Electromagnetic Absorption in the Copper Oxide Superconductors

Electromagnetic Absorption in the Copper Oxide Superconductors

SELECTED TOPICS IN SUPERCONDUCTIVITY Series Editor: Stuart Wolf

Naval Research Laboratory Washington, D.C. CASE STUDIES IN SUPERCONDUCTING MAGNETS Design and Operational Issues Yukikazu Iwasa ELECTOMAGNETIC ABSORPTION IN THE COPPER OXIDE SUPERCONDUCTORS Frank J. Owens and Charles P. Poole, Jr. INTRODUCTION TO HIGH-TEMPERATURE SUPERCONDUCTIVITY Thomas P. Sheahen THE NEW SUPERCONDUCTORS Frank J. Owens and Charles P. Poole, Jr. QUANTUM STATISTICAL THEORY OF SUPERCONDUCTIVITY Shigeji Fujita and Salvador Godoy STABILITY OF SUPERCONDUCTORS Lawrence Dresner

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Electromagnetic Absorption in the Copper Oxide Superconductors Frank J. Owens Army Armament Research Engineering and Development Center Picatinny, New Jersey and Hunter College of the City University of New York New York, New York

and

Charles P. Poole, Jr. Institute of Superconductivity University of South Carolina Columbia, South Carolina

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Preface to the Series

Since its discovery in 1911, superconductivity has been one of the most interesting topics in physics. Superconductivity baffled some of the best minds of the 20th century and was finally understood in a microscopic way in 1957 with the landmark Nobel Prize-winning contribution from John Bardeen, Leon Cooper, and Robert Schrieffer. Since the early 1960s there have been many applications of superconductivity including large magnets for medical imaging and high-energy physics, radio-frequency cavities and components for a variety of applications and quantum interference devices for sensitive magnetometers and digital circuits. These last devices are based on the Nobel Prize-winning (Brian) Josephson effect. In 1987, a dream of many scientists was realized with the discovery of superconducting compounds containing copper --oxygen layers that are superconducting above the boiling point of liquid nitrogen. The revolutionary discovery of superconductivity in this class of compounds (the cuprates) won Georg Bednorz and Alex Mueller the Nobel Prize. This series on Selected Topics in Superconductivity will draw on the rich history of both the science and technology of this field. In the next few years we will try to chronicle the development of both the more traditional metallic superconductors as well as the scientific and technological emergence of the cuprate superconductors. The series will contain broad overviews of fundamental topics as well as some very highly focused treatises designed for a specialized audience.

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Preface

In 1987 a major breakthrough occurred in materials science. A new family of materials was discovered that became superconducting above the temperature at which nitrogen gas liquifies, namely, 77 K or –196°C. Within months of the discovery, a wide variety of experimental techniques were brought to bear in order to measure the properties of these materials and to gain an understanding of why they superconduct at such high temperatures. Among the techniques used were electromagnetic absorption in both the normal and the superconducting states. The measurements enabled the determination of a wide variety of properties, and in some instances led to the observation of new effects not seen by other measurements, such as the existence of weak-link microwave absorption at low dc magnetic fields. The number of different properties and the degree of detail that can be obtained from magnetic field- and temperature-dependent studies of electromagnetic absorption are not widely appreciated. For example, these measurements can provide information on the band gap, critical fields, the H–T irreversibility line, the amount of trapped flux, and even information about the symmetry of the wave function of the Cooper pairs. It is possible to use low dc magnetic field-induced absorption of microwaves with derivative detection to verify the presence of superconductivity in a matter of minutes, and the measurements are often more straightforward than others. For example, they do not require the physical contact with the sample that is necessary when using four-probe resistivity to detect superconductivity. Also, there is no limit on the form of the samples required for electromagnetic absorption studies since sintered and granular materials, crystals, and thin films are all equally acceptable. The purpose of this volume is to provide an introduction to electromagnetic absorption measurements in superconductors, with an emphasis on the new superconducting materials, showing the variety of basic properties that can be delineated by such measurements. The volume is not intended to be a detailed review of all vii

viii

PREFACE

the work done in the area, but rather an introduction to the field supplemented by an outline of the theory and discussions of relevant experimental results. The focus is on qualitative aspects and experimental measurements rather than on detailed theoretical considerations so that the reader can obtain a basic understanding and appreciation of the wealth of information provided by electromagnetic absorption measurements, as well as insights into the mechanisms of absorption. Thus the references cited are not meant to be comprehensive lists of work in the field but collections of representative articles. The level of presentation is such that the volume can be used as a supplementary text for a graduate course in solid state physics, materials science, or superconductivity. The book is intended to be self-contained in that it starts with an elementary introduction to superconductivity, with an emphasis on those properties that are germane to understanding electromagnetic absorption of the superconducting state. Then we provide an overview of the properties of the copper oxide and fullerene superconductors, followed by a chapter on experimental techniques and another on electromagnetic absorption in the normal state. Our attention then turns to microwave absorption in a zero magnetic field and in low magnetic fields; this is followed by an explication of the role played by vortex motion. Absorption in the infrared and optical regions is then treated, and the book ends with a discussion of applications.

Acknowledgment One of us (CPP) would like to thank his son Michael for drawing several of the more difficult figures. Frank J. Owens and Charles P. Poole, Jr.

Contents

Chapter 1. The Superconducting State 1.1. Zero Resistance . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1. The Superconducting Gap . . . . . . . . . . . . . . . 1.1.2. Cooper Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. The Meissner Effect . . . . . . . . . . . . . . . . . . . . . . . 1.2.1. Magnetic Field Exclusion . . . . . . . . . . . . . . . 1.2.2. Temperature Dependencies . . . . . . . . . . . . . . . 1.2.3. Applied and Internal Fields . . . . . . . . . . . . . . 1.2.4. Type I and Type II Superconductors . . . . . . . . . . 1.2.5. Quantization of Flux . . . . . . . . . . . . . . . . . . 1.2.6. Vortex Configurations . . . . . . . . . . . . . . . . . 1.2.7. Flux Creep and Flux Flow . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 2 4 7 7 12 14 17 23 24 25 29

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31 31 34 36 38 39 40 42 44 47 49 50 51 54

Chapter 2. The New Superconductors 2.1. The Copper Oxides . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. Lanthanum and Neodymium Superconductors . . . . . . . 2.1.2. The Yttrium Superconductor . . . . . . . . . . . . . . . . . 2.1.3. Bismuth and Thallium Superconductors . . . . . . . . . . . 2.1.4. Mercury Superconductors . . . . . . . . . . . . . . . . . . 2.1.5. Infinite-Layer Phases . . . . . . . . . . . . . . . . . . . . . 2.1.6. Ladder Phases . . . . . . . . . . . . . . . . . . . . . . . . 2.2. General Properties of Copper Oxide Superconductors . . . . . . . . 2.2.1. Commonalities of the Cuprates . . . . . . . . . . . . . . . 2.2.2. Energy Bands . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3. Cooper Pair Binding . . . . . . . . . . . . . . . . . . . . . 2.3. Perovskite Superconductors . . . . . . . . . . . . . . . . . . . . . 2.4. Carbon-60 Superconductors . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

. . . . . . . . . . . . . .

CONTENTS

X

Chapter 3 . Experimental Methods and Complementary Techniques 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. 3.9. 3.10. 3.11.

Radio Frequency Measurements Using LC Resonant Circuits . . . . . . Microwave Measurements Using Cavity Resonators . . . . . . . . . . Electron Paramagnetic Resonance . . . . . . . . . . . . . . . . . . . . Paramagnetic Probes . . . . . . . . . . . . . . . . . . . . . . . . . . . Stripline and Parallel Plate Microwave Resonators . . . . . . . . . . . . Nuclear Magnetic Resonance . . . . . . . . . . . . . . . . . . . . . . . Nuclear Quadrupole Resonance . . . . . . . . . . . . . . . . . . . . . Muon Spin Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . Positron Annihilation . . . . . . . . . . . . . . . . . . . . . . . . . . . Mössbauer Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . Photoemission and X-Ray Absorption . . . . . . . . . . . . . . . . . .

Chapter 4 . Electromagnetic Absorption in the Normal State 4.1. Metallic State . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Electrical Conductivity .. . . . . . . . . . . . . . . . . 4.1.2. Surface Resistance . . . . . . . . . . . . . . . . . . . . 4.1.3. Power Dissipation . . . . . . . . . . . . . . . . . . . . 4.1.4. Temperature Dependencies . . . . . . . . . . . . . . . . 4.2. Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Metal-Insulator Transition . . . . . . . . . . . . . . . . . . . . 4.4. Antiferromagnetic Transition . . . . . . . . . . . . . . . . . . . 4.5. Ferromagnetic Transition . . . . . . . . . . . . . . . . . . . . . 4.6. Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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. . . . . . . . . . .

. . . . . . . . . . .

Chapter 5 . Zero Magnetic Field Microwave Absorption 5.1. Electromagnetic Absorption and the Two-Fluid Model . . . . . . . . . . 5.2. Surface Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Electromagnetic Absorption in the BCS Theory . . . . . . . . . . . . . 5.4. Copper Oxides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1. Penetration Depth Measurements . . . . . . . . . . . . . . . . 5.4.2. Surface Resistance Measurements . . . . . . . . . . . . . . . . 5.4.3. Penetration Depth Measurements as a Probe of s- and d-Wave Symmetry . . . . . . . . . . . . . . . . . . . . . 5.4.4. Electromagnetic Absorption Due to Fluctuations . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 59 62 65 66 68 69 69 70 70 71

75 75 78 79 81 82 83 83 84 89 94

95 97 99 102 102 105 107 107 112

Chapter 6. Low Magnetic Field-Induced Microwave Absorption 6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.2. Properties of Low Magnetic Field Absorption Derivative . . . . . . . . 114

CONTENTS

xi

6.3. Properties of Low-Field Direct Microwave Absorption . . . . . . . 6.4. Origin of Low Magnetic Field Derivative Signal . . . . . . . . . . 6.4.1. Loops and Josephson Junctions . . . . . . . . . . . . . . . 6.4.2. Absorption Mechanism . . . . . . . . . . . . . . . . . . . . 6.5. Magnetic Field Absorptionin Alternating Applied Fields . . . . . . 6.6. Low Magnetic Field Derivative Signal as a Detector of Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

120 125 125 128 132

. . 136 . . 137

Chapter 7 . Electromagnetic Absorption Due to Vortex Motion 7.1. Theory of Electromagnetic Absorption Due to Vortex Dissipation . . 7.1.1. Penetrating Fields . . . . . . . . . . . . . . . . . . . . 7.1.2. Flux Creep, Flux Flow, and Irreversibility . . . . . . . . 7.1.3. Coffey–Clem Model . . . . . . . . . . . . . . . . . . . 7.2. Experimental Results . . . . . . . . . . . . . . . . . . . . . . . 7.2.1. RF Penetration Depth Measurements . . . . . . . . . . 7.2.2. Microwave Bridge Measurements . . . . . . . . . . . . 7.2.3. Strip Line Resonator Measurements . . . . . . . . . . . 7.3. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . References .. . . . . . . . . . . . . . . . . . . . . . . . . . .

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139 139 140 142 145 145 148 153 156 156

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159 160 163 164 165 166 166 169 169 172 174

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175 178 178 181 184 186 188

Chapter 8. Infrared and Optical Absorption 8.1. 8.2. 8.3. 8.4. 8.5. 8.6. 8.7. 8.8. 8.9. 8.10.

Absorption in the Infrared . . . . . . . . . . . . . . . . . . . Detecting Molecular and Crystal Vibrations . . . . . . . . . . Soft Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . Dielectric Constant and Conductivity . . . . . . . . . . . . . Reflectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . Kramers–Kronig Analysis . . . . . . . . . . . . . . . . . . . Drude Expansion . . . . . . . . . . . . . . . . . . . . . . . . Plasma Oscillations . . . . . . . . . . . . . . . . . . . . . . . Energy Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . Absorption at Visible and Ultraviolet Frequencies . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 9 . Applications 9.1. 9.2. 9.3. 9.4. 9.5. 9.6. 9.7.

Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . Delay Lines . . . . . . . . . . . . . . . . . . . . . . . . . . Stripline Resonators . . . . . . . . . . . . . . . . . . . . . . Cavity Resonators . . . . . . . . . . . . . . . . . . . . . . . Transmission Lines . . . . . . . . . . . . . . . . . . . . . . Superconducting Antennae . . . . . . . . . . . . . . . . . . Infrared and Optical Sensors . . . . . . . . . . . . . . . . .

. . . . . . .

xii

CONTENTS

9.8. Magnetometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

1 The Superconducting State

This chapter presents a brief overview of the properties of the superconducting state, with an emphasis on those characteristics that are germane to the absorption of electromagnetic energy in this state.

1.1. ZERO RESISTANCE Let us consider conductivity before we consider superconductivity. In a metal, the valence electrons are relatively weakly bound to the atoms of the lattice, so they detach themselves from these atoms, become delocalized in an energy state called the conduction band, and wander around the lattice. The application of a sufficiently strong electric field E will cause the conduction electrons at the top of the conduction band, a region called the Fermi surface, to move through the lattice. The electric field exerts a force F = –eE on the electrons, where e is the electronic charge, and in the absence of any resistance, the electron velocity should increase continuously. However, this does not happen because the electrons collide with the vibrating atoms as well as with impurity atoms and defects in the lattice, and are scattered out of the path of flow. The result is that the electrons acquire a limiting velocity v, producing a current density J = nev, where n is the volume density of electrons. The metal acquires a resistance R as a result of this scattering of the electrons. Because the atoms are vibrating about their equilibrium positions in the lattice, and at higher temperatures the vibrating atoms have larger amplitudes of oscillation, the probability for electrons to scatter from them increases, and hence the resistance increases with the temperature, as shown in Fig. 1.1 for sodium. The resistance of a metal does not become zero as the temperature is lowered to absolute zero, even though the lattice vibrations freeze out, but instead R approaches a limiting value. This constant resistance at low temperature is called the residual resistance, and is a result of scattering from imperfections and defects in the lattice. In a superconductor, however, there is a transition temperature Tc at 1

2

CHAPTER 1

Figure 1.1. Temperature dependence of the resistance of sodium normalized to its value at 290 K.

which the resistance to direct current flow and low-frequency ac current flow becomes zero. Figure 1.2 is a plot of the dc resistance normalized to its room temperature value versus temperature for the superconductor Hg0.8Pb0.2Ba2Ca2Cu3O8+x, which reaches zero resistance at 130 K, the highest temperature of any superconductor at ambient pressure (1).

1.1.1. The Superconducting Gap Prior to the development of the theory of superconductivity, experiments had shown that the carriers of current in the superconducting state have a charge 2e, which is twice the electron charge. This means that the electrons at the Fermi surface of the metal, which carry the current, are bound in pairs called Cooper pairs (2). The existence of these bound pairs in the superconducting state alters the energy band structure by introducing a gap in energy, with the normal conduction electrons above the gap and the Cooper pairs below. In a metal, the top occupied band is not full, and the Fermi level demarcates the energy of the uppermost filIed state in the band. In the superconducting state, the presence of bound electron pairs implies that the energy gap D is located at the Fermi level. The magnitude of this superconducting gap corresponds to the binding energy of the electron pairs. It is the energy

THE SUPERCONDUCTING STATE

3

Figure 1.2. Temperature dependence of the resistance of the Hg-Pb-Ba-Ca-Cu-O superconductor normalized to its value at 295 K. (After Iqbal et al., Ref. 1.)

difference between the normal electrons and the bound electron pairs at the Fermi level. As we will see later, this gap plays an important role in the absorption of electromagnetic radiation in the superconducting state. If the incident electromagnetic radiation has a photon energy less than the gap, there will be no absorption at 0 K. In fact, studies of the reflection of microwave and infrared (IR) radiation provide evidence for the existence of this energy gap in the superconducting state. A measurement of the frequency dependence of the reflection of infrared light from a superconductor below Tc allows a determination of the energy gap. Figure 1.3 shows a plot of the fraction of light reflected from the elemental superconductor indium at 3.39 K in the superconducting state relative to its normal state reflection (3). The reduction of the reflection of infrared photons at 10 cm–1 starts where the radiation begins to induce excitations across the superconducting gap; in effect, each absorbed IR photon breaks a Cooper pair. The Bardeen–Cooper–Schrieffer (BCS) theory in the weak coupling limit predicts that the band gap Eg (0) = 2D (0) at absolute zero is related to the transition temperature Tc by the expression (4) Eg(0)=2D (0) = 3.52kBTc

(1.1)

4

CHAPTER 1

Figure 1.3. Fraction of electromagnetic energy reflected from indium in the superconducting state as a function of the frequency of the radiation. (After Richards and Tinkham, Ref. 3.)

As the temperature is raised above absolute zero, the superconducting gap decreases in magnitude in the manner illustrated in Fig. 1.4, which presents a plot of Eg(T)/Eg(0) versus reduced temperature T/T c for the element tantalum (5). The temperature dependence of the gap follows an approximate (Tc – T)½ relation, which is the well-known mean field result for the order parameter in a second-order phase transition. This suggests that the superconducting transition is second order.

1.1.2. Cooper Pairs One of the major problems in the development of an understanding of superconductivity is explaining how two negatively charged electrons can be bound into pairs despite the repulsive electrostatic Coulomb force between them. In 1956, a year before the discovery of the BCS theory, L. N. Cooper showed how lattice phonons could produce a binding of two electrons near the Fermi level (2). The effect of isotopic labeling on the transition temperature provided aclue to the nature of the mechanism. Measurements of the transition temperatures of several isotopes of the element mercury had demonstrated that the transition temperature shifts to lower values as the mass of the Hg nucleus increases. More specifically, the experiment showed that the shift of Tc is proportional to (m)–½ where m is the mass of the mercury isotope (6). Since the spring constant k of a vibrating mass is given by k = mw2, the vibrational frequency w = (k/m)½ measured by IR spectroscopy is also inversely proportional to the square root of the atom’s mass. This result provided a critical piece of evidence supporting the role of lattice phonons in

THE SUPERCONDUCTING STATE

5

Figure 1.4. Temperature dependence of the reduced energy gap Eg(T)/Eg(0) versus reduced temperature T/T c for superconducting tantalum. (After Townsend and Sutton, Ref. 5).

superconductivity. The BCS theory, which appeared in 1957, proposed that the Cooper pair binding arises from a phonon coupling mechanism. A classical (i.e., nonquantum mechanical) description can be used to to obtain some insight into how lattice phonons can cause binding of the electrons into pairs. Because the valence electrons have detached themselves from atoms to move freely through the lattice, the atoms of the metal have acquired a positive charge. When the conduction electrons move past these positively charged atoms, the atoms are attracted to the electrons and there is a slight shift in the positions of the atoms toward the passing electrons. This situation is illustrated in Fig. 1.5. This distorted region is slightly more positively charged than the rest of the lattice and it follows the electron as it moves through the lattice. This more positive region may attract a distant electron and cause it to follow the distortion as it moves through the lattice, in effect forming a bound electron pair. An alternative way to view the process is as an exchange of phonons between two electrons, with one electron emitting a phonon which is then absorbed by another electron. We say that the interaction between the electrons is transmitted by the phonon. The process is represented diagrammatically in Fig. 1.6. The binding energy of the two electrons is on the order of 10–4 eV, and the separation of the electrons is about 10 3 Å, which is about 300 lattice spaces. Thus the quantum mechanical wavelength of the Cooper pairs is much longer than the diameters and

6

CHAPTER 1

Figure 1.5. Illustration of how a conduction electron moving through a lattice distorts the lattice along its path.

spacings of the atoms of the solid. As a result, the Cooper pairs do not “see” the atoms of the lattice and are not scattered by them. The spins of the electrons of the pair are oppositely aligned, so a bound Cooper pair has zero spin and is a boson. This means that at absolute zero all Cooper pairs will be in the ground state and have the same energy and therefore the same wavelength. Thus not only is the wavelength of the pairs very long, but all the pairs have the same wavelength. Further, it turns out that the phase of the wave of every pair is the same as that of any other pair. Thus the Copper pairs have a phase coherence analogous

Figure 1.6. Illustration of the phonon exchange process between two electrons which binds them into a Cooper pair.

THESUPERCONDUCTINGSTATE

7

to the waves of light produced by a laser. In other words, the motion of the pairs in the lattice is correlated. It is this remarkable property of the quantum mechanical wave describing the Copper pairs that accounts for their movement through the lattice without scattering and the resulting zero resistance of the superconducting state. The BCS theory explains how these circumstances reduce the energy of the super electrons below the energy gap.

1.2. THE MElSSNER EFFECT 1.2.1. Magnetic Field Exclusion The second major characteristic of the superconducting state, in addition to that of zero resistance, is called the Meissner effect. If a superconducting material is cooled below its transition temperature in an applied magnetic field B0 B0 = µ 0H0

(1.2)

where µ0 is the permeability of free space and H0 is the magnetic intensity; the magnetic flux density Bin within the bulk of the material will be expelled below the transition temperature Tc (7). This behavior is most commonly observed by measuring the temperature dependence of either the magnetization M or the dimensionless susceptibility χ

χ = M/H

in

(1.3)

of the sample. These various quantities have the following relationships inside the superconductor (1.4) (1.5) where the meter-kilogram-second (mks) system of units is used [in the centimetergram-second (cgs) system µ 0 = 1 and x is replaced by 4pχ]. For a perfect superconductor, the internal field Bin = 0 and the dimensionless susceptibility χ = -1.This means that we have for the magnetization or magnetic moment per unit volume M = –Hin. The material, in effect, behaves like a perfect diamagnet. Figure 1.7 shows the results of a measurement of the temperature dependence of the magnetization for a single crystal of Y-Ba-Cu-O. The effect occurs because the applied magnetic field B0 causes the surface current density J shown in Fig. 1.8 to flow in the proper direction to produce a dc magnetic field that cancels the internal field and makes Bin = 0 inside the bulk of the superconducting sphere.

8

CHAPTER 1

Figure 1.7. Temperature dependence of the magnetization M of a single crystal of a Y-Ba-Cu-O superconductor.

Although magnetic flux is excluded from the bulk, it can penetrate the surface layers of the superconductor. Fritz London (8) used the two-fluid model of superconductivity and Maxwell’s equations to explain the Meissner effect and flux penetration into surface layers. The two-fluid model envisions the superconducting state as having a mixture of normal electrons and superconducting electrons, with

Figure 1.8. Shielding current flowing around the surface of a superconducting sphere in an applied dc magnetic field.

THE SUPERCONDUCTING STATE

9

the latter fraction increasing as the temperature is lowered in the range below Tc. In order to describe the Meissner effect, London postulated that for a superconductor, (1.6) where m is the effective mass of the electron, ns is the volume density of Cooper pairs, and e is the electron charge. Since curl J = 0 inside a superconductor, this relation (1.6) ensures that there will be no magnetic field inside the bulk of the material. If we make use of the inhomogeneous Maxwell curl equation with the electric field term ∂D/∂ t set equal to zero curl B = µ 0J

(1.7)

we can take the curl of this expression and use it to eliminate curl J from Eqs. (1.6) and (1.7) to obtain the Helmholtz differential equation ∇2 B = B/l2

(1.8)

where B denotes the internal field Bin and l, called the London penetration depth, (1.9) is a measure of the extent of the penetration of the magnetic field B into the material. The values of l for the elemental superconductors are typically in the range of 10–6 cm (9). We can also take the curl of Eq. (1.6) and eliminate curl B from Eqs. (1.6) and (1.7) to obtain the Helmholtz equation for the current density (1.10) The one-dimensional solutions of these two equations for a direction x perpendicular to the surface (1.11) (1.12) show that that the magnetic field B and the current density J are confined to a thin surface layer of thickness λ. Figure 1.9a shows how an external field B0 enters the superconductor at the surface and then decays exponentially through the surface layer to zero far inside. For this geometry Eq. (1.11) becomes (1.13)

10

CHAPTER 1

Figure 1.9. (a) Experimental decay of a dc magnetic field inside a Type I superconductor for the case l << a, and (b) the current density inside a superconductor for the same case. The figure is drawn for the thick sample case l < a.

where l << a. The presence of this field B induces a current density J through the Maxwell curl relation (1.7). The applied field B0 and the internal field B(x) are both in the z direction, so from Eq. (1.7) we find that J(x) flows in the y direction with the value (1.14) which depends on the slope dB/dx of the internal magnetic field. The equation for J (x) analogous to Eq. (1.13) is (1.15)

THE SUPERCONDUCTING STATE

11

and this current density also decays to zero in the surface layer, as shown in Fig. 1.9b. Note how the current density reverses direction on the two sides of the material. This occurs because the derivative dB/dx of Eq. (1 .14) reverses sign. To provide a physical picture for the relationship between B (x) and J(x) in Eqs. (1.13) and (1.15) consider a superconducting slab of length L, width L, and thickness 2a placed in a uniform magnetic field B0 with the orientation shown in Fig. 1 .10. The applied field B0 and the internal field B(x) are both in the z direction, and Figs. 1.9a and 1.9b are slices through the slab in the z-x plane at the position y = 0. We see from the figure that the current density encircles the figure in planes perpendicular to the applied field direction. The current density flows in opposite directions on opposite sides of the slab. We have been discussing the usual case in which the sample thickness a is large with respect to the penetration depth l. Films of superconductors are often made thin enough so that the thickness is less than l, with the result that external fields penetrate even for a Type I superconductor. Figure 1.11 shows the current density Jy(x) inside a thin film for which a << l.

Figure 1.10. Flat superconducting slab with thickness 2a much less than the penetration depth l.

12

CHAPTER 1

Figure 1.11.Current density Jy (x) inside a superconductor for a << l L. Note that for this case the magnitude of Jy (x) decreases linearly with x.

1.2.2. Temperature Dependencies The density of superconducting electrons ns has the temperature dependence sketched in Fig. 1.12 (1.16) and inserting this expression into Eq. (1.9) provides the temperature dependence for the penetration depth l shown in Fig. 1.1 3 (1.17) where l0 = l(0) is the penetration depth at absolute zero given by Eq. (1.9) with ns = ns(T). Besides l there is another important length parameter called the coherence length x, which is a characteristic of the superconducting state. The boundary between a superconductor in contact with a normal metal is not sharp because the density of superconducting pairs increases in a gradual manner from the surface into the bulk of the superconductor. The coherence length x is a measure of the distance over which this density of the Cooper pairs changes to its final value. We will see in the next section that the relative magnitudes of x and l determine the behavior of a superconductor in a magnetic field.

THE SUPERCONDUCTING STATE

Figure 1.12. Temperature dependence of the density ns (T) of superconducting electrons in the superconducting state below Tc.

Figure 1.13.Temperature dependence of the penetration depth l(T) below Tc.

13

14

CHAPTER 1

1.2.3. Applied and Internal Fields Equation (1.5) shows how the internal fields Bin and Hin in a material are related to each other through the magnetization M and the susceptibility x of the material. The question arises as to how these fields are related to the applied field B0 =µ 0H0 for a negative susceptibility in the range -1< x < 0. We will examine this for the case of a sample in the shape of an ellipsoid with an applied magnetic field oriented either along its symmetry axis or perpendicular to this axis. In both cases the internal fields in the ellipsoid given by Eq. (1.5) are parallel to the applied field. For a general sample shape, or for an ellipsoid in an applied field with an arbitrary orientation, the internal fields can have different directions than the applied field, and they can also vary in strength from place to place in the sample. It is instructive to treat ellipsoids because a long cylindrical sample can be approximated by an elongated prolate ellipsoid, and a thin film of circular cross section can be approximated by a flat oblate ellipsoid. In the ellipsoid under discussion, the internal magnetic fields Bin and Hin are related to the applied fields B0 =µ 0H0 through a demagnetization factor N in accordance with the two expressions (1.18a) (1.18b) where N is a dimensionless geometrical coefficient with the range of values 0 £ N £ 1. This factor determines the extent to which the magnetization of the sample contributes to the internal magnetic fields Bin and Hin. The parameter N is a geometrical coefficient because it depends on the ratio c/a of the ellipsoid's semimajor axis c along the symmetry direction to its semimajor axis a perpendicular to this direction. The demagnetization factors N|| and N^ for the applied field B0 aligned, respectively, along directions parallel and perpendicular to the symmetry axis satisfy the normalization condition (1.19) Figure 1.14 shows the dependence of the demagnetization factors on this c/a ratio, where c > a for a prolate ellipsoid, and c < a for an oblate ellipsoid. A general ellipsoid has three principal directions and three demagnetization factors with the condition nx + ny + nz = 1. Eliminating the magnetization M between Eqs. (1.18a) and (1.18b) gives the expression (1.20)

THE SUPERCONDUCTING STATE

15

Figure 1.14. Dependence of demagnetization factors parallel (N ||) and perpendicular (N ^) to the symmetry axis on c/a ratio for a prolate (c > a) and an oblate (c < a) ellipsoid.

where N can be either the parallel or the perpendicular demagnetization factor. With the aid of Eqs. (1.3) to (1 .5) we can write the internal fields in terms of the applied field (1.2), the susceptibility and the demagnetization factor (1.21a)

(1.21b)

(1.21c) where we recall that B0 = µ 0H0 for the applied field. The B field has the property of having lines that are continuous during the passage from one medium to the next, whereas the H field lines can be discontinuous. We see from Fig. 1.15 that these B lines are continuous. The prolate, oblate, and spherical shapes all expel some flux, but for the applied field along the symmetry axis c, the prolate one does so to a greater extent and the oblate one to a lesser extent than the sphere for which N = 1/3. It is clear from Fig. 1.15 that the B field far from the ellipsoid is uniform with the value B0 and inside the ellipsoid is also uniform with the value given by Eq.

16

CHAPTER 1

Figure 1.15.Magnetic flux through a prolate ellipsoid, a sphere, and an oblate ellipsoid in a uniform magnetic field.

(1.21a). For the special case of a sphere for which N = 1/3, this inside field has the value (1.22) The field outside the sphere B out is the vector superposition of the applied field B0 and a dipole field Bdipole arising from the magnetic moment µ = MV of the sphere (1.23) The radial Br and azimuthal Bθ components of this outside field depend on the angle q and fall off with distance as 1 /r 3 in the following manner (10)

THE SUPERCONDUCTING STATE

17

(1.24a)

(1.24b)

where q is the angle with respect to the applied field direction.

1.2.4. Type I and Type II Superconductors

We have been discussing “perfect” superconductors for which χ = -1. More generally there are two types of superconductors characterized by the GinsburgLandau parameter k = l/x ––

(1.25)

––

with k < 1/√ 2 forType I andk > 1/√ 2 for Type II superconductors. The two types are also differentiated by their behavior in a dc magnetic field. In Type I, as the external magnetic flux density Bapp is increased, it does not penetrate the bulk of the material in the superconducting state until a field is reached called the criticalfield Bc , above which the superconducting state no longer exists. The situation is illustrated in Fig. 1.16. In a Type II superconductor there is no magnetic field Bin inside the material until a field Bc1 is reached, referred to as the lower critical field. At Bc1 flux density begins to penetrate the sample but does not remove all of the superconducting state. As the field is further increased to a value Bc2 referred to as the upper criticalfield, more flux density continues to penetrate the sample. At Bc 2 the superconducting state is totally removed. The copper oxide superconductors, which can be superconducting as high as 133 K, are Type II superconductors. For a Type II superconductor Bc2 can be quite large. Upper critical fields of several hundred tesla (T) have been reported for high-temperature superconductors (see, e.g., Ref. 11). In addition to the lower and upper critical fields, a Type II superconductor has what is called a thermodynamic critical field Bc given by (1.26) where k is defined by Eq. (1.25). The upper critical magnetic field Bc2 is temperature dependent and often has a dependence that can be described by: (1.27) where B0 is the critical magnetic field at absolute zero. This temperature dependence is sketched in Fig. 1.17. The slope of the line at Tc has the typical value

18

CHAPTER 1

TYPE I SUPERCONDUCTOR

TYPE II SUPERCONDUCTOR

Figure 1. 16. Plot of magnetic field Bin inside a Type I and a Type II superconductor as a function of external applied magnetic field.

