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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.
v
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Preface
Superconductivity is a striking physical phenomenon that has attracted the attention of
physicists, chemists, engineers, and also the nontechnical public. The theory of super-
conductivity is considered difficult. Lectures on the subject are normally given at the
end of Quantum Theory of Solids, a second-year graduate course.
In 1957 Bardeen, Cooper, and Schrieffer (BCS) published an epoch-making mi-
croscopic theory of superconductivity. Starting with a Hamiltonian containing electron
and hole kinetic energies and a phonon-exchange-pairing interaction Hamiltonian, they
demonstrated that (1) the ground-state energy of the BCS system is lower than that of
the Bloch system without the interaction, (2) the unpaired electron (quasi-electron) has
an energy gap
∆
0
at 0 K, and (3) the critical temperature T
c
can be related to ∆
0
by
2∆
0
= 3.53 k
B
T
c
, and others. A great number of theoretical and experimental investiga-
tions followed, and results generally confirm and support the BCS theory. Yet a number
of puzzling questions remained, including why a ring supercurrent does not decay by
scattering due to impurities which must exist in any superconductor; why monovalent
metals like sodium are not superconductors; and why compound superconductors, in-
cluding intermetallic, organic, and high-T
c
superconductors exhibit magnetic behaviors
different from those of elemental superconductors.
tures of both electrons and phonons in a model Hamiltonian. By doing so we were able
Recently the present
authors extended the BCS theory by incorporating band struc-
to answer the preceding questions and others. We showed that under certain specific
conditions, elemental metals at low temperatures allow formation of Cooper pairs by the
phonon exchange attraction. These Cooper pairs, called the pairons, for short, move as
free bosons with a linear energy–momentum relation. They neither overlap in space nor
interact with each other. Systems of pairons undergo Bose–Einstein condensations in two
and three dimensions. The supercondensate in the ground state of the generalized BCS
system is made up of large and equal numbers of
± pairons having charges ±2e, and
it is electrically neutral. The ring supercurrent is generated by the
± pairons condensed
at a single momentum q
n
= 2π
n
L
–1
, where L is the ring length and n an integer. The
macroscopic supercurrent arises from the fact that
± pairons move with different speeds.
Josephson effects are manifestations of the fact that pairons do not interact with each
other and move like massless bosons just as photons do. Thus there is a close analogy
between a supercurrent and a laser. All superconductors, including high-T
c
cuprates, can
be treated in a unified manner, based on the generalized BCS Hamiltonian.
vii
viii
PREFACE
Because the supercondensate can be described in terms of independently moving
pairons, all properties of a superconductor, including ground-state energy, critical tem-
perature, quasi-particle energy spectra containing gaps, supercondensate density, specific
heat, and supercurrent density can be computed without mathematical complexities. This
simplicity is in great contrast to the far more complicated treatment required for the phase
transition in a ferromagnet or for the familiar vapor–liquid transition.
The authors believe that everything essential about superconductivity can be pre-
sented to beginning second-year graduate students. Some lecturers claim that much
physics can be learned without mathematical formulas, that excessive use of formulas
hinders learning and motivation and should therefore be avoided. Others argue that
learning physics requires a great deal of thinking and patience, and if mathematical
expressions can be of any help, they should be used with no apology. The average
physics student can learn more in this way. After all, learning the mathematics needed
for superconductor physics and following the calculational steps are easier than grasping
basic physical concepts. (The same cannot be said about learning the theory of phase
transitions in ferromagnets.) The authors subscribe to the latter philosophy and prefer
to develop theories from the ground up and to proceed step by step. This slower but
more fundamental approach, which has been well-received in the classroom, is followed
in the present text. Students are assumed to be familiar with basic differential, integral,
and vector calculuses, and partial differentiation at the sophomore–junior level. Knowl-
edge of mechanics, electromagnetism, quantum mechanics, and statistical physics at the
junior–senior level are prerequisite.
A substantial part of the difficulty students face in learning the theory of supercon-
ductivity lies in the fact that they need not only a good background in many branches
of physics but must also be familiar with a number of advanced physical concepts such
as bosons, fermions, Fermi surface, electrons and holes, phonons, Debye frequency, and
density of states. To make all of the necessary concepts clear, we include five preparatory
chapters in the present text. The first three chapters review the free-electron model of a
metal, theory of lattice vibrations, and theory of the Bose–Einstein condensation. There
follow two additional preparatory chapters on Bloch electrons and second quantization
formalism. Chapters 7–11 treat the microscopic theory of superconductivity. All basic
thermodynamic properties of type I superconductors are described and discussed, and
all important formulas are derived without omitting steps. The ground state is treated
by closely following the original BCS theory. To treat quasi-particles including Bloch
electrons, quasi-electrons, and pairons, we use Heisenberg’s equation-of-motion method,
which reduces a quantum many-body problem to a one-body problem when the system-
Hamiltonian is a sum of single-particle Hamiltonians. No Green’s function techniques
are used, which makes the text elementary and readable. Type II compounds and high-T
c
superconductors are discussed in Chapters 12 and 13, respectively. A brief summary and
overview are given in the first and last chapters.
In a typical one-semester course for beginning second-year graduate students, the
authors began with Chapter 1, omitted Chapters 2–4, then covered Chapters 5–11 in that
order. Material from Chapters 12 and 13 was used as needed to enhance the student’s
interest. Chapters 2–4 were assigned as optional readings.
The book is written in a self-contained manner so that nonphysics majors who want
to learn the microscopic theory of superconductivity step by step in no particular hurry
PREFACE
ix
may find it useful as a self-study reference. Many fresh, and some provocative, views
are presented. Researchers in the field are also invited to examine the text.
Problems at the end of a section are usually of a straightforward exercise type
directly connected to the material presented in that section. By solving these problems,
the reader should be able to grasp the meanings of newly introduced subjects more firmly.
The authors thank the following individuals for valuable criticism, discussion, and
readings: Professor M. de Llano, North Dakota State University; Professor T. George,
Washington State University; Professor A. Suzuki, Science University of Tokyo; Dr. C.
L. Ko, Rancho Palos Verdes, California; Dr. S. Watanabe, Hokkaido University, Sapporo.
They also thank Sachiko, Amelia, Michio, Isao, Yoshiko, Eriko, George Redden, and
Brent Miller for their encouragement and for reading the drafts. We thank Celia García
and Benigna Cuevas for their typing and patience. We specially thank César Zepeda and
Martin Alarcón for their invaluable help with computers, providing software, hardware,
as well as advice. One of the authors (S. F.) thanks many members of the Deparatmento
de Física de la Facultad de Ciencias, Universidad Nacional Autónoma de México for
their kind hospitality during the period when most of this book was written. Finally we
gratefully acknowledge the financial support by CONACYT, México.
Shigeji Fujita
Salvador Godoy
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Contents
Constants, Signs, and Symbols
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv
Chapter 1.
Introduction
1.1.
Basic Experimental Facts
1
1.2.
Theoretical Background
9
1.3.
Thermodynamics of a Superconductor
12
1.4.
Development of a Microscopic Theory
19
1.5.
Layout