO sample in which fluorine had been implanted using beams of 180-kV fluorine ions. Table 12.2 lists some examples of reports that fluorine affected superconductivity and in some instances raised the transition temperature above that of the same material without fluorine.

12.4. METASTABILITY

One of the common factors of the observations listed in Table 12.1 is the lack of stability of superconductivity. It typically disappears after a month when the current at which resistance is measured is increased or after cycling the temperature several times from cold to warm and then back to cold. Researchers who have thought about the question of higher temperature superconductivity have suggested that materials which become superconducting at higher temperature may be metastable. A metastable material is at a minimum of energy for the system, but this minimum is not the lowest possible one. A simple example of a metastable system


is a small ball, such as a golf ball, balanced on top of a larger ball, say, a basketball. As long as the two balls are not disturbed, the small ball remains on top of the large one; the slightest movement however causes the golf ball to fall off. In supercon-ductivity metastability may result from hole doping. If a synthesis of a material produces a quantity of holes appropriate for superconductivity, it is possible for the hole distribution to change over time, forming regions of high and low concentra-tion. The redistributed hole concentrations may be outside the range required for superconductivity. Such a process can lead to a time-dependent disappearance ofsuperconductivity, as occurred in many of the nonreproducable observations. Products of a chemical reaction may also be metastable, with slow continuing reactions forming new materials. Thus if superconductivity involves a metastable product, it can slowly disappear with time.

12.5. LADDER PHASES

While not meeting the standards of scientific proof, taken collectively these sightings of higher temperature superconductivity suggest the existence of materi-als with higher transition temperatures. Besides the necessary copper, the element strontium is generally present in materials reported to have very high transition temperatures. This is particularly intriguing in view of recent theoretical predictions of much higher temperature superconductivity in the so-called ladder phases whose composition is strontium (or perhaps calcium) and copper oxide. The rectangular lattice arrangement of oxygen atoms is the same as in cuprates and ladder phases, but what differs is the placement of copper on oxygen planes. In cuprates half of the sites between oxygens are occupied by copper ions, and the formula for the planes is CuO2; in the ladder phases two-thirds of the sites are similarly occupied,corresponding to the formula Cu2O3, as shown in Fig. 12.2. It is clear from Fig. 12.2b why these materials are called ladder phases.

The CuO2 plane of the cuprates (see Fig. 12.2a) is clearly two-dimensional;physicists describe the dimensionality of the ladder phase plane of Fig. 12.2b as quasi-one-dimensional. If no oxygen atoms connected the ladders, the structure would be purely one-dimensional. However because the ladders are connected, they are not purely one-dimensional nor two-dimensional, like the CuO2 planes in Fig. 12.2a that characterize copper oxide superconductors; therefore we adopt the term quasi-one-dimensional.

A whole family exists of these phases with the general formula Srn–1Cun+1O2n.T. M. Rice* theoretically investigated these phases and predicted that compounds where n = 3,7, 11 . . . will display superconductivity when lightly doped with holes; this “should occur on a separate but higher temperature scale.” Calculations predict

*E. Dagotto and T. M. Rice, Science 618, 271 (1996).

CHAPTER 12172

a)

b)

Figure 12.2. Arrangement of copper and oxygen atoms in (a) CuO2 plane of cuprates, and (b) Cu2O3

plane of “ladder phase.”

that the holes pair up to form singlet states, i.e., states in which electron spins onthe two holes of a Cooper pair are aligned in opposite directions. The triplet state corresponding to when both electron spins point in the same direction has a slightly greater energy. This difference in energy, referred to as the spin gap, is in effect thesuperconductinggap. Ladderphases of the Srn–1Cun+1O2n series canbe synthesizedonly by high-pressure methods, and evidence has been found for the existence ofthis energy gap but to date no indication of superconductivity has been found.

Two other phases whose copper and oxygen planes have a ladder structure can be synthesized without using high pressure, namely, Sr1.17CU2O3.42 andCaCu2O3. In the former material, the lattice contains alternating rows of simple one-dimensional CuO2 chains, columns of strontium atoms, and Cu2O3 planeshaving a ladder structure. The other material, CaCu2O3, does not have CuO2 chains,only ladder-like planes, and these planes are not flat but puckered; that is they have peaks and valleys. Severe puckering is known to inhibit superconductivity in


cuprates. Perhaps compounds related to these ladder phases will be the high-temperature superconductors of the future.

FLUCTUATIONS

Known copper oxide superconductors are approximately two-dimensional;that is their electrical conductivity is concentrated in the copper oxide planes shown in Fig. 12.2a. This is believed to be an important factor in their higher transition temperatures. Thus it is tempting to speculate that one-dimensional systems have even higher temperature transitions. The discovery of superconductivity in reduced dimensional organic materials, although at much lower temperatures, sheds some light on this issue. The organic compound TTF-TCNQ is an example of a crystal containing one-dimensional chains consisting of molecules shown in Fig. 12.3. The chain structure of crystals is clear from the arrangement of molecules shown in Fig. 12.4. At room temperature this material is one of the best-known organic conduc-tors. However when the temperature dependence of conductivity is measured below room temperature, unusual behavior is observed: There is a very strong increase in conductivity followed by an abrupt decrease near 60 K, as shown in Fig. 12.5. The increase in conductivity as the temperature is lowered is believed to be associated with superconducting fluctuations, which are easier to observe in one-dimensionalsystems. Such fluctuations can exist for short times above the bulk transition temperature. They occur in small regions of the sample with dimensions close to the size of a Copper pair, and they may be considered precursors to superconduc-tivity. However in the case of the TTF-TCNQ system, the material does not become superconducting as the temperature is lowered further; instead a metal-to-insulatortransition occurs, and conductivity of the sample decreases, as shown in Fig. 12.5. This behavior is quite prevalent in one-dimensional systems, but no one-dimensionalsuperconductors are known. In fact it is a generally accepted belief that superconduc-

Figure 12.3. Molecular structures ofthe TTF and TCNQ molecules.

12.6. DIMENSIONALITY AND

174 CHAPTER 12

Figure 12.4. Crystal structure of TTF-TCNQ showing parallel one dimensional chains of the two constituent molecules ar-ranged along the c axis.

Figure 12.5. Plot of the electrical resistance versus the temperature in a crystal of TTF-TCNQ showing a minimum near 60 K. [Adapted from M. J. Cohen et al., Phys. Rev. B 10, 1298 (1974).]


tivity cannot exist in one dimension. However when organic materials are engi-neered to be less one-dimensional, for example by appropriately doping the latticewith molecules lodged between the chains, then these organic materials becomesuperconducting,albeit at quite low temperatures.