(1.28a)

as indicated by the dashed line in the figure. If we equate Bc2 to the Pauli limiting field, i.e., the magnetic field BPauli –– defined by equating the magnetic energy to the superconducting gap energy (2√2µ BBPauli = Eg) and make use of the BCS gap ratio Eg = 3.53 kBTc, we obtain the slope of -1.83T/K given by Eq. (1.28a), a value close to that observed with many cuprates. In practice this critical field slope is anisotropic, with typical cuprate single crystals exhibiting slopes less than -1.83T/K for the applied field aligned along the c direction and greater than this value for B0 aligned in the ab plane. A typical cuprate value for the lower critical field slope at Tc is (1.28b) and this can also exhibit anisotropies. Type I superconductors have critical field slopes dBc/dT of between –15 and –50 mT/K, which is near the geometric mean

THE SUPERCONDUCTING STATE

19

Figure 1.17.Temperature dependence of critical magnetic field Bc(T) relative to its value Bc(0) at 0 K.

of the slopes ofEqs. (1.28a) and (1.28b). For the elemental superconductor lead we have, for example, dBc /dT = –22.3 mT/K. The existence of a critical magnetic field that removes the superconducting state implies that there is an upper limit to the current density that can be carried by the superconductor. This is a direct consequence of the fact that a current produces a magnetic field. The critical current density Jc and the thermodynamic critical field Bc are related by (1.29) where µ 0 is the permeability. This result for Jc is close to the depairing current density (1.30) which is the current density that causes Cooper pairs to break up. The temperature dependence of the critical current density is obtained from Eq. (1.29) with the aid of Eqs. (1.17) and (1.27) (1.31)

20

CHAPTER 1

and this is plotted in Fig. 1.18. If a magnetic field Bapp having a magnitude between Bc1 and Bc2 is applied to a Type II material in the superconducting state and then reduced to zero, some magnetic flux remains trapped inside the superconductor. The dashed line in Fig. 1.16b intersects the ordinate axis Bin at the value of this trapped field. In effect the superconductor remembers it was exposed to a magnetic field and if held below Tc behaves like a magnet. For applied fields below Bc1, a Type II superconductor totally excludes magnetic flux and is said to be in the Meissner state, while for fields between Bc1 and Bc2 there is partial flux penetration and it is said to be in the mixed state. In the mixed state there are thin tube like normal regions in which the magnetic flux is present. A somewhat overly simplified picture of these thin threads of flux, called vortices, is illustrated in Fig. 1.19. This figure is drawn for the special case of a thin film in which the film thickness is comparable to the vortex diameter. With a bulk superconductor, the vortex length far exceeds its diameter. It is instructive to consider some characteristics of the region near the surface of a Type II superconductor. The density of superconducting electrons or Cooper pairs ns(x) is zero at the surface (x = 0) and gradually increases as one moves inward until it reaches the bulk value ns (∞) far inside, as shown in Fig. 1.20a. The distance inward from the surface where ns(x) becomes appreciable in magnitude is called the coherence length x, and this fundamental length parameter is indicated in the

Figure 1.18. Temperature dependence of the critical current density Jc (T) relative to its value J c (0) at 0 K.

THE SUPERCONDUCTING STATE

Figure 1.19. Illustration of the passage of magnetic field lines B through vortices in the mixed state of a Type II superconducting thin film.

figure. If there is a magnetic field B0 in the region outside the superconductor, then the field inside B (x) decreases with distance until it becomes zero far inside. The fundamental length parameter l for the decay of B(x) inside from the surface is called the penetration depth, and it is also indicated in Fig. 1.20a. This figure is drawn for a Type I superconductor for which the Ginzburg–Landau parameter –– –– k = l/x has the value k < 1/√ 2 The case for a Type II superconductor k > 1 / √ 2 is sketched in Fig. 1.20b, and for the cuprates we have k ≈ 100. In order to understand why some superconductors are Type II and some are Type I, let us consider the free energy at the interface between a normal metal and a superconductor. In a zero dc magnetic field, the Gibbs free energy per unit volume of a material in the superconducting state Gs(T,0) is lower than that of the normal state Gn(T,0) because of the ordering of the electrons into Cooper pairs, When a magnetic field B0 is applied, the free energy per unit volume is increased by an amount B20/2µ0 because the magnetization of the superconductor is opposite to the direction of the applied dc magnetic field. However, at the interface between a superconductor and a normal metal, the system must be in equilibrium, and the free energies per unit area G* must be equal on both sides of the interface, i.e., G*n = G*s. The surface energy per unit area due to the ordering of the electrons into pairs can be approximated as B2cx/2µ0 and the corresponding magnetic surface

21

22

CHAPTER 1

NORMAL

SUPERCONDUCTING

a)

Figure 1.20. (a) Dependence of the magnetic field B and the density of superconducting electrons n s on the distance x inside a Type II superconductor at a normal metal/superconductor interface, and (b) change in the magnetic and electron ordering contributions to the Gibbs free energy G at the same interface.

energy per unit area is approximately B2cl/2µ0, to give for the net free energy per unit area at the surface (1.32) The relative magnitudes of x and l will determine whether the surface energy is positive or negative. In a Type I superconductor, x is greater than l and the surface energy is positive. The formation of normal regions in the bulk of the superconductor would increase the normal superconducting surface and hence further increase the free energy, which would not be energetically favorable. When l is greater than x, the surface energy is negative, and the appearance of normal regions in the bulk of the superconductor would reduce the free energy, making it energetically favorable to form the tubular normal regions (vortices) through which magnetic flux can thread.

THE SUPERCONDUCTING STATE

23

1.2.5. Quantization of Flux In the mixed state, the flux penetrating the material in each vortex has the value F0 given by h/2e = 2.07 × 10–15 weber, where a weber is a tesla (meter)2. In other words, the flux in the mixed state is quantized. Increasing the applied magnetic field increases the number of vortices in the superconductor, and this causes the flux density Bin to increase in discrete increments determined by the value of F0. The quantization of flux is a direct result of the coherence of the wave functions describing the Cooper pairs. To obtain the expression for F0 mentioned above, consider a small superconducting ring of radius R cooled to the superconducting state and carrying a supercurrent of density J. Since the wave function of every Cooper pair is in phase with that of every other pair, an exact integral multiple N of the wavelength l of the pair is required to fit around the circumference 2pR of the ring; otherwise the waves would not be coherent. In effect, the current in the ring is quantized with the quantization condition (1.33) Since l = h/P where P is the momentum of a Cooper pair, we have in effect a Bohr like momentum quantization condition for the current in the ring. (1.34) The energy of a current flowing in a loop can be written in terms of the current I and the flux F through the loop, (1.35) Since the current I for n electrons moving around the loop with velocity v is nve/2pR, where e is the electron charge, Eq. (1.35) becomes (1.36) The energy of n electrons moving around the ring can also be written (1.37) where m is mass of the electron, P = 2mv is the momentum of a Cooper pair which contains two electrons. Comparing Eqs. (1.36) and (1.37) gives the momentum P (1.38) Substituting this into Eq. (1.34) and using the flux quantization condition F = NF0 provides the expression for the unit quantum of flux.

24

CHAPTER 1

(1.39) which has the numerical value given at the beginning of this section.

1.2.6. Vortex Configurations A vortex has a core of normal material of radius x where the magnetic field is the strongest; outside this core the magnetic field decreases with distance r from the core. This surrounding field is appreciable in magnitude for r < l, and drops to a very small value for r>> l, in accordance with Fig. 1.21. We see from Fig. 9.13 of Ref. 12 that for the cuprates, which have k >> 1 and hence l >> x, almost all of the magnetic flux lies outside the core. As a result the penetration depth l may be looked upon as the effective radius of a vortex. More quantitatively, the magnetic field strength outside the core falls off with distance in accordance with the expression (1.40) where K0(r/l) is the zero-order modified Bessel function. At large distances, r >> l; the function K0(x) has the asymptotic behavior e-x/(2x/p)1/2; and this gives for the magnetic field far outside the vortex core (1.41)

Figure 1.21. Distance dependence of the magnetic field B(r) around an individual vortex, showing how the core has the radius x and the flux distribution has the radius l.

THE SUPERCONDUCTING STATE

25

This expression clarifies our assertion that for l >> x the penetration depth l is the effective radius of the vortex. This distance dependence of B(r) is sketched in Fig. 1.21. The factor F0/2pl2 in Eq. (1.41) is related to the lower critical field through the expression (1.42) where k = l/x. The upper critical field has a similar expression, with the coherence length replacing the penetration depth (1.43) At the lower critical field, the vortices are separated by the approximate distance d @ l, and at the upper critical field, the density of vortices is so high that the cores, which have the radius x, are almost touching.

1.2.7. Flux Creep and Flux Flow In a mixed state below a certain value of the temperature and applied magnetic field, the vortices are symmetrically arranged in a hexagonal pattern called an Abrikosov lattice (13). The intersection points of the vortices with a plane perpendicular to the applied field display this hexagonal lattice, as illustrated in Fig. 1.22a. For a given magnetic field there will be some temperature at which this regular arrangement becomes disordered as portrayed in Fig. 1.22b. The vortex lattice is said to have melted and the vortices can move around in a random fashion that resembles the motion of the molecules of a two-dimensional liquid. This phase is called the vortex liquid phase. The temperature at which the vortex lattice melts depends on the magnetic field strength. It is found that the dc magnetic field B and the temperature Tat which the melting occurs are related by the expression (14) (1.44) The exponent q is usually close to 213, and measurements carried out on various cuprates have provided values in the range from 112 to 3/4. The B-Tline described by this equation is called the irreversibility line because below it magnetic properties such as magnetization as a function of dc magnetic field display hysteresis. The magnetization is not reversible in the sense that when the dc field is decreasing, at a certain point the value of the magnetization is different from what it is when the dc field is increasing at that same point. Above the B-T irreversibility line, the

26

CHAPTER 1

a

b

Figure 1.22.Arrangement of vortices, viewed from the top, in the superconducting state of a Type II superconductor: (a) for the ordered vortex lattice state and (b) for the melted fluid vortex state.

magnetic properties are reversible. Figure 1.23 shows a determination of the irreversibility line in YBa2Cu3O7-x (1 1). If a current density J is sent through a superconductor at some angle relative to the applied dc magnetic field, there will be a force per unit length F acting on each vortex given by F = J × F0

(1.45)

where the fluxon vector F0 is aligned in the vortex direction and represents the quantized flux present there. If this force is strong enough, the vortices may move. This movement of vortices introduces an effective resistance to the current flow in the superconducting state. The force [Eq. (1.45)] on each vortex does not change when the applied field increases because the quantum of flux F0 confined by each vortex remains the same. It is the density of vortices that increases with the applied field, which means that the total force arising from all the vortices increases, and as a result the resistance to current flow increases with increasing strength of the applied dc magnetic field. The increase in resistance will also depend on whether

THE SUPERCONDUCTING STATE

27

Figure 1 .23. Plot of the irreversibility line for a single crystal of YBa2Cu3O7-x. The magnetic properties are reversible in the flux liquid phase above this line; they exhibit hysteresis in the flux solid phase of pinned vortices below this line. (After Worthington et al., Ref. 11 .)

the vortices are in the lattice or the liquid phase. Vortex motion occurs more easily in a liquid, so the resistance is higher for this phase. The melted vortex fluid phase can be modeled as a two-dimensional liquid in which a moving vortex experiences a frictional retarding force per unit length Fdrag = hvv proportional to its velocity vv and a viscous drag coefficient h. In equilibrium Fdrag equals the driving force (1.45), and for J perpendicular to F0 we have (1.46) A regular array of vortices in a superconductor produces an average magnetic field B in the material, and when this array moves, it generates an average electric field E given by E = vv × B

(1.47)

Since the resisitivity r is defined by r = E/J

(1.48)

28

CHAPTER 1

the velocity vv can be eliminated between Eqs. (1.46) and (1.47) to provide an expression for the resistivity induced by the moving vortices for the case in which B and J are perpendicular to each other (15). ρ=BΦ0/η

(1.49)

This simple model predicts that at constant temperature below Tc the resistivity will increase linearly with the strength of the applied dc field in the fluid vortex phase. In the vortex lattice state, the vortices are more strongly bound at their lattice sites and cannot move as easily as in the fluid state. They are considered to be trapped in potential wells and movement can only occur by a thermally activated hopping of bundles of vortices. Thus their velocity can be described by (16) (1.50) where U0 is the height of the energy barrier that has to be overcome in order for flux bundles to hop. In the vortex lattice phase, a larger force is required to induce changes in the resistance of the superconductor. The change in resistivity is obtained by combining Eqs. (1.47), (1.48), and (1.50) to give for the vortex lattice state (1.51) which is also linear in the applied field. Figure 1.24 shows a measurement of resistance versus temperature in zero magnetic field and a field of 10 tesla for the

Figure 1.24.Measurement of resistance normalized relative to the room temperature resistance Rn plotted against temperature in a zero applied field and in a magnetic field of 10 T for the n = 3 Hg-Ba-Ca-Cu-O superconductor (T. Datta, unpublished).

THE SUPERCONDUCTING STATE

29

HgBa2Ca2Cu3O10 (17) superconductor. As we will see later, the movement of flux in a Type II superconductor has a pronounced effect on the electromagnetic absorbing properties of the superconductor below Tc.

References 1. Z. Iqbal, T. Datta, D. Kirven, A, Lungu, J. C. Barry, E J. Owens, A. G. Rinzler, D. Yang, and F. Reidinger, Phys. Rev. B49,12331 (1994). 2. M. L. Cooper, Phys. Rev. 104, 11289 (1956). 3. P. L. Richards and M. Tinkham, Phys. Rev. 119,575 (1960). 4. J. Bardeen, M. L. Cooper, and J. Schrieffer, Phys. Rev. 108, 1175 (1957). 5. P. Townsend and J. Sutton, Phys. Rev. 128, 591 (1962). 6. C. A. Reynolds, B. Serin, W. H. Wright, and L. B. Nesbitt, Phys. Rev. 78 ,487 (1950). 7. W. Meissner and R. Ochsenfeld, Naturwiss. 21,787 (1933). 8. E London and H. London, Proc. Roy. Soc. (London) A149 72 (1935). 9. A. L. Schawlow and G. E. Devlin, Phys. Rev. 113, 120 (1959). 10. C. P. Poole, Jr., H. A. Farach, and R. J. Creswick,Superconductivity, Academic Press, San Diego (1995). 11. T. K. Worthington, W. J. Gallagher, andT. R. Dinger, Phys. Rev. Lett. 59, 1160 (1987). 12. C. P. Poole, Jr., H. A. Farach, and R. J. Creswick, Superconductivity, Academic Press, San Diego (1995). 13. A. A. Abrikosov,J. Phys. Chem. Solids 2 199 (1957). 14. K. A. Müller, M. Takashige, and J. G. Bednorz, Phys. Rev. Lett. 58, 1143 (1987). 15. J. Bardeen and M. J. Stephen, Phys. Rev. 140A, 1197 (1965). 16. P. W. Anderson,Phys. Rev. Lett. 9,309 (1962). 17. T. Datta, Z. Iqbal, and D. Kirven (unpublished).

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2 The New Superconductors

2.1. THE COPPER OXIDES 2.1.1 Lanthanum and Neodymium Superconductors Prior to 1986, the highest transition temperature to the superconducting state was 23.2 K in the niobium-germanium alloy compound Nb3Ge. In late 1986 Alex .. Müller and George Bednorz of the IBM laboratory in Zurich, Switzerland, reported evidence of superconductivity at 30 Kin a new class of materials, lanthanum copper oxides (1). The observation was reported for the compound La2CuO4 doped with small amounts of barium, with the formula La2–xBax CuO4. The optimum x for superconductivity was about 0.2. Substitution of strontium for barium having the composition La1.85Sr0.15CuO4 increased the transition temperature to about 40 K (2). Since then, a number of materials in the copper oxide or cuprate family have been discovered that superconduct at temperatures as high as 133 K at atmospheric pressure; Table 2.1 gives some examples of these. Figure 2.1 shows the tetragonal unit cell of the La-Sr-Cu-O superconductor. An important feature of the structure is that the Cu2+ and O2– ions form a CuO2plane in the unit cell, with each Cu2+ ion bonded to four oxygens and each O2– ion bonded to two coppers in the plane. This particular planar arrangement of coppers and oxygens, sketched in Fig. 2.2, is a common feature of all the superconducting cuprates. The La-Sr-Cu-O compound has one plane per formula unit, and it is perfectly flat. This material is a poor metal in the normal state, with a resistivity three orders of magnitude greater than that of copper. It conducts electricity by the hopping of holes in the CuO2 planes. Below Tc, the supercurrent flows in these planes and is carried by Cooper pairs formed from mobile holes. The fact that the conduction process is largely confined to the CuO2 planes means that many of the properties of this material, such as resistivity, critical current, critical magnetic fields, and penetration depth are anisotropic. 31

32

CHAPTER 2

Table 2.1.Progress in Raisingthe Transition Temperatures ofCuprate Superconductors atAmbient Pressure Material La2–xBaxCuO4 YBa2CU3O7–x Bi2Sr2–xPbxCa2Cu3O10 TI2Ba2Ca2Cu3O10 HgBa2Ca2Cu3O8

Tc (K)

Year

Reference

30 90 113 125 133

1986 1987 1988 1988 1994

1 4 5 6 7

The concentration of holes in the copper oxide planes of La2CuO4 is increased by replacing some of the La3+ by Sr2+ or Ba2+. For example, in La1.8Sr0.2CuO4, the average charge on the copper in the planes is +2.2, which means that 20% of the coppers are in the 3+ valence state. The Cu2+ ion has 9 electrons in its outer 3d orbital, one electron short of a full shell. The absence of this one electron needed to fill the shell constitutes a hole. The Cu3+ ion has 8 electrons in the 3d orbital, 2 electrons short of a filled shell, thereby providing two holes that can contribute to the conduction process. Thus increasing the Sr2+ content in La2–xSrxCuO4 i ncreases the hole concentration in the copper oxide planes and hence the conductivity. The undoped La2CuO4 compound is an antiferromagnetic insulator with a Neél temperature for the onset of antiferromagnetism of 341 K. As the Sr content is increased, the Neél temperature decreases. In the region of Sr concentration where the material is antiferromagnetic, it is not a superconductor, but it becomes one at

Figure 2.1. Unit cell of La-Sr-Cu-O showing the conducting layers and binding layers.

THE NEW SUPERCONDUCTORS

33

Figure 2.2. Arrangement of coppers and oxygens in the CuO2 plane of a cuprate superconductor.

Sr concentrations where the antiferromagnetic phase no longer exists. However, there is evidence that short-range antiferromagnetic order persists even in the superconducting phase, and in this phase the superconducting transition temperature depends on the Sr concentration. These features have been summarized in the so-called universal phase diagram for copper oxide superconductors presented in Fig. 2.3. This diagram shows how the Néel temperature and the superconducting transition temperature depend on the strontium concentration and thus on the hole content.

Figure 2.3. Universal phase diagram of La-Sr-Cu-O showing the dependence of the transition to the antiferromagnetic state and to the superconducting state on the concentration of Sr.

34

CHAPTER 2

The majority of the copper oxide superconductors carry current by means of holes, but there are some in which the current is carried by electrons (3). When neodymium copper oxide, Nd2CuO4, which has a structure that is similar to but not identical with that of La2CuO4, is doped with cerium Ce4+ in the Nd3+ sites, it adds one electron to the copper oxide planes. The resulting compound becomes superconducting at 30 K with the Cooper pairs formed from electrons.

2.1.2. The Yttrium Superconductor The discovery that generated the extensive news coverage in 1987 was the observation of superconductivity in YBa2Cu3O7–x at 90 K, 13 degrees above the temperature at which liquid nitrogen boils (4). The unit cell shown in Fig. 2.4 is orthorhombic, with the b axis slightly longer than the a axis, and it contains two

Figure 2.4. Orthorhombic unit cell of YBa2Cu3O7–x .

THE NEW SUPERCONDUCTORS

35

copper oxide planes lying adjacent to each other perpendicular to the c axis. However, there are also present chains of Cu-O parallel to the b axis formed by sharing an oxygen at a common corner. Doping of the copper oxide planes with holes is achieved by reducing the oxygen content from the stochiometric value of 7.00. The material with lower oxygen levels has oxygen vacancies in the chains at the common corner. In other words, as the oxygen content is decreased, some connections between adjacent copper ions in the chains are removed. The oxygen content of the planes is unaffected. As the oxygen is removed, the chains become more positive and this in turn induces an increase in the hole concentration in the copper oxide planes. This reduced oxygen content has a strong influence on the properties of the material. For x greater than about 0.7, the unit cell is tetragonal and the material is antiferromagnetic, with the Néel temperature decreasing as the oxygen is further reduced. For x less than about 0.7, the unit cell is orthorhombic and the material is no longer antiferromagnetic, but it becomes superconducting, with the superconducting transition temperature increasing to slightly above 90 K as x is reduced toward the optimum value of approximately 0.1. These properties are summarized in Fig. 2.5.

Figure 2.5. Dependence of the antiferromagnetic and the superconducting transition temperatures on the oxygen content in YBa2Cu3O7–x.

36

CHAPTER 2

The replacement of copper in the planes by a transition ion such as Ni or by Zn reduces the transition temperature, and with high enough doping the superconductivity disappears. The deterioration of the superconductivity is greater with the addition of Zn than with Ni, hinting at the possible importance of the magnetic nature of the copper ion in the mechanism of superconductivity. Most rare earth elements such as Nd or Gd can be totally substituted for Y without any effect on the superconductivity. For example, the Gd3+ ion has a net magnetic moment µ = 9.2 µ B and the Gd sublattice undergoes a transition to an antiferromagnetic phase at 2.2 K. Substituting other magnetic rare earth ions such as Dy, Er, Ho, and Nd for Y has the same effect as in the Gd case, representing an unusual situation where the same material can be an antiferromagnet and a superconductor at the same time.

2.1.3. Bismuth and Thallium Superconductors In January 1988, H. Maeda and his co-workers (5) discovered a new family of copper oxide superconductors having the general formula Bi2Sr2Can–1CunO2n+4, where n refers to the number of copper oxide planes per formula unit. Figure 2.6 shows the bismuth structure of the n = 2 and n = 3 phases. Growth of the n = 3 phase was facilitated by partial substitution of Pb for Bi, giving a transition temperature of 113 K. Not long after the discovery of the bismuth superconductors, A. Hermann and Z. Sheng (6) synthesized a family of superconductors containing thallium with

Bi,Pb 2223

Bi,Pb 2212

Figure 2.6. Crystal structures of two Bi-Pb-Ba-Ca-Cu-O superconductors.

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37

the formula Tl2Ba2Can–1CunO2n+4. This series of compounds is isomorphic with the bismuth series, with T1 substituting for Bi, and Ba replacing Sr; the structure of this family drawn from a different perspective is shown in Fig. 2.7. The alternating binding layers and conducting layers with their copper oxide planes are indicated, and these will be discussed later. We see from Table 2.1 and Fig. 2.8 that the transition temperature increased to 125 K as the number of copper oxide planes per conducting layer (i.e., per formula unit) was increased to 3, but then it decreased with the addition of more planes. This is generally true for the copper oxide superconductors, including the more recently discovered mercury materials, which are discussed in the next section. In both of these families no change in stoichiometry is necessary to achieve hole doping of the copper oxide planes.

Tl2Ba2Ca2Cu3O10 Figure 2.7. Structures of the T1-Ba-Ca-Cu-O superconductors. Members of the Bi-Sr-Ca-Cu-O family of superconductors have the same structures. (Adapted from Torardi et al., Ref. 21.)

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Figure 2.8. Transition temperature as a function of the number of copper oxide planes in the conduction layers of the Hg, T1, and Bi superconductors.

2.1.4. Mercury Superconductors In the spring of 1993 a new superconductor series, HgBa2Can–1CunO2n+2, was synthesized. The n = 3 compound with a transition temperature to the superconducting state of 133 K has the tetragonal unit cell sketched in Fig. 2.9, with alternating binding and conducting layers, and the three copper oxide planes of the conducting layer separated from each other by layers of calcium ions (7). It has been found that some oxygen atoms are located in the interstitial site in the Hg layer, shown at the top of the figure by the half-shaded circle, and that these oxygens are more weakly bound to their sites than the other oxygens in the material. Since the properties of the superconductor such as the transition temperature depend on the oxygen concentration, which affects the hole concentration in the copper oxide planes, this material displays some instability with respect to oxygen content. However, it has been found that substitution of Pb for one-third of the Hg atoms makes this material more stable (8). Unlike most of the copper oxide superconductors, applying pressure of up to 140 kbar increases the transition temperature to 147 K, halfway to room temperature, as shown in Fig. 2.10. Pressure can also be produced chemically in this material by replacing the Ba2+ ion by the smaller Sr2+, but unfortunately when this was done, it was found that the transition temperature decreased to 127 K. Another interesting property of the mercury compound is that the application of magnetic fields up to 10 T at 77 K does not cause the resistance to increase, suggesting that this material may more strongly pin flux at 77 K than

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39

Figure 2.9. Unit cell of the Hg-Ba-Ca-Cu-O, n = 3 superconductor. The half-shaded circle at the top indicates that some of these sites can be partially occupied with oxygen.

the other copper oxide superconductors. Furthermore, it has also been found that the material can be grain aligned. Both of these results suggest that Hg-Ba-Ca-CuO may be able to carry more current at 77 K than the other copper oxide superconductors, and perhaps with its fabrication into wires, superconducting magnets that operate at liquid nitrogen temperature will be possible.

2.1.5. Infinite-Layer Phases In 1993 superconductivity was discovered in a series of compounds having the general formula Srn+1CunO2n+1+x (9). The n = 1 compound Sr2CuO3.1 with Tc = 70 K has the same structure as La2CuO4, but with a large number of oxygen vacancies in the Sr2O2 layers. These oxygen-deficient layers constitute charge reservoir layers that serve to dope the copper oxide planes. The n = 2 compound Sr3Cu2O5+x has two copper oxide layers between the Sr2O2 layers and a Tc of 100 K. The limit of the series is the infinite-layer phase SrCuO2 whose structure is shown in Fig. 2.11.

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Figure 2.10. The transition temperature of the n = 3, Hg-Ba-Ca-Cu-O superconductor as a function of applied pressure.

However, recent work shows that the infinite-layer phase may not be superconducting, and that the observed superconductivity, which is always a small fraction of the sample for these materials, may be associated with intergrowths of Srn+1CunO2n+1+x. These compounds require high pressure to synthesize, and the results have not been extensively studied or reproduced in other laboratories.

2.1.6. Ladder Phases Recent theoretical calculations of the properties of the so-called copper oxide ladder phases, most of which have the general formula Srn–1CunO2n–1, have suggested that these materials could be high-temperature superconductors (10). As shown in Fig. 2.12, the structure of the copper oxide planes differs from that of the known copper oxide superconductors. Essentially the structure consists of ladderlike chains connected by oxygen bonds. Dagotto and Rice (10) have predicted that compounds in the series having oxygen contents ON of N = 3,7, 11 . . . . (i.e., n = 2,4,6, . . . ) will be frustrated quantum antiferromagnets and have spin gaps. This means that pairs of S = ½ Cu2+ ions can couple to form singlet states, with the spin gap being the energy difference between the ground singlet and the excited triplet

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41

Figure 2.12. Arrangement of the copper and oxygen atoms in the ladders of the so-called ladder phase LaCuO2.5 materials. The horizontal Cu-O distances are 2.29 Å and the vertical Cu-O distances are 1.95 Å.

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state of the pair. The pairing occurs across the rungs of the ladder. If the exchange coupling between Cu2+ions parallel to the ladders, J||, is smaller than the J^ between the ladders, then the spins on adjacent ladders can pair to form singlet states that lower the energy of the system. It is further predicted that when it is lightly doped with holes, the spin gap will remain open and the materials may become high-temperature superconductors. However, if J|| is approximately equal to J^, charge density waves may exist along the chain and decay more slowly than superconducting correlations. For superconducting correlations to dominate, J^ must be larger than J||. Azuma et al. (1 1) have been able to synthesize these materials under high pressure and using nuclear magnetic resonance and magnetic susceptibility measurements, have shown that SrCu2O3 has a spin gap but that Sr2Cu3O5 does not, which is in agreement with the predictions of Rice et al. However, to date there have been no observations of superconductivity in these materials, perhaps because of the difficulty in doping the planes with holes under high-pressure synthesis. There have, however, been some tentative hints that superconductivity may exist in these materials, in some instances perhaps even toward room temperature. For example, very large surface resistance drops have been observed in another ladder phase type of material, Sr14–xCaxCu24O41+d (12), which does not belong to the general class Srn–1CunO2n–1 discussed earlier. The effects are very sensitive to oxygen exposure and occur in the outer regions of the sample, within about one microwave penetration depth of the surface. The very large surface resistance drops were followed by subsequent large increases in surface resistance. The effects were attributed to superconducting fluctuations followed by a metal-to-insulator transition, as discussed in Sect. 5.4.4.

2.2. GENERAL PROPERTIES OF COPPER OXlDE SUPERCONDUCTORS The properties of the individual copper oxide superconductors are summarized in Table 2.2, which is borrowed from Ref. 13. Some are tetragonal (T) and some are orthorhombic (O), but close to tetragonal. Some, such as the yttrium and mercury compounds, are the aligned type (A) with one formula unit per unit cell, and others, such as the lanthanum, bismuth, and thallium compounds, which are denoted by S to indicate staggered positions, are body centered with two formula units per unit cell. In the aligned compounds, all of the copper ions in successive conduction layers lie one above the other, while in the body-centered compounds, the copper ions in the planes alternate in position (i.e., are staggered) in successive conduction layers, as may be seen from a close scrutiny of Fig. 2.7. We see from the table that all of the cuprates have the cell dimension a0 @ 3.85 Å, but they differ in c0, which increases with the number of atoms in the unit cell.

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43

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2.2.1. Commonalities of the Cuprates In addition to the properties listed in Table 2.2, there are a number of characteristics that are common to all of these materials. These common features are important because they suggest the directions to go to raise Tc and provide the background needed for finding a theoretical explanation of the origin of the superconductivity in the cuprates. The most important commonality is the presence of conduction layers and binding layers that alternate in position along the c-direction, as sketched in Fig. 2.13. They have the following features and play the following roles: 1. The conduction layers consist of copper oxide planes separated by layers of calcium ions, or in the case of Y-Ba-Cu-O, separated by yttrium ions, as shown in Fig. 2.14. They exhibit little variation from compound to compound. Electrical conduction in both the normal and the superconducting states takes place mainly by hole hopping, which is confined to the copper oxide planes of the conduction layers. In the neodymium compound, the conduction is via electron hopping. 2. The binding layers exhibit greater variations in constitution, and their common cases are sketched in Fig. 2.15. These binding layers structurally support

Figure 2.13. Layering scheme of the copper oxide superconductors. Figures 2.14 and 2.15 sketch the conduction layers and the binding layers, respectively, of individual cuprates.

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45

Figure 2.14. Sketches of the conduction layers of several individual copper oxide superconductors.

the conduction layers and also serve as charge reservoirs that provide the proper amount of hole doping in the copper oxide planes. We see from the figure that the CuO chains of the yttrium compound are structurally in the binding layer, but nevertheless they are responsible for some of the current flow. Figure 2.1 1 shows that the infinite-layer phase lacks binding layers. There are also some magnetic commonalities among the various cuprates. They are:

• •

the presence of magnetic copper ions evidence for short-range antiferromagnetic correlations within the planes

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Figure 2.15. Sketches of the binding (charge reservoir) layers of several individual copper oxide superconductors.

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•

47

dependence of the superconducting transition temperature Tc and the Néel temperature TN of nonsuperconducting-related phases on the hole concentration

2.2.2. Energy Bands Let us discuss the implications of these properties. The fact that the current is largely confined to the copper oxide planes means that the band structure of the material can be approximated by a two-dimensional model. Figure 2.16 shows the energy versus density of states for a simple two-dimensional array of copper and oxygen atoms where each copper atom is bonded to four nearest-neighbor oxygen atoms. The top occupied band is formed from the highest occupied molecular orbitals (HOMO) of the atoms, and the first unoccupied band is formed from the first empty orbitals of the atoms. In this simple model, the energies of the filled lower band EL(K) and the empty upper band EU(K) are, EU(K) = EU+ 2tU(cosKxa + cosKya)

(2.1)

EL(K) = EL+ 2tL(cosKxa + cosKya)

(2.2)

Figure 2.16. Illustration of density of states for two-dimensional copper oxide planes showing the van Hove singularities. The highest occupied and lowest unoccupied molecular orbital regions, HOMO and LUMO, respectively, are indicated.