12.7. ROOM-TEMPERATURE SUPERCONDUCTIVITY

Let us conclude Chap. 12 by returning to the question will there ever be aroom-temperature superconductor?Experimentalhints, USOs, and speculationsbysome theorists suggest that it is a possibility and perhaps worth seeking. But whatstrategies should we use to search for higher temperature superconductors?

In 1992 a diverse group of researchers at a two-day workshop in Bodega Bay,California, considered the issue of making much higher temperature superconduc-tors. T. H. Geballe,*who attended this workshop, summarized some guidelines that emerged from discussions:

Materials should be layered, with one or more planes capable of electron orhole conduction; these are the CuO planes in copper oxide superconductors.

Localized states in the conduction plane should have a net spin of one-half;copper in copper oxides superconductors has a net spin of one-half.

In between the layers of conducting planes, there should be layers of otherions that act as charge reservoirs to dope conducting planes.

The concentration of charge carriers in conducting planes should be suchthat the system is close to a metal insulator transition. In copper oxidesuperconductors, there is generally a hole concentration where the materialis not superconducting but can undergo a metal insulator transition.

In undoped material, for example LaCuO4 without Sr, material shoulddisplay antiferromagnetic ordering.

In analogy with organic superconductors, superconductingcandidates couldbe one-dimensional materials displaying superconducting fluctuations andmetal insulator transitions. Either subjecting possible superconductors to pres-sure or doping them with appropriate additives to increase dimensionality,could help.

*T. H. Geballe, Science 259, 1550 (1993).

Appendix A

Units and Conversion Factors

Å

ºCcm

Kµmnm

mT

Angstrom unit; a very small unit of length equal to 10–10 m or 10–8

cm (0.00000001 cm) or 3.94 x 10–9 in.Celsius temperature degree; ºC = 5(ºF – 32)/9 centimeter; one hundredth of a meter; 100 cm = 1 mkelvin temperature degree; K = ºC + 273.2 micrometer or micron; subunit of length equal to 10–6 mnanometer; a very small unit of length equal to 10–9 m or 10–7 cm(0.0000001 cm) or 3.94 x 10–8 in.millitesla; subunit of magnetic field B; 1000 mT = 1 T

177

Appendix B

Symbols and Abbreviations

AB

Bc

Bcl

Bc2c

°CCo

cm centimeter; unit of length

amp or ampere; unit of electric current magnetic field symbol; unit is Tesla (T) critical magnetic field (occasionally Hc)lower critical field (occasionally Hcl)upper critical field (occasionally Hc2)speed of light in a vacuum; 3 x 108 m/secdegrees Celsius (centigrade); unit of temperature quantum of circulation in a superfluid; value: h/mHe

e charge of an electron E

FhJ

JcK Kelvin; unit of temperature kB

m meter; unit of lengthNeR

sec second; unit of time T

Tc critical temperature V

κ

electric field; unit ofmeasurement: V/m

force unit of measurement: newton (N) Planck's constant; 6.63 x 10–34 J-secelectric current density; unit of measurement: A/cm2

critical current density; unit of measurement: A/cm2

Boltzmann's constant; value: 1.38 x 10–23 J/K

number of valence electrons in an atomresistance; unit of measurement: ohm (Ω)

Tesla; unit of magnetic field

volt; unit of electric voltageξ coherence length; radius of a vortex core (xi)

Ginzburg Landau parameter; κ = λL/ξ (kappa)

Eg energy gap

179

180 APPENDIX B

λL

x susceptibility (chi)

penetration depth, depth of penetration of magnetic field into super-conductor, radius of a vortex (lambda)

magnetic flux; unit ofmeasurement: T-m2 (phi)ΦΦ

flux quantum; value: h/2e = 2.07 x 10–5 T-m2 (phi)o

Appendix C

ChemicaI Elements

Symbols and atomic numbers of elements mentioned in the text follow. Transitiontemperatures are given for superconducting elements.

Ag silver, 47AlAu gold, 79

B boron, 5Ba barium, 56BeBi bismuth, 83Br bromine, 35C carbon, 6

Ca calcium, 20CdCe cerium, 58Cl chlorine, 17Cr chromium, 24Cs cesium, 55Cu copper, 29

F fluorine,9GaGe germanium, 32

He helium, 2Hg

I iodine, 53InIr

aluminum, 13; Tc = 1.18 K

beryllium, 4; Tc = 0.026 K

cadmium, 48; Tc = 0.5 K

gallium, 31; Tc = 1.1 K

H hydrogen, 1

mercury, 80; Tc = 4.2 K

indium, 49; Tc = 3.4 Kiridium, 77; Tc = 0.1 K

181

APPENDIX C182

K potassium, 19 LaLi lithium, 3

MoN nitrogen, 7

Na sodium, 11 NbNi nickel, 28 0 oxygen, 8

PbPd palladium, 46 Pt platinum, 78

Rb rubidium, 37 Rh rhodium, 45

S sulphur, 16 Se selenium, 34 Si silicon, 14 SnSr strontium, 38 TaTcTe tellurium, 52 TiTlU uranium, 92 VWY yttrium, 39

ZnZr

lanthanum, 57; T c= 6.3 K

molybdenum, 42; T c= 0.9 K

niobium, 41; Tc = 9.3 K

lead, 82; Tc = 7.2 K

tin, 50; Tc = 3.7 K

tantalum, 73; Tc = 4.5 K technetium, 43; Tc = 7.8 K

titanium, 22; Tc = 0.4 K thallium, 8 I; Tc = 2.4 K

vanadium, 23; Tc = 5.4 K tungsten (wolfram), 74; Tc = 0.02 K

zinc, 30; Tc = 0.85 K zirconium, 40; Tc = 0.6 K

Appendix D

Glossary

A-15 compound

Alloy Substance composed of two or more metals intimately mixed.

Ammeter Instrument for measuring electric current.

Ampere (A) Unit of electrical current; equals 1 C per sec.

Anisotropy Variation of properties along different directions.

Angstrom (Å) Unit of length; equal to 10–8 cm.

Annealing Heating generally followed by gradual cooling for the purpose of

Antiferromagnetic ordering An ordered magnetic state in which magnetic atom

Atomic number of an atom Number of protons in its nucleus.

BCS Presently accepted theory of superconductivity originated by J. Bardeen, L.

Bearing Device that reduces friction between two metal parts that move over

Binary compound A compound containing two elements.

Binder Plasticized polymer designed to hold magnetic particles together; e.g., plastic magnets are iron particles in a binder.

Borocarbide Compound containing the elements boron (B) and carbon (C), such as RM2B2C, where M is usually the element Ni and R is a rare earth element.

Bose–Einstein condensation Passage of a system of boson particles, such asCooper pairs or helium atoms, to their lowest energy state where they exhibitspecial properties, like superconductivity or superfluidity.

An A3B binary chemical compound with a 3: 1 ratio of atoms;structure shown in Fig. 6.5.

preventing or removing internal stress.

spins are alternately in the up and down directions.

N. Cooper, and J. R. Schrieffer.

each other.

Brittle Easily broken; fractures if stressed beyond a certain limit.

183

184 APPENDIX D

Buckminsterfullerene Same as fullerene.

Ceramic Product made from clay or related material.

Celsius Sometimes called centigrade (C); a temperature scale related to Fahren-heit (F) through the expression C = 5(F – 32)/9.

Chalcogenide One of four elements—oxygen (O), sulphur (S), selenium (Se), and tellurium (Te)—in Row 6 of the periodic table.

Chevrel compound Generallyaternarycompound AxMo6X8, whereX is sulphur(S), selenium (Se), or tellurium; A can be almost any element, and x has a value from 1 to 2.

Cold working Working a metal at or below its recrystallization temperature.

Composite Made of separate or disparate elements or parts.

Conductor A material that conducts electricity well, i.e., a material of lowresistivity.

Cooper pair Composed of two electrons; the charge carrier of the electric super current in a superconductor.

Coulomb (C)

Covalent bonding Chemical bonding scheme in which the involved atoms share electrons, commonly found in organic molecules.

Critical current density (Jc) Highest current density that can flow through asuperconducting material; expressed as A/cm2.

Criticalcurrent(Ic) Highestcurrent thatcan flow through aparticularsupercon-ducting wire; expressed as amperes.

Critical magnetic field (Bc) Highest possible field sustained by a Type I super-conductor(higherfieldsdrive thematerial normal); expressedinTesla(T). TypeII superconductors have lower Bcl and upper Bc2 critical fields.

Critical surface Surface in B, J, T (magnetic field, current density, temperature) coordinate system below which a material is superconducting.

Critical temperature (Tc) Temperature below which a material superconducts;expressed in Kelvin degrees (K).

Cryogenic Adjectivesignifying lowtemperature.

Cryotron Type of superconducting switch.

Current density (J) Electric current per unit cross section; usually expressed as

Delocalization Continual wandering throughout a material of electrons that carry

Dewar Container for holding a low-temperature liquid, such as nitrogen or

Unit of electrical charge.

amperes per square centimeter.

for example an electrical current.

helium.

APPENDIX D 185

Diamond

Doping

A form of carbon with a crystal structure in which each atom has four nearest neighbor carbons arranged in a regular tetrahedron.

Adding a small amount of one atom to replace another in a compound,such as replacing 10% of the lanthanum by strontium in the superconductor

Mass that an electron seems to have when certainexperiments are carried out; effective mass sometimes differs from the actualmass.

Electromagneticpropulsion Propulsion involving interactions of electric andmagnetic fields.

Energy gap Separation in energy of level containing superelectrons and level containing normal state electrons.

Epitaxial film Thin-film single crystal in the case of the cuprates made with copper oxide planes parallel to the surface.

Extinction Decrease in light intensity due to scattering from dust for example inits path.

Extrude To thrust out, force orpress out a viscous liquid; to form a material likea metal to a desired cross section by forcing it through a die.

Extrusion Act ofextruding or something that has been extruded.

Face-centered cubic at the vertices and in the centers of the faces; sometimes abbreviated fcc.

Factor of ten notation 0.0013 = 1.3 x 10–3, 343 = 3.43 x 102.

Flux (Φ) Quantity of magnetic field or magnetic field times cross sectional area through which the field passes; expressed in Tesla times meter squared (Tm2).The term magnetic flux is synonymous with flux.

Flux creep Very slow motion of magnetic flux when pinning forces are strong; see flux flow.

Flux flow Faster motion of magnetic flux when pinning forces are weak; see fluxcreep.

Flux melting Conversion of array of rigidly fixed vortices to array of randomly moving or wandering vortices.

Fluxquantum Φo

Fullerene Carbon compound in which each carbon atom is bonded to three

Gap

Generator Device for converting mechanical energy into electricity.

Geodesic dome

(La0.9Sr0.1)2CuO4.Effective mass of electron

Crystal structure in which the unit cell is a cube with atoms

=h/2e; amount of magnetic flux in a vortex.

others; applied especially to the 60 atom compound C60.

Separation in energy between normal and superconducting states.

Architectural dome invented by R. Buckminster Fuller.

186 APPENDIX D

Graphite Form of carbon with a structure in which the atoms are arranged in sheets with hexagonal patterns stacked one above the other.

Grain alignment Arrangements of grains or microcrystals so that most of theiratomic planes are aligned in the same general direction.

Gyromagnetic ratio Ratio of angular frequency to magnetic field at which anelectron or nucleus absorbs radio frequency energy when present in a magnetic field.

Heavy electron Electron that acts as though it has a large mass.

Heavy fermion compound Compound withheavy electrons.

Hexagonal close packed Crystal structure in which layers of identical atoms inhexagonal arrays are stacked one above the other with the third, fifth, etc., layers aligned with the first layer, and the fourth, sixth, etc., layers aligned with the second layer but displaced relative to the first.

Hole Absence of an electron in an otherwise full electron band; a hole carries electric current like a positive charge.

Homopolar generator Device forgenerating electricity by rotating a metal diskbetween magnet poles.

Infrared (ir) Light beyond the visible range in which the wavelength is greater and the frequency is less than that of visible light.

Infrared spectroscopy Measurement of infrared light frequencies corresponding to molecular vibrations.

Insulator A poor conductor of electricity.

Intercalation Insertion of atoms or ions between layers of materials, such as graphite.

Interference Interaction of two or more waves to form regions of brightnesswhere wave components add and regions of darkness where they subtract orcancel each other.

Intermetallic compound Alloy whose ratio of component atoms is expressed in terms of integers, such as A2B3 or A3B.

Ion Electrically charged atom, such as the doubly charged ion Cu2+ ofthe copperatom or negatively charged fluorine (F–).

Irreversibility Failure of a superconductor to retrace its magnetic state when an applied magnetic field is first increased and then decreased in value (or vice versa).

Isotope Two isotopes of an element have the same number of protons but a different number of neutrons in the nucleus.

APPENDIX D 187

Isotope effect

Josephson effect Tunneling of Cooper pairs with zero-applied voltage.

Josephson junction Two superconductors separated by a barrier through which

Kelvin (K) Temperature scale related to Celsius (C) by the equation K=C + 273.2.

Lattice vacancy Missing atom in a regular lattice.

Laves phase Class of metallic AB2 compounds, some of which superconduct.

Levitation

Long-range order Magnetic state in which the alignment of spins extends overthe entire lattice.

Magnetic field Magnetic flux per unit area; measured in Tesla (T).

Magnetic moment Small bar magnet associated with an electron, atom, or

Massspectrometer Instrument for determining the mass ofan atom or molecule.