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where EU and EL are the energies of the free atom orbitals and tL and tU are the corresponding overlap integrals. The peak in the density of states near the middle of each band is common to two-dimensional systems and is known as a van Hove singularity (14). It is a result of the many combinations of Kx a and Ky a giving the same energy. A number of attempts to explain superconductivity in the copper oxide superconductors draw on the existence of this van Hove peak in the density of states near the Fermi level. This large density of states at EF means that many single electron states at the top of the conduction band are available to form Cooper pairs. In the weak coupling form of the BCS theory Tc is given by Tc = 1.14(Eg/kB)exp[–1/VD(EF)]

(2.3)

where the activation energy V contains the interaction between the holes forming the Cooper pairs and D(EF) is the density of states at the Fermi level. Note that a larger D(EF) means a higher Tc. Because of the reduced dimensionality, the band gap is anisotropic, having values approximately 1.75kBTc parallel to the c-axis and 2.7kBTc parallel to the copper oxide planes. The energy gap in the ab plane is substantially higher than that predicted by weakly coupled BCS theory, corresponding to the limit VD(EF) << 1, which means that the Cooper pairs are more strongly

Figure 2.17. Plot of Bc2 versus temperature near Tc for Y-Ba-Cu-O for the magnetic field parallel to the c-axis (O) and perpendicular to the c-axis (D), respectively (after Welp et al., Ref. 15).

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49

bonded than in the low-temperature metallic superconductors. A stronger V in Eq. (2.3) also would provide a higher Tc. However, Eq. (2.3) may not be applicable in the stronger coupling limit where the exponential factor exp[–1/VD(EF)] is replaced by the expression { 2sinh[–1/VD(EF)]}–1. The separation between the holes of the Cooper pairs in the copper oxide planes is typically on the order of 20 Å, which is significantly smaller than in the low-temperature metallic superconductors, where this distance can be as large as 10,000 Å. Since the Ginzburg–Landau parameter k = l/x is on the order of 100, the copper oxide superconductors, as discussed earlier, are very much Type 11. This stronger binding of the Cooper pairs also means that the upper critical magnetic fields and critical current densities will be larger. Figure 2.17 shows the results of a measurement of Bc2 near Tc parallel to c and perpendicular to c in YBa2Cu3Ox. At absolute zero the value of Bc2 is estimated to be several hundred tesla, but this cannot be measured reliably. Another consequence of the two-dimensional nature of the copper oxides is the anisotropy of the electrical transport both in the superconducting and in the normal states. The conductivity parallel to the planes is about 2 to 4 orders of magnitude greater than that perpendicular to the planes. Figure 2.18 shows measurements of the resistance perpendicular to and parallel to the c-axis in YBa2Cu3O7.

2.2.3. Cooper Pair Binding The nature of the force that binds the holes into Cooper pairs remains unclear and has been a subject of intense research. At present two models are receiving the most attention. One suggests that the charge pairing is mediated by vibrations of the highly polarizable oxygen ions. This idea, which is a BCS-like mechanism, in conjunction with the existence of a van Hove singularity in the density of states near the Fermi level, is at present considered by some as a possible mechanism. Because of the intimate relationship between the existence of antiferromagnetism and superconductivity exemplified in Figs. 2.3 and 2.5, as well as the existence of short-range antiferromagnetic correlations of the Cu spins in the superconducting state, magnetic pairing mechanisms are also being investigated. It has been proposed that magnetic interactions between antiferromagnetic fluctuations of the Cu spins and the charge carriers cause the pairing interactions. The idea is that as a charge carrier moves by, it disrupts the magnetic spin of one ion, which flips the spin of a neighbor, thus attracting a second charge carrier of the opposite spin. This model requires the wave function describing the Cooper pair to have d-symmetry. At present a great deal of effort is being made to determine the symmetry of the Cooper pairs, and much evidence has accumulated in favor of d-wave pairing. Electromagnetic absorption measurements constitute one approach used to investigate this issue. This is discussed in greater detail in Sect. 5.4.3.

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Figure 2.18. Resistivity versus temperature parallel (a) and perpendicular(b) to the copper oxide planes in Y-Ba-Cu-O. Note the change in the ordinate scales.

2.3. PEROVSKITE SUPERCONDUCTORS The compound Ba1–xKxBiO3–yforms for x > 0.25 and crystallizes in the cubic perovskite structure illustrated in Fig. 2.19. It was found to be a superconductor in 1988 and has Tc @ 40 K for x @ 0.4 (16). This compound has the variable-valence bismuth ion and utilizes oxygen vacancies to achieve charge compensation. Its structure has similarities to those of the cuprates, but it lacks copper. The distorted perovskite BaPb1–xBix03 is tetragonal, and in 1975 was found to superconduct for x in the concentration range 0.3 ≥ x ≥ 0.05 with Tc up to 13 K (17). Bednorz and Muller knew about this work and referred to their "metallic, oxygen-deficient . . . perovskite-like mixed valence copper compound" samples in their pioneering 1986 article.

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51

Figure 2.19. Cubic unit cell of Ba1–xKxBiO3–y perovskite.

2.4. CA RBON -6 0 SUPERCONDUCTORS In 1990 the synthesis of a new molecule consisting of 60 carbon atoms bonded to each other in a spherical soccer ball-like arrangement was reported (18). The molecule, whose structure is shown in Fig. 2.20, has 12 pentagonal and 20 hexagonal faces symmetrically arranged to form a molecular ball, which is now known as fullerene after the architect Buckminister Fuller, who invented the geodesic dome. These ball-like molecules can form a crystal lattice having the face-centered cubic structure shown in Fig. 2.21. In the lattice each C60 molecule is separated from its nearest neighbors by 10 Å (center-to-center distance of the molecules). There is empty space in the unit cell between the C60 molecules, making it possible to put small alkali atoms such as potassium in tetrahedral and octahedral interstitial positions, as shown by the black circles in Fig. 2.21. The alkali atoms ionize to form cations such as K+ by transferring electrons to the fullerene molecules. The transferred electrons become delocalized among the conjugated or alternating single-double bond arrangement of the fullerenes, and they become charge carriers by hopping between adjacent fullerene molecules to make the material a conductor. In 199 1 A. F. Hebard and co-workers (1 9) at Bell Telephone Laboratories made the surprising discovery that alkali-doped C60 crystals superconduct. For example, the K3C60 material is superconducting at 18 K, and other alkali atoms doped into the lattice also make it superconduct, The transition temperature has been shown to depend on the size of the cation (20), which increases in the order K < Rb < Cs. The large cations increase the volume of the unit cell, and it is found that the transition temperature correlates with this volume, as shown in Fig. 2.22. One of the important issues that arose shortly after the discovery of superconductivity in these materials was the nature of the mechanism responsible for it. A critical experiment for testing whether the mechanism is BCS is to measure the effect of isotopic labeling on the transition temperature. C60 made from the carbon-

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Figure 2.20. Illustration of the C60 (fullerene) soccer ball-like molecule.

Figure 2.2 1. Structure of A3C60 showing small alkali atoms in the tetrahedral and octahedral interstitial sites around the large C60 molecules arranged on a face-centered cubic (FCC) lattice.

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53

Figure 2.22. Transition temperature T c of A3C60 superconductors plotted against the lattice parameter (adapted from Hebard, Ref. 20).

Figure 2.23. Magnetization as a function of temperature for K312C60 (♦) and K313C60 (O) superconductors (adapted from Chen and Lieber, Ref. 21).

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13 isotope is 8.3% heavier than C60 made from the normal carbon-12, and so the phonon-mediated BCS theory predicts a downward shift of the transition temperature in K313C60 compared with K312C60. This was checked by measuring the temperature dependence of the magnetization in both materials. The results presented in Fig. 2.23 (21) indicate that the isotopically labeled material becomes superconducting 0.4 K below that of the nonlabeled material. Unfortunately the result does not unambiguously confirm the BCS mechanism since the theory predicts that the downward shift should be 0.8 K. Because the alkali atoms can diffuse out of the lattice, the bulk form of these materials as first synthesized had to be kept in sealed tubes under a slight excess of inert gas pressure. This meant that four-probe resistance measurements could not be made on the materials. Instead, surface resistance drops as well as magnetization measurements of the type presented in Fig. 2.22 were used to determine the onset of superconductivity (19).

References 1. J. G. Bednorz and K. A. Müller, Z Phys. B64,189 (1986). 2. S. Uchida, H. Takagi, K. Kisho, K. Kitazawa, K. Fueki, and S. Tanka, Jpn. J. Appl. Phys. 26, L443 (1987). 3. C. C. Almasan and M. B. Maple in Chemistry of High Temperature Superconductors, C. N. R. Rao, ed., World Scientific, Singapore (1991). 4. M. Wu, J. Ashburn, C. Torng, P. Hor, R. Meng, L. Gao, Z. Huang and C. W. Chu, Phys. Rev Lett. 58, 908 (1987). 5. H. Maeda, Y. Tanaka, M. Fukutomi, and T. Asano, Jpn. J Appl. Phys. 27, L209 (1988). 6. Z. Sheng and A. Herman, Nature 332, 55 (1988). 7. A. Schilling, M. Catoni, J. D. Guo and H. R. Ott, Nature 363, 565 (1993). 8. Z. Iqbal, T. Datta, D. Kirven, A. Lungu, J. C. Barry, F. J. Owens, A. G. Rinzler, D. Yang and E Reidinger, Phys. Rev. B49, 12322 (1994). 9. Z. Hori, M. Takano, M. Azuma and Y. Takeda, Nature 364, 315 (1993). 10. E. Dagotto and T. M. Rice, Science 271, 618 (1996). 11. M. Azuma, Z. Hiroi, M. Takano, K. Ishida and Y, Kitaoka, Phys. Rev. Lett 73, 3463 (1994). 12. F. J. Owens, Z. Iqbal and D. Kirven, Physica C267, 147 (1996). 13. C. P. Poole, Jr., H. A. Farach and R. J. Creswick, Superconductivity, Academic Press, San Diego (1995). 14. L. VanHove, Phys. Rev. 89, 1189 (1953). 15. U. Welp, M. Grimsditch, H. You, W. K. Kwok, M. Fang, G. W. Crabtree and J. Z. Lin, Physica C161, 1 (1989). 16. L. F. Mattheiss, E. M. György, and D. W. Johnson, Phys. Rev. 837, 3745 (1988). 17. A. W. Sleight, J. L.Gilson, and P. E. Bierstedt, Solid State Commun. 17, 27 (1975). 18. H. W. Kroto, J. R. Heath, S. C. O'Brien, R. E Curl, and R. E. Smalley, Nature 318, 162 (1986). 19. A. F. Hebard, J. Rossinsky, R. C. Haddon, D. W. Murphy, S. H. Glarum, T. T. M. Plastra, A. P. Ramirez and A. R. Kortan, Nature 350, 320 (1991). 20. A. F. Hebard, Physics Today 48 26, Nov. (1992). 21. C. C. Chen and C. Lieber, J. Am. Chem. Soc 114,3141 (1992). 22. C. C. Torardi, M. A. Subramanian, J. C. Calabrese, J. Gopalakrishnan, K. J. Morrissey, T. R. Askew, R. B. Flippen, U. Chowdhry and A. W. Sleight, Science 240,631 (1988).

3 Experimental Methods and Complementary Techniques

The main purpose of the remaining chapters of this book is to survey and explain electromagnetic absorption in superconductors in the microwave region of the spectrum, and to a lesser extent in the radio frequency (RF), infrared, and optical regions. This chapter introduces the reader to some of the experimental methods that are employed to carry out those electromagnetic absorption studies that are of the nonresonant type. Radio frequency and microwave techniques have also been employed to study resonant absorption in superconductors, studies that are generally termed magnetic resonance, and this chapter also reports on some of these results. Standard nuclear magnetic resonance (NMR) and electron paramagnetic resonance (EPR) spectrometers, particularly the latter, can be adapted for carrying out the nonresonant absorption experiments, and most microwave absorption studies involve using adapted EPR spectrometers. The essence of a magnetic resonance experiment is the measurement of the precession of nuclear or electronic spins in a strong magnetic field. Other particles with spin that have been employed to probe superconductors are muons, positions, and inelastically scattered neutrons, and some of this work is described here. A great deal of research has also been carried out on superconductors using frequencies beyond the infrared, such as X-rays, and this work is commented upon briefly. Thus the remainder of the book covers radio frequency, microwave, infrared, and optical absorption in superconductors, and this chapter reports on techniques for carrying out this work as well as techniques that complement this work.

3.1. RADIO FREQUENCY MEASUREMENTS USING LC RESONANT CIRCUITS Radio frequency electromagnetic absorption in the superconducting state can be studied by a technique originally developed by Schawlow and Devlin (1) in 55

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which the sample is located in the coil of an LC circuit and the change in the frequency of the RF oscillator during the absorption process is measured. Figure 3.1 shows a schematic of the original experimental method. Clover and Wolf (2) have described a simple inexpensive marginal oscillator that can be used for this measurement. The sample is placed in a cylindrical quartz tube and a coil wound around the tube forms the inductance part of the LC tank circuit of an RF marginal oscillator that operates in the kilohertz to megahertz range. The coil containing the tube may be mounted on the cold tip of a cryogenic Dewar or immersed in a cryogenic fluid as shown in Fig. 3.1. For dc field measurements, the coil is placed between the poles of an electromagnet with its axis, and hence its RF B field, perpendicular to the dc magnetic field direction. When the dc magnetic field is increased with the sample in the superconducting state, the penetration depth of the RF field into the sample increases, This changes the effective permeability of the sample and hence the inductance of the coil because the inductance of a coil is proportional to the permeability µ of the material inside it. Since the oscillation frequency w is given by (3.1)

Figure 3.1. LC oscillator apparatus for measuring the penetration depth. The sample is contained in a coil that forms part of the LC circuit of an RF oscillator.

EXPERIMENTAL METHODS AND COMPLEMENTARY TECHNIQUES

57

and the change in inductance DL is small, the result is a shift in the oscillation frequency Dw of the magnitude (1) (3.2) For a cylindrical coil, the change in the penetration depth l is related to the change in the frequency by (1) (3.3) where r is the radius of the sample rod and A is the cross-sectional area between the rod and the coil. One advantage of this method is that it allows a determination of the frequency dependence of the dc magnetic field-dependent part of the absorption. The frequency dependence of the RF penetration depth is obtained by measuring Dw/w0 versus B for different choices of w0. For this geometry with the coil axis perpendicular to the dc magnetic field, the electric current that circulates around the coil has components that are both parallel to and perpendicular to this dc field. The vortices in the sample are parallel to the dc field and they experience the Lorentz force J × B arising from the perpendicular component of the current density J. The vortices that are set in motion by the Lorentz force experience a viscous retarding force, and the result is energy dissipation. To study this vortex-induced dissipation, it is desirable to enhance the component of the current perpendicular to the dc field, and this can be done by using a rectangular wire-wound inductor having a width much greater than its thickness, and aligning the dc magnetic field perpendicular to the large surface of the inductor. For an LC circuit containing a rectangular inductor of dimensions a,b, the frequency shift of the resonator is related to the penetration depth by (3) (3.4) where a0 and b0 are the dimensions of the sample, we is the frequency for the empty inductor, and w0 is the initial frequency with the superconductor in place in the coil and the applied magnetic field B0 = 0. In an experiment, the frequencyω is measured for various values of B0 and T. This method can also be used to determine the temperature dependence of the RF power absorption at zero field or in a constant dc magnetic field. Figure 3.2 shows, as an example, a measurement of the absorption of Y-Ba-Cu-O powder in a cylindrical coil at 10 MHz in which the frequency increases with the temperature. An explanation for this absorption in the normal state is given in Chap. 4, and that in the superconducting state is given in Chapters 5 and 7. The measured frequency shift must be corrected for the frequency shift of the empty coil. It is especially important to make this correction when using samples

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Figure 3.2. Measurement of the temperature dependence of the frequency shift at 10 MHz for the cylindrical coil of an LC circuit filled with apowder sample of Y-Ba-Cu-O.

having a small fraction of superconducting component, or samples that do not fill the volume of the coil. Figure 3.3 shows the temperature dependence of this frequency shift for the empty coil used to obtain the data presented in Fig. 3.2. This shift is mainly a result of the temperature dependence of the resistivity of the wire used to form the coil since expansion and contraction effects would tend to be negligible. A typical coil made of 10-mil copper wire capable of resonating at 350 Hz can have a resistance decrease from 6 ohms to 1 ohm on cooling from room temperature to liquid nitrogen temperature. Since the frequency w of an inductancecapacitance-resistance (LCR) circuit is given by (3.5) we see how a decrease in the resistance R produces the increase in the frequency shown in Fig. 3.3. One approach to dealing with this problem is to use two identical coils, one of which contains the sample, and measure the difference in frequency or ratio of the frequencies of the coils as a function of temperature. The use of phase-sensitive detection can enhance the sensitivity of the system, and there are a number of different ways to accomplish this. One approach modulates the oscillator frequency and utilizes a phase-sensitive detector to compare its phase with that of a reference signal of the same initial frequency. When the frequency of the oscillator changes because of a change in the penetration depth,

EXPERIMENTAL METHODS AND COMPLEMENTARY TECHNIQUES

59

Figure 3.3. Temperature dependence of the shift in the resonant frequency of the empty coil used to collect the data plotted in Fig. 3.2. The observed slight shift in frequency is due to the temperature variations of the resistivity of the coil.

the phase of the modulation with respect to the reference changes, and the phasesensitive detector produces a dc output voltage proportional to this phase difference.

3.2. MICROWAVE MEASUREMENTS USING CAVITY RESONATORS The method most commonly used to measure electromagnetic absorption in high-temperature superconductors in the gigahertz frequency range employs a microwave bridge and a cavity resonator (4). A standard x-band electron paramagnetic resonance spectrometer is easily adapted for making this measurement at 9.2 GHz. The absorption of microwaves as a function of the dc magnetic field is determined by measuring the reflection coefficient G(B,T) of a resonant cavity containing the sample in the superconducting state. Near optimal coupling where the cavity is impedance matched to the waveguide, the changes in G(B,T) are proportional to changes in the surface resistance of the sample. The experimental arrangement, shown in Fig. 3.4, consists of a microwave bridge with the cavity containing the sample mounted in one arm of the bridge, The microwave power source, either a klystron or a Gunn diode, is tuned initially to the resonant frequency of the cavity. This is accomplished by slightly varying the frequency of the klystron by mechanically tuning the size of the klystron cavity or by adjusting the voltage of the Gunn diode until there is a dip in the microwave

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Figure 3.4. Microwave bridge and cavity resonator arrangement for measuring absorption at 9.2 GHz.

energy reflected to the detector arm of the bridge as indicated by a drop in the diode current. For greater sensitivity, the magnetic field can be modulated and phase-sensitive detection employed. The microwave interaction with the sample produces a change in the cavity frequency called dispersion, and a change in the quality factor or Q of the cavity called absorption. Usually an automatic frequency control (AFC) system is employed to ensure that the frequency of the microwave source remains locked on the resonant frequency of the cavity so the frequency change that occurs during absorption is not detected. The absorption by the sample changes the power reflected from the cavity to the arm of the bridge containing the diode detector. The change in the dc current measured across this diode is directly proportional to the microwave power absorbed by the sample. Ordinarily the sample is located in the center of a TE102 rectangular resonant cavity or in the center of a TE011 cylindrical resonant cavity where the RF magnetic field strength is strongest and the RF electric field is zero. Another approach is to replace the end wall of the cavity by the superconducting material (5). This method is suited to thin films having a surface area large enough to be made into the wall of a cavity. In both cases the change in the Q of the cavity can be measured as the sample temperature is lowered below Tc.

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To measure the effect of a magnetic field on the microwave absorption, the microwave cavity is placed between the poles of an electromagnet as indicated in Fig. 3.4. The sample may be cooled by containment in a double-walled quartz glass tube inserted through a hole in the microwave cavity. The temperature is regulated by the flow of cold nitrogen or helium gas through the tube in conjunction with a regulated heater and diode sensor interfaced with a controller. This method of measurement is quite sensitive, but because the klystron must be tuned to the resonant frequency of the cavity, frequency-dependent measurements are not feasible. One can, of course, carry out measurements with instrumentation operating at different microwave frequency bands. Cavities have also been used to determine the surface resistance in the upper megahertz frequency range (6). Figure 3.5 illustrates a cavity system for such a measurement. The resonator consists of a cylindrical copper cavity 10 cm in diameter and 80 cm long for measurements below 600 MHz. The sample is contained in a long thin quartz tube that lies along the axis of the cylinder. The RF

Figure 3.5. Cavity for measuring absorption in the 600-MHz frequency range (from Delayen and Bohn, Ref. 6).

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Figure 3.6. Microwave bridge for measuring the transmission of microwave energy through a sample material.

is coupled into the cavity at the top, and at the side is a pickup probe where changes in the RF due to absorption in the sample are detected. The arrangement allows a measurement of the dc field, temperature, and frequency dependence of the absorption. Transmission of electromagnetic radiation through samples can be measured by the experimental arrangement shown in Fig. 3.6 (7). In this case no microwave cavity is used, but the sample is situated in the waveguide between the microwave source and the detector. The measurement compares the detector current when the microwaves are sent though the arm of the system containing the sample with that obtained when the microwaves are routed through the empty arm. This method, however, is not as sensitive as the cavity method.

3.3. ELECTRON PARAMAGNETIC RESONANCE Electromagnetic absorption in the gigahertz range arising from precessing electron spins can be studied using an electron paramagnetic resonance spectrometer. Examples of spins that produce this resonance absorption are unpaired electrons in free radicals, first transition series ions such as Cu2+ (3d9) and Mn2+ (3d5), which have unfilled 3d-electron shells, and the rare earth ion gadolinium, Gd3+ (4f7), which has a half-full 4f-electron shell. Especially favorable for detection are those transition metal ions with odd numbers of electrons. Free radicals associated with defects or radiation damage can be detected. The spectrometer with the block diagram presented in Fig. 3.7 contains the microwave bridge–electromagnet system sketched in Fig. 3.4 (8) together with

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Figure 3.7. Block diagram of a 100-kHz modulated electron paramagnetic resonance spectrometer equipped with automatic frequency control (AFC) and phase-sensitive detection.

some highly sophisticated electronic components. In this experiment the dc magnetic field is slowly scanned with the microwave frequency held constant. Resonance absorption occurs when the condition hv = gbB

(3 .6)

is satisfied, where h is Planck’s constant, v is the microwave frequency, g is the dimensionless Landé factor and b is the Bohr magnetron. For the x-band (9.2 GHz) and g @ 2, the EPR absorption occurs for fields near 0.32 T. The presence in acuprate superconductor of a Cu2+ EPR absorption signal in the range g ≅ 2.05 to g ≅ 2.27

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is indicative of the presence of a nonsuperconducting fraction such as the “green phase” Y2BaCuO5, which is often present in poorly made Y-Ba-Cu-O samples. In well-made superconducting samples, the Cu2+ ions are “EPR silent” so no absorption is detected. The scanning dc magnetic field is generally ac modulated at a frequency of 100 kHz. This modulation is supplied by an RF generator through a pair of coils mounted on the side walls of the microwave cavity. The modulation of the absorption results in a time-varying ac output signal at the crystal diode which changes phase by 180 degrees at the peak of the absorption signal, as shown in Fig. 3.8. Phase-sensitive detection is employed which compares the phase of this output with that of a reference signal, thereby causing the derivative of the absorption to be recorded. This reduces noise and enhances the sensitivity. As we discuss later, electron paramagnetic resonance has been used to study nonresonant low magnetic field-induced absorptions at fields of tens of millitesla in the copper oxide superconductors. In the case of nonresonance absorption, the shape of the detected signal depends upon the slowly increasing dc magnetic field and the superimposed ac magnetic field modulation. Modulation effects are particularly pronounced at high modulation amplitudes. The effect of the modulation can be eliminated by directly detecting the change in absorption. Such direct absorption signals from superconductors are easier to interpret, but the sensitivity is lower, Conventional EPR, on the other hand, is invariably carried out using modulated scanning fields.

Figure 3.8. Illustration of how ac magnetic field modulation affects the signal at the detector.

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Figure 3.9. Shift of the ESR signals from paramagnetic markers located on the side and end of a YBa2Cu3O7– d sample from their superimposed position (A) above Tc to different field positions (B,C) below Tc. The separation of the lines is proportional to the susceptibility [From Farach et al., Ref. 9].

3.4. PARAMAGNETIC PROBES The magnetic field inside a superconducting sample can be probed by placing a free radical marker at the face of a specimen normal to the magnetic field direction and another at a face of the specimen that is parallel to the external magnetic field (9). In the superconducting state, the two markers experience different local magnetic fields, so the resonant positions of the lines shift in the manner shown in Fig. 3.9. The observed shift occurs because the free radicals respond to the surface field, and this differs from the applied field B0 in accordance with Eq. (1.21a) and the boundary conditions Bin = Bsurf perpendicular to and Hin = Hsurf parallel to the

Figure 3.10. Temperature dependence of the susceptibility of YBa2Cu3O7–d determined by the ESR probe method of Fig. 3.9 [From Farach et al., Ref. 9].

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surface. Thus the observed shift in line position is a measure of the magnitude of the internal field Bin within the sample, and assuming that the demagnetization factor N is known, this permits the temperature dependence of the susceptibility x to be evaluated from Eq. (1.21a), with the results presented in Fig. 3.10. This probing technique can also be carried out by NMR, using for example, a silicone oil coating.

3.5. STRIPLINE AND PARALLEL PLATE MICROWAVE RESONATORS We have discussed microwave experiments that employ standard waveguide resonant cavities. Other types of resonators are also available for use. For example, stripline resonators have been employed to study the dc magnetic field dependence of the surface resistance of superconductors (10). A typical resonator, shown in Fig.

Figure 3.11. A stripline resonator (a) for measuring the surface resistance of a deposited superconducting film strip. The resonator is located (b) between two ground planes. [From Revenaz et al., Ref. 10].

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3.11, consists of a patterned thin film strip of a superconductor deposited on a substrate such as LaA1O3. The strip ofsuperconducting transmission line, patterned by photolithography, has a length of one-half of the wavelength of the microwave energy at the resonant frequency. Overtone resonances occur at multiples of the characteristic resonant frequency, wn = nw0. Thus measurements at overtone frequencies allow a determination of the frequency dependence of the surface resistance Rs. The quality factor Q and the resonant frequency w0 are measured and the surface resistance is calculated from these values with the aid of the expression: Rs = Gw0/D(l/d)Q

(3.7)

where G is a geometric factor, D(l/d) is a correction factor calculated from the current distribution in the stripline, l is the microwave penetration depth, and d is the film thickness. Magnetic field-dependent studies are carried out by applying the dc magnetic field perpendicular to the stripline, as shown in Fig. 3.1 1, which places B parallel to the c-axis of the superconductor. This method is only applicable to materials that can be made into thin films. Another resonator method uses the plates of the resonator as the superconducting sample to be studied (11). The two parallel plates are separated by a thin

Figure 3.12. A parallel plate resonator for measuring the surface resistance of superconductors that form the plates in the resonator. (From Taber, Ref. 11]).

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dielectric spacer about 75 µm thick using the arrangement shown in Fig. 3.12. The samples, which are the parallel plates, are pressed together by dielectric posts. The container is made of brass and the inner surfaces are gold plated. Two coupling probes are shown at the top of the container. These coupling probes are made by soldering 50-ohm microstrips to the ends of rigid coaxial cable. The vertical position of these probes can be varied to control the amount of coupling. Microwave energy is coupled in through one probe and out through the other.

3.6. NUCLEAR MAGNETIC RESONA NCE Nuclear magnetic resonance absorption occurs when a nucleus with a nonzero nuclear spin I in an applied magnetic field Bo is radiated with the frequency ω, which satisfies the same resonant condition [Eq. (3.6)] as EPR, but in the NMR case a different notation is used w = gB0

(3.8)

where gb = hg. The gyromagnetic ratio g, sometimes called the magnetogyric ratio, is characteristic of the particular nucleus, and it is normally three orders of magnitude less for NMR than it is for the EPR case. Typical NMR measurement frequencies range from about 60 to 400 MHz. The isotopes of TI and Y are particularly favorable for NMR because they have a nuclear spin I = 1/2, so they lack a quadrupole moment and their lines are not broadened by noncubic crystalline electric fields. The dominant isotope of oxygen, 16O, which is 99.76% abundant, and the dominant isotope of carbon, 12C, which is 98.9% abundant, both have I = 0 so they do not exhibit NMR. The importance of NMR arises from the fact that the value of g is sensitive to the local chemical environment of the nucleus, and it is customary to report the chemical shift d (3.9) which is the extent to which g deviates from gR of a reference sample, where for proton reference samples, gR/2p is close to 42.576 MHz/T. Chemical shifts are small, and are usually reported in parts per million (ppm). Cuprates enriched with the I = 7/2 oxygen isotope 17O provide spectra that distinguish the oxygens at the different sites in the Cu-O ligand, and the NMR spectra of the isotope 63Cu in natural abundance differentiate the copper ions at different positions in the lattice (12–16). NMR studies of 89Y nuclei in YBa2Cu3O7–d provided information on the penetration depth and the homogeneity of the magnetic field inside the superconductor. NMR spectroscopy using the 1.1 % abundant 13C isotope, which has nuclear spin I = 1/2

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has been instrumental in confirming the structures of fullerenes such as C60 and C70 (13).

3.7. NUCLEAR QUADRUPOLE RESONANCE A nucleus with spin I = 1/2 has an electric quadrupole moment, and the crystalline electric fields at an atomic site with symmetry less than cubic split the nuclear spin levels in a manner that depends on the site symmetry. These spacings are measured experimentally by the technique of nuclear quadrupole resonance (NQR), which requires no applied magnetic field and operates at frequencies similar to those used for NMR. Table VI-14 of Ref. 17 lists the point symmetries for some of the atomic sites in several high-temperature superconductors. NQR can provide information on the site symmetries of the various Cu sites in the cuprates, and can probe changes in symmetry at these sites that result from variations in the oxygen content (16, 18). When changes in temperature produce nonuniform compression of the different Cu sites, this is reflected in the NQR spectrum.

3.8. MUON SPIN RELAXAT/ON The negative muon µ– acts in all respects like an electron and the positive muon µ like a positron except that they have a mass 206.77 times larger. When polarized muons are implanted into a sample located in a magnetic field, their precession at the frequency gµ/2p = 135.5 MHz/T provides a microscopic probe ofthe distribution of local magnetic fields; in particular, the width of the muon spin relaxation (µSR) signal from a superconductor provides an estimate of this field distribution and the penetration depth l, (16, 19). As an example, the temperature dependence of the penetration depth lab(T) was measured by Harshman et al. (19) in a single crystal of YBa2Cu3O7–d with an 11 T applied magnetic field B0 aligned parallel to the c-axis (θ = 0). This orientation of B0 automatically gives values of the lab(T) component, and a fit to the expression +

(3.10) provided the zero Kelvin temperature value lab(0) = 141.5 nm. Next, measurements of the angular dependence of the average internal magnetic fields for the applied field B0 oriented at different angles q relative to the c-direction provided a determination of the effective mass ratio m*c /m*ab > 25. Since the penetration depth is proportional to the square root of the effective mass through the expression (3.11)

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where e* = 2e, and fGL is a temperature-dependent parameter in the Ginzburg–Landau theory, we obtain lc/ lab > 5, which gives lc > 700 nm.

3.9. POSITRON ANNIHILATION In positron annihilation spectroscopy (PAS), a sample is irradiated by a radioactive source such as 22NaCl, which emits high-energy (545 keV) electrons with positive charges e+ called positrons and a 1.28 MeV g-ray. When the positron enters the solid, it rapidly loses most of its kinetic energy and approaches thermal energy, ≈ (3/2)kBT≈ 0.04 eV, in the short time of 0.001 to 0.01 ns. After thermalization, the positron diffuses like a free particle, although its motion is correlated with nearby conduction electrons, until it encounters an electron e– and annihilates in about 0.1 ns, producing two 0.5 1 MeV g rays in the process e+ + e– ⇒ g + g

(3.12)

The electron moves much faster than the positron, and momentum balance causes the two g-rays that move off in opposite directions to make a slight angle with respect to each other, The angular correlation of this annihilation radiation (ACAR) is one of the important parameters that is measured. The positron lifetime τ is the time delay between the emissions of the 1.28 MeV and 0.51 MeV g-rays. The positron is sensitive to the details of the local electronic environment, and these are reflected in its mean lifetime t, its angular correlation, and its Doppler broadening parameters S and W (16, 20). These parameters exhibit discontinuities at the transition temperature. The positron annihilation characteristics are determined by the overlap of the positron and electron densities, and positron density plots provide estimates of the electron distributions between the various atoms in the superconductor. A two-dimensional angular correlation technique called 2D-ACAR samples the anisotropy of the conduction electron motion and provides information on the topology of the Fermi surface.

3.10. MÖSSBAUER RESONANCE Mössbauer resonance measures gamma rays emitted by a recoilless nucleus when it undergoes a transition from a nuclear ground state to a nuclear excited state. For 57Fe, the gamma ray has an energy of 14.4 keV and the line width is typically 5 × 10–9 eV, The g-ray can shift in energy, called an isomer shift, or its spectrum can split into a multiplet by the hyperfine interaction from the nuclear spin, by crystal field effects, or by the quadrupole interaction. Line broadening and relaxation provide additional information. These factors are sensitive to the chemical environment of the nucleus in the lattice. Mössbauer workers frequently quote energy shifts in velocity units, mm/s.

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In a typical experiment, one of the atoms of a superconductor, such as Cu, Y, or T1, is partially replaced by a small concentration of a nucleus such as 57Co, 57Fe, 151Eu, or 119Sn, which is favorable for Mössbauer studies. Sometimes the replacement is 100%, as in the compound EuBa2Cu3O7–d. The partial substitution can have the effect of lowering the transition temperature, particularly when Cu is being replaced. The spectra provide information on the valence state of the nucleus (e.g., Fe2+ or Fe3+), whether it is high spin (e.g., S = 5/2) or low spin (e.g., S = 1/2), the identity of the dominant substitutional site [e.g., Cu( 1) or Cu(2)], etc. Antiferromagnetic ordering due to the presence of Fe has been observed.

3.11. PHOTOEMISSION AND X-RAY ABSORPTION Photoemission spectroscopy (PES) measures the energy distribution of the electrons emitted by ions in various charge and energy states. These electrons have energies characteristic of particular atoms in particular valence states. To carry out this experiment, the material under study is irradiated with ultraviolet light (UPS) or X-rays (XPS), and these incoming photons eject electrons from atomic energy levels. The emitted electrons, called photoelectrons, have a kinetic energy KE which is the difference between the photon energy hvph and the ionization energy Eion required to remove an electron from the atom, as follows KE = hvph – Eion

(3.13)

The detector measures the kinetic energy of the emitted electrons and since hvph is known, the ionization energy is determined from Eq. (3.13). Each atomic energy state of each ion has a characteristic ionization energy, so the measured kinetic energies provide information about the energy levels of the atoms. In addition, many ionization energies are perturbed by the surrounding lattice environment, so this environment is probed by the measurement. PES can furnish spectra of the outer or valence electrons, and also of the inner electron energy levels called core levels. In addition, PES can provide information on the energy bands and the density of states (16, 21, 22). It is also possible to carry out the reverse experiment, called inverse photoelectron spectroscopy (IPS), in which the sample is irradiated with a beam of electrons and the energies of the emitted photons are measured. When UV photons are detected, the method is sometimes called Bremsstrahlung isochromat spectroscopy (BIS). A related experiment is electron energy loss spectroscopy (EELS), in which one measures the decrease in energy of the incident electron beam (23). Another technique called Auger electron spectroscopy involves a radiationless transition in which an X-ray photon generated within an atom does not leave the atom as radiation, but ejects an electron from a higher atomic level.

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An energetic photon is capable of removing electrons from all occupied atomic energy levels that have ionization energies less than the photon energy. When the photon energy drops below the largest ionization energy corresponding to the K level, then the n = 1 electron can no longer be removed, and the X-ray absorption coefficient abruptly drops. It does not, however, drop to zero because the X-ray photon is still energetic enough to knock out electrons in the L (n = 2), M (n = 3), etc., levels. The abrupt drop in absorption coefficient is referred to as an absorption edge; in this case it is a K-absorption edge, Acronyms are used: for example, X-ray absorption spectroscopy (XAS), X-ray absorption near edge structure (XANES), X-ray absorption fine structure (XAFS), and extended X-ray absorption fine structure (EXAFS) spectroscopy. Another way to obtain absorption edges, called electron energy loss spectroscopy, is to irradiate a thin film with a beamofmonoenergetic electrons with energies of, for example, 170 keV. When the electrons pass through the film, they exchange momentum with the lattice and lose energy by exciting or ionizing atoms, and an electron energy analyzer determines the energy Eabs that is absorbed. This energy equals the difference between the kinetic energy KE0 of the incident electrons and the kinetic energy KEsc of the scattered electrons Eabs = KE0 – KEsc

(3.14)

When the intensity of the scattered electrons is plotted as a function of the absorbed energy, then peaks are found at the binding energies of the various electrons in the sample.

References 1. A. L. Schawlow and G. E. Devlin, Phys. Rev. 113, 120 (1959). 2. R. B. Clover and W. P. Wolf, Rev. Sci. Instrum 41,617 (1970). 3. V. A. Gasparov, R. Huguenin, D. Pavvuna, and J. van der Mass, Solid State Comm. 69, 1147 (1989). 4. E J. Owens, A. G. R. Rinzler, and z. Iqbal, Physica C233,30 (1994). 5. A. M. Portis, D. W. Cooke, E. R. Gray, P. N. Arent, C. L. Bohn, J. R. Delayen, C. T. Roach, M. Heine, N. Klein, G. Müller, S. Orbach, and H. Piel, Appl. Phys. Lett. 58, 308 (1991). 6. J. R. Delayen and C. L. Bohn, Phys. Rev. 40, 5151 (1989). 7. A. T. Wijerante, G. L. Dunifer, J. T. Chen, and L. E. Wenger, Phys. Rev. B37, 615 (1988). 8. C. P. Poole, Jr., Electron Spin Resonance, Wiley, New York(1967,1983). 9. H. A. Farach, E. Quagliata, T. Mzoughi, M. A. Mesa, C. P. Poole, Jr. and R. Creswick, Phys. Rev. B41, 2046 (1990). 10. S. Revenaz, D. E. Oates, D. Labbe-Lavigne, G. Dresselhaus, and M. S. Dresselhaus, Phys Rev. B50, 1178 (1984). 11. R. C. Taber,Rev. Sci. Instrum. 61, 2200 (1990). 12. C, H. Pennington and C. P. Slichter, in Physical Properties of High Temperature Superconductors, D. M. Ginsberg, ed., Chap. 5, World Scientific, Singapore (1990). 13. R. E. WalstedtandW. W. Warren, Jr., Science 248, 1082(1990).

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14. D. BrinkmannandM. Mali, inNMRBasicPrinciplesandProgress, Vol. 31, p. 2, Springer-Verlag, Berlin (1994). 15. R. D. Johnson, D. S. Bethune, and C. S. Yannoni,Acc. Chem. Res. 25, 169 (1992). 16. C. P. Poole, Jr., H. A. Farach, and R. C. Creswick, Superconductivity,AcademicPress, NewYork (1995). 17. C. P. Poole, Jr., T. Datta, and H. A. Farach, Copper Oxide Superconductors, Wiley, New York (1988). 18. P. K. Babu andJ. Ramakrishna, Supercond. Rev. 1,75 (1992). 19. D. R. Harshman, L. F. Schneemeyer, J. V. Waszczak, G. Aeppli, R. J. Cava, B. Batlogg, L. W. Rupp, Jr., E. J. Ansaldo, and D. LI. Williams, Phys. Rev. B39,851 (1989). 20. S. J. Wang, S. V. Naidu, S. C. Sharma, D. K. De, D. Y. Jeong, T. D. Black, S. Krichene, J. R. Reynolds, and J. M. Owens, Phys. Rev. B37,603 (1988). 21. Z. X-. Shen, W. E. Spicer, D. M. King, D. S. Dessau, andB. O. Wells, Science 267, 343 (1995). 22. H. M. Meyer and J. H. Weaver, in Physical Properties of High Temperature Superconductors, D. M. Ginsberg, ed., Vol. 2, Chap. 6, World Scientific, Singapore (1990). 23. N. Nücker, H. Romberg, M. Alexander, and J. Fink, in Studies in High Temperature Superconductors, A. V. Narlikar, ed., Nova Sci. Publ., New York (1992).

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4 Electromagnetic Absorption in the Normal State

In this chapter we begin by presenting some background material on the resistivity of metals. Then we discuss the properties of the surface resistance of metals and semiconductors in the normal state and show how measurements of the surface resistance as a function of temperature and magnetic field can be used to explore a range of phenomena, such as metal–insulator transitions, ferromagnetic and antiferromagnetic ordering, and giant magnetoresistive effects.

4.1. METALLIC STATE 4.1.1.

Electrical Conductivity

When a potential difference exists between two points in a conducting wire, an electric field E is established in the wire that exerts the force F = –eE on the conduction electrons of the charge –e. A typical electron is accelerated, undergoes a collision, is accelerated again, collides again, etc. The collision time or average time between collisions is denoted by t; typical values for metallic elements are listed in Table 4.1 (1). The result is that the electron moves forward at an average velocity vav, and the n electrons per unit volume produce the current density J J = nevav

(4.1)

This current density is proportional to the electric field through Ohm’s law J= σ 0 E = E/ρo

(4.2)

where s0 is the dc electrical conductivity and its reciprocal r0 is the dc resistivity. The conductivity is proportional to the collision time t 75

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77

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(4.3) where m is the mass of the electron. When a harmonically varying electric field E=E0 exp(–iwt) acts on the conduction electrons, the conductivity s assumes the form (4.4) withthereal andimaginary parts s1 ands2 (4.5) We see from Table 4.1 that for good metals t @ 2 ×10–13 s at 77 K and t @ 4 × 10–14 s at 273 K. In the microwave region, w/2p = 1010 Hz, so ωτ << 1 and s1 >> s2, making the electrical conductivity real. The data in Table 4.1 show that t decreases as the temperature is increased. Far below and far above the Debye temperature QD, the relaxation time has the following respective limiting temperature dependencies (4.6a) (4.6b) which determine the temperature dependencies of the resistivity.

4.1.2. Surface Resistance In a waveguide or resonant cavity made from a perfect conductor, the boundary conditions require that at the walls the H field be parallel and the E field be perpendicular to the surface. If the walls are made from good but not perfect conductors, then in addition to the transverse field Ht there will be a small transverse field Et at the surface, and the ratio of these fields gives the complex surface impedance Zs (4.7) where Rs is the surface resistance and Xs is the surface reactance. For a metal with a complex electrical conductivity (4.5), the surface impedance Zs (w) is given by (4.8)

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In the normal metallic state at microwave frequencies s1 >> s2, and with the aid of the identity (4.9) we see that the surface resistance Rs equals the surface reactance Xs in accordance with the expression (4.10) where we write σinstead of s1 here and in Eqs. (4.12) and (4.13). The electric and magnetic fields Et and Ht at the surface penetrate into the metal and decay exponentially inside. For the magnetic field component, the field inside Hin is given by expression (4.11) and a similar expression can be written for Ein. The quantity d given by (4.12) is called the skin depth, and it is a measure of how far the fields penetrate into the metal. Figure 4.1 shows how the skin depth and the surface resistance depend on the frequency for the metal copper and the cuprates in the normal state. The factor ix/d in Eq. (4.11) indicates that the exponential decay exp(–x/d) is accompanied by a phase shift. The quantity d is also a measure of the thickness of the surface layer of the metal within which the energy dissipation occurs. Comparing Eqs. (4.10) and (4.12) provides the especially simple expression (4.13) for the surface resistance.

4.1.3. Power Dissipation At the surface of the metal, the magnetic field Ht immediately outside induces a current per unit width Ks directly inside given by Ks = n × Ht

(4.14)

where n is a unit vector pointing outward from the surface so Ks\ is perpendicular to Ht, Near the surface inside, there is a current density J(x, t) = J(x) exp(iwt), which decays with distance in accordance with the expression

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Figure 4. 1. Frequency dependence of the skin depth d and the surface resistance Rs of copper and the cuprate superconductors (HTSC) in their normal state.

(4.15) in analogy with Eq. (4.11). The surface current per unit width is obtained by an integration (4.16) (4.17) For a sample of width w, the total current IT induced by a uniform field Ht at the surface is (4.18) Figure 4.2 helps to explain the relationships (4.18) between J, Ks and IT , The power loss per unit area dP/dA arising from the surface current Ks is given by (4.19) Thus we see that the energy absorbed when electromagnetic radiation is incident on a metal is proportional to the surface resistance. Equations 4.10 and 4.15 provide

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Figure 4.2. Surface layer of thickness d, width w, and length L showing the directions of the current flow (4.18) and of the magnetic field Ht at the surface.

the basis for explaining many of the surface resistance effects in metals and semiconductors.

4.1.4. Temperature Dependencies The resistivity r of a metal is the sum of a contribution from impurity scattering ri and a contribution from photon scattering rph(T)

(4.20) The temperature dependence of the resistivity arises from the photon scattering term rph, and at high temperatures, i.e., above the Debye temperature QD, with the aid of Eqs. (4.3) and (4.6b), r (T) can be written (4.21) The presence of the temperature-independent ri term explains why the resistivity of the sample does not go to zero at 0 K. Above room temperature, impurity scattering is small compared with phonon scattering, so the resistivity becomes linearly dependent on the temperature and thus the surface resistance depends on the temperature as T1/2. This T1/2 temperature dependence of Rs is typical for the cuprates in their normal state. At low temperatures, far below the Debye temperature QD, we have from Eq. (4.6a), t ⇒ T–3. When we include an additional phonon scattering correction factor proportional to T2, the resistivity has the low-temperature dependence (4.22)

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an expression often referred to as the Bloch T5 law. Thus the surface resistance will depend on the temperature as (4.23)

an expression applicable to classical superconductors in their normal states at sufficiently low temperatures.

4.2. SEMICONDUCTORS In an intrinsic semiconductor the resistivity depends on temperature as (4.24)

where D is the band gap. The surface resistance in a semiconductor, from Eq. (4. 10), will thus have the temperature dependence (4.25)

Figure 4.3 shows a measurement of the temperature dependence or the surface resistance of silicon at 9.2 GHz. A surface resistance measurement in an intrinsi-

Figure 4.3. Temperature dependence of the surface resistance of silicon.

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cally semiconducting material can be used to determine the band gap of the material from the slope of a plot of lnRs versus 1/T. If the material is doped, the situation is more complicated because donor and acceptor ionization energies are much less than the band gap and hence dominate the conductivity process for semiconductors.

4.3. METAL-INSULATOR TRANSITION A metal–insulator transition in which there is an abrupt increase in the resistivity of the sample as the temperature is lowered can be observed by a temperature-dependent surface resistance measurement. Figure 4.4 shows the temperature dependence of the surface resistance through the semiconductor-to-insulator transition in nickel sulfide (NiS) at 264 K . This is an example of an antiferromagnetically induced semiconductor-to-metal transition because both the susceptibility and the resistivity increase as the temperature is lowered through the transition (2). We see from the figure that Rs undergoes an abrupt rise from the lower value in the semiconducting state above the transition to a higher value in the insulating state below the transition.

4.4. ANTI FERROMAGNETIC TRANSlTlON Typically an antiferromagnetic transition is accompanied by a pronounced kink in the susceptibility at the Neé1 temperature TN, as shown in Fig. 4.5 for MnO,

Figure 4.4. Temperature dependence of the surface resistance of NiS showing the metal insulator transition close to 260 K.

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Figure 4.5. Temperature dependence of susceptibility x of MnO showing the rise in x with decreasing temperature above TN = 116 K, followed by a drop in the antiferromagnetic state below TN. (Adapted from Nagaimiya et al., Ref. 3.)

which becomes antiferromagnetic at TN = 116 K (3). Since the permeability µ and the susceptibility χ are related bv the expression, (4.26) the surface resistance [Eq. (4.10)] for a magnetic material becomes (4.27) with the result that the permeability and hence the surface resistance display discontinuities at TN. Figure 4.6 shows a measurement of the temperature dependence of the surface resistance at 9.2 GHz in MnO, which exhibits an anomalous peak at 116 K, where the material becomes antiferromagnetic (4).

4.5. FERROMAGNETIC TRANSITION In a ferromagnetic transition, the magnetization M(T) of the sample increases at the Curie temperature, Tc. The magnetization in the ferromagnetic phase has been found empirically to depend on temperature as (5)

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Figure 4.6. Measurement of surface resistance of MnO through the antiferromagnetic transition at TN = 116 K.

(4.28) where M(0) is the magnetization at 0 K and c is a constant. Since by definition χ(T) = M(T)/H, and the permeability and the susceptibility are related by Eq. (4.26), the permeability µ(T) below Tc has a dependence on temperature given by (4.29) where χ0 = M(0)/H is a constant. The temperature dependence T> QD of the surface resistance from Eqs. (4.10) and (4.21) in the ferromagnetic state will then be (4.30) Thus the onset of the ferromagnetic state will be indicated by a rapid increase in the surface resistance of the sample. Figure 4.7 shows the temperature dependence of the surface resistance of gadolinium, which is ferromagnetic below 293 K. In the ferromagnetic state the permeability is dependent on the magnetic field strength, increasing rapidly at low dc magnetic fields, reaching a maximum, and then decreasing somewhat. Figure 4.8 shows that the permeability of gadolinium

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Figure 4.7. Increase in the surface resistance of gadolinium (Gd) with the decrease in temperature below the transition to the ferromagnetic state near 293 K.

Figure 4.8. Plot of relative permeability µ/µ0 of Gd versus dc magnetic field in the ferromagnetic state at 77 K (from Urbain et al., Ref. 6).

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at 77 K plotted against the applied dc magnetic field exhibits this behavior (6). The maximum in the permeability is temperature dependent, shifting to higher magnetic field values at lower temperatures. At constant temperature, the field dependence of the surface resistance will reflect the magnetic field dependence of the permeability. Figure 4.9 shows the surface resistance versus magnetic field strength at 184 K in gadolinium (4). Since Rs is proportional to µ1/2 the initial rise of the surface resistance to its maximum value will generally be nonlinearly dependent on the dc magnetic field strength. Thus an electron spin resonance spectrometer can be used to observe the derivative of the field dependence of electromagnetic absorption. Figure 4.10 shows half of the derivative signal obtained in the ferromagnetic state of Gd recorded at 184 K for upward and downward sweeps of the dc magnetic field (4). The derivative is centered about zero field, and the other half is obtained by reversing the dc field sweep direction. Note that there is hysteresis because on the downward sweep the maximum of the derivative occurs at a lower field. The onset of the ferromagnetic state can also be monitored by measuring the amplitude of the derivative signal at a constant dc magnetic field as a function of temperature. Figure 4.1 1 presents the results of such a measurement in Gd showing that the onset of the existence of the derivative signal occurs at the Curie temperature (4). Since derivative detection is

Figure 4.9. Surface resistance of Gd versus dc magnetic field in the ferromagnetic state at 184 K.

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Figure 4.10. Low field derivative of microwave absorption by Gd versus dc magnetic field in the ferromagnetic state at 184 K for upward and downward sweeps of the magnetic field.

Figure 4.71. Temperature dependence of the derivative signal in Gd through the ferromagnetic transition at 290 K showing the onset of the signal as the field is decreased through the transition.

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89

very sensitive, this technique is a useful tool for studying weakly ferromagnetic materials such as Gd2CuO4 and EuTbCuO4 (7,8).

4.6. MAGNETORESISTANCE Magnetoresistance refers to a phenomenon in which a dc magnetic field causes either an increase or a decrease in the resistance of the sample. Recently there has been a great deal of interest in the subject because of the discovery of materials that display very large magnetoresistive effects referred to as giant or colossal magnetoresistance (CMR). These materials may have a number of application possibilities, such as in devices in magnetic recording heads or sensing elements in magnetometers. The perovskite-like material, LaMnO3, in which La3+ is partially replaced with ions having a valence of 2+ such as Ca, Ba, Sr, Pb, and Cd, has been shown to exhibit very large magnetoresistive effects. The substitution of the 2+ ion for the 3+ ion results in a mixed valence system of Mn3+/Mn4+, thereby creating mobile charge carriers. The unit cell of the crystal is sketched in Fig. 4.12. The system La0.67Ca0.33MnOx displays more than a thousand fold change in resistance with the application of a 6-T dc magnetic field. Figure 4.13 shows how the resistivity of a thin film of the material exhibits a pronounced decrease with increasing values of the dc magnetic field (9). The temperature dependence of the resistivity also

Figure 4.12. Unit cell of the orthorhombically distorted perovskite structure of LaMnO3.

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Figure 4.13. Magnetoresistance behavior of epitaxial La-Ca-Mn-O films (after Jin et al., Ref. 9).

Figure 4.14. Temperature dependence of the resistivity in sintered samples of La-Ca-Mn-O (after Radaeilli et al., Ref. 10).

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Figure 4.15. dc magnetic field dependence of surface resistance of bulk La-Sr-Mn-O in the ferromagnetic state at room temperature.

displays unusual behavior, as shown in Fig. 4.14, which plots the resistivity versus the temperature through the Curie temperature (10). Besides displaying large magnetoresistive effects, La1–xSrxMnO3 undergoes a variety of magnetic transitions that depend on the value of x. For x between 0.2 and 0.5, the material becomes ferromagnetic, with the value of the Curie temperature depending on x. While these systems have not yet been extensively studied by surface resistance methods, such methods should provide a convenient way to investigate the phenomenon (1 1,12). The surface resistance exhibits very large changes in the presence of a dc magnetic field. Figure 4.15 shows a measurement of the dc field dependence of the fractional change in the surface resistance at 9.2 GHz at room temperature in La0.8Sr0.2MnO3. Figure 4.16 presents a measurement of the temperature dependence of the surface resistance showing the onset of the ferromagnetic transition in this sample near 380 K (1 1). Note that there is an initial drop in the surface resistance, followed by an increase. This is in contrast to the temperature dependence of the bulk resistivity, which decreases at the Curie temperature. This difference occurs because the surface resistance depends on two

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Figure 4.16. Temperature dependence of the surface resistance of bulk La-Sr-Mn-O showing the drop in surface resistance at the onset of the ferromagnetic state at 380 K.

parameters, namely, the permeability and the resisitivity. The initial decrease in the surface resistance is due to the temperature-dependent decrease in bulk resisitivity, but as the ferromagnetic alignment progresses, the permeability starts to increase, thus causing an increase in the surface resistance, which is reflected in the appearance of the cusplike behavior shown in Fig. 4.16. The dependence of the surface resistance on both the permeability and the resistivity explains why Rs has a much stronger dependence on magnetic field in the ferromagnetic phase than the bulk resistivity. This strong dependence of the surface resistance on the magnetic field in these materials may have application potential in microwave devices. The ferromagnetic transition can also be detected by the presence of the low field cusp in the microwave absorption derivative signal shown for this material in Fig. 4.17, and its temperature dependence shown in Fig. 4.18 (1 1). These results show that measurements of the dependence of the surface resistance on the temperature and on the dc magnetic field provide a method for characterizing the properties of materials displaying colossal magnetic resistance. The penetration depth at RF frequencies, measured by the LC coil method discussed in Chap. 3, has also been shown to be strongly dependent on the dc magnetic field. Figure 4.19 shows a plot of the measured frequency shift versus dc magnetic field at 10 MHz in La0.7Sr0.3MnO3 at room temperature (13). Equation (3.3) shows that this frequency shift Dw/w0 is proportional to the penetration depth.

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Figure 4.17. Derivative of the microwave absorption versus the dc magnetic field in the ferromagnetic state of bulk La-Sr-Mn-O at room temperature.

Figure 4.18. Temperature dependence of the derivative of the microwave absorption in bulk La-Sr Mn-O showing the onset of the signal at the Curie temperature.

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Figure 4.19. dc magnetic field dependence of frequency shift at 10 MHz in bulk La0.7Sr0.3MnO3 at room temperature. This shift is proportional to the penetration depth.

References 1. C. P. Poole, Jr., H. A. Farach, and R. J. Creswick, Superconductivity, Academic Press, San Diego (1995). 2. J. T. Sparks and T. Komoto, Rev. Mod. Phys. 40, 752 (1968). 3. T. Nagaimiya, K. Yosida, and R. Kubo, Adv. Phys. 4, 1 (1955). 4. E J. Owens (unpublished). 5. B. E. Argyle, S. Charap, and E. W. Pugh, Phys. Rev. 132, 2051 (1963). 6. G. Urbain, P. Weiss, and F. Trombe, Compt. Rend. 200, 2132 (1935). 7. M. D. Sastry, J. K. S. Ajayakumur, R. M. Kadam, G. M. Phatak, and R. M. Iyer, Physica C 170, 41 (1990). 8. B. Oseroff, D. Rao. E. Wrigth, D. C. Vier, S. Hultz, J. D. Thompson, Z. Fisk, S. W. Cheung, M. F. Hundley, and M. Tovar, Phys. Rev. B41,1934 (1990). 9. S. Jin, M. McCormack, T. H. Tiefel, and R. Ramesh, J. Appl. Phys. 78, 6929 (1994). 10. P. G. Radaeilli, D. E. Cox, M. Marezio, S. W. Cheong, P. E. Schiffer, and A. P. Ramirez, Phys. Rev. Lett. 75, 4488 (1995). 11. E J. Owens, J. Phys. Chem. Solids 58, 1311 (1997). 12. S. E. Lofland, S. M. Bhagat, S. D. \ Y. M. Muskovskii, S. G. Karabashev, and A. M. Balbashov, J. Appl. Phys. 80, 3592 (1996). 13. F. J. Owens, J. Appl. Phys. 82, 3054 (1997).

5 Zero Magnetic Field Microwave Absorption

In this chapter we discuss the absorption of microwave radiation in the superconducting state in the absence of any applied magnetic field.

5.1 ELECTROMAGNETIC ABSORPTION AND THE TWO-FLUID MODEL As the temperature of a superconducting sample is lowered below Tc there is a pronounced reduction in the surface resistance and thus in the absorption of electromagnetic energy. Figure 5.1 shows a representative microwave measurement of the surface resistance at 9.2 GHz in Hg-Pb-Ba-Ca-Cu-O (1). The earliest attempts to explain this reduction in electromagnetic absorption in superconductors were based on the expression (Eq. 1.9) for the penetration depth derived from the London theory using the two-fluid model. Qualitatively, this model envisions the superconducting state as involving a mixture of normal nn and superconducting ns electrons, where the sum nn + ns equals the total number of conduction electrons. We now know that the superelectrons are Cooper pairs. These superconducting charge carriers are not scattered by phonons, they experience no resistance and they cannot absorb electromagnetic radiation. As the temperature is lowered below Tc , the number of superconducting charge carriers increases; the number of normal electrons decreases; and consequently the London penetration depth given by Eq. (1.9) decreases and the sample becomes less absorbing to electromagnetic radiation. In this model the temperature dependence of the reduction in the penetration depth is given by Eq. (1.17). Measurements of the temperature dependence of the penetration depth in tin by Pippard (2) show that it could be described well by Eq. (1.17) at low temperatures but deviated from this dependence as T approached Tc. 95

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At the time of the development of the theory, researchers did not know about the superconducting gap. The presence of a gap opening up with a temperature dependence of (T – Tc)1/2 significantly enhances the absorption process near the transition because at some temperature near Tc, depending on the energy hv of the incident radiation, the gap will equal the energy of the radiation and absorption will occur because of excitations across the gap. The importance of the role of the energy gap in the absorption of electromagnetic radiation in the superconducting state is seen in measurements of the energy or frequency dependence of the surface resistance at constant temperature. Figure 5.2 is a plot of the surface resistance in aluminum at T/Tc = 0.7 in the superconducting state normalized to a value above Tc and plotted versus hw/kBTc. The sharp increase in the surface resistance just above hw/kBTc = 2.5 occurs because the photons of the incident radiation are causing transitions across the gap, in effect breaking up Cooper pairs to form normal carriers which can then absorb electromagnetic energy. This kind of measurement can be used to obtain the value of the superconducting gap.

Figure 5.1. Temperature dependence ofsurface resistance through the transition to the superconducting state in a granular pellet of Hg-Pb-Ba-Cu-O.

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Figure 5.2 Frequency dependence of the ratio of the microwave absorption determined surface resistance in the superconducting state to that in the normal state in aluminum at a constant temperature of 0.83 K, where Tc = 1.18 K.

5.2. SURFACE IMPEDANCE In this section we discuss the theory of electromagnetic absorption in the “local limit” where the current density J is determined by the local value of the vector potential A. This limit applies when the penetration depth is greater than the coherence length, corresponding to the condition for Type II superconductivity. Since the copper oxides are Type II superconductors, with k = l/x >> 1, a description of electromagnetic absorption in the local limit applies. The surface impedance Zs of a metal or superconductor, which was introduced in Sect. 4.1.2, is a complex quantity with real and imaginary parts Zs = Rs + iXs

(5.1)

where Rs is the surface resistance and Xs is the associated reactance. The complex electrical conductivity s = s1 + is2

(5.2)

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is related to the surface impedance bv the expression (3,4) (5.3) We showed in Sect. 4.1.2 that for a metal in the normal state s1 >> s2, which gives (5.4) The reason s1 is greater than s2 is that normal electrons have very low inertia, responding promptly to an ac field, and thus the current induced by the ac electromagnetic field is close to being in phase with the ac field, and there is little inductive reactance. For superelectrons s1 is very large, but superelectrons have such a pronounced inductive response that s1 << s2 in the microwave region below Tc. If Eq. (5.3) is expanded in a power series for this limit, we obtain the approximate expression (5.5) Recalling Eq. (4.3) for the dc conductivity so of normal state electrons, we make the assumption that an analogous expression applies for superelectrons (5.6) and we write t = 1/w to obtain s and Zs from Eqs. (5.2) and (5.3), respectively (5.7) (5.8) where use was made of Eq. (1.9) for the penetration depth. This gives for the resistive and the reactive parts of the surface impedance (5.9a) (5.9b) The surface resistance Rs measures the absorption due to electronic excitations and the thermally activated dissociation of Cooper pairs, and the surface reactance Xs is the inductive response of the superconducting condensate. Thus a measurement of the surface reactance allows a direct determination of the penetration depth l, and a measurement of both the surface resistance and reactance provides a deter-

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mination of s1. Once the penetration depth is known, the superfluid density ns can be calculated from Eq. (1.9), namely (5.10)

5.3. ELECTROMAGNETIC ABSORPTION IN THE BCS THEORY There are two features of the BCS theory that bear strongly on the nature of electromagnetic absorption in the superconducting state. One aspect, which has been mentioned above, is the existence of the superconducting energy gap. For a photon hv of energy less than the gap at 0 K, D(0), the absorption process will be enhanced near Tc when the gap has closed to a value comparable to hv, allowing excitations across the gap. At low temperatures (T < 1/2 Tc), the temperature dependence of the penetration depth, which by Eq. (5.10) is inversely proportional to the square root of the superfluid density ns, can be described by thermally activated behavior in the BCS formalism, i.e, (5) (5.11) and, (5.12) The second important characteristic of the BCS model that has a major influence on electromagnetic absorbing properties in the superconducting state is the existence of singularities in the density of states at the edge of the superconducting gap. In the weak coupling limit discussed in Sect. 2.2.2, the density of states Ns (E) in the superconducting state is given by (6) (5.13) where Nn(E) is the density of states in the normal state. Figure 5.3 gives a schematic of the density states in a superconductor. Near Tc as the superconducting gap closes and approaches the energy of the incident photon, the photon energy overlaps the energy region where there is a rapid increase in the density of states, and thus an increase in the number of superfluid electrons available for excitation. This results in a peak in the temperature dependence of the conductivity s1 near Tc as shown in Fig. 5.4. This peak is known as the coherence peak. In the temperature region between 1/2 Tc and Tc, the BCS formalism does not yield simple analytical expressions for the temperature dependence of l, Rs, and s. However, computergenerated numerical solutions have been obtained for a given set of parameters which will be shown to illustrate the BCS predictions (7). Figure 5.5 presents a plot

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SUPERCONDUCTOR

Figure 5.3, Density of states in the neighborhood of the superconducting gap in a BCS superconductor.

Figure 5.4. Real part of the conductivity s1 of a BCS superconductor plotted against the temperature assuming a Tc of 90 K (after Bonn and Hardy, Ref. 4).

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Figure 5.5. Plot of l2(0)/l2(T) versus temperature for a BCS superconductor (after Bonn and Hardy, Ref. 4).

Figure 5.6. Plot of the logarithm of surface resistance versus temperature for a BCS superconductor (after Bonn and Hardy, Ref. 4).

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of l2(0)/l2(T) versus temperature calculated on a BCS model. The existence of an energy gap leads to a leveling off, with l(T) ⇒ l(0) at low temperature, as shown in the figure. The surface resistance shown in Fig. 5.6 exhibits a rapid drop just below Tc due to the onset of screening by the supercurrents, and exponential behavior at lower temperatures. Near Tc , the coherence peak is not evident in the temperature dependence of the surface resistance because it is dominated by the rapid increase in the penetration depth as the temperature approaches Tc from below.

5.4. COPPER OXIDES 5.4.1. Penetration Depth Measurements The penetration depth and surface resistance have been measured in high-quality single crystals of Y-Ba-Cu-O, and ordinarily reports from different research groups are in agreement with each other. Figure 5.7 shows a representative result for a high-quality single crystal of YBa2Cu3O6.95 for the ab orientation of the

Figure 5.7. Measured dependence of l2(0)/l2(T) on the temperature for a single crystal of YBa2Cu3O6.95.

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orthorhombic unit cell (8). The most striking feature of the data is that at low temperature they do not display the leveling-off behavior of Fig. 5.5 that is predicted by the BCS theory; rather, l2 (0)/l2(T) is linearly dependent on temperature below about ½Tc. At much lower temperatures there is a deviation from the linear dependence on T, with an observed dependence closer to T2. The crossover temperature T* from the T to the T2 behavior depends on the impurity content of the crystal. Measurements have also been made on epitaxial films where the crossover to the T2 dependence occurs at a higher temperature, near 30 K (9). The difference between the crystals and the films has been attributed to a higher impurity content of the films. The effect of impurity content on the functional dependence of the penetration depth has been investigated in single crystals. Figure 5.8 shows a plot of the l(T)/l (0) in single crystals in which some of the copper ions have been replaced by zinc, corresponding to the formula YBa2(Cu1–xZnx)3O6.95 (10). The data show that the crossover temperature from the T dependence to the T2 dependence increases as the impurity content increases. Because in the copper oxides the superconducting current flows primarily in the ab plane, the penetration depth is expected to be anisotropic. The measurement of the temperature dependence of l2(0) /l2(T) parallel and perpendicular to the c-axis presented in Fig. 5.9 shows that l^ c(T) > l || c(T) (11). Another interesting feature of the penetration depth data is their behavior near Tc. Figure 5.10 is a plot of l3(0)/l3(T) versus temperature near Tc. The almost straight line indicates a cubic dependence of 1/λ on temperature, which is consistent with critical behavior of a three-dimensional superfluid (12).

Figure 5.8. Temperature dependence of the penetration depth l(T) of Zn-doped Y-Ba-Cu-O for 0.31% Zn for 0.15% Zn (◊), and for no Zn ( ) present (from Bonn et al., Ref. 10).

•

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Figure 5.9. Plot of l2(0)/l2(T) versus the temperature parallel to the c-axis (•) and parallel to the a-axis (D) of Y-Ba-Cu-O (from Hardy et al., Ref. 11).

Figure 5.10. Plot of l3(0)/l3(T) of Y-Ba-Cu-O versus the temperature in the neighborhood of T c (from Kamal et al., Ref. 12).

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The most significant result of these studies is that the temperature dependence of l(T) for the copper oxide superconductors does not follow the predictions of the phonon-mediated BCS theory at low temperatures. Does this then mean the mechanism of superconductivity in the copper oxides is not phonon based? The answer to this question requires further research. There are other explanations for the linear dependence of 1/l2(T) on temperature below Tc/2. For example, it has been shown that if some of the copper oxide layers are not superconducting but are sandwiched between layers that do superconduct, a linear temperature dependence of 1/l2(T) could be expected. We will return to this question later in the context of the s-wave versus d-wave question.

5.4.2. Surface Resistance Measurements As seen in Eq. (5.8), the surface resistance depends on s1. We know from Eq. (4.5) that s1(T) can be written in the form (5.14) where the scattering time t depends on the temperature in accordance with Eq. (4.6), and so clearly Rs will depend on the impurity content because impurities affect the scattering time t. In fact, substitution of impurities will decrease the scattering time and thus decrease the microwave losses. Somewhat contrary to intuition, samples with higher concentrations of impurities will have lower surface resistance in the superconducting state. Measurements of the surface resistance in the copper oxides generally confirm the importance of the role of impurities. Figure 5.11 shows that the surface resistance in the ab plane of a high-quality pure single crystal of YBa2CU3O6.95 exhibits a rapid drop at Tc followed by a broad peak at intermediate temperatures and then a decrease at lower temperatures (8). The figure also shows that the doped crystal has a lower surface resistance and lacks the broad peak. The most prominent feature reported by different workers is the broad peak at 40 K. Measurements of the temperature dependence of l do not exhibit such a peak so it is concluded that this peak must be associated with s1. Since this peak does not occur near Tc, it is clearly not associated with the singularity in the density of states near Tc that is responsible for the coherence peak in BCS superconductors. The broad peak has been attributed to a temperature-dependent increase in the quasi-particle lifetime. If t in Eq. (5.14) is temperature dependent then a peak can appear because of the combined effect of the normal fluid lifetime that increases with temperature and the normal fluid density that decreases with temperature. It is known that the impurity content strongly affects t, and the lower curve in Fig. 5.11 shows how the substitution of 0.75% Ni for Cu in Y-Ba-Cu-O significantly lowers the surface resistance in the region of the broad peak. The prediction of Eq. (5.9) that the surface resistance in the superconducting state increases with the frequency of measurement is confirmed by the data plotted in Fig. 5.12 (10).

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Figure 5.11. Temperature dependence of the surface resistance of a pure crystal of Y-Ba-cu-O showing a broad peak below Tc and in a doped crystal (∆ ) in which 0.75% of the Cu ions are replaced by Ni (from Bonn et al., Ref. 8).

Figure 5.12. Temperature dependence of the logarithm of the surface resistance in the ab plane of a (from Bonn et al., Ref. crystal of Y-Ba-Cu-Oat two differentfrequencies: 34.8 GHz ( )and 3.8 GHz 10).

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107

5.4.3. Penetration Depth Measurements as a Probe of s- and d-Wave Symmetry One of the important unresolved questions involving the cuprates is the nature of the wave function describing the hole Cooper pairs. For a standard BCS superconductor, the wave function is isotropic, corresponding to an s-state (L = 0). Some of the proposed mechanisms for pairing of the holes, such as those of a magnetic type, require the Cooper pair wave function to be d-like, having L = 2. BCS s-wave pairing results in an exponentiallly activated temperature dependence of many dynamic and thermodynamic parameters. One such parameter is the penetration depth, and at low temperature it reflects changes in the density of pairs that are respcnsible for screening electromagnetic fields. In s-wave weak coupling BCS theory, the change in the penetration depth with temperature is given by Eq. (5.11). However, for d-wave pairing, the penetration depth has a different dependence on temperature (1 1), and for Dl = l(T) – l(0) we have (5.15) where D0 is the maximum value of the gap that occurs at T = 0. Thus for d-wave pairing, the penetration depth is predicted to have a linear dependence on temperature well below Tc whereas by contrast s-wave pairing is not linear. These predictions have motivated detailed measurements of the temperature dependence of l(T) at low temperatures in order to address the question of s-wave versus d-wave pairing. A difficulty with relying on penetration depth measurements to clarify the nature of the pairing is the strong dependence of l(T) on the impurity content, as demonstrated by the Y-Ba-Cu-O single-crystal data presented in Fig. 5.8. Theoretical treatments of systems having d-wave gaps and strong impurity scattering yield a penetration depth that depends on T2 below a crossover temperature T*, above which the dependence is linear (13). The data in Fig. 5.8 do show a linear dependence of l on the temperature for T greater than a temperature T*, and a nonlinear dependence below T* that is in qualitative agreement with these predictions. However, the strong influence of impurities on the temperature dependence does not make this unequivocal evidence for d-wave pairing.

5.4.4. Electromagnetic Absorption Due to Fluctuations The transformation to the superconducting state in the absence of an applied magnetic field is a second-order phase transition. Common to second-order transitions is the existence of pretransitional fluctuations. A somewhat oversimplified way to view superconducting fluctuations is to consider them as small transitory droplets of superconductivity with the radius of a coherence length that form and dissipate in a short time. Superconducting fluctuations above Tc manifest themselves by small decreases in the resistance or susceptibility occurring above the

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main transformation to the superconducting state. Their existence can be seen clearly in the temperature dependence of the resistance of the Hg-Pb-Ba-Ca-Cu-O superconductor shown in Fig. 1.2. The main transition to the superconducting state denoted by the half value of the resistance is 133 K; however, a slight deviation due to fluctuations from the linear decrease of the normal state can be seen as high as 150 K. The lower the dimensionality of the system sustaining superconductivity, the higher the temperature above Tc at which fluctuations appear. Just as the bulk resistance starts to deviate from its normal decrease above Tc, so it is with the surface resistance. In the fluctuation regime the surface resistance Rs has been shown to depend on the temperature as (3) Rs~(T–Rc)”

(5.16)

where the exponent n depends on the dimensions of the system, expected values being 1/2 for a three-dimensional system, 1 for a two-dimensional system, and 3/2 for a one-dimensional system. Surface resistance measurements have not been widely employed to study fluctuational effects, but they may be a very useful tool for this because direct electrical contact with the sample is not required. Small deviations in the resistance near Tc can be masked by surface contact resistance effects. Lehoczky and Bruscoe (14) studied the temperature dependence of the surface resistance above Tc at 24 GHz in films of lead whose thickness was less than the coherence length, making them in effect two-dimensional superconduc-

Figure 5.13. Plot of surface resistance (normalized to normal state surface resistance) versus reduced temperature, T/Tc, in films of lead with thickness less than the coherence length, making them essentially two dimensional (from Lehoczyy and Briscoe, Ref. 14).

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Figure 5.14. Sketch of the unit cell of the one-dimensional organic conductor TTF-TCNQ showing a TCNQ molecule C6H4(C3N2)2 in the center and TTF molecules (C3S2H2)2 along the cell edges [from Owens and Poole, Ref. 17, p. 174].

tors. Figure 5.13 shows Rs/Rsn versus T above Tc in lead, showing evidence for fluctuations. It is interesting that superconducting fluctuations were observed by microwave absorption in the mid-1970s in the one-dimensional organic conductor TTF-TCNQ (15). The unit cell shown in Fig. 5.14 consists of one-dimensional chains of donor and acceptor molecules. Measurements of the microwave absorption at 10 GHz in these materials showed large drops in the absorption to a minimum in the vicinity of 50 K, followed by a subsequent rapid increase, as shown in Fig. 5.15 (15). Although there was some controversy concerning the origin of these surface resistance cusps at the time, it is now generally believed that they are a result of superconducting fluctuations followed by a charge density wave-driven metal insulator transition. It is widely accepted that purely one-dimensional systems cannot sustain conventional superconductivity because of the existence of charge density wave instabilities, but apparently they do show fluctuations as though they want to become superconducting. In fact, the organic material TTF-TCNQ was eventually made to be a superconductor by inserting additional molecules into the lattice, which increased the interaction between the chains, thus raising the dimensionality above one. Clearly this provides an approach for synthesizing higher temperature

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Figure 5.15. Surface resistance (normalized to the value R n at 300 K) versus temperature of the organic superconductor TTF-TCNQ (from Cohen et al., Ref. 15).

superconductors. One might look for one-dimensional materials that display superconducting fluctuations and then seek appropriate dopants to enhance the interaction between the chains. For example, the ladder phase materials discussed in Chap. 2 contain copper oxide ligands that are of a dimension between one and two. Indeed, large microwave absorption cusps similar to those observed in TTF-TCNQ have been noticed at temperatures as high as 280 K (16) in the ladder phase-type material Sr14–xCaxCu24O41+d. The structure of this material consists of alternating parallel planes and chains of copper oxide which have the ladder structure shown in Fig. 2.12. Figure 5.16 shows a sharp drop in surface resistance in the metallic region between 230 K and 270 K where the superconducting fluctuations occur, the onset of the metal insulator transition at the 230 K minimum, and a rise in Rs in the insulator phase below 230 K. The onset of the drop in the surface resistance shifts to lower temperature with increased calcium content x, as shown in Fig. 5.17 (16). Superconductivity has been observed in this material when it has the very high calcium content of x = 13.6. The Tc for this calcium content, the solid circular point in Fig. 5.17, lies nicely on the plot of the temperature of the surface resistance minimum versus calcium content, suggesting that these surface resistance drops could well be associated with superconductivity.

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Figure 5.16. Plot of the temperature dependence of the surface resistance RS in the ladder phase Sr14–x CaxCu24O41.

Figure 5.17.Plot of the temperature of surface resistance drops in the ladder phase material Sr14–x CaxCu24O41 , versus calcium content x. The large dark circle at the lower right designates the superconducting transition temperature of the sample with x = 13.6.

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References 1. E J. Owens, A. G. Rider and Z. Iqbal, Physica C233, 30 (1944). 2. A. B. Pipard, Proc. Roy. Soc. London A216, 547 (1953). 3. M. Tinkham, Introduction to Superconduclivily, McGraw-Hill, New York (1996). 4. D. A. Bonn and W. N. Hardy in Physical Properties of High Temperature Superconductors, Vol. V, D. M. Ginsberg, ed., World Scientific, River Edge, N.J. (1996). 5. D. C. Mattis and J. Bardeen, Phys. Rev. B111, 412 (1958). 6. J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev. 106, 162 (1957). 7. J. Halbritter, Z. Phys. 266, 209 (1977). 8. D. A. Bonn, P. Dosanjh, R. Liang and W. N. Hardy, Phys. Rev. Lett. 68, 2390 (1992). 9. Z. Ma, R. C. Taber, L. W. Lombardo, A. Kapitulnik, M. R. Beasley, P. Merchant, C. B. Eom, S. Y. Hou, and J. M. Phillips, Phys. Rev. Lett. 71, 781 (1993). 10. D. A. Bonn, S. Kamal, K. Zhang, R. Liang, D. J. Baar, E. Klein and W. N. Hardy, Phys. Rev. B50, 4051(1994). 11. W. N. Hardy, D. A. Bonn, R. Liang, S. Kamal and K. Zhang, in Proc. Seventh Int. Symp. on Superconductivity(1966). 12. S. Kamal, D. A. Bonn, N. Goldenfeld, P. J. Hirschfeld, R. Liang, and W. Hardy, Phys. Rev. Lett. 73,1845(1994). 13. D. J. Scalapino, Physics Reports 350, 331 (1995). 14. S. L. Lehoczyy and C. V. Briscoe, Phys. Rev. Lett. 23, 695 (1969). 15. M. J. Cohen, L. B. Coleman, A. F. Garito, and A. J. Heeger, Phys. Rev. B10, 1298 (1974). 16. F. J. Owens, Z. Iqbal, and D. Kirven, Physica C267, 147 (1996). 17. E J. Owens and C. P. Poole, Jr. The New Superconductors, Plenum Press, New York (1996).

6 Low Magnetic Field-Induced Microwave Absorption

6.1. INTRODUCTION When a dc magnetic field is applied to a material in the superconducting state, the surface resistance increases. From the point of view of the absorption mechanism, three regions of dependence on the applied magnetic field can be distinguished. Figure 6.1 is a generalized plot of the surface resistance behavior as a

Figure 6.1. Generalized diagram of the effect of an applied magnetic field B on the surface resistance R s showing three major regions in which the increase in Rs arises from different mechanisms. 113

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function of the applied field. Initially there is a very rapid increase at low fields, in the vicinity of the lower critical field Bcl, followed by a more slowly increasing absorption as a function of the dc field strength. At higher fields, above the irreversibility line, the dissipation increases somewhat more strongly with dc magnetic fields. This chapter is concerned with the nature of the absorption at the lowest applied fields.

6.2. PROPERTIES OF LOW MAGNETIC FIELD ABSORPTION DERIVATIVE The absorption in the low dc magnetic field limit was first observed as a derivativeof the absorption versusdcmagneticfieldusing anelectronparamagnetic resonance spectrometer (1-5). The intensity of the derivative increased as the temperature was lowered below Tc. The field position of the center of the derivative was independent ofthe microwavefrequency, indicatingthatthe signal did notarise from a resonant absorption. The existence of a derivative signal having a phase opposite to a resonant signal implied a strong nonlinear increase in absorbed microwave energy as afunction ofincreasing magnetic field atquite low fields, less than 10 mT. There had actually been an observation of the derivative in the organic superconductor (TMTSF)2ClO4 by a French group in 1983, but no further studies were made nor was a mechanism proposed (6). This low-field microwave absorption (LFMA) in the copper oxides has been the subject of investigation by numerous groups. Many studies of its properties have been reported and several mechanisms have been proposed to explain it. Here we review the properties of the absorption and discuss the mechanism responsible for it. Because the absorption increases nonlinearly with magnetic field, the derivative of the absorption versus dc field can be readily detected with an EPR spectrometer. A comprehensive list of early observations is contained in Ref. 7. Figure 6.2 is a typical result showing one half of the absorption derivative signal plotted against increasing magnetic field in a ceramic sample of the N = 3 superconductor Bi1.5 Pb0.5Sr2Ca2Cu3O10 for sweeps of the magnetic field from zero to 6 mT and back at two different temperatures (8). The other half of the derivative signal is obtained by sweeping in the negative B direction. The modulation for this observation was 1 mT. The two derivatives shown in Fig. 6.2 illustrate many of the properties of the absorption. The figure shows that the intensity increases as the temperature is lowered below Tc. Note that this is in contrast to the surface resistance, which decreases as T is lowered below Tc. Figure 6.3 is a plot of the temperature dependence of the intensity of the derivative signal in this material. These data are fairly representative of the temperature dependence observed in pure single-phase samples. As the temperature is lowered, the negative peak of the line shifts to higher

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Figure 6.2. Low dc magnetic field microwave absorption derivative signal for the N = 3, Bi-Pb-Sr-CaCu-O superconductor for upfield and downfield scans at two different temperatures.

Figure 6.3. Temperature dependence of the microwave absorption derivative intensity measured at the dc magnetic field of 1 mT in N = 3, Bi-Pb-Sr-Ca-Cu-O material.

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magnetic field and the line width increases. Figure 6.4 shows a plot of the line width and field at the peak versus temperature. The LFMA measurements that we have described were all made by sweeping to the same magnetic field at each temperature, in this case 6 mT. This is necessary because the peak of the line and its width are both affected by the magnetic history of the sample and by the method of application of the dc magnetic field, such as whether it is applied before or after the sample is cooled below Tc. The derivative of the absorption signal displays a hystersis that is evident in Fig. 6.2, which presents a sweep upfield from zero followed by a return to zero field. We see from the figure that the signal during the return downfield sweep is weaker than it was for the initial upfield sweep. This hysteresis depends on the maximum range of the sweep, and it is more pronounced for larger applied fields. The magnitude of the hysteresis is defined as the shift in the magnetic field position of the negative peak for an upfield scan followed by a downfield one. Figure 6.5 shows how this magnitude depends on the maximum value of the magnetic field applied to the Bi1.5Pb0.5Sr2Ca2Cu3O10 sample at 77 K. There is also hysteresis in the line width in that it appears broader on the return sweep. It is also noted that on the sweep up after the application of a magnetic field, the peak of the line occurs at a higher

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Figure 6.4. Plot of line width ( ) and field position at the peak of the microwave absorption derivative signal plotted against temperahue for the N = 3, Bi-Pb-Sr-Ca-Cu-O superconductor.

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Figure 6.5. Magnitude of microwave absorption derivative signal hysteresis versus maximum dc magnetic field applied at 77 K for the superconductor N = 3, Bi-Pb-Sr-Ca-Cu-O,

magnetic field position and this upward shift increases with larger maximum applied fields. Figure 6.6 shows this effect. The magnitude of the upward shift corresponds to the magnitude of the hysteresis, for a sweep up to a given field and back. The hysteresis for a sweep to a constant magnetic field also increases as the temperature is lowered below Tc. The low-field microwave absorption displays a time dependence after the removal of a magnetic field (9). This measurement is made by cooling the sample below Tc in a zero magnetic field and then applying a field of 100 mT. The field is then removed and the LFMA is rapidly recorded on the down scan for different elapsed times after the removal of the field. Figure 6.7 shows a representative result for the high-temperature superconductor Tl0.5Pb0.5Sr2CaCu2O7 at 77 K. The intensity grows as a function of time, and the position of the line shifts to the lower field. We will see later that many of the properties of the absorption are a result of the flux-trapping behavior of the sample, and measurements of this type can clarify the nature of this flux trapping. The absorption derivative signal has also been observed in crystals and thin films (10–12). In the case of films, the properties of the absorption depend to some extent on the film quality. At temperatures close to Tc , films that are not perfectly

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Figure 6.6. Magnetic field position of the center of the absorption derivative signal of the N = 3, Bi-Pb-Sr-Ca-Cu-O superconductor at 77 K plotted against the maximum of the dc field sweep applied before recording the scan of the derivative signal.

crystalline produce absorption derivatives that resemble those from composites. The major difference in the films is that the absorption signal depends on the orientation of the applied field with respect to the normal to the surface of the film. Figure 6.8 is a plot of the intensity of the absorption at 77 K versus the orientation of the applied magnetic field with respect to the normal to a film of YBa2Cu3O7–x deposited on an MgO single crystal. The c-axis of the orthorhombic cell is perpendicular to the film surface. The intensity decreases as the dc magnetic field is rotated away from the perpendicular to the surface. This particular measurement was made on a film in which most of the crystals are arranged so that c is perpendicular to the substrate but in which there is some misalignment about the c-axis (12). In a high-quality film, no low dc magnetic field-induced derivative signal is detected when the dc magnetic field is parallel to the surface of the film. It is also observed in the films that the magnitude of the hysteresis depends on the orientation of the dc magnetic field, being larger when the magnetic field is perpendicular to the film surface. Figure 6.9 shows a plot of the hysteresis at 77 K versus the applied field in a film of YBa2Cu3O7–x for the dc field parallel and perpendicular to the c axis. Results similar to these have been observed in single crystals of the cuprates at high modulations and at temperatures close to Tc.

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Figure 6.7. One half of the microwave absorption derivative signal in TI-Pb-Sr-Ca-Cu-O at 77 K for an upfield scan (top) and downfield scans made immediately (0 s) and 78 s after the removal of a 100-mT dc magnetic field.

At low temperatures and low modulations, fine structure is sometimes observed superimposed on the broad lines from crystals, and in some instances well-defined sharp lines are observed instead of a broad line (13–15). Figure 6.10 shows a spectrum obtained for a single crystal of YBa2Cu3O7–x by Vichery et al. (15). This spectrum is observed at low magnetic fields using modulations of 1.25 µT. The separation between adjacent lines has the constant value of 7 µT. The spacing between the lines depends on the orientation of the dc magnetic field as (cos q)–1. The q = 0 orientation gives the minimum separation, and does not correspond to any significant direction in the crystal. Sometimes several sets of lines distinguished by different spacings between them and different directions in which q = 0 were observed. This was attributed to the presence of Josephson junction loops with planes oriented in several particular directions.

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Figure 6.8. Intensity of microwave absorption derivative signal in a thin film of Y-Ba-Cu-O at 77 K versus orientation of dc magnetic field with respect to the perpendicular to film surface.

6.3. PROPERTIES OF LOW-FIELD DIRECT MICROWAVE A BSORPTlON The previous section provided details on the LFMA signal obtained when the scanning magnetic field is modulated. It is not clear how to sort out the effects due to the scanning applied field and those arising from the modulation. To obtain a clearer understanding of the microwave absortion process, the modulation was eliminated and direct microwave detection was employed (16,17). To accomplish this, a 9.3-GHz home-built superheterodyne spectrometer was employed, and detection was accomplished at the 30-MHz intermediate frequency by mixing the main 9.3-GHz signal with that from a Gunn diode local oscillator. A digital signal analyzer was employed to average signals obtained from repeated magnet scans. A typical signal obtained from a zero field-cooled (ZFC) granular Y-Ba-Cu-O at 5 K during a magnet scan from –2 mT to +2 mT is presented in Fig. 6.1 1. The small shift of the minimum from zero is due to a small hysteresis of the magnet. Cycling the applied field over a range of less than 2 mT caused the absorption curve to repeat itself. However, if the applied field was increased above the critical field Bcl , the absorption signal exhibited hysteresis, which was attributed to the penetration into and pinning of flux in the bulk of the sample. Subsequent low-field sweeps

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Figure 6.9. Effect of orientation of the dc magnetic field with respect to Y-Ba-Cu-O film surface on hysteresis of derivative signal at 77 K: parallel and perpendicular ( ) to the surface.

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exhibited no hysteresis. When the sample was field cooled (FC), the minimum in the absorption shifted to the value Bm in the direction of the cooling field in the manner illustrated in Fig. 6.12. The observed shift in the minimum is proportional to the cooling field up to BFC of about 40 mT, and above this value the curve levels off in the manner illustrated in Fig. 6.13. Thus the trapped flux is proportional to BFC for small cooling fields, and it saturates for high cooling fields. To test the behavior of the remnant magnetization of the field-cooled samples, the samples were reduced to zero field as before, rotated through an angle q relative to the cooling field direction, and then scanned in the field. Figure 6.14 presents absorption signals obtained for the rotation angles q = 0°, 90°, and 180°, and Fig. 6.15 shows how the minimum in the absorption curves, Bmin, has a cosine dependence on the angle B min(q) = Bm cos q

(6.1)

The dependence of the absorption signal χ'' plotted in Figs. 6.1 1,6.12, and 6.14 on the applied fieldB and the angle θ could be approximated by the following empirical equation

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Figure 6.10. Low magnetic field multiline microwave absorption derivative signal obtained for a single crystal of Y-Ba-Cu-O. (From Vichery et al., Ref. 15.)

(6.2) where from Fig. 6.13 Bm ≈ 2 mT for BFC = 4 mT. The parameter x”0 establishes the zero for the ordinate scale, and the factor A converts field values to susceptibility units. The empirical constant B0 accounts for the rounding at the bottom of the absorption curves on Figs. 6.1 1,6.12, and 6.14.

Figure 6.11.Direct microwave absorption signal for a zero field-cooled sample of granularY-Ba-Cu-O. (From Pertile et al., Ref. 17.)

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Figure 6.12. Direct microwave absorption signals of field-cooled granular Y-Ba-Cu-O scanned from 0 to 8 mT. Spectra are shown for field coolings carried out at BFC = 8, 10, and 12 mT, (From Pertile et al., Ref. 17.)

Figure 6.13. Dependence of the field Bm for minimum absorption on the field cooling field B FC at the temperature of 10 K. (From Pertile et al., Ref. 17.)

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Figure 6.14. Direct absorption spectra of Y-Ba-Cu-O taken after 2-mT field cooling and measured at the angles q = 0°, 90°, and 180° relative to the field cooling direction. (From Mzoughi et al., Ref. 16.)

Figure 6.15. Angular dependence of the minimum field B min normalized relative to B min (0) = Bm on the angle q relative to the field cooling direction for Y-Ba-Cu-O cooled in 2 mT to 5 K. The data fit a plot of cos q. (From Mzoughi et al., Ref. 16.)

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6.4. ORIGIN OF LOW MAGNETIC FIELD DERIVATIVE SIGNAL 6.4. 1. Loops and Josephson Junctions Consider a small loop of superconducting wire with a cross-sectional area A of 1 µm2 held at a temperature below Tc. If a dc magnetic field is applied perpendicular to the loop, as shown in Fig. 6.16a, and slowly increased in magnitude, the magnetic flux F = BA through the loop will not increase continuously, but will increase in quantum steps F0 in the manner shown in Fig. 6.16b, where F0= h/2e is the quantum of flux. When the applied flux BA equals an integral number of fluxons nF0, there is a jump in the flux F through the loop. This jump takes place in a time Dt on the order of 10–12 s and produces a voltage V = –dF/dt = –F0/Dt in the loop, which in turn produces a current. Because Dt is small, the induced current will be large, and can exceed the critical current of the superconductor. This will momentarily remove the superconducting state from the loop and there will be a pulse of normal current. If microwave radiation were incident on the loop as the dc magnetic field was increased, every time a flux jump occurred the pulse of normal current would result in increased microwave absorp-

Figure 6.16. (a) Square current loop of area A equal to 1 µm2. (b) Flux passing through this loop as a function of the applied dc magnetic field. (c) Surface resistance determined by microwave absorption versus applied field for the same loop.

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tion. The LFMA spectrum as a function ofdc magnetic field would look like a series of sharp lines, as shown in Fig. 6.16c. The separation between the lines would be F0/A corresponding to the loop area A = F0/DB

(6.3)

Note the resemblance of this spectrum in Fig. 6.16c to that shown in Fig. 6.10 for a superconducting single crystal at low applied magnetic fields. This suggests that micron-sized current loops with the areaA given by Eq. (6.3) may be the cause of the absorption lines in Fig. 6.10. Setting DB = 7 µT and F0 = 2.07 mT(µm)2, we obtain for the area of the loop A = 300 (µm)2

(6.4)

which is a reasonable value. In order to understand how these loops can arise, it is necessary to grasp the nature of a Josephson junction. A Josephson junction consists of a thin insulating material, about 10 to 20 Å thick, sandwiched between two superconducting metals, as sketched at the top of Fig. 6.17. In many instances the insulating layer is a thin oxide coating on an evaporated metal film. A voltage is applied to the junction and it is then cooled below the transition temperature of the superconductor, When the voltage is turned

Figure 6.17. Illustration of Josephson junction (top) formed by insulating layer between two superconductors and (bottom) curtent-voltage characteristics of the junction.

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off, it is observed that a dc current continues to flow through the junction as though the sandwich were one continuous slab of superconductor. Cooper pairs pass readily through the nonsuperconducting insulating layer without breaking up. The phenomenon is the result of tunneling in which quantum mechanically there is some probability that a Cooper pair can traverse the insulator even if its energy is less than the barrier height between the superconductor and the insulator. The effect is very pronounced in the superconducting sandwich because of the large wavelength of the Cooper pairs, on the order of the thickness of the insulator, and the fact that all pairs have the same wavelength and are in phase with each other. The bottom part of Fig. 6.17 shows the current-voltage characteristics for the junction. No current flows across the insulator until the applied voltage V is such that 2Ve is equal to the superconducting gap of the superconductor. Now consider a current loop formed by two Josephson junctions as shown in Fig. 6.18. It turns out that the relative phase of the waves in the lower and upper superconducting wires can be changed by the application of a magnetic field perpendicular to the plane of the loop. Figure 6.19 shows how the current through the junction is changed as the strength of this applied magnetic field is increased. The points where there is no current are the points where the currents on each side of the loop are a half wavelength out of phase. The current in a Josephson junction is very sensitive to a small magnetic field, and magnetic field measuring devices (i.e., magnetometers) have been designed based on this effect. They are called SQUID magnetometers, the name being an acronym for superconducting quantum interference device. Note that there is a maximum dc magnetic field, BJc2, above which no current flows. For higher applied fields, the two superconductors are decoupled.

Figure 6.18. Current loop containing by two Josephson junctions, each formed by an insulator connecting two superconducting wires.

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Figure 6.19. Current flowing in Josephson junction loop of Fig. 6.18 as a function of dc magnetic field applied perpendicular to the loop.

6.4.2. Absorption Mechanism Consider a superconducting current loop enclosing an area A on the order of 1 µ 2 located in a perpendicular applied magnetic field, and assume that the loop contains a Josephson junction. The current I flowing across the junction depends on the relative phases Q1 and Q2 of the Cooper pairs on each side of the junction in accordance with the expression: (6.5) When the applied field B0 is increased, the flux through the loop F = B0A increases in discontinuous steps because of flux quantization in the superconducting state. Thus as the field increases, flux jumps occur when BA = nF0. This in turn causes a phase jump given by 2pF/F0. These jumps occur in a very short time, on the order of 10–12 s, and produce voltage pulses given by (6.6) These jumps produce a current in excess of the critical current for the junction, meaning that in effect they produce normal current flow. If microwaves are incident

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on the loop, then there will be energy absorption by the normal current produced by the flux jump. Xia and Stroud (1 8) have calculated the derivative spectrum of the absorption as a function of increasing field for this situation, and the result is shown in Fig. 6.20. The spectrum is a series of equally spaced sharp lines and the magnetic field separation between adjacent lines is F0/A. If the magnetic field is rotated away from the normal to the loop, the separation of the lines will change as F0/Acosa where a is the angle between the field and the normal to the loop. The striking similarity of the predicted spectrum to the sharp line spectra obtained in the single crystals, as well as the observed angular dependence of the line separations, indicates that flux jumping through superconducting loops is the likely cause of the microwave absorption at low magnetic fields. The absorption only occurs at low magnetic fields because of the low values of the upper critical field BJc2 for Josephson junctions. In a composite of YBa2Cu3O7–x, the average BJc2 of the junctions has been estimated to be about 6 mT at 77 K (19). This mechanism is also the cause of the broad line observed in composite samples where, instead of every loop having a single area and orientation, there is a distribution of loop areas and orientations with respect to the applied magnetic field. It has been shown that the shape of the derivative of the absorption as a function of applied magnetic field can be accounted for by assuming that the distribution of loop areas is determined by a Boltzmann function of the energy BAI of a loop of area A carrying a current density I in a magnetic field B (20): f =f0exp(–BAI/kBT)

(6.7)

Figure 6.20. Calculated multiline microwave absorption derivative spectrum, dl/dH, for a 1 µm2 superconducting loop as a function of the strength of a dc magnetic field applied perpendicular to the loop (from Xia and Stroud, Ref. 18).

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The line shape can be simulated by a sequence of narrow derivative lines centered at each magnetic field of the sweep for each area A of the distribution that meets the flux jump condition (6.3). The height of each derivative line is made proportional to f. Figure 6.21 shows the simulated derivative signal in the Bi1.5Pb.5Sr2Ca2Cu3O10 superconductor at 77 K. The temperature dependence of the line intensity at a given field is proportional to both the current I flowing in the loop and to the distribution function. Thus the intensity will depend on the temperature as I(T) exp[–BAI(T)/k BT]

(6.8)

and close to Tc the current I(T) will vary with temperature as (Tc – T)1/2. Although the model can account for the observed sharp line spectra in single crystals, the broad line in composites, and the temperature dependence of the intensity, nevertheless it does not account for other properties of the absorption, such as the field and temperature dependence of its line width, position, and hysteresis. These latter effects are believed to be manifestations of the flux trapping behavior of the materials. The hysteresis of the field position of the center of the derivative on the up- and down-sweep for a given applied field is a result of flux being trapped in the sample. The difference in the field position for up- and down-sweeps measures the remanent magnetic field in the sample. Thus the hysteresis is a direct measure of the trapping of flux in the sample, and its value will depend on whether the sample is cooled below Tc in a magnetic field or in a zero field, being larger for field cooling. As shown in Fig. 6.5, the hysteresis increases at a constant temperature with the magnitude of the maximum magnetic field applied to the sample, but it eventually levels off or saturates for high fields.

•

Figure 6.21. Simulation (-------) and experimental data ( ) for half of the derivative of a microwave absorption signal versus dc magnetic field for polycrystalline N= 3, Bi-Pb-Sr-Ca-Cu-O at 77K assuming a Boltzmann distribution of loop areas as described in the text.

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In some instances it increases to a maximum and then decreases, as shown in the case of the thin films in Fig. 6.9. The upward shift of the magnetic field position of the center of the line as a function of the size of the previously applied field at constant temperature is also a result of flux trapping effects.The magnitude of the upward shift for a given previously applied field has been shown to equal the size of the hysterersis for a sweep up and back down to that field. One can use the hysteresis and the upward shift of the line to measure Bc1 for the sample at a given temperature. It is at the field Bc1 that the hysteresis or upward shift of the line is first observed. The broadening of the derivative as a function of a previously applied magnetic field is not well understood, but is probably related to flux trapping. It is noted that the broadening begins at the same magnetic field where the upward shift and hysteresis first appear. The broadening may be aresult of a change in the distribution of loop areas with trapped flux. Current loops formed by Josephson junctions have upper critical magnetic fields BJc2 above which superconducting current cannot flow in the loop. Since there is a distribution of loop areas, there will be a distribution of critical fields. Magnetic fields associated with trapped flux may exceed the critical fields of some of the junctions and drive those junctions normal, thereby altering the distribution of Josephson loops. The temperature dependence of the field position and the width and hysteresis of the derivative reflect the temperature dependence of the ability of the sample to trap flux. The temperature dependence of the magnetization of a Type II superconductor is given by (21) M = Jc(kBT/EA) ln(t)

(6.9)

where EA is an activation energy and t is the time. It has been shown that Jc/EA in YBa2Cu3O7–x depends on the temperature as (1 – T/T c) (22). In measuring hysteresis, the experiment can be carried out so that the time to sweep up and back in a field is constant for a sweep to a given magnetic field. Under these conditions the temperature dependence of the hysteresis and the line shift should depend on the temperature as T( 1 – T/T c). The time dependence of the hysteresis and the absorption intensity after the removal of a dc magnetic field are a result of flux decay, which can be described by Eq. 6.9 (9). The dependence of the magnitude of the hysteresis on the orientation of the dc magnetic field in crystals and films is a reflection of the anisotropy of flux trapping, The data in Fig. 6.9 indicate that more flux is trapped when the dc magnetic field is parallel to the c-axis of the orthorhombic unit cell of Y-Ba-Cu-O than in the ab plane orientation. This result is in agreement with magnetization measurements in single crystals which show that the pinning barriers are larger in the c direction (22).

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6.5. MAGNETIC FIELD ABSORPTION IN ALTERNATING APPLIED FIELDS A difficulty in studying the properties of low magnetic field absorption in the derivative mode is that the imposed magnetic field modulation may also cause microwave absorption in addition to that caused by the scanning dc magnetic field and as will be shown, these absorptions are not independent. Figure 6.22 is a plot of the intensity of the derivative signal versus the amplitude of the modulation in a polycrystalline sample of YBa2Cu3O7–x showing quite different behavior from a resonant signal. The presence of a discontinuity between curved and linear regions is not understood. Figure 6.23 shows that the temperature dependence of the absorption depends on the modulation amplitude, but it is not clear why there is such a difference between the low and high modulation behaviors. An RF magnetic field may induce absorption in a superconductor through a number of processes. As discussed by Halbritter (23) in the context of the two-fluid model, there will be resistive losses because some ac current is carried by nonpaired holes or electrons that are present in the superconducting state. There can be dissipation due to ac-induced flux oscillation, especially involving intergranular flux, and losses due to the flux jump process discussed earlier.

Figure 6.22. Intensity of a microwave absorption derivative signal versus the amplitude of an ac applied magnetic field at 85 K in a ceramic sample of Y-Ba-Cu-O (adapted from Blazey and Huhler, Ref. 11).

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Figure 6.23. Temperature dependence of microwave absorption derivative signal in Y-Ba-Cu-O at a low (- - - - - ) and at a high (––––) modulation amplitude (adapted from Blazey and Huhler, Ref. 11).

When the frequency of the ac magnetic field was increased from 25 kHz to 100 kHz, there was an increase in the 9.2-GHz microwave energy absorbed in the superconducting state at 77 K, as shown in Fig. 6.24 for modulation amplitudes up to 4 mT. This increase in absorption was only about 20% for an ac modulation amplitude of 0.5 mT, whereas for ac amplitudes of 3 to 4 mT, the absorption at 100 kHz was twice that at 25 kHz. The data also showed that for both frequencies the absorption intensity increased nonlinearly with the peak-to-peak amplitude of the ac magnetic field. No dc magnetic field was present during this measurement. The temperature dependence of this ac microwave absorption is shown in Fig. 6.25, measured at a constant amplitude and plotted as the difference in the absorption at zero ac field P(0) and that at an ac field of 2.5 mT peak-to-peak amplitude P(Bac) in order to separate out the temperature dependence of the microwave absorption in a zero magnetic field. This decrease in P(Bac) – P(0) with temperature provides evidence for the nature of the mechanism giving rise to ac magnetic field-induced absorption. A microwave absorption arising from intergranular flux motion or normal carrier flow would increase as the temperature approached Tc from below, not decrease as observed here. However, an absorption arising from ac-field induced flux jumps through current loops formed by weak links decreases

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Figure 6.24. Microwave absorption intensity versus the magnetic field modulation amplitude in polycrystalline Y-Ba-Cu-O at 77 K for 25 kHz (D) and 100 kHz modulation frequencies. The units on the y-axis are arbitrary.

as the temperature approaches Tc from below, which is similar to the case of a slowly increasing dc magnetic field, as discussed in the previous section. Consider a single superconducting loop formed by a weak link and having its area A perpendicular to the ac magnetic field. The current flowing in the loop depends on the relative phase difference Dq of the waves of the Cooper pairs on each side of the link: I = I0 sin DQ

(6.10)

The ac magnetic field perpendicular to the loop causes time-dependent changes in the relative phase given by (6.11) whereF(t) is the time-dependent flux through the loop due to the ac magnetic field, ω is the frequency, F0 = h/2e is the quantum of flux, and B0 is the amplitude of the ac field. Because the flux through the loop is quantized, there are phase jumps at the instant the applied ac flux becomes an integral multiple of the quantum of flux. The flux jumps occur in short times, on the order of 10–12 s, and produce voltage pulses given by (see Eq. (6.6))

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Figure 6.25. Temperature dependence of the difference between the microwave absorption signal at 2.5 mT peak-to-peak field modulation amplitude and the zero field absorption, P(Bac) – P(0). The results are for polycrystalline Y-Ba-Cu-O at 77 K. The units on the y-axis are arbitrary.

(6.12) These voltage pulses produce current pulses whose magnitude exceeds the critical current of the loop formed by the weak link. In the ac field case, the flux jumps repeat periodically, depending on the frequency of the ac field, with more jumps per unit time at higher frequencies. For each weak link there is a magnetic field BJc2 above which the link cannot exist. The magnitude of BJc2 is temperature dependent, decreasing as the temperature approaches Tc from below. Thus at a given magnetic field, the number of effective weak links decreases as the temperature increases, resulting in a decrease in the microwave absorption with temperature. Further evidence that the ac magnetic field-induced absorption is a result of this mechanism comes from the effect of a small dc magnetic field on the ac absorption. Figure 6.26 gives the results of such a measurement showing that the difference in the microwave absorption, P(Bac) – P(0), decreases with increasing

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Figure 6.26. Dependence of the difference between the microwave absorption signal at 2.5 mT peak-to-peak field modulation amplitude and the zero ac field absorption, P(Bac) – P(0), on an applied dc field. The data are for polycrystalline Y-Ba-Cu-O at 77 K. The units on the y-axis are arbitrary.

dc magnetic field. The critical dc magnetic field BJc2 is related to the area of the loop A by (24) (6.13) Because there is a distribution of loop areas, as the dc magnetic field is increased, it exceeds an increasing number of effective critical fields for loops, and the number of weak links is reduced. Thus the decrease in P(Bac) – P(0) with increasing dc magnetic field is consistent with the ac field-induced absorption arising from flux jumps in loops formed by weak links.

6.6. LOW MAGNETIC FIELD DERIVATIVE SIGNAL AS A DETECTOR OF SUPERCONDUCTIVITY The appearance and growth of the intense low dc magnetic field derivative signal with the onset of superconductivity allows the use of an electron paramag-

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netic resonance spectrometer as a sensitive detector of the presence of superconductivity. It can be used to search for small fractions of a superconducting compo. nent embedded in various types of fabricated materials, such as sintered pellets, thin films, and crystals. It has been estimated that this method can detect superconducting fractions as small as 1 µg. It requires no physical contact with the sample and is therefore particularly convenient for samples that must be maintained in sealed tubes because of sensitivity to moisture or lack of stability in air. For example, when the material A3C60 was first synthesized in bulk, it had to be stored in a sealed tube under an overpressure of an inert gas to prevent the alkali substituent from diffusing out of the material, and thus four-probe resistance measurements were not feasible. The LFMA in conjunction with a susceptibility measurement of the Meissner effect was used as evidence of superconductivity (25). However, there are some pitfalls in using the LFMA signal as an indicator of superconductivity. In Chap. 4 we saw that a low dc magnetic field derivative signal can be obtained in a ferromagnetic material, and its intensity grows as the material enters the ferromagnetic phase. There are, however, some very clear differences between the LFh4A obtained from a superconductor and that from a ferromagnet. In a ferromagnetic material, the signal intensity depends on the frequency as w1/2, whereas in the superconductor it is independent of the frequency. The hysteretic effect is another distinguishing feature. In the ferromagnetic case, the derivative signal generally appears at a lower dc magnetic field when sweeping downward from a given field than when sweeping upward. For a superconductor, the reverse is true. Thus when a derivative microwave absorption signal is observed at a low dc magnetic field, it is necessary to make sure that it is not arising from a ferromagnetic phase. One clear way of distinguishing between these two possibilities is to measure the direct surface resistance against temperature. In the ferromagnetic case, it increases as the material is cooled below the transition temperature and becomes ferromagnetic, while in the superconducting case, there is a decrease upon cooling below the transition temperature Tc. An additional advantage is that the low-field microwave absorption method is more sensitive than the direct surface resistance method for detecting the presence of superconductivity.

References 1. K. W. Blazey, K. A. Müller, J. G. Bednorz, W. Berlinger, G. Amoretti, E. Buluggiu, A. Vera and F. C. Matacotta, Phys. Rev. B36,7241 (1987). 2. D. Shaltiel, J. Genossar, A. Grayevesky, Z. H. Kalman, B. Fisher, and N. Kaplan, Solid State Commun. 63, 987 (1987). 3. C. Rettori, D. Davidov, I. Belaish, and I. Felner, Phys. Rev. B36, 4028 (1987). 4. J. Stankowski, P. K. Kahol, N. S. Dalal, and J. S. Moodera, Phys. Rev. B36,7126 (1987). 5. R. Dumy, J. Hautala, S. Ducharme, B. Lee, 0. G. Symko, P. C. Taylor, D. J. Zheng, and J. A. Xu, Phys. Rev. B36, 2361 (1987). 6. J. M. Delrieu, N. S. Sullivan, and K. Bechgaard, J. Physique C3,1033 (1983). 7. F. J. Owens, Synthetic Metals 33, 225, (1989).

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8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

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F. J. Owens, Phys. Status Solidi B162, 565 (1990). F. J. Owens and Z. Iqbal, Physica C238, 171 (1990). J. T. Suss, W. Berlinger, A. M. Portis, and K. A. Müller, Solid State Comm 71, 929 (1989). K. W. Blazey and A. Huhler, Solid State Commun. 72, 119 (1989). F. J. Owens (unpublished). K. W. Blazey, A. M. Portis, K. A. Muller, J. G. Bednorz, and E Holtzbcrg, Physica C153, 56 (1988). A. Dulcic, R. H. Crepeau, and J. H. Freed, Physica C160, 223 (1989). H. Vichery, E Beuneu, and P. Lejay, Physica C159,823 (1989). T. Mzoughi, H. A. Farach, E. Quagliata, M. A. Mesa, C. P. Poole, Jr., and R. J. Creswick, Phys. Rev. B46,1130 (1992). A. Pertile, O. A. Lopez, H. A. Farach, R. J. Creswick, and C. P. Poole, Jr., Phys. Rev. B52, 15475 (1995). T. Xia and D. Stroud, Phys. Rev. 39, 4792 (1989). A. D. Vedeshwar, Solid State Commun. 74 , 23 (1990). F. J. Owens, Physica C171,25 (1990). P. W. Anderson, Phys. Rev. Lett. 9,309 (1962). Y. Yeshurun and A. P. Malozemoff,Phys. Rev. Letters 60, 2202 (1988). J. Halbritter, Z Physik 266, 209 (1974). M. Tinkham and C. J. Lobb, in Solid State Physics, H. Ehrenreich and D. Tumbull, eds. Vol. 42, p. 91, Academic Press, New York(1989). A. F. Hebard, M. J. Rosslinsky, R. C. Haddon, D. W. Murphy, S. H. Glarurn, T. T. M. Palstra, A. P. Ramirez, and A. R. Korton, Nature 350,600 (1991).

7 Electromagnetic Absorption Due to Vortex Motion

In this chapter the effect of vortex motion on the surface resistance in the superconducting state of the cuprates is discussed. It is shown how the nature of the vortex dynamics affects electromagnetic dissipation by radio frequency and microwave radiation.

7.1. THEORY OF ELECTROMAGNETIC ABSORPTlON DUE TO VORTEX DISSIPATION 7.1.1, Penetrating Fields The application of a dc magnetic field B0 perpendicular to the surface of a Type II superconductor causes vortices F0 to form inside. The further application of an RF or microwave field Hµ(t) parallel to the same surface induces via Eq. (4.14) an oscillating surface current Ks(t) = n× Hµ, where n is a unit vector pointing outward from the surface. Inside the surface there is an associated current density J(x, t), in accordance with Eqs. (4.15) to (4.18), which exerts the Lorentz force F (t) = J × F0 on the vortices. Figure 7.1 shows the geometric arrangement of these vectors. Since the force F(t) is perpendicular to the vortex axis, it sets the vortex into a transverse oscillation that can propagate further into the material than the current density J, in effect increasing the effective penetration depth. The transverse motion will be retarded by a viscous damping force that dissipates energy and hence contributes to RF and microwave absorption. Vortices that are pinned can oscillate in their potential wells. The dynamics of the oscillation as well as the details of the dependence of the surface resistance on the dc magnetic field depend in turn on the vortex state—whether it is a solid or liquid; the transition from the solid to the liquid state can be observed from these studies. 139

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Figure 7.1. Geometry used to model vortex-induced electromagnetic dissipation in superconductors.

7.1.2. Flux Creep, Flux Flow, and Irreversibility At low temperatures the vortices remain fixed in position on the regular two-dimensional hexagonal lattice sketched in Fig. 1.22a. This lattice state is analogous to a two-dimensional solid configuration, and the pinning centers that are present hold the vortices more strongly to their lattice sites. When the temperature is raised, the vortices undergo random thermal motions about their equilibrium positions on the hexagonal lattice, and if the temperature is raised high enough, the thermal energy will dominate and cause the vortex lattice to transform to a two-dimensional liquid or fluid state characterized by continuous vortex motion. This transformation is often referred to as a pinning-depinning transformution; the flux solid phase is called the flux pinned state; and the flux liquid is called the flux flow state. The transformation from the solid to the liquid phase can be brought about by either raising the temperature or by increasing the strength of the applied magnetic field. The relationships between these phases are easily depicted on a plot of applied field B0 versus temperature T called a magnetic phase diagram; a representative diagram is depicted in Fig. 7.2. We see from the figure that there is a Meissner state of perfect flux expulsion below the temperature-dependent lower critical field Bc1(T), and the material becomes normal (nonsuperconducting) above the temperature-dependent upper critical field Bc2(T), The flux solid and flux liquid phases are separated by the temperature-dependent depinning field B*( T), also called the depinning line or flux melting line flux melting line = B*(T)

(7.1)

which plays an important role in explanations of microwave absorption in superconductors, and is indicated in Fig. 7.2. The same B* line can be considered as a

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Figure 7.2. Magnetic phase diagram of a Type II superconductor showing the Meissner, flux solid, and flux liquid phases, together with the lower critical field Bc1(T), the upper critical field B C2(T), and the depinning temperature B*(T) lines that separate the phases. The values of the critical temperature Tc and critical fields B c1 and B c2 at T = 0 are indicated.

field-dependent depinning temperature T*(B). In other words, a point on this line at the particular position B', T' can be designated by the notation B*(T') or by the notation T*(B'). When a transport current flows through a superconductor in the presence of a magnetic field B0 applied perpendicular to the current direction, the vortices experience the Lorentz force per unit length given by F/L = J× F0

(7.2)

and if this force exceeds the pinning forces, then the vortices begin to move in a direction perpendicular to both J and F0. When the pinning forces dominate, there can be a very slow motion called flux creep, and when the Lorentz force (7.2) dominates, there is a faster motion of the vortices called flux flow. The latter faster motion is characteristic of the flux liquid phase. In a typical case there can be a range of pinning strengths, and flux creep might occur when the Lorentz force unpins some vortices and sets them in motion, but their motion is hindered by encounters with others that remain in place at pinning sites. The situation can be quite complicated to describe as successively higher fields unpin more and more vortices. Another factor to take into account is the magnetic field-dependent irreversibility temperature Tin. This is the temperature above which field-cooled and zero field-cooled magnetization and susceptibility data superimpose, and below which ZFC measurements lie below FC ones. A plot of the applied field dependence of Tin(B0) is called the irreversibility line. The presence of irreversibility manifests

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itself in the appearance of hysteresis effects. In other words, the susceptibility, magnetization, and other properties have values that depend on the previous magnetic history of the sample, such as the order in which applied magnetic fields and temperatures have been varied.

7.1.3. Coffey–Clem Model Coffey and Clem (1) have developed a unified theory to explain the effect of vortex oscillation on the surface impedance Zs of Type II superconductors. Their model takes into account the effects of the flux pinned and the flux flow states. It includes coupling of the supercurrent density to vortex displacements, dynamic flux creep effects, and normal-like quasi-particle excitations. The result of their theory is a complex penetration depth which is a function of the dc magnetic field, ac frequency, and temperature. Fortunately, however, most of the measurements of the dc magnetic field dependence of microwave absorption which have been made at low fields near Tc can be accounted for by using the limiting cases of weak and strong pinning in the Coffey–Clem model. Here we present a simplified version of the theory that does not include all the effects mentioned above, but yields expressions that correspond to weak and strong pinning limits, and in addition provides some insight into the physics of the process. The movement of vortices at the velocity v brought about by the time-dependent microwave field Hµ(t) produces an electric field E(t, x) through the v × B0 interaction of the vortices with the applied dc magnetic field B0. The surface impedance Zs is defined by Eq. (4.7) as the ratio of E(t, x) to Hµ(t, x) at the surface where x = 0: (7.3) where Rs is the surface resistance and Xs is the surface reactance. The microwave absorption is proportional to the surface resistance, which is the real part of Zs. For the geometry of Fig. 7.1, the incident RF magnetic field Hµ(t) penetrates exponentially into the superconductor in accordance with the expression (7.4) The microwave magnetic field amplitude Hµ , is assumed to be much smaller than the dc applied field Hµ< < B0/µ0, and the penetration depth λ is assumed to be greater than the spacing between vortices. The time-varying microwave field produces a current in the y direction which can be obtained from Eq. (7.4) using Maxwell's curl relation (1.7). ∇×H = J, yielding: (7.5)

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where µ0 is the permeability of free space. This current produces a time-varying force J × F0 on the vortices in the z direction, as indicated in Fig. 7.1, given by (7.6) where F0 = h/2e is the quantum of flux. The movement of the vortices along z is described as an oscillation at their pinning sites in apotential having a force constant K, or as a flow in the liquid phase having a viscosity h . The governing equation of motion for the position z of an individual vortex in both the liquid and the solid phases is then (7.7) Assuming a solution of the form,

(7.8) and substituting into Eq. (7.7) to evaluate the constant z0, we obtain for the vortex velocity (7.9) The surface impedance can be evaluated from Eqs. (7.3), (7.4), and (7.9), noting from Fig. 7.1 and Eq. (7.3) that E is perpendicular to both v and B0, which gives the magnitude E = |v × B0| = vB0, so the surface impedance has the form (7.10) The surface resistance is the real part of Zs , giving (7.11) In this model the penetration depth of the microwaves l depends on whether the superconductor is in the creep or flow region. Campbell (2) has shown that in the creep region l is given bv (creep region)

(7.12)

and since in this region K2 >> h2w2, the surface resistance is (creep region)

(7.13)

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In the vortex liquid state where h2w2 >> K2, the penetration depth has been shown to be (3) (flow region)

(7.14)

to give for the surface resistance (flow region)

(7.15)

Thus the theory predicts that in the creep and flow regions of the vortex state, the microwave absorption increases as B01/2 but has a different slope in each region as well as a different frequency dependence. The model also predicts that the temperature dependence of the absorption will be different in each region. In the flux creep region, the temperature dependence of Eq. (7.13) will be via h/K3/2. Taking the temperature dependence of h as given by (1–T/T c) and that of K as [ 1 – (T/Tc)2]2 leads to the following temperature dependence of Rs (1) (creep region)

(7.16)

In the flux flow region, the temperature dependence is through h, which is given by (1 – T/T c), meaning that the surface resistance (7.15) will depend on Tas

Figure 7.3. Magnetic field dependence of surface resistance (log scale) versus reduced temperature at 1.5 T for Y-Ba-Cu-O calculated from the Coffey–Clem model (Ref. 1). Curve a is with no magnetic field present; curve b has vortices present but there is no motion; and curve c allows flux creep and flux flow.

ELECTROMAGNETIC ABSORPTION DUE TO VORTEX MOTION

(flow region)

145

(7.17)

Thus the theory predicts that the temperature, frequency, and dc magnetic field dependencies of vortex-induced electromagnetic dissipation will be different in the vortex fluid and vortex solid phases, and therefore it should be possible to observe the vortex solid-to-liquid transition through surface resistance measurements. Experimental verification of these predictions is discussed in the following sections. The complete Coffey–Clem model, which includes the local supercurrent density modifications arising from vortex displacements, does not yield simple analytical expressions relating surface resistance to dc magnetic field and temperature such as Eqs. (7.11), (7.13), and (7.15) to (7.17). However, it is possible to calculate the surface resistance for a specific set of material parameters. Figure 7.3 gives the surface resistance versus reduced temperature at an applied field of 1.5 T for parameters appropriate to Y-Ba-Cu-O (1). The curve labeled a is for zero dc magnetic field; curve b has vortices present but not flux creep; and curve c has vortices that are undergoing flux creep and flux flow.

7.2. EXPERIMENTAL RESULTS 7.2.1. RF Penetration Depth Measurements In the region of applied dc magnetic field and temperature where the vortices are pinned, the dependence of the penetration depth on the field is given by Eq. (7.12). In the liquid vortex state where the vortices are not pinned, the dependence of the penetration depth on field and frequency is given by Eq. (7.14). Thus the transition between the flow and pinned regimes should be manifested by a change in the slope of a plot of the penetration depth versus B01/2 when the applied field B0 equals the depinning field B* at that temperature. Equations (7.12) and (7.14) also predict that the frequency dependence of the penetration depth should be different in each phase, being independent of frequency in the pinned regime but having a 1/w1/2 dependence in the flow regime. Wu and Sridhar (4) reported the first magnetic field-dependent studies of the penetration depth in a cuprate for dc magnetic fields up to 0.1 T using single crystals ofYBa2Cu3Oy. Representative results are presented in Fig. 7.4 for the magnetic field parallel to the c-axis at 9.7 K. Notice that Dl = 0 in the Meissner state until the critical field Bcl(||)is reached at 10 mT (100 G). The break at the lower critical field Bc1 is sharp, and provides a much clearer signature of this critical field than magnetization measurements. A measurement of the field dependence of the penetration depth at a number of different temperatures allows a determination of the temperature dependence of the lower critical field, which was found to follow the BCS prediction. The measured field dependence at constant temperature was consistent with Eq. (7.12), and the temperature dependence of the pinning force

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Figure 7.4. Change in penetration depth Dl versus magnetic field for a single crystal of Y-Ba-Cu-O at 9.7 K for the applied field parallel to the c-axis (from Wu and Sridhar, Ref. 4).

constant was shown to be [(1 – (T/T c)2]2. The measurements were not made at a high enough dc magnetic field or temperature to observe the predicted cross-over to the liquid vortex state. The first observation of the melting of the vortex lattice in a cuprate determined from the dc magnetic field dependence of the penetration depth was in a Bi-Pb-SrCa-Cu-O superconductor in which the transition occurs at lower magnetic fields than in Y-Ba-Cu-O (5). Figure 7.5 is a plot of the frequency shift Df from 7.9903 MHz versus a dc magnetic field at 77 K in the superconductor Bi-Pb-Sr-Ca-Cu-O. We know from E2q. (3.3) that Df/f0 is proportional to the change A1 in the RF penetration depth; hence the scale for Dl added to the right side of Fig. 7.5. Figure 7.6 is a plot of the frequency shift versus B01/2 at two frequencies in the megahertz range at 77 K in Bi-Pb-Sr-Ca-Cu-O. The data show that the penetration depth depends on B01/2 and that there is a change in slope at the depinning field B* which marks the transition from the pinned to the depinned phase of the vortices. Equations (7.12) and (7.14) indicate that the slope of the dependence of the penetration depth on B01/2 in the flux flow regime should be larger than in the pinned region, contrary to the experimental results in Fig. 7.6. The reason for this discrepancy is not clear, but perhaps is associated with increased misalignment of vortices from the direction of the applied dc magnetic field in the fluid vortex phase, which

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Figure 7.5. Shift of frequency Df in LC coil experiment versus dc magnetic field at 77 K in superconducting Bi-Pb-Sr-Ca-Cu-O.

would have the effect of lowering the dissipation. This conclusion is supported by other work which demonstrates that there is some dissipation even when the RF current is parallel to the dc magnetic field (6). The frequency dependence of the dc magnetic field-dependent penetration depth in the pinned region can be determined by measuring the slope of Df/f0 versus B1/2 for different frequencies f0 below the depinning field B*. The measurement shows the penetration depth to be independent of frequency, which is in agreement with Eq. (7.12). However, the same measurement above B* shows that the penetration depth decreases as the frequency increases, as predicted by Eq. (7.14), but the studies have not been detailed enough to quantitatively verify the 1/w1/2 dependence. Figure7.7 shows how the slope of Df/f0 versus DB1/2 above B* shifts to lower values at higher frequencies. Figure 7.8 shows how the depinning field B* decreases in value when the frequency increases. This result is in agreement with theoretical predictions of the frequency dependence of the irreversibility line. Experimental measurements of the irreversibility temperature Tirr in Bi-Sr-Ca-Cu-O at constant dc magnetic field show that Tirr increases with increasing frequency (7,8). Since B* and Tirr are related by the equation (7)

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Figure 7.6. Frequency shift Dfin LC coil versus square root of applied dc magnetic field at 7.99 MHz and 4.343 MHz ( ) in Bi-Pb-Sr-Ca-Cu-O at 77 K. Each vertical arrow indicates the change in slope at the field B*.

•

(7.18) where A and the exponent q are constants, the field B* would be expected to decrease with increasing frequency, as observed.

7.2.2. Microwave Bridge Measurements We have seen that in the superconducting state the application of a dc magnetic field increases the surface resistance of the sample, and this is shown in Fig. 7.9 for Hg0.7Pb0.3Ba2Ca2Cu3O8+x (9). The theoretical treatments of the dc field-enhanced surface resistance due to vortex oscillation discussed earlier show that the surface resistance depends linearly on B01/2 in both the strong and the weak pinning regimes, but the slope is different in each region. In Fig. 7.10 the microwave absorption is plotted versus B01/2 at 77 K for both an increasing and decreasing dc magnetic field. The data show that the microwave absorption depends linearly on B01/2 and that there is achange of slope at the depinning field B*. On the downsweep ofthe dc magnetic field, there is an onset of hysteresis in the data starting at B*, clearly demonstrating that B* corresponds to the irreversibility transition. Below B*, flux is more strongly pinned in the sample, while above it, the vortices are more free to move. Similar

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Figure 7.7. Frequency dependence of slope of the frequency shift Df/f0 versus (DB)1/2 above B* at 77 K in Bi-Pb-Sr-Ca-Cu-O.

Figure 7.8. Plot of frequency dependence of B *, the dc magnetic field at which the transition from the pinned to the depinned vortex fluid phase occurs.

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Figure 7.9. Dependence of the surface resistance R s at 9.2 GHz on the dc magnetic field at 122 K in the N = 3, Hg-Pb-Ba-Ca-Cu-O superconductor.

results have been obtained in other copper oxide superconductors such as the T1-Ba-Ca-Cu-O and Bi-Sr-Ca-Cu-O types (10,11). Figure 7.11 is a plot of the surface resistance versus B01/2 at two different temperatures showing the change in slope, which shifts to a lower field value at the higher temperature. A measurement of the field dependence of the microwave absorption at a number of different temperatures allows a determination of the temperature dependence of B* and thus provides a plot of the B-Tirreversibility line at 9.2 GHz. This irreversibility line furnishes the temperature at which the vortex lattice melts for each applied field. Figure 7.12 is a plot of B* versus (1 – T/T c), and the line through the data is a fit to Eq. (7.18) with A = 367 mT, the exponent q = 0.724, and Tirr set equal to T to make the plot. Generally the irreversibility line determined from the field dependence of the surface resistance gives a smaller B-T reversible region than a determination by bulk techniques such as magnetic susceptibility. This may occur because in the surface resistance determination, the electromagnetic energy is only probing the surface regions of the sample where the defect structure may not reflect the situation in the bulk. Pinning is known to be strongly dependent on the character and number of defects in a sample. It also has been pointed out that the measurement of the B-T

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151

Figure 7.10. Change in the surface resistance versus the square root of the dc magnetic field at 77 K for increasing and decreasing (O) magnetic field showing that the onset of irreversibility occurs at the field B* where the slope changes. The data are for the N = 3, Hg-Ba-Ca-Cu-O superconductor.

irreversibility line depends on the frequency of the probing radiation, which would mean that ac and dc susceptibility measurements of the B-T line would yield different results (7,8). There are some variations in the behavior of the dc field dependence of the dissipation in different superconductors. In the TI-Ba-Ca-Cu-O and Bi-Sr-Ca-Cu-O materials, the dependence of the surface resistance on the dc magnetic field is weaker in the flux flow state, in contrast to the Hg0.7Pb0.3Ba2Ca2Cu3O8+x material, and is not in agreement with the prediction of the theoretical model (10,11). As in the case of the RF penetration depth measurements discussed earlier, this could be due to increased misalignment of vortices in the fluid phase in these materials, but further studies are needed to clarify the issue. Measurements of magnetoelectromagnetic absorption on single crystals allow investigation of the anisotropy of the dissipation. In single crystals of Bi2Sr2CaCu2O8+x , for example, magnetic field-induced electromagnetic absorption is observed to be larger when the dc magnetic field is parallel to the c-axis of the unit cell, i.e., perpendicular to the copper oxide planes (12). This is consistent with

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Figure 7.11. Change in surface resistance versus square root of dc magnetic field at 97 K and at 91 K (O) for the N = 3, Hg-Pb-Ba-Ca-Cu-O superconductor. These results show that the magnetic field B* where the slope changes, indicated by vertical bars, depends on the temperature.

magnetic field-dependent resistivity data which show lower pinning barriers to flux motion when the field is parallel to the c-axis. While the plots of the microwave absorption versus B01/2 shown in Fig. 7.1 1 appear to suggest that the depinning transition is first order, detailed studies of the dc magnetic field dependence of the surface resistance at small field intervals in the transition region reveal a deviation from the B01/2 dependence. Figure 7.13 shows the results of such a study at constant temperature in the Bi-Sr-Ca-Cu-O superconductor, revealing the existence of anomalous behavior at the transition. The three points at the beginning and the three points at the end of the curve provide the slopes below and above the transition, respectively. The curvature in between is not resolved on broader scans of the type presented in Fig. 7.1 1. This anomaly has been attributed to a magnetic field-dependent distribution of pinning barriers in the material, and the existence of a critical depinning barrier U* (13). Those vortices arriving at sites having barriers less than U* at a given temperature will be in the fluid phase, while those with barriers greater than U* will be in the solid phase. The possibility of a distribution of pinning barriers has been indicated by a number of experiments, such as magnetic relaxation studies (14). A distribution of critical

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153

Figure 7.12. The temperature dependence of the irreversibility field at 9.2 GHz for the superconductor N = 3, Hg-Pb-Ba-Ca-Cu-O. The ordinate is the magnetic field B* for the crossover between reversible and irreversible behavior.

pinning barriers U*, could account for the anomaly shown in Fig. 7.13 at the depinning transition.

7.2.3. Strip Line Resonator Measurements The stripline resonator method enables surface resistance determinations at much higher frequencies and over a broader range of frequencies than the RF oscillator method. However, since the stripline of the resonator is deposited as a thin film, the measurements can only be made on films. Figure 7.14 illustrates a measurement of the dc magnetic field dependence of the surface resistance in YBa2Cu3Ox strips at 4.3 K and 13.8 GHz (15). The data, which were taken at much higher magnetic fields than previous measurements, show a nearly linear dependence of the surface resistance on the dc magnetic field strength. A measurement at 1.24 GHz gives a weaker dependence, with Rs varying with the applied field approximately as B00 8. These results are not in agreement with the predictions of the theory for the strong pinning limit that might be expected to be applicable at 4.3 K in this material. It is possible that the disagreement is aresult of the field dependence of the pinning force constant K, which may be more in evidence at higher magnetic field measurements of these studies. From Eq. (7.1 1) a linear dependence of the surface resistance on dc magnetic field would imply that the force constant depends

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Figure 7.13. Plot of surface resistance versus applied B1/2 measured at very small field intervals in the depinning transition region of single crystals of Bi-Sr-Ca-Cu-O at 77 K, showing the deviation from a B1/2 dependence on the applied field in this region.

Figure 7.14. Surface resistance versus dc magnetic field at 4.3 K and 13.8 GHz in Y-Ba-Cu-O films (from Revenaz et al., Ref. 15).

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155

Figure 7.15. Frequency dependence of surface resistance of Y-Ba-Cu-O films at 4.3 K measured in zero dc magnetic field ( ) and a 3.5 T field. (O) (from Revenaz et al., Ref. 15).

•

on the field as B0–1/3. Another possibility is that at the higher magnetic field strengths, where the vortex density is greater, the vortex–vortex interactions affect the dynamics. The simple theory outlined at the beginning of this chapter does not take this into account.

Figure 7.16. Temperature dependence of the surface resistance in film strips measured at different magnetic field strengths of 0.5 T, 1 T, and 3 T (from Revenaz et al., Ref. 15).

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Figure 7.15 presents a measurement of the frequency dependence of the surface resistance in a zero dc magnetic field and in a field of 3.5 T at 4.3 K. In such a dc field at 4.3 K, the material is likely to be in the pinned state, and therefore the contribution of vortex dissipation to the surface resistance obtained by the difference between Rs at B = 0 and Rs at B = 3.5 T at a given frequency should increase with frequency, as predicted by Eq. 7.13. Although there is considerable scatter the data seem to show that the two lines converge, suggesting that there is a frequency decrease. Figure 7.16 is a plot of the surface resistance at 1.46 GHz versus temperature measured at three different magnetic field strengths (15). The data show a change in the dependence of Rs on temperature from a weak somewhat linear behavior at low temperature to a much stronger dependence at higher temperatures. The crossover region shifts to a lower temperature at higher magnetic fields, suggesting that it may be associated with the vortex solid-to-liquid transition. However, further quantitative analysis of the data and perhaps additional measurements would be needed to verify this,

7.3. CONCLUDING REMARKS Measurements of the dc magnetic field dependence of the surface resistance in the superconducting state can provide useful information about the flux pinning behavior of superconductors, particularly in the surface regions of materials. In the case of thin films, the method should provide a way to characterize their properties, and because the film thickness can be compared with the penetration depth of the radio frequency, the results may be comparable to bulk measurements. Most of the dc magnetic field-dependent measurements of the surface resistance have been made at relatively low dc fields near Tc using either the LC coil or the microwave cavity method. The results can generally be accounted for by the Coffey–Clem model in the weak and strong pinning limits, which predicts that Rs has the applied field dependence B01/2. There is therefore a need for further work at higher dc magnetic fields where the density of the vortices is higher and vortex–vortex interactions may play a role in determining the dynamics. The stripline resonator measurements, which are limited to thin films and have generally been carried out at higher dc magnetic fields, show some agreement and some disagreement with the theory. There is a need for a more quantitative comparison between theory and experiment at higher magnetic fields.

References 1. M. W. Coffey and J. R. Clem, Phys. Rev. Lett. 67, 386 (1991). 2. A. M. Campbell, J. Plays. C2, 1492 (1969). 3. W. Tomasch, H. A. Blackstead, S. T. Roggiero, P. J. McGinn, J. R. Clem, K. Shen, J. W. Weber, and D. Boyne, Phys. Rev. B37, 9864 (1988). 4. D. Wu and S. Sridhar, Phys. Rev. Lett. 65, 2074 (1990).

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5. F. J. Owens, Solid State Commun. 81, 97 (1992). 6. E. K. Moser, W. Tomasch, P. J. McGinn, and J. Z. Liu, Physica C176, 235 (1991). 7. A. P. Malozemoff, T. K. Worthington, Y Yeshurun, F. Holtzbergmand P. H. Kes, Phys. Rev. B38, 7203 (1988). 8. Y.Yeshurun,A.P.Malozemoff,T.K.Worthington,M.Yandropski,L.Krvsin-Elbank,F.Holtzberg, T. R. Dinger, and G. V. Chandrashekar, Cryogenics 29, 258 (1989). 9. E J. Owens, A. G. Rinzler, and Z. Iqbal, Physica C233, 30 (1994). 10. F. J. Owens, Physica C178, 456 (1991). 11. F. J. Owens, Physica C195, 225 (1992). 12. F. J. Owens, Z. Iqbal, and E. Wolf, J. Phys. Condens. Matter 4, 205 (1992). 13. F. J. Owens, J. Phys. Chem. Solids 55, 167 (1994). 14. M. Reissner and W. Steiner, Supercond. Science Techn. 5, S367 (1992). 15. S. Revenaz, D. E. Oaks, D. Labbe-Lavigine, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B50, 1178 (1994).

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8 Infrared and Optical Absorption

The preceding few chapters have been concerned mainly with microwave absorption and to a lesser extent with radiofrequency absorption. In this chapter the frequency range is extended to the infrared, visible, and ultraviolet (UV) (1). In this region the absorption is explained in terms of the imaginary part of the dielectric constant, and the Kramers–Kronig technique is useful for the data analysis. From a spectroscopic viewpoint, infrared absorption is concerned with vibrational transitions and normal modes. In addition, the energy gaps of high-temperature superconductors are in the infrared region, so infrared absorption in the neighborhood of the gap energy Eg = 2D hv = Eg

(8.1)

can give us information on these gaps.

8.1. ABSORPTION IN THE INFRARED When the intensity of light I0 is incident on a sample, it is partly transmitted, It, and partly reflected, It, so the amount Ia that is absorbed is given by Ia = I0 – Ir – It

(8.2)

as shown in Fig. 8.1. Transmission spectrometers measure It, generally when Ir is small, while reflectance spectrometers measure Ir, generally when It is small. Either way, the spectrometer provides the frequency dependence of the ratio Ia/I0, and a maximum in Ia indicates the center of an absorption line. In a single-beam measurement, the absorption Ia itself is determined, and with a double-beam technique, the absorption of a sample is measured relative to that of a reference material. Superconductors tend to be opaque at infrared and visible frequencies, so reflectance techniques apply. The reflection Ir is measured and the absorption Ia is 159

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Figure 8.1. Absorbed (Ia), transmitted (It), and reflected (Ir) components of incident light beam of intensity I0

deduced from Eq. (8.2) by assuming It = 0. At higher frequencies, in the X-ray region, the radiation can penetrate and transmission detection of It is generally employed. Infrared spectroscopists report their results in several different energy units, and if we had standardized this chapter by, for example, converting all energies to joules, it would be difficult to compare the results presented here with those found in the literature. The appropriate conversion factors are 100 cm–1 = 0.0124 eV = 3 THz

(8.3)

where the velocity of light c = 2.9979 × 1010 cm/s is the conversion factor between reciprocal centimeters and hertz.

8.2. DETECTING MOLECULAR AND CRYSTAL VIBRATIONS Thermal energy causes atoms in molecules and solids to vibrate about their equilibrium positions, and the frequencies of the normal modes of vibration are ordinarily in the infrared region (2,3). These normal modes involve coherent oscillations of atoms in the unit cell relative to each other at a characteristic frequency with the center of gravity preserved. They have a maximum frequency vmax that may be estimated by equating the energy hvmax with the Debye energy kBQD (8.4) where QD is the Debye temperature, or vmax may be estimated from the well-known formulal minvmax = vs, which provides the expression (8.5)

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where vs is the velocity of sound in the material, and the distance a between atoms is related to the shortest vibrational wavelength through a ≈ lmin/2. Incident infrared radiation can induce transitions between these vibrational states. These transitions typically fall in or near the range 1012 to 1013 Hz, which is within the frequency band of infrared spectrometers. In infrared spectroscopy, an IR photon hv is absorbed directly to induce the vibrational transition (1), while in the case of Raman spectroscopy, an incident optical photon of frequency hvinc is absorbed and another optical photon hvemit is emitted, with the energy of the n = 1 vibrational level hv related by the difference (8.6) where vinc > vemit for what is called a Stokes line and vinc < vemit for an anti-stokes line. The wavelength of hvemit is measured by the Raman spectrometer. The fundamental vibrational energy levels have the energies (8.7) where the vibrational quantum number nv = 0, 1,2,3, . . , is a positive integer, and v0 is the characteristic frequency for aparticular vibrational mode. Transitions occur for the condition (8.8) and the lowest frequency transition with n’v = nv = 1 is called a fundamental band. The frequency v of the radiation that induces the transition of Eq. (8.8) can come either directly from an incident infrared photon hv, or in the Raman case, indirectly from an optical photon hvinc, which induces a transition upward to a virtual level followed by the emission of another optical photon hvemit due to a downward transition to the final vibrational level n’v. Infrared spectral lines are caused by a change in the electric dipole moment µD µD = dq

(8.9)

of the molecule where q is the charge and d is the charge separation. Raman lines appear when the incoming radiation field brings about a change in the polarizability P, which is defined as the ratio of the induced dipole moment µ ind to the electric field E of the incident radiation. (8.10)

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Figure 8.2a. Raman spectra of twin free YBa2Cu3O7 recorded with the laser beam directed along the c-axis and the indicated polarizations. The x-axis is the baseline for the lower spectrum, and the dashed lines indicate the baselines of the three upper spectra. (From McCarty et al., Ref. 4.)

These measurement techniques are complementary because some vibrational transitions are IR active and some are Raman active, meaning that some of them can be detected by direct infrared absorption and others only by the Raman process. In conventional Raman spectroscopy, an incident unpolarized laser beam simultaneously excites many Raman active modes. Polarized light enhances some of these modes and diminishes or eliminates others, and a variety of directions and polarizations can be employed to sort out and identify the modes, as illustrated by the data for twin free YBa2Cu3O7 presented in Fig. 8.2 (4). This occurs because the selection rules for the transition are affected by the symmetry of the molecule or by the symmetry of the unit cell of the lattice. Isotopic substitutions are helpful for identifying modes, such as enrichments with the abundant isotope 65Cu and the rare isotope 18O at particular lattice sites.

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Figure 8.2b. Raman spectra of twin free YBa2Cu3O7 recorded with the laser beam propagating in the x,y plane. Note the scale factor change for the two middle spectra. (From McCarty et al., Ref. 4.)

8.3. SOFT MODES A second-order structural phase transition is one in which the low- and high-temperature crystal structures differ by only small lattice displacements, with no abrupt change in configuration taking place. In many cases the order parameter changes gradually with the temperature, and exhibits the behavior (Tc – T)1/2 in the neighborhood of the transition temperature Tc. The force constant can become zero at Tc for one of the modes of vibration called a soft mode, and for this mode the frequency also ideally drops to zero at Tc. Most vibrational modes increase in frequency as the temperature is lowered, so soft modes can be identified by a decrease in frequency as the temperature is lowered at the approach to Tc from below. Well below the transition, the frequency begins to increase again, and sometimes there is a split into two modes because of a lowering of the lattice symmetry below Tc . The phase transition from the normal to the superconducting

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Figure 8.2c. Raman spectra of twin free YBa2Cu3O7 recorded with the laser beam directed along the c-axis, and the indicated polarizations selected to enhance different modes than Fig. 8.2a. Note the scale factor change for the lower Ag mode spectrum. (From McCarty et al., Ref. 4.)

state is second order in zero field and first order in the presence of a magnetic field. In the former case there is no latent heat; in the latter case there is a latent heat; and both cases exhibit a discontinuity in the specific heat. This superconductivity phase transition can involve a change in crystal structure in which individual atoms undergo very small shifts in position.

8.4. DIELECTRIC CONSTANT AND CONDUCTIVITY We saw in previous chapters that electrical conductivity plays a crucial role in determining the characteristics of microwave absorption in superconductors, and the same can be said for infrared absorption. This is because, as we show in the next section, the reflectance of light arises from the dielectric constant ε, which has real and imaginary parts, e' and e", respectively

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(8.11) and the imaginary part e" depends on the conductivity s. To see this for a conductor, we write the definition of the electric flux density D assuming a high enough frequency so vibrational effects on the dielectric constant can be neglected, i.e., assuming e = e0 D = e0E

(8.12)

and we substitute this and Ohm’s law J= sE

(8.13)

into the inhomogeneous Maxwell curl equation (8.14) for the case of a harmonic time dependence exp(iw t) to obtain

(8.15) Comparing this with Eq. (8.11) provides us with the identification (8.16)

8.5. REFLECTIVITY The reflectance or reflectivity R is the fraction of light reflected from the sample R = Ir/I0,

(8.17)

At frequencies where there are no absorption bands, the dielectric constant e = e' is real, and for this case the reflectance at normal incidence is given by (8.18) an expression that is easily solved for e'. For oblique incidence, the reflectivity depends on the angle. If we assume that the same expression applies to the case of a complex dielectric constant, we obtain

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(8.19) in which the square root factor (ε' + iε")1/2 may be written (8.20) with d defined through what is called the loss tangent (8.21) This means that different choices of e' and e" give the same value of R, so e’(w) and e"(w) are not uniquely determined by measurements of the reflectivity. The next section describes a way to resolve this dilemma.

8.6. KRAMERS–KRONIG ANALYSlS A method called Kramers–Kronig analysis has been developed to obtain the conductivity from reflectivity measurements. It is based on the fact that the frequency-dependent quantities e' (w) and e"(w) are not independent of each other, but are linked together through the Kramers–Kronig relations (8.22a)

(8.22b) where the integration is carried out over w'. With the aid of these expressions, the frequency dependence of e'(w) and e"(w) can be extracted from measurements of the reflectivity.

8.7. DRUDE EXPANSlON It is often possible to fit the frequency-dependent complex dielectric constant e(w) determined by a Kramers–Kronig analysis to an expression containing Drudelike terms as follows (8.23)

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where fi is the oscillator strength and the relaxation times ti are responsible for the broadening of the resonances. This expression can be written in terms of its real and imaginary parts (8.24) where the factors A'i and A"i are real with the following frequency dependencies (8.25a)

(8.25b) For the usual limit of narrow lines, witi >> 1, these expressions simplify to (8.26a)

and (8.26b)

Figure 8.3. Normalized line shape of the dielectric constant e = e' + iε" showing the real part ε' called dispersion, which is antisymmetric and passes through zero in the center where w = wi and the imaginary part e" called absorption, which reaches a maximum at w= w i.

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Figure 8.4. Infrared spectrum of an Nd2CuO4 single crystal at 10 K showing (a) reflectance and (b) the imaginary part of the dielectric constant e" obtained from a Kramers–Kronig analysis using the value e∞ = 6.8. (From Crawford et al., Ref. 5).

corresponding to Lorentzian line shapes. The sketches of these functions in Fig. 8.3 show that the real (A'i, dispersion) and imaginary (A"i , absorption) parts produce resonant lines centered at wi. The limiting dielectric constant for large w, denoted by e∞ in Eq. (8.23), is obtained from a fit to the data, so it is a limiting value for the range of frequencies under investigation, rather than the ultimate limit e0 of free space. The summation terms of Eq. 8.23 are Lorentz oscillator types that account for features arising, for example, from rotational, vibrational, or electronic processes. Many experimentalists measure the infrared reflectivity and then report their results asplots of e" = Im[ε(ω)] versus the frequency, while others present plots of the high-frequency conductivity s = we" versus the frequency. We see from a comparison of Figs. 8.4a and 8.4b (5) that e" plots (as well as s plots) are superior to reflectance plots for the determination of the positions wi and widths 1/ti of individual absorption lines arising from the summation terms of Eq. (8.24). This is

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because the reflectance plotted in Fig. 8.4a is a mixture of absorption and dispersion and hence it cannot provide the resonant frequencies wi with any precision. Some review articles are available on infrared (6,7) and Raman (8.10) spectroscopic studies of superconductors.

8.8. PLASMA OSCILLATIONS At very high frequencies, the conduction electrons of a metal act like a plasma, that is, an electrically neutral ionized gas in which the negative charges are mobile electrons and the positive charges are the background lattice of cations. When the frequency of an incident electromagnetic wave is much higher than the frequencies w i of all the Drude terms of the summation in Eq. 8.23, then the contributions of these Drude terms can be neglected and the frequency-dependent dielectric constant e(w)becomes (8.27) where the plasma frequency w p

(8.28) is the characteristic frequency of oscillation of the conduction electron plasma. To obtain Eq. (8.27) we dropped the damping factor iw/tp and made the identification e∞ = fp= e0 in (8.23). This high-frequency plasma dielectric constant ε(ω) is associated with oscillations of the conduction electrons relative to the positive charge background set up in the plasma by the incoming electromagnetic wave.

8.9. ENERGY GAP Tunneling and vibrational spectroscopy are complementary ways to determine the energy gap of a superconductor. In this section we say a few words about the infrared determination of gaps; the alternative tunneling method has been described elsewhere (1). For a superconductor at absolute zero, we expect light with frequencies v lower than Eg/h to be reflected to a greater extent than frequencies v > Eg/h, as in the case of a normal metal. Above absolute zero, these latter frequencies can excite quasiparticles and induce aphotoconductive response. Figure 8.5 shows low-temperature experimental data R(T)/R0 for the reflection of infrared radiation at frequencies below the gap value Eg ≈ 70 cm–1, and the drop in reflectivity for frequencies above

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Figure 8.5. Infrared reflectance spectra showing the abrupt change in transmission at the energy gap for the superconductor Ba0.6K0.4BiO3. The upper figure plots data for the temperatures T= 11, 14, 17, and 21 K, and the lower figure demonstrates the suppression of the low-frequency reflectivity enhancement by successively increasing applied magnetic fields Bapp = 0, 1,2, and 3 T. (From Schlesinger et ai., Ref. 11.)

this value for the cubic perovskite superconductor Ba0.6K0.4BiO3 (1 1). Similar reflectivity results have been obtained for many other superconductors. Figure 8.5a shows how increasing the temperature decreases the frequency at which the reflectivity undergoes a sharp drop in value. This is explained by the temperature dependence Eg(T) of the energy gap, which often has the form (8.29) where Eg = Eg(0), with the BCS value given by Eg = 3.52 kBTc. Figure 8.5b shows how increasing the applied magnetic field produces the same effect as increasing the temperature. This occurs because for many superconductors the critical magnetic field Bc(T) depends on the temperature through the expression

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171

(8.30) where Bc = Bc(0). Equation 8.30 can be inverted by considering T the critical temperature Tc(B) as a function of the applied field (8.31) where we write B in place of Bc(T) and Tc is the critical temperature for B = 0. It is now clear that Tc(B) decreases as the applied field increases. In the presence of a magnetic field, the gap equation (8.29) becomes (8.32)

Figure 8.6. Temperature dependence of the normalized microwave surface resistivity r/rn of aluminum (upper figure) for microwave frequencies in the range 12 to 80 GHz , where r n is the normal state surface resistivity. Each curve is labeled with its equivalent kBTc value. The plot of the normalized resistivity r/rn at the reduced temperature T/Tc = 0.7 versus kB/Tc (lower figure) exhibits a break at the energy 2.6 kBTc corresponding to the energy gap Eg = 2.6 kBTc (From Biondi and Garfunkel, Ref. 12.)

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a formula that is only valid for T < Tc(B). This expression clarifies that the energy gap decreases when either the temperature or the magnetic field increases. Increasing either T or B also decreases the Cooper pair concentration ns, which causes the reflectivity peak below the gap to decrease in magnitude, as in the figures. The infrared method of measuring energy gaps can be applied at much lower frequencies to elemental and other classical superconductors that have much lower transition temperatures Tc. As an example consider the temperature dependence of the normalized microwave resistivity r(T)/rn of aluminum, which is shown in Fig. 8.6 (upper figure) for five microwave frequencies v in the range from 12 to 80 GHz (12). Each curve is labeled by its microwave photon energy hv expressed in the units kBTc where Tc = 1.2 K for aluminum, and a temperature of 1 K is equivalent to 20.84 GHz. The three lowest curves extrapolate to zero resistivity, which indicates that superelectrons are not excited above the gap, and the two upper curves extrapolate to a finite resistivity, which is indicative of the presence of excited quasi-particles, The lower figure shows a plot of the microwave resistivity of each frequency at the temperature of T = 0.7 Tc versus the energy. We see that the slope of the curve is small up to the energy 2.6 kBTc, and larger beyond this point, indicating a gap energy of Eg ≈ 2.6 kBTc, a value somewhat less than the BCS prediction, Eg ≈ 3.53 kBTc. The more rapid rise in resistivity beyond this point is attributed to the superelectrons that have become excited to the quasi-particle state.

8.10. ABSORPTION AT VISIBLE AND ULTRAVIOLET FREQUENCIES Visible (13,000 to 25,000 cm–1, or 1.6 to 3.0 eV) and ultraviolet (3.1 to 40 eV) radiation has been employed to detect crystal field split electronic energy levels in insulating solids containing transition ions, and to determine energy gaps in semiconductors as well as the locations of impurity levels within these gaps. The response of metals to incident optical radiation depends on the plasma frequency wp (Eq. 8.28), which lies in the near infrared region for high-temperature superconductors and in the ultraviolet for many good conductors such as alkali metals. As an example of the interaction of optical radiation with superconductors in the normal state we examine the optical reflectance (reflectivity) of the series of La2–xSrxCuO4 compounds prepared for the composition range from x= 0 to x = 0.34 (13). The broad spectral scan, up to 37 eV, that is shown in Fig. 8.7 exhibits three reflectivity edges. The highest frequency edge near 30 eV falls off as 1 / w4, and it was attributed to excitations involving valence electrons. The midfrequency band from 3 to 12 eV was assigned to interband excitations from oxygen 2p valence bands to La 5d/4f orbitals, with the semiconductor La2CuO4 having an optical energy gap of about 2 eV. The low-frequency edge is absent in the x = 0 insulating compound and it is very high, off the scale on Fig. 8.7, in the two doped conductors.

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Figure 8.7. Optical reflectivity with the E vector polarized in the ab plane for La2–xSrxCuO4 single crystals for three compositions x. (From Uchida et al., Ref. 13.)

Figure 8.8. Frequency dependence of the optical conductivity s(w) of La2–xSrxCuO4 obtained from a Kramers–Kronig analysis of the reflectivity spectra of Fig. 8.7 for the E vector polarized in the ab plane. Results for several compositions x are shown. (From Uchida et al., Ref. 13.)

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A Kramers–Kronig analysis carried out for the reflectance spectra of Fig. 8.7 provided the conductivity spectra presented in Fig. 8.8 for the low-energy range from 0 to 4 eV (8). We see that s(w) at these low frequencies increases continuously with the level of doping x, being low for the insulators (x = 0,0.2,0.6), high for the superconductors (x = 0.1, 0.15, 0.2) and highest for the nonsuperconducting metal (x = 0.34). Recall that La2–x,SrxCuO4 is a hole superconductor. A similar set of spectra obtained for the electron superconductor Nd2–x,CexCuO4–y exhibited the same dependence of the low-frequency conductivity on x as in the hole case.

References 1. C. P. Poole, Jr., H. A. Farach, and R. I. Creswick, Superconductivity, Academic Press, San Diego (1995). 2. C. Kittel, Introduction to Solid State Physics, Wiley, New York (1996). 3. N. W. Ashcroft and N. D. Mermin, Solid State Physics, Saunders, Philadelphia (1976). 4. K. F. McCarty, J. Z. Liu, R. N. Shelton, and H. B. Radousky, Phys. Rev. B41, 8792 (1990); B42, 9973 (1990). 5. M. K. Crawford, G. Bums, G. V. Chandrashekhar, F. H. Dacol, W. E. Farneth, E. M. McCarron, III, and R. J. Smalley, Phys. Rev. B41, 8933 (1990). 6. A. P. Litvinchuk, C. Thomsen, and M. Cardona, in Physical Properties of High Temperature Superconductors, Vol. 4, Chap. 6, World Scientific, Singapore (1994). 7. T. Timusk and D. B. Tanner, in Physical Properties of High Temperature Superconductors, Vol. 1, Chap. 7, World Scientific, Singapore (1994). 8. M. Cardona, Physica C 185C–189C, 65 (1991). 9. E. Faulques and R. E. Russo, in Applications of Analytical Techniques to the Characterization of Materials, (D. L. Perry, ed.), p. 59, Plenum, New York (1991). 10. C. Thomsen and M. Cardona, in Physical Properties of High Temperature Superconductors, Vol. 1, Chap. 8, World Scientific, Singapore (1989). 11. Z. Schlesinger, R. T. Collins, J. A. Calise, D. G. Hinks, A. W. Mitchell, Y. Zheng, B. Dabrowski, N. E. Bickers, and D. J. Scalapino, Phys. Rev. B40, 6862 (1989). 12. M. A. Biondi and M. P. Garfunkel, Phys. Rev. 116, 853 (1959). 13. S. Uchida, T. Ido, H. Takagi, T. Arima, Y. Tokura, and S. Tajima, Phys. Rev. B43,7942 (1991).

9 A PPL ICATIONS

In this chapter we give an overview of some of the many applications of superconductors that are based on their electromagnetic absorbing properties.

9.1. THIN FILMS In the microwave region of the spectrum, many of the devices that employ superconductivity, such as delay lines, involve a microwave signal passing through strips of superconducting material. It is important therefore that these materials have low surface resistance in order to reduce the loss on transmission through the strip. The main advantage of using superconductors is that below Tc they have significantly lower surface resistance than other materials. A thin film is a layer of material a few lattice parameters thick, typically 350 nm, deposited on another material called a substrate. Superconducting thin films of Y-Ba-Cu-O have been made which have a surface resistance of 0.1 milliohms at 77 K and 10 GHz. The surface resistance of copper under the same conditions is 8.7 mohms. The resistance of the material also depends on its form. Figure 9.1 presents a plot of the frequency dependence of the surface resistance at 77 K of Y-Ba-Cu-O in bulk form, and also in thick and thin films of the material (1). Data for copper at 77 K are included for comparison. The graph shows that below about 10 GHz at 77 K, the thin films of Y-Ba-Cu-O have a significantly lower surface resistance than copper. Figure 9.2 shows an apparatus used to make thin films of Y-Ba-Cu-O. The component starting materials Y2O3, BaCO3, and CuO are contained in three heated holders located inside an evacuated chamber, along with the substrate on which the film is to be deposited. The holders are heated to temperatures high enough (1000°C and above) to allow the materials in them to evaporate. The substrate is maintained at a lower temperature, typically around 375°C. The evaporated materials condense on the substrate to form YBa2Cu3O7. Since the superconducting properties of 175

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Figure 9.1. Plot of surface resistance versus frequency for various forms of superconducting Y-Ba-Cu0 compared with copper at 77 K (from Porch, Ref. 1).

Figure 9.2. Vacuum chamber system for vapor deposition of thin films on a heated substrate. The starting materials for forming the film are placed in the heated holders.