MBE Molecular beam epitaxy; method ofpreparing a superconductor by depos-

Meissner effect Expulsion of magnetic field from a superconductor.

Microwaves Electromagnetic radiation with a wavelength from 30 cm–0.3 mm

Mixed state Superconducting state of partial-flux exclusion.

MRI Magnetic resonance imaging; application of NMR to produce images of the

Nanostructure

Néel temperature (TN ) Temperature below which a material becomes antiferro-magnetic.

Neutron Neutral particle in the nucleus of an atom.

NMR see nuclear magnetic resonance.

Normal state Nonsuperconducting state.

Nuclear magnetic resonance Interaction of the magnetic moment of a nucleus

Ohm (Ω) Unit of resistance.

Ohm’s law Relationship V = IR between voltage (V ) and resistance (R) in a wire

Optical extinction Reduction of light intensity by transmission through matter.

Shift in the superconducting transition temperature Tc arising from the presence of different isotopes.

Cooper pairs can tunnel from one side to the other.

Suspension of a magnet in space above a superconductor.

nucleus that has a spin.

iting one layer of atoms at a time.

and a frequency from 1000–1,000,000 MHz.

human body and other things.

Structure in the size range of tens of angstroms.

with an applied magnetic field that involves the absorption of energy.

carrying an electrical current (I).

188 APPENDIX D

Organic molecule Molecule containing the element carbon.

Orthorhombic crystal structure Having a rectangular unit cell with three

Pairing mechanism Factors responsible for holding the two electrons of a

Periodic table

Permeability (µ) Parameter that determines magnetism in a material.

Perovskite Crystal structure of the mineral calcium titanate (CaTiO3); the mineral itself is also called perovskite; term often used for a material with a structure similar to that of CaTiO3.

Persistent current Current that flows in a superconductor without loss of energy for an indefinite period of time.

Phonon Localized sound vibration or particle of sound.

Photon Particle of light.

Pinning Holding in place or restraining the motion of a vortex.

Polymer Long-chain organic molecule formed by joining together many individ-

Proton Positively charged particle in the nucleus of an atom; see atomic number

Radar Acronym for radio detection and ranging.

Rail gun Propulsion in this gun via interaction between current through projectile and magnetic field from current in rails.

Raman spectroscopy Determination of infrared vibrational frequencies bymeasuring frequency shifts of visible light scattered from the sample.

Rare earth One of 14 metallic elements in the periodic table with atomic numbers from 57–70.

Resistivity (ρ) Property of a metal that measures its ability to carry an electriccurrent; unitisohm-centimeter(Ω -cm). Goodconductorshavelowresistivities.

Resistance (R) Property of a wire that measures its dissipation of heat when carrying an electrical current; resistance equals resistivity (ρ)of the metal timesthe length (L) ofthe wire dividedby the cross-sectional area (A ) of the wire; i.e.,R = ρ L/A.

Secondsound Type of sound in a superfluid caused by variations in the densityof normal and superfluid components.

Short-range order Magnetic state in which spin alignment extends over only limited regions of the lattice.

SMES Superconducting magnetic energy storage.

mutually perpendicular sides unequal in length: a b c.

Cooper pair together.

Systematic arrangement of elements in tabular form.

ual small molecules called monomers.

of an atom.

APPENDIX D 189

Sodium chloride (NaCI) Chemical name of common table salt.

Sol gel Method of preparing a superconductor by evaporating a solution contain-ing starting materials to a viscous mass.

Specific heat Quantity of heat energy that must be added to a material to raise its temperatureby 1 oC.

Spin angular momentum Amount of rotational motion associated with an electron or nucleus due to its possession of spin.

Spin lattice relaxation time Time required for a magnetic moment that absorbs radio frequency energy to pass that energy to lattice or molecular vibrations in the surrounding medium.

Spinel Mineral MgAl2O4 or a compound having the same structure.

SQUID Acronym for superconducting quantum interference device; a supercon-ducting loop containing two or more Josephson junctions that responds to very small changes in a magnetic field.

Stoichiometry Ratios of elements in a chemical compound; the compound A2B3

is stoichiometric but A1.9B3.1 is not. Styrofoam Stiff foam-blended plastic made from polystyrene, which is a poly-

mer built up from styrene C8H8 units.

Substrate Nonsuperconducting material on which a superconducting thin film isgrown.

Superconducting gap Separation in energy between a state containing Cooper pairs and a state containing normal unpaired conduction electrons.

Superfluidity

Supersonic Moving more rapidly than the speed of sound.

Ternary compound A compound containing three elements.

Tesla (T)

Tetragonal crystal structure of its sides equal in length (a = b), but not equal to the third dimension c.

Thin film width.

Transition element See Fig. 2.2.

Trapped flux Magnetic flux retained by a superconductor after removing exter-nally applied magnetic field.

Tublene

Twinning

Flow of a liquid without resistance nor heat dissipation.

Unit of magnetic field.

Structure having a rectangular unit cell with two

Superconducting film whose thickness is much less than its length and

Carbon tubes whose walls have a graphite structure.

Growth pattern of an orthorhombic crystal in which some regions have interchanged a and b axes relative to other regions.

190 APPENDIX D

Two-fluid model Interpenetrating fluids of normal electrons and superelectrons in a superconductor and interpenetrating regions of normal fluid and superfluid atoms in a superfluid.

Type I superconductor Superconducting element with low transition tempera-ture Tc and low critical current density Jc values and only one critical field Bc.

Type II superconductor Superconductor that totally expels and excludes mag-netic flux below the lower critical field Bcl but does so only partially in the rangefrom Bcl to the upper critical field Bc2; all superconductors except elements are Type II.

Ultraviolet (uv) Light beyond the visible range, where the wavelength is less and the frequency is greater than that of visible light.

Unit cell Small volume of atoms in a lattice that repeats throughout space to generate a crystal structure.

Van der Waals force Relatively weak force between molecules due to shifts intheir electric charge distributions.

Voltmeter Device for measuring voltage.

Vortex Magnetic flux tube in a Type II superconductor containing one quantum of flux Φo and a rotation tube in a superfluid containing one or more quanta ofcirculation Co.

Vortex lattice Regular arrangement of vortices in a hexagonal pattern.

Vortex fluid Array of vortices capable of motion because it lacks sufficient pinning.

Suggested Reading

CHAPTER 1 Dahl, P. F., History of Superconductivity Chap. 12 (AIP: 1992). Gleick, J., “In the trenches of science,” New York Times Magazine (Aug. 16, 1987). Hazen, R. M., The Breakthrough (Sumter Press, 1988).Simon, R., and A. Smith, Superconductors Chap. 13 (Plenum: New York, 1988). Vidali, G. Superconductivity, the Next Revolution Chaps. 3 and 9 (Cambridge University Press: New

York, 1993).

CHAPTER 2

Davis, H. M., “Low-temperature physics,” Scientific American (June, 1949).Kittel, C., Introduction to Solid-State Physics Chaps. 7 and 8 (Wiley: New York, 1995). Martin, D. H., Magnetism in Solids (MIT: 1967). Tanner, B. K. Introduction to the Physics of Electrons in Solids (Cambridge University Press: New York,

Wannier, G., “Nature of solids,” Scientific American (Dec., 1952). 1995).

CHAPTER 3 Bishop, D. J., P. L. Gammel, and D. A. Huse, “Resistance in high-temperature superconductors,”

Clark, J. “SQUIDs,” Scientific American (Aug., 1994). Davis, H. M., “Low-temperature physics,” Scientific American (June, 1949). Ginsburg, V. L., “Brief history of superconductivity,” Superconductivity 5:1 (1992). Kittel, C., Introduction to Solid-State Physics, Chap. 11 (Wiley: New York, 1995). Kresin, V. Z., and S. A. Wolf, Fundamentals of Superconductiviry, Chap. 1. (Plenum: New York, 1990). Langenberg, D. N., D. J. Scalapino, and B. N. Taylor, “Josephson effect,” Scientific American (June,

Matthias, B. T., “Superconductivity,” Scientific American (Nov., 1957). Parks, R. D., “Quantum effects in superconductors,” Scientific American (Oct., 1965). Rose-Innes, A. C., andE. H. R. Rhoderick, Introduction to Superconductivity, Chaps. 1 and 2 (Pergamon

Scientific American (Feb., 1994).

1966).

Press: Oxford, 1994).

191

192 SUGGESTED READING

CHAPTER 4Lifshitz, E. M. “Superfluidity,” Scientific American (June, 1958).London, F. Superfluids Vol. 2 (Wiley: New York, 1954).Mendelssohn, K., Quest for Absolute Zero Chap. 10. (McGraw-Hill: New York, 1966). Tilley, D. R., and J. Tilley, Superfluidity and Superconductivity (Adam Hilger: Boston, 1986).

CHAPTER 5Cooper, L. N., “Theory of superconductivity,” Am. J. Physics 28:91 (1960). Ginsburg, D. M., “Experimental foundations of BCS theory,” Am. J. Physics 30:433 (1962). Kresin, V. Z., and S. A. Wolf, Fundamentals of Superconductivity Chap. 7 (Plenum: New York, 1990). Rose-Innes, A. C., and E. H. Rhoderick, Introduction to Superconductivity Chap. 9. (Pergamon: 1994).Simon, R., and A. Smith, Superconductors Chaps. 2 and 3 (Plenum: New York, 1988).

CHAPTER 6 Chaikin, P. M., and R. L. Greene, “Superconductivity and magnetism in organic solids,” Physics Today,

Lynton, E. A., Superconductivity Chaps. 1 and 7. (Methuen: London, 1969). Poole, C. P., H. A. Farach, and R. J. Creswick, Superconductivity (Academic: New York, 1995).

(May, 1966).

CHAPTER 7Curl, R. F., and R. E. Smalley, “Fullerenes,” Scientific American (Oct., 1991).Hebard, A. F., “Superconductivity in doped fullerenes,” Physics Today (Nov., 1992).Huffman, D. R., “Solid C60,” Physics Today (Nov., 1991).Smalley, R. E., “Great balls of carbon,” Sciences (Mar./Apr., 1991).

CHAPTER 8Adrian, F. J., and D. O. Cowan, “New superconductors,” Chemical Engineering News (Dec. 21, 1992).Batlogg, B., “Physical properties of high Tc superconductors,” Physics Today (June, 1991). Bishop, D. J., P. L. Gammel, and D. A. Huse, “Resistance in high-temperature superconductors,”

Burns, G., High-Temperature Superconductivity, an Introduction. (Academic: New York, 1992).Cava, R. J., “Superconductors beyond 1-2-3,” Scientific American (Aug., 1990).

Poole, C. P., H. A. Farach, and R. J. Creswick, Superconductivity (Academic: New York, 1995).Poole, C. P., T. Datta, and H. A. Farach, CopperOxide Superconductors. (Wiley: New York, 1988).Properties of High-Temperature Superconductors. MRS Bulletin Vol. 15 (June 15, 1990). Yam, P., “Trends in superconductivity,” Scientific American (Dec., 1993).


CHAPTERClarke, J., “SQUIDs, brains, and gravity waves,” Physics Today

9(Mar., 1986).

Kunzler, J., and M. Tanenbaum, “Superconducting magnets,” Scientific American (June, 1962).

SUGGESTED READING 193

Larbalestier, D., G. Fisk, B. Montgomery, and D. Hawksworth, “High field superconductivity,” Physics

Larbalestier, D., “Critical currents and magnetic applications of high Tc superconductors,” Physics

Pake, G. E., “Magnetic resonance,” Scientific American (Aug., 1958). Sheahen, T. P., Introduction to High-Temperature Superconductivity (Plenum: New York, 1994).Wherli, F. W., “Origins and Future of Nuclear Magnetic Resonance Imaging,” Physics Today (June,

Wolsky, A. M., R. F. Giese, E. J. Daniels, “New superconductors, prospects for applications,” Scientific

Today (Mar., 1986).

Today (June, 1991).

1992).

American (Feb., 1989).

CHAPTER 10 Applications of High-Temperature Superconductivity. MRS Bulletin, vol. 17 (Aug., 1992). Larbalestier, D., “Critical currents and magnetic applications of high Tc superconductors,” Physics

Sheahen, T. P., Introduction to High-Temperature Superconductivity (Plenum: New York, 1995).Simon, R., “High Tc Thin Films and Electronic Devices,” Physics Today (June, 1991).Wilson, M. N., Superconducting Magnets (Clarendon: Oxford, UK, 1983).

Today (June, 1991).

CHAPTER 11

Geballe, T. H., and J. K. Hulm, “Superconductors in electrical power technology,” Scientific American

High-Temperature Superconductivity in Perspective. US Congress, Office of Technology Assessment

Kolm, H. J., and R. D. Thornton, “Electromagnetic flight,” Scientific American (Oct., 1973). Matisoo, J., “Superconducting Computer,” Scientific American (May, 1980).Sheahen, T. P., Introduction to High-Temperature Superconductivity (Plenum: New York, 1994).Stix, G., “Air trains,’’ Scientific American (Aug., 1982). Wolsky, A. M., R. F. Giese, and E. J. Daniels, “New superconductors, prospects for applications,”

(Nov., 1980).

OTA-E-440. GPO, (Apr., 1990).


CHAPTER 12 Geballe, T. H., “Paths to higher temperature superconductors,” Science 259: 1550 (1993). Geballe, T. H., “Searching for superconductivity above the present limit,” Physica C 209:13 (1993). Geballe, T. H., “Why we don’t have a room-temperature superconductor—yet,” Materials Research

Ginsburg, V. L., and D. A. Kirzhnits, eds., High-Temperature Superconductivity Consultants Bureau,

Kivelson, S. A., and Emery, V. J., “Strategies for finding superconductivity in conducting polymers,”

Little, W. A., “Superconductivity at room temperature,” Feb. (1965).

Society Bulletin (July, 1992).

New York (English translation, 1982; Russian edition, 1977).

Synthetic Metals 65:249 (1994).

Index

A-I5 Bandcompound, 65, 71, 72, 79 magnet, 125 energy, half full, 13notation, 78 table, 80 Barium, 76unit cell, 79

Abrikosov. 37 lattice, 36, 38

Abrikosov. A., 36 phonon, 64Absolute zero, 23Alignment. grain, 107, 141, 142Alkali, 11 Bearing

Alkaline earth, 11Allen, J. F., 49Alloy, 72, 76

binary, 76, 79 NbSn. 97NbTi, 78. 125NbV, 76, 77 NbZr, 76,77TcMo, 78 Binder, 140

Aluminum, 7, 26, 27 organic,141superconducting, 148, 149

Ammeter, 41,42, 67 116, 119Amphoteric, 11AND gate, 152 Anisotropy, 82, 107 Antiferromagnetic, 20, 2 1, 1I9

borocarbide, 84 valence state, 83 lanthanum compound, 106 ordering, 82, 175

energy, 13, 61

Bardeen, J., 2, 56, 58

Batlogg, B., 4Bcc, body centered cubic, 75BCS, 60, 63, 64, I65

phonon mediated, 91, 92theory, 2, 56

ball, 156, 157frictionless, 156, 157superconducting, 156, I57

doped fullerene, 89, 90

Beauvais, L., 116Bednorz, J. G., 2–4, 83, 111BEDT-TTF charge transfer organic. 82 Benzene, 91.94Beryllium, 9, 11Bierstedt, P. E., 83

Binding layer. 99-101, 103, 104, 109, 110,

infinite layer phase, 117, 118

compound, 101, 104, 113, 114, 143 compound, magnet, 126

Bismuth, 4, 76

BiSrCaCuO, 101; see also Bismuth, compoundBlood vessel, 132 Boiling line, 44 Bondi, M. A., 149

Apraphul, 15 1Argon,11

195

196

Borocarbide, 72, 84Boron, 11Boronitride, 72, 84Bose gas, 54Bose, S. B., 52Bose–Einstein

condensation, 52 statistics, 52

Boson, 53, 54 slave, 66

Brain, 137 MRI, 132, 133

Buckminsterfullerene, 72, 89; see also Fuller-ene

Buckyball, 71 Bulk synthesis, 170 Bundle of filament wires, 128 Butuzov, S. Y., 168

Cadmium. 27, 58 Caesium, 74, 76 Calcium, 108

between planes, 98 titanate, 83, 84

bonding, 92 dioxide, 23

Carbon, 11, 58, 87-89, 91, 92

forms of, 88 Casimir, H. B. G., 55 Celsius, 22 Centigrade, 22; see also CelsiusCentrifugal force, 50Ceramic, 108, 109, 140Cerium, 108Chain, Cu-O, 172 Chalcogen, 11, 83Chalcogenide, 83

Charge reservoir layer, 100, I75Chaudhari, P., 1Chevrel phase, 72, 80

crystal structure, 81 Chip, 156 Chlorine, 13, 19, 159 Chromium, 21, 76Chu, C. W. (Paul), 3, 4, 116, 168 Circulation, 5 1

quantized, 52 Coherence length, 55, 107Cold rolling, 141 Cold working, 128

Chevrel phases, 83

INDEX

Compass, 129Computer, 15 1-153

PC, 154superconducting, 154, 155

Conductionhole, 106 layer, 99, 100, 102-104, 109, 110, 1I9layer, infinite layer phase, 117, 118plane, 175

Cooper pair, 54, 58, 60, 61, 63, 65, 66. 87, 148

hole type, 104, 106Cooper, L. N., 2. 56, 58 Copper, 7, 1 1, 12

crystal structure, 14 oxide, 7 oxide planes, 98-100valence change, 105, 107, 112

Cornell, E. A., 54Coulomb, 19 Counterion, 82 Critical current, 31, 32

density, 40 in magnetic field, 142 in wire, 141

measurement, 41 table, 27 upper, 4 1, 124, 125

Critical field, 31, 39, 40, 65, 107

Critical surface, 40, 41, 98 Crow, J. W., 154Cryocooler, 24 Cryogenic

fluid, 4 1liquid, 22

Cryotron, 161 Crystal structure, 13 Cube, 92, 93 Cuprate

layered structure, 98 properties, 1 I9

density, 14, 16 electric, 15, 16persistent, 126

Current

Cytoplasm, 132

Datta, T., 4Decoration, 38 Delay line, 150Delocalization, 93

INDEX

Dewar, 23 figure, 24Sir James, 23

Dewdney, A. R., 152Diamagnetism, 19Diamond, 7, 88, 89Digit, 153 Dimensional, 119 Dirac, P. A. M., 52DNA, 129 Dodecahedron, 92, 93 Doping, hole, 104 Ductility, 78

Earth’s magnetic field, 18 Einstein, A., 52Electric

current, 14 field, 16-18power, 134

Electromagnetic propulsion, 159, 160 Electron

delocalized, 93 donor, 82 heavy, 81 hopping, 105 motion, 105 pair, 61 superconductor, 108

Electron–phonon interaction, 56, 57, 66, 87 Energy

allowed, 8 dissipation, 144 gap, 149, 172 level, 8, 53 storage, 134 storage by magnet, 129

Epitaxial film, 107, 146 Exciton, 66Extinction, 85 Extrusion, 140

Fahrenheit, 22 Faraday’s law, 157 Fcc, face centered cubic, 75 Fermi

level, 61 liquid, 54 surface nesting, 66

Fermi, E., 52Fermi–Dirac, 52

197

Fermion, 53heavy, 81

Ferrimagnetic, 20, 21Ferrite, 21Ferromagnet, 122Ferromagnetic, 20, 21Ferromagnetism, 20Filament

wire, 128 Film, 139, 140, 143, 144

epitaxial, 107, 146thin, 144, 145

First sound, 49, 50 Flow with zero resistance, 52 Fluorine, 11, 169, 170Fluoroscopy, 134 Flux

confinement, 34 creep, 144 flow, 147 flow, eliminating, 143, 144liquid, 37-39, 143, 144 loop, 159 magnetic, 34 movement, 39, 40 movement, eliminating, 143, 144quantum, 50 slippage, 127 solid, 37-40trapped, 34, 35, 154, 155

Fluxoid, 19Fluxon, 56 Force

centrifugal, 50 in magnet, 127 Lorentz, 39

Fountain effect, 49Fullerene, 85, 89

alkali doped, 89, 93 isotopic shift, 58 octahedral site, 89 structure, 89 tetrahedral site, 89 with 76 carbon atoms, 93

Future, 165

Gallium, 27 Gao, L., 116

band, 62 superconducting, 62

Gap

198

Garfunkel, M. P., 149Gate, 152Geballe, T. H., 168, 175 Generator, 159, 162Geodesic dome, 88, 89Germanium, 76Gillson, J. L., 83Ginzburg, V. L., 55, 167GL theory, 55Gold, 11, 12 Gor’kov, L. P., 56Gorter, C., 55Grain, 139, 140, 142

Graphite, 85, 88

Halide,11Hawser, K. H., 132, 133Hcp, hexagonal close packed, 75Heavy electron, 71

compound, 8 1superconductor, 72, 82

alignment, 107, 141, 142

Heavy Fermion compound, 81Hebard, A. F., 92Helium, 74

circulation, 5 1 element, 9, 11fountain effect, 49I and II, 45isotopes, 44, 54 lambda point, 46liquefaction, 25liquid, 23, 25rotating superfluid, 50specific heat, 46superfluid, 43, 44, 47, 54superfluid phase diagram, 44surface film, 46viscosity, 45, 48

Hermann, A. M., 4, 113Hexagon, 90, 95 HgBaCaCuO. 101 ; see also Mercury, com-

High temperature superconductors, 97Hole

pound

conduction, 106 hopping, 105, 119motion, 105

Homopolar generator, 159Hopfengartner. B., 146Hopping hole, 107, 119

INDEX

Hot spot, 127, 128Huang, Z. J., I16Huffman, D. R., 85,86Hullet, R. G., 54Hydrogen, 8 , 9 , 11

energy levels, 9NMR, 129shell, 12

Icosahedron, 92, 93Iijima, S., 96Implantation, 170Indium, 27Infinite layerphase, 116, 117

Infrared, 62, 147Intercalation, 91Interference, 69Intergranular, 141 Interstellar dust, 85, 86Invertor, 135Iron, 2 I Isotope, 57, 87

effect, 55, 56effect fullerene, 91, 92effect table, 58 mercury, 57

crystal structure, 118

Izumov, Yu. A., 77, 8 I

Jones, H., 49Josephson

effect, 66junction, 67, 68, 136, 154, 156

Josephson, B. D., 66

Kalbitzer, H. R., 132, 133Kapitsa, P. L., 44Kelvin, 22Kizhnits, D. A., 167Kitazawa, K., 3Kratschmer, W., 85, 86Kroto, H., 87Krypton, 74 Kunz, J. E., 125Kurmaev, E. Z., 77, 81

Ladder phase, 172 Lambda

line, 44point, 46

INDEX

Landau, L., 55Lanthanum compound, 101, 104, 107

phase diagram, 106structure, 109

Larmor frequency, 129, 131Laser, 142

ablation, 145Laves phase, 72, 80Layering scheme, 102 Lead, 7, 27, 58, 74, 16

substitution, 115Levitation, 30, 156–158, I62

train, 158 Lifshitz, E. M., 48Lin, J. G., 116Liquid

Fermi, 54interpenetrating, 55quantum, 54

Lithium, 9, 11, 54, 74, 76Little, W. A., I65Liver, 132 Load, 135London

equations, 29model, 55penetration depth, 29, 55

London, F., 28, 55London, H., 28, 55 Lorentz, force, figure, 39

Maeda, H., 113Maglev, 157Magnet

air core, 122atomic, 19conventional, 121highest field, 124 iron core, 121–123levitation, 30MRI, 129saturation, 123 SMES, 129superconducting, 33, 124, 125, 142 wire, 78, 98

Magneticdecoration, 38flux, 19 flux quantum, I9phase diagram, 38, 39

199

Magnetic field, 17, 18channeling, 37of earth, 18high, 98of refrigerator magnet, 19

Magnetic Resonance Imaging, 121, 130; see

Magnetism, 17 Magnetometer, 29, 121, 136

SQUID, 69Malleable, 142Manganese, 20, 21

also MRI

antiferromagnetism, 2 1oxide, 21

effective, 82 spectrometer, 87, 88

Matthias value, 77Matthias, B. T., 76, 166, 167Maxwell’s equations, 28MBE, 169 McMillan, W. L., 65, 66Meisner

Mass

effect, 28, 55effect determination, 30state, 38, 39

Meissner, A., 28Melt textured growth, 141, 142Melting line, 44, 143, 144Memory cell, 154 Menag, R. L., 116Mercury, 4, 7, 26, 27, 58, 76

compound, 101, 104, 114, 115compound, lead substitution, 115compound, pressure dependence, 116compoundstructure, 110isotopes, 57 liquid, 22 transition temperature, 34, 97 zero resistance, 27

Metal-insulator transition, 175Metastability, 170MgO, 11, 150Michel, C., 2, 83Microcrystal, 139, 141Microwave, 148–150

Miedema

Miedema, A. R., 76

energy, 149

empirical rules, 76

200

Millitesla, 18 Molecular beam epitaxy, 169Molybdenum, 58, 75, 77

MRI, 127, 134 carbide MoC, 80

brain, 132, 133 fluoroscopy, I34human head, 133 installation, 130 origin of name, I31

MRI, acronym for Magnetic Resonance Imag-

Muller, K. A., 2–4, 1 1, 83

NaCI, 11, 13crystal structure, 14 NMR, 131

Nanotube, 96Naphthalene, 94 Nb, see Niobium, 2 Mb3Ge, 2, 25, 97 NbTi, 163

ing, 121, 130

magnet, 125, 126 SMES, 136

Neodymium, 108 compound, 101, 104

Neon, 74 Nesbit, Z. B., 57 Neuron, 129, I37 Neutron, 57 Newton, unit of force, 18Newton’s law, 15, 160 Nickel, 84

substitution, 112Niobium, 2, 7, 26, 27, 74–76

transition temperature, 97 Niobium–titanium magnet, 125, 126 Nitrogen, 11

liquid, 1, 23 NMR, 129

spectrometer, 130 spectrum, 13 1

NMR

NMR, acronym for Nuclear Magnetic Reso-

Nuclear Magnetic Resonance, see NMR, 129Nucleus, 11

Ochsenfeld, R., 28 Octahedron, 92, 93

nance, 129

INDEX

Oersted, H. C., 17Ohm, 15, 16 Ohm’s law, 15, 16, 18One dimensional

chains, 173, 174system, 173

Onnes, H. K., 5, 25–27, 124 Optic fiber, 134 Optical extinction, 85 OR gate, 152 Order parameter, 55 Organic

charge transfer superconductor, 82 superconductor, 72

Orthorhombic, 144 Osmium, 58 Outage, 134 Oxygen, 11, 19, 81, 83, 98

atmosphere, 11 3

Pake, G. E., 130 Paramagnetic, 20 Paramagnetism, 20 Pauli exclusion principle, 53 Penetration depth, 32, 107

London, 29 Pentagon, 90, 95 Periodic table, 9

figure, 10 superconductors, 73

Permeability of free space, 32 Perovskite, 83

cubic potassium, 84structure, 84, 109 superconductor, 72, 108

Persistent current, 126 Phase diagram

lanthanum compound, 106magnetic, 31, 38, 39, 142-144

Phenomenological theory, 56 Phonon, 57 Phosphorus NMR, 129 Photon, 53, 86Pile of disks, 48 Pinning, 37, 127, 144

center, 144 in magnet wire, 128

Plaskii, V. I., 168Plasma, 132 Plasticizer, 140 Platonic solid, 92

INDEX

Polarization, 66 Polonium, 81 Polonium, 165 Polyhedron,95Polymer, 140 Polypropylene, 166 Poole, C. P., Jr., 37, 123, 126 Poole, M. A., 45 Potassium, 74

cubic perovskite, 84 doped fullerene. 91

Pressure, 76, 97, 116 high pressure synthesis, 172

Propulsioncoil, 158 electromagnetic, 159, 160

Protein, 129 Proton, 57

NMR, 129, 130 Puckering, I 10, 172

Quantumliquid, 54 mechanics, 54, 67 theory, 59

Quench, 127

Radar, 148 Radiation, 147

shield, 149 Radon, 74 Rail gun, 161 Rare earth, 71Rare gas, 11 Raveau, B., 83 Red giant, 87 Refrigerator,142Resistance, 16

contact, 28 four probe measurement, 28leads, 147 temperature dependence, 40

Resistivity, 7, 15, 16table, 7

Resonant structure, 93, 94 figure, 94

Reynolds, C. A., 57 Rhenium, 58 Ribbon, 14 1, 142

201

Room temperature superconductivity, 165

Rubidium, 54, 74 Ruthenium, 58

Saturation of magnet, 123, I24 Scattering, 60 Schrieffer. J. R., 2. 56, 58, 63 Second sound, 49, 50 Selenium, 80, 83

Chevrel phase, 80 Serin, B., 57, 168 Sheng, Z. Z., 113 Silicon, 76 Silver, 11, 12

tube, 140 Sintering, 140 Slave boson, 66 Sleight, A. W., 83 Smalley, R., 87 SMES, 135, 136, 162

SMES, acronym for Superconducting Mag-

Soccer ball, 85 Sodium, 13, 19, 74, 159

resistance, 16, 17 Sound, first, 49, 50 Sphere, 35 Spin, 19, 129

bag, 66 flip, 133 gap, 172 -lattice relaxation time, 133 -spin relaxation time, 133

guidelinesfor, 175

diagram of system, 135

netic Energy Storage, 135

Spinel, 83 Sputtering, 145, 146Square planar, 110 SQUID,69, 136, 137

magnetometer, 29 superconducting quantum interference de-

vice, 29

columns, 172 infinite layer phase, 118 in lanthanum compound, I06 titanate, 78, 83

Submarine, 137, 160Substrate, 140, 144, 145, 150 Sulphur, 19, 83

Strontium, 76

202 INDEX

Sun, Y. Y., 116Superconducting Magnetic Energy Storage

Transition temperature (cont.)valence electrons, 75

(SMES), 135 Transmission line, 162Superconductivity, explanations of, 55 TTF-TCNQ, 173, 174 Superconductor resistance, 174

classical, 71 Tublene, 95, 96electron, 108 Tumor, 133, 134room temperature, 165 Tungsten, 75, 78 Type I, 33 Twinning, 144Type 11, 33, 65 Two dimensional system, 173

Supercurrent, 59 Two fluid model, 56 establishing, 127 of helium, 47, 55

Superelectron density, 55 of superconductor, 55 Superfluid, 44 Type I superconductor, 33

motion, 45 Type II superconductor, 33 Superfluidity, 43 Superleak, 48, 49Surgeon, 134 Ultraviolet, 85, 86

Susceptibility, 79, 92 Suspension coil, 158 Sverkhprovodimost’, Russian word for super-

Switch, 153, 154, 156 Switchingunit, 135Synthesis, 115

Tanaka, S., 4Tanenbaum, M., 125 Tantalum, 27 carbide, 80 Tape, 140, 142, 143 nitride, 80 TCNQ organic salt, 82 oxide, 80 Technetium, 74, 75,77, 80, 84

Chevrel phase, 80 atom, 16, 17 Temperature lattice, 60

measurement, 22 Viscosity, 43 scale, 22 superfluid helium, 45

Tesla, 18 water, 133Tetrahedron, 92, 93 Thallium, 4, 58, 76

compound, 101, 102, 104, 113, I 14 Thin film, 37 Vortex, 34–36

superconducting, 76 figure, 36 Tilley, D. R., 51 Tilley, J., 5 1 Tin, 27, 58 TIBaCaCuO, 101; see also Thallium, com-

Train, levitated, 158 units, 36Transistor, 20, 21, I54 Transition temperature Wave, 59, 60, 67

layer dependence, 114 -length, 65 table, 72 phase, 63, 64, 68

Unidentified Superconducting Objects (USO),

USO, acronym for Unidentified Supercon-168, 175

conductivity, 167 ducting Object, 168

Vacuum deposition, 145 Valence, 10, 11

Van der Walls force, 91 Vanadium, 27, 76

electron, 71, 73, 75, 76, 79

Vibration

Voltage, 15, 16, 18 Voltmeter, 41, 42 Vonsovskii, S. V., 77, 81

helium, 50, 51 pinning, 37 quantum, 36, 50, 51 repulsion, 37

pound superfluid, 50, 51

INDEX

Wave (cont.)water, 59

Weber, 36 Wilkes, J., 49Wire, 139–141, 143

magnet, 78, 98NbTI, 139, 140, 143 read, 155sensing, 155write, 155

Wolfram, 78Woodstock of Physics, 4 Wright, W. H., 57Wu. M. K., 3, 40, 112

203

Xenon, 74Xue, Y. Y., 116

YBaCuO, 143; see also Yttrium, compoundYttrium, 3, 4, 98

compound, 100, 101, 104, 111, 112

Zhong, Zhao, 4 Zinc, 26, 27, 58

Zirconia, 142 Zirconium, 58, 76

substitution, 112