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177

Y-Ba-Cu-O depend on its oxygen content, oxygen gas is introduced into the chamber after the film has been deposited. The film is generally annealed (heated) in an oxygen atmosphere at 850°C for a number of hours to achieve the proper oxygen stoichiometry and thereby convert the material to a 90 K superconductor. The substrate must be a single crystal whose lattice parameters are compatible with Y-Ba-Cu-O. Single crystals of MgO cut with an exposed [001] plane have been used. Besides having very low surface resistance in the superconducting state, the films also display very sharp resistance drops at the superconducting transition, as shown in Fig. 9.3 (2). As we will discuss later, these sharp drops can be the basis for infrared and optical sensors. The films also have high critical current densities, on the order of 106A/cm2. There are other ways to make thin films, such as sputtering and laser ablation. In the sputtering technique, positive ions such as argon are accelerated in a vacuum chamber to impinge on a target of the material to be made into a film, as illustrated in Fig. 9.4. The incident argon ions knock atoms out of the target material, and these atoms migrate to the cooler substrate where they condense to form the film. In the laser ablation method, a high-power pulsed laser beam incident on the target material, causes the surface layers of the target to evaporate. The film is formed when this evaporated material condenses on the cooler substrate.

Figure 9.3. Temperature dependence of the resistance of a thin film of Y-Ba-Cu-O showing the very sharp drop at the transition temperature (from Hopfengärtner et al., Ref. 2).

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Figure 9.4. Illustration of a device for making films by the sputtering process.

9.2. DELAY LINES A delay line is usually a long microstrip or a coplanar line that is deposited on a substrate. It is used to delay in time one incoming signal with respect to another by sending one signal through a longer path. Figure 9.5 illustrates a delay line. It is essential that the deposited strip have as low a surface resistance as possible in order to minimize attenuation as the signal passes through the strip. Since superconductors below Tc have lower surface resistance than other materials, superconducting delay lines constitute an important application. Delay lines made of Y-Ba-Cu-O films and operating at 77 K, such as the one in Fig. 9.5, have been found capable of delaying a signal at 10 GHz for a nanosecond with only a few decibels of loss. Both the thickness and the width of the strip influence the loss. Typical substrates used in delay lines are around 10µ thick, so they are essentially thin films themselves. Fig. 9.6 compares the loss as a function of frequency for delay lines made of copper and Y-Ba-Cu-O maintained at 77 K (3,4). The figure shows that the use of superconducting film greatly reduces the loss of the delay line.

9.3. STRIPLINE RESONATORS A resonator is a structure that can sustain an oscillatory electromagnetic field at a number of discrete frequencies. The resonant frequencies depend on the geometry of the system. As in the delay lines, it is important to minimize losses in

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Figure 9.5. Delay line made by depositing a thin film strip of a high-temperature superconductor on a substrate such as MgO.

Figure 9.6. Comparison of the loss versus the frequency at 77 K of a delay line made of copper and another made of a Y-Ba-Cu-O superconductor (after Hammond, Ref. 3).

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resonators. These losses primarily arise from the surface resistance of the material from which the resonator is constructed. Resonators are used in microwave circuits as filters to separate out specific frequencies from a broadband incoming signal. An important parameter used to characterize the resonator is the quality factor Q, which is defined as Q0 = 2p (energy stored)/(energy dissipated per cycle)

(9.1)

A resonator made of a low surface-resistance material has a higher Q and hence less power loss. Figure 9.7 shows a stripline ring resonator made by depositing a material such as copper or a superconductor on a substrate with dimensions such that it resonates at 3.4 GHz. Figure 9.8 compares the Q of ring resonators made of copper and Y-Ba-Cu-O at 77 K deposited on sapphire (3). The graph shows that below 10 GHz the ring resonator made of the superconductor has a Q that is more than an order of magnitude better than copper at the same temperature. Figure 9.9 shows another kind of stripline resonator in the form of a meander line of Y-Ba-Cu0 deposited on LaAlO3 (4). The superconducting strip is one half the wavelength of the fundamental frequency f0, which in this case is 1.24 GHz. The strip is clamped between copper plates and a lower loss dielectric material. The surface resistance of this strip at 4.2 K is 2 × 10–5 ohms, which is quite low. Bandpass filters are used in receiving signals. It is desirable that the receiver have a band width comparable to the band width of the incoming signal and that it

Figure 9.7. Illustration of a ring resonator made by depositing an annular strip of Y-Ba-Cu-O on a substrate. The dimensions were chosen so resonance occurs at 3.4 GHz (after Hammond, Ref. 3).

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Figure 9.8. Comparison of the Q of Cu and Y-Ba-Cu-O resonators at 77 K plotted against the frequency (from Hammond, Ref. 3).

reject frequencies outside this range. The filter should pass frequencies within the band width with minimum dissipation and distortion. Bandpass filters are constructed by depositing on a substrate a series of strips of materials having low surface resistance, with each strip having a geometry that resonates at a different frequency. Figure 9.10 shows a common type of bandpass filter. Figure 9.11 compares the performance at 77 K of a filter made of copper with one made of Y-Ba-Cu-O as a function of frequency. Again because of the lower surface resistance, there is much less loss in the superconducting filter than there is in the copper one.

9.4. CAVITY RESONATORS A cavity resonator is usually a cylindrical or rectangular metal container whose dimensions are multiples of the guide half-wavelength of the microwave radiation that is coupled into it to form a standing wave pattern. Cavity resonators are important components in microwave circuitry, and they are used as filters to separate out a specific frequency. They are also employed as the main element in feedback oscillators. When the exciting power source is turned off, the standing electromagnetic field inside the cavity decays in time because of losses due to leakage, owing to the finite conductivity of the cavity walls and the presence of any dissipative dielectric material. Quality factors add as reciprocals, so the overall Q arising from these three factors is (5)

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Figure 9.9. Illustration of a stripline resonator made in the form ofa meander line: (a) overall side view and (b) top view of center conductor transmission stripline (from Oates et al., Ref. 4).

(9.2) The Q due to the conductivity losses is given by (9.3) where, from Eq. (4.10), Rs = (wµ 0/2s)1/2

(9.4)

and G is a parameter related to the geometry of the cavity. Constructing the walls of the cavity from a superconducting material and operating at a temperature below Tc can significantly increase Qc For example, making the cavity walls from a thin film of Y-Ba-Cu-O, which has a surface resistance at 77 K of 10–5 ohms at 1 GHz, would increase the Qc by a factor of 103 over a cavity made with silver walls.

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Figure 9.10. Illustration of a bandpass filter made by depositing a series of thin microstrips of Y-Ba-Cu-O of different geometries on a substrate.

As discussed in Chap. 3, cavities in which one wall has been replaced by a superconducting thin film are often used to make surface resistance measurements on superconductors. If such a cavity with dielectric material excluded (1/Qe ≈ 0) is strongly undercoupled so the leakage losses can be neglected (1/QL ≈ 0), the overall Q then becomes

Figure 9.11. Comparison of the transmission loss at 77 K of a bandpass filter made of Cu (———) and another made of Y-Ba-Cu-O (- - - - -) (from Hammond, Ref. 3).

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Figure 9.12. Illustration of a microwave cavity used to measure the surface resistance of a thin superconducting film by making one wall of the cavity from the film.

(9.5) where QTC is the contribution to the Q of the total cavity interior, excluding the extra wall, and QWc is the contribution to Q from the superconducting wall. The surface resistance is obtained by measuring the overall Q of the cavity as a function of temperature to determine QWc(T) and then making use of Eq. (9.3) to determine Rs, where the factor G is now a parameter specific to the superconducting wall. Figure 9.12 shows an example of a cavity used to measure the surface resistance of superconducting films (1).

9.5. TRANSMISSION LINES One kind of superconducting microwave transmission line, shown in Fig. 9.13, consists of two infinitely long plates of superconducting material, usually with films deposited on the inner surfaces of the substrates separated by a small distance h, and with the plates having a width w >> h. Again, the low surface resistance of the superconducting plates reduces the losses on transmission. The velocity cz of the

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Figure 9.13. A microwave transmission line made of two infinitely long parallel superconducting films with the close spacing h, and width w >> h. The films of thickness t are deposited on the inside surfaces of the substrate plates. The figure shows the films but not the substrates.

signal down the transmission line depends on the spacing h between the plates, as shown in Fig. 9.14. The reduced velocity of the signal c2/c0, where c0 is the velocity of light in free space, decreases with decreasing spacing because of the increased penetration of the signal into the film, which increases the inductance but not the capacitance. In addition, the attenuation of the signal also increases when the spacing decreases, as shown in Fig. 9.15.

Figure 9.14. Plot of the reduced signal velocity cz/c0 versus the spacing h/l of the plates of Fig. 9.13 for two different film thicknesses t > l (———) and t = l (- - - - -) (adapted from Lancaster, Ref. 14).

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Figure 9.75. Plot of the attenuation coefficient versus the spacing h/l of the plates of Fig. 9.13 for two different film thicknesses t > l (———) and t < l (- - - - -) (adapted from Lancaster, Ref. 14).

9.6. SUPERCONDUCTING ANTENNAE Antennae are geometrical arrays of conducting material arranged in such a way so as to effectively radiate electromagnetic energy (transmitting operation), or to pick up an incoming signal (receiving operation). An antenna with dimensions d comparable to the wavelength l of the radiation constructed from conventional metal conductors can be quite efficient, and making one from superconductors does not greatly improve its performance. For smaller sizes, d << l , the efficiency decreases due to ohmic losses, and making a small antenna from superconducting materials improves its efficiency. One of the important issues in antenna design is the matching of the antenna impedance to the source impedance in order to optimize the transfer of radiation between the source and the antenna. This matching becomes more difficult with a smaller antenna made from superconductors because of the reduction in impedance. Making the matching circuitry from superconducting materials can help. Figure 9.16 shows a small dipole antenna made by depositing a superconducting film on a substrate (6). A small antenna of this kind has a large capacitive reactance and a small inductive reactance. The low value of the reactance in the superconducting

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Figure 9.16. Dipole antenna made by depositing Y-Ba-Cu-O strips on a substrate.

state causes the antenna to have a high Q. Figure 9.17 plots the Q versus the efficiency for a small dipole antenna of the kind shown in Fig. 9.16. Certain transmission situations require high signal directionality. A superdirective antenna array using high-temperature superconductors has been demonstrated (7). It consists of two dipoles half a wavelength long placed side by side a distance d apart, as shown in Fig. 9.18. The signal is fed into the two dipoles at the bottom of the figure from a coaxial cable and capacitor, which form a single stub tuner-matching circuit at the bottom. This allows the dipoles to have equal and opposite currents as required for enhanced directivity.

Figure 9.17. Comparison of a plot of Q versus efficiency at 77 K for a dipole antenna made of copper (- - - -) and one made of Y-Ba-Cu-O (———) (adapted from Lancaster, Ref. 14).

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Figure 9.18. Sketch of a two-element highly directive antenna array made of superconducting materials. The double dipole radiator is shown at the top, and the source coaxial cable with its associated single stub tuner is sketched at the bottom.

9.7. INFRARED AND OPTICAL SENSORS There are many applications of the electromagnetic absorbing process in parts of the electromagnetic spectrum other than the microwave region, and this section describes infrared and optical applications. An infrared detector is a device that indicates how much infrared radiation is present. An efficient radiation detector can make use of the sharp change in the resistance of a Y-Ba-Cu-O film, shown in Fig. 9.3, in the neighborhood of its transition temperature Tc = 90.5 K. This resistance change occurs over the narrow temperature range of about 0.3 K. Suppose that the temperature is maintained at precisely 90.7 K so that the resistance of the material is half of its value above Tc. If infrared radiation strikes the superconductor, it will heat the material and raise its temperature, and this temperature change will be reflected by an increase in resistance, thereby providing a measure of the amount of radiation that is present. The drop in resistance at the transition is so sharp that a temperature change of 0.0001 K results in a detectable change in resistance. An illustration of a device based on such a principle is shown in Fig. 9.19, The main part of the detector is a superconducting film deposited on a substrate such as mica, and this lies on a copper block in contact with liquid nitrogen. Tiny wire leads are attached to the superconductor to measure its resistance, and the film is enclosed in a vacuum container to isolate it from its environment. A small heater on the block maintains the temperature at a constant value near Tc. When infrared radiation strikes the film, its resistance increases; this is detected by applying a small current and measuring the voltage across the resistance leads. One could also design an optical and IR detector based on the sharp drop of the surface resistance of a superconductor shown in Fig 5.1 (8). Figure 9.20 shows what such a detector might look like. It would have a source of microwaves

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Figure 9.1 9. Sketch of an infrared and optical light sensor employing a thin film of Y-Ba-Cu-O.

Figure 9.20. Block diagram of an optical sensor that employs the sharp change in microwave surface resistance at the transition temperature illustrated in Fig. 9.3 as the radiation detection mechanism.

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provided, perhaps, by a Gunn diode located in a microwave bridge of the type sketched in Fig. 3.4. The superconducting sample held below Tc would be located in the cavity, which would be provided with an aperture and focusing lenses to direct the light to the sample, as shown.

9.8. MAGNETOMETERS There have been a number of proposals to use the dc magnetic field dependence of electromagnetic absorption as the sensing element in magnetic field detectors (9, 10). In principle, magnetometers employing field-dependent microwave absorption could approach the sensitivity of SQUID magnetometers. In Chap. 6 we saw that a very low-strength dc magnetic field applied to a single crystal of Y-Ba-Cu-O in the superconducting state produces a very sharp derivative signal that can be detected by an electron paramagnetic resonance spectrometer. This spectrum is a result of flux jumps arising from intrinsic weak links in the materials formed at twin boundaries in the crystal, The separation of the lines in the spectrum shown in Fig. 6.10 is less than 0.1 mT, so in effect this system could be the basis of a reasonably sensitive magnetometer, albeit an expensive one. A somewhat less expensive magnetometer can be designed around the effect of a dc magnetic field on the direct absorption of microwave energy in the superconducting state using a microwave bridge arrangement analogous to the infrared one sketched in Fig. 9.20. In this application, a cavity without an aperture containing the superconducting sample becomes the magnetic sensor. Figure 9.21 is a plot of the change in the diode current versus the dc magnetic field for an epitaxial film of Y-Ba-Cu-O on an MgO substrate acting as a sensor (1 1). Below 1 mT (i.e. < 10 G) this magnetometer would have a sensitivity of 5 µA per millitesla. A magnetometer that senses a dc magnetic field by the change in the frequency of an RF LC circuit in which the coil contains the sample could also be quite sensitive (12). The device would consist of an LC oscillator and a frequency meter as shown in Fig, 3.1. Clover and Wolf (13) have described a simple tunnel diode oscillator employing a germanium IN37 12 diode powered by a 1.4-V Duracell battery which can be built for less than $100. The magnetic sensing element of the instrument is the coil of the oscillator LC circuit wound around a thin quartz cylindrical holder containing the as-sintered sample of the superconductor. The strength of the dc magnetic field is determined by measuring the frequency shift of the oscillator. Figure 9.22 shows the shift in frequency as a function of dc magnetic field for a sintered sample of the 133-K Hg-Ba-Ca-Cu-O superconductor located in the coil of the LC circuit. Below 50 mT this magnetometer would have a sensitivity of 7 × 10–4 mT/Hz. In principle, with a stable oscillator and an accurate measurement of frequency, this could be a very sensitive magnetometer.

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Figure 9.21. Plot of the change of detector current in a microwave bridge as a function of magnetic field applied to the cavity of the bridge containing a thin film of Y-Ba-Cu-O. This result shows that such a bridge could be the basis of a magnetometer.

Figure 9.22. Frequency shift of an LC circuit versus applied dc magnetic field strength in which the coil is filled with the 133-K superconductor Hg-Ba-Ca-Cu-O at 77 K. These data demonstrate that such an arrangement could be developed into a sensitive magnetometer.

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References 1. I. A. Porch, Microwave Surface Impedance of YBaCuO, PhD Thesis, University of Cambridge (1991). 2. B. Hopfengärtner, B. Hensel, and G. Saemann-Ischenko, Phys. Rev. B44, 741(1991). 3. R. Hammond, Supercurrents 6, July (1989). 4. D. E. Oates, A. C. Anderson, D. M. Sheen, and S. M. Ali, IEEE Trans. Microwave Theory and Techniques39, 1522 (1991). 5. C. P. Poole, Jr., Electron Spin Resonance, Chap. 5, Wiley (New York) 1983; reprint edition, Dover, NewYork(1997). 6. G. C. Liang, R. S. Withers, B. E Cole, S. M. Garrison, M. E. Johansson, W. S. Ruby, and W. G. Lyons, IEEE Trans. Appl. Superconductivity 3, 3037 (1993). 7. Y. Huang, M. J. Lancaster, T. S. M. MacLean, Z. Wu, and N. McN. Alford, Physica C180, 267 (1991). 8. F. J. Owens, Patent pending. 9. B. E Kim, J. Bohandy, F. J. Adrian, and K. Moorani, Appl. Phys. Lett. 56, 2037 (1990). 10. S. Tyagi and M. Barsoum, Supercond. Sci. Technol. 1,20 (1988). 11, E J. Owens, unpublished. 12. F. 1. Owens, Appl. Superconductivity 2,51 (1994). 13. R. B. Clover and W. P. Wolf, Rev. Sci. Instr. 41, 617 (1970). 14. M. J. Lancaster, Passive Microwave Applications of High Temperature Superconductors, Cambridge Univ. Press, Cambridge (1997).

Index

Abrikosov lattice, 25,26 Absorption, 60 direct detection, 64 modulation detection, 64 X-ray edge, 72 AI, 171 Antenna, 186, 187 dipole, 188 Antiferromagnetic transition, 83 Antiferromagnetism, 83 La-Sr-Cu-O,32 MnO, 84 ordering, 75 phase diagram, 33 temperture dependence, 84 Y-Ba-Cu-O, 35 Antistokes’ line, 161 Argon, 178 Attenuation coefticient, 186 Auger electron, 7 1 Automatic frequency control (AFC), 60, 63 Ba-K-Bi-O, 50 infrared, 170 structure, 51 Band conduction, 1 diagram, 47 Bandpass filter, 183 BCS theory Cooper pair, 7 electromagnetic absorption, 99 penetration depth, 105 weak coupling, 3, 48

Bessel function, modified, 24 Binding layer, sketch, 44,46 Bi-Pb-Sr-Ca-Cu-O microwave absorption, 1 14–1 17, 130 penetration depth, 146–148 structure, 36 transition temperature, 32 Bi-Sr-Ca-Cu-O microwave absorption, 154 structure, 36 transition temperature, 32, 38 Bohr magneton, 63 Bremsstrahlung, 71 Carbon 60: see Fullerene, 5 1 Cavity resonator, 59, 18 1, 184 600 MHz figure, 61 Chain of Cu-O atoms, 35 Charge density wave, 109 Charge transfer layer, 46 Chemical shift, 68 Co, 71 Coffey–Clem model, 142, 156 complete, 145 Coherence length, 12 peak, 99, 102 Collision time, 75 Colossal magnetoresistance, 89 Conduction layer aligned, 42 sketch, 44, 45 193

194

Conductivity AC, 78 BCS, 100 complex, 78 electrical, 75 optical, 173 temperature dependence, 100 Cooper pair, 2,4 binding, 4,49 binding mechanism, 5 breakup, 3, 19,96 Josephson junction, 127 phase, 134 phase coherence, 6 phonon mechanism, 5 wave function, 7.23, 107 Copper oxide plane aligned, 42 figure, 33, 37, 38 number, 37,38 staggered, 42 transition temperature, 38 Copper oxides, see cuprate, 42 Critical current density, 9, 10 Josephson, 128 penetration,10, 11 temperature dependence, 19, 20 Critical field lower, 17, 25 slope, 18 thermodynamic, 17,19 upper, 17,25 upper-figure, 48 Crossover temperature, 103, 107 Cu, 62, 63, 64,68, 69, 71, 162 Cuprate, 42 commonalities, 44, 45 general properties, 42 Hg, 2 individual compound properties, 43 orthorhombic, 42 tetragonal, 42 Curie temperature, 84, 87,91 Current-transport, 141 Debye energy, 160 temperature, 78,81, 160 Delay line, 175, 178, Y-Ba-Cu-O, 179 Demagnetization factor, 14

INDEX

Density of superelectrons, 13 Density of states, 47, 48, 99 figure, 100 Depairing current density, 19 Depinning barrier, 152 field, 146, 147 line, 140 temperature, 141 transition, 152 Dewar, 56 Diamagnet, perfect, 7 Dielectric constant complex, 164, 165, 167 conductivity, 164, 165 Dimensionality, 109 Dipole antenna, 187, 188 field of sphere, 16 Dispersion, 60 Double beam, I59 Drude model, 166 D-wave pairing, 105 Effective mass, 69 Electromagnetic absorption, 99, 107 due to vortex motion, 139 Electron conduction, 34, 75 scattering, 1 superconductor, 174 valence, 1 Electron paramagnetic resonance: see EPR, 62 Electron spin resonance: see ESR, 62 Ellipsoid, 14, 15 oblate, 14, 15 prolate, 14, 15 Energy band, figure, 47 Energy gap, 2, 169 A1, 171 infrared, 2, 171 microwave, 2 temperature dependence, 17 1 Energy surface, 2 1, 22 EPR, 55 bridge block diagram, 63 silent, 64 spectrometer, 190 ESR see EPR, 55 Eu, 71 Experimental techniques, 55

INDEX

Fe, 70, 71 Fermi level, 47 surface, 1 Ferromagnetic transition, 84 Ferromagnetism, 84 microwave absorption, 88 ordering, 75 Fieldcooling(FC),121, 123, 124, 130 Fluctuations, 107–109 Flux creep, 25, 113, 140, 141 expulsion, I5 flow,25, 113, 140, 141, 151 jump, 133, 135 melting line, 140 penetration, 8 quantization, 23 solid, 141 trapping,20, 117, 121, 131 Fluxon, 26 Free energy, Gibbs, 2 1, 22 Fullerene doped-figure, 52,53 magnetization, 54 structure figure, 52 transition temperature, 53, 54 Fundamental band, 161 Gap anisotropy, 48 energy, 2 superconducting, 96, 169 Ta, 5 Gd, 36,62, 85, 86, 87 Geodesic dome, 53 g-factor, Landé, 63 Giant magnetoresistance, 75, 89 Gibbs free energy, 21,22 Ginzburg–Landau parameter, 2 1, 49 theory, 70 Green phase, 64 Gunn diode, 59, 120 Gyromagnetic ratio, 68 Helmholtz equation, 9

195

Hg-Ba-Ca-Cu-O ( cont.) magnetometer, I9 1 pressure, 40 resistivity temperature dependence, 28 structure, 39 transition temperature, 32, 38, 40 Hg-Pb-Ba-Ca-Cu-O, 38 frequency shift, 150 microwave absorption, 95, 96, 148– 153 resistivity, 3 surface resistance, 108 Hole conduction, 34 Hysteresis, 120, 142 LFMA, I14 microwave absorption, 130 modulation, 114 Impedance of surface, 78 In, 4 Infinite layer phase, 41 figure, 41 Infrared, 55 absorption, I59 active, 162, 163 Ba-K-Bi-O, 170 detector, 188, 189 energy gap, 2 In, 4 Nd-Ce-Cu-O, I68 sensor, 188, 189 spectroscopy, I61 spectrum, 168 Y-Ba-Cu-O, 189 Irreversibility, 140, 151 line, 25,26, 141, 147, 153 Y-Ba-Cu-O,26,27 Isotope effect, 4 Josephsonjunction, 119, 125, 126 loop, 128 pair of, 127 Y-Ba-Cu-O, 129 Josephson loop, 134 Klystron, 59 Kramers–Kronig analysis, 159, 166, 173, 174

Hg, 4

Hg-Ba-Ca-Cu-O figure, 39 magnet, 190, 19 1

La-Ba-Cu-O,32 transition temperature, 32 La-Ca-Mn-O, 90

196

Ladder phase, 40, 4 1, 110 exchange coupling, 42 figure, 41 Laser ablation, 177 La-Sr-Cu-O, 174 antiferromagnetism, 32 phase diagram, 33 transition temperature, 32 unit cell, 3 I, 32 visible region absorption, 172–1 74 La-Ca-Mn-O,89,90 La-Sr-Mn-O, 89, 91–94 Latent heat, 164 LC oscillator, 56 LCR circuit, 58 LFMA, I13 alternating fields, 132 Bi-Pb-Sr-Ca-Cu-O, 114–1 17, 130 Bi-Sr-Ca-Cu-O, 154 derivative, I14 direct detection, 122, 123 Hg-Pb-Ba-Ca-Cu-O,95,96, 148, 150–1 53 hysteresis, 114, 117, 13I indicator of superconductivity, 136, 137 mechanism, 128 modulation, 114, 133 origin, 125 sharp line structure, 122 sweep range, 116 temperature dependence, 115 thin film, 117, 118 time dependence, 1 17, 131 Y-Ba-Cu-O, 118-124, 131-146,153–155 London penetration depth, 9 Lorentz force, 26, 139 Lorentzian shape, I68 Low magnetic field induced microwave absorption, see LFMA, 113 Magnet, 190, 191 Magnetic field, 14 internal, 14–17 slope, 18 Magnetic moment of sphere, 16 Magnetization ferromagnetism, 85 remnant, 121 temperature dependence, 8, 85 Magnetoelectromagnetic absorption, 151 Magnetogyric ratio, 68 Magnetometer, 127, 190, 191

INDEX

Magnetoresistance, 89 magnetic field dependence, 90 temperature dependence, 90 Magnetoresistivity, 75 Maxwell curl equation, 9, 165 equations, 8 Meander line, 180 Meissner effect, 7,8 region, 113 state, 20, 141 Melting line, 140, 141 Metal-insulator transition, 75, 83, 109 Metallic elements, table, 76, 77 MgO, 118 Microwave bridge, 60 bridge block diagram, 62 energy gap, 2, 171 measurement, 59 Microwave absorption: see LFMA, 1 I3 angular dependence, 120 explanation, 140 ferromagnetism, 88, 93 multiline derivative spectrum, 129 zero field, 95 Mn, 62 MnO, 83 antiferromagnetism, 84 surface resistance, 85 Modulation, 64 Mossbauer resonance, 70 Muon spin relaxation, 69 Nb–Ge transition temperature, 31 Nd, 36 Nd-Ce-Cu-O,34, 174 electron conduction, 34 Kramers–Kronig, I68 structure, 34 Néel temperature, 32, 83 Ni, 36, 105, 106 NiS, 83 Normal mode, 160 Nuclear quadrupole resonance, 69 Ohm’s law, 75,165 Optical, 55 conductivity, 173 detector, 188

INDEX

Optical (cont.) reflectivity, 173 sensor, 188, 189 Organic conductor, 109 Oscillator, feedback, 181 Parallel plate resonator, 66, 67 Paramagnetic marker, 65 probe, 65 Penetration depth, 1 1 anisotropy, 69 BCS, 101 creep region, 143 Bi-Pb-Sr-Ca-Cu-O, 146–148 flow region, 144 frequency dependence, 147 frequency shitt, 149 London, 9 low temperature, 102 magnetic field dependence, 21, 92 measurement, 57, 145 temperature dependence, 13,95, 10 1, 102, 103, 104, 107 Y-Ba-Cu-O,69, 102–104, 145, 146 Permeability Gd, 85 Gd temperature dependence, 86 Perovskite cubic, 50, 170 figure, 51 structure, 89 Phase jump, 134 Phase diagram La-Sr-Cu-O,33 magnetic, 140 universal, 33 Y-Ba-Cu-O,35 Phase sensitive detector, 58, 63 Phonon exchange, 6 scattering, 8 1 Photoemission, 71 Pinning barrier, 152 force, 153 line, 140 vortex, 146 strong, 153 Plasma frequency, I69

197

Plasma (cont.) oscillation, 169 Polarized light, 162 Positron annihilation, 70 Power dissipation, 79 Pressure, 40 Quality factor Q, 60, 180, 184 addition by reciprocals, 182 Radio frequency, 55 Raman active mode, I62 spectroscopy, I6 I Y-Ba-Cu-O, 162–164 Reactance of surface, 79 Reflectance, 159 Reflectivity, 165 complex, 166 Remnant magnetization, 121 Resistance drop at Tc, 3 residual, 1 surface, 78 zero, 1 Resistivity anisotropy, 50 frequency dependence, 176 Hg-Ba-Ca-Cu-O, 29 Hg-Pb Ba-Ca-Cu-O,3 moving vortices, 28 semiconductor, temperature dependence, 82 temperature dependence, 8 1 Resonant cavity, 59 600 MHz figure, 61 Resonator cavity, 181, 184 parallel plate, 66, 67 Q comparison, 181 ring, 180 stripline, 67, 153, 156, 178, 182 superconducting, I80 Y-Ba-Cu-O, 180, 181 Scattering electron,1 phonon, 8 1 Screening supercurrrent, 102 Selection rule, 162 Semiconductor, 82 resistivity, 82

198

Sensor detector, 188 infrared, 188, 189 optical, 189 Skin depth, 79,80 Sn, 71 Soft mode, 163 Sputtering, 177, 178 SQUID, 127 Stokes' line, 161 Stripline resonator, 67 measurements, I53 Structure, 43, 76 Bi-Pb-Sr-Ca-Cu-O,36 Hg-Ba-Ca-Cu-O, 39 La-Sr-Cu-O,31, 32 Nd-Ce-Cu-O,34 TI-Ba-Ca-Cu-O,37 Y-Ba-Cu-O, 34 Substitution, 36 Substrate, 175 Superheterodyne spectrometer, 120 Surface energy, 2 1, 22 Surface impedance, 78,97,98, 143, 144 Surface reactance, 79 Surface resistance, 78 BCS,101 creep region, 143, 144 depinning transition, 154 energy gap, 171 ferromagnetic, 85, 87, 91, 92 flow region, 144, 145 fluctuations, 108 frequency dependence, 80, 97 Gd, 86 Hg-Pb-Ba-Ca-Cu-O, 108 magnetic field dependence, 87, 91, 113 measurement, 105 microwave, 171 MnO, 85 semiconductor, 82 temperature dependence, 86, 92, 96, 101, 110, 111 Y-Ba-Cu-O, 105–107, 175–177 Susceptibility definition, 7 microwave, 65 temperature dependence, 84 Y-Ba-Cu-O,65 S-wave pairing, 105

INDEX

Ta, 4, 5 TCNQ, 109 Temperature dependence density of superelectrons, 12 magnetization, 7 penetration depth, 12 resistivity, 28, 50 Thin film, 175 preparation, 177 resistivity, 177 TI, 68, 7 I TI-Ba-Ca-Cu-O structure, 36,37 transition temperature, 32, 38 TI-Pb-Sr-Ca-Cu-O, 117, 119 TMTSF, 14 Transition temperature BCS relation, 48 number of copper oxide planes, 38 pressure dependence, 40 Transmission line, 184, 185 Transmittance, 159 Transport current, 141 TTF-TCNQ, 109 surfaceresistance, 110 Twin free crystal, 162, 163, 164 Two fluid model, 8.95 Type I superconductor, 17 Type II superconductor, 17 Ultraviolet absorption, 172 Valence electron, 1 Van Hove singularity, 48 Vapor deposition, 176 Velocity, reduced, I85 Vibrational state, 160, 161 Visible absorption, I72 La-Sr-Cu-O, 172-174 Vortex, 20 bundle, 28 density, 26 distance dependence, 24 drag coefficient, 27 energy barrier, 28 equation of motion, 143 fluid, 113 force, 26 lattice, 25 liquid, 25 melting, 25, 146

INDEX

Vortex (cont.) motion, 139 oscillation, 139 radius, 24 solid, 113 thermal hopping, 28 two dimensional liquid, 27 Vortex-vortexinteraction, 155 Wave function Cooper pair, 107 D-wave, 107 S-wave, 107 Weak link, 133, 134 X-ray absorption, 71 Y-Ba-Cu-O, 102 antenna, 187 antiferromagnetism, 35 delay line, 179

199

Y-Ba-Cu-O (cont.) infrared, 189 irreversibility line, 26, 27 LFMA, 118–124,131–135, 144 _146, 153–155 Josephson junction, 129 magnetization temperature dependence, 7 magnetometer, 190, 191 oxygen content, 177 penetration depth, 69, 102-104,145, 146 phase diagram, 35 Raman spectrum, 162, 164 resistivity temperature dependence, 50 resonator, 180, 181 surface resistance, 105–1 07, 175–1 77 susceptibility, 65 transition temperature, 32 twin free, 162, 163, 164 unit cell, 34 Zero field cooling (ZFC), 120, 122 Zn, 36, 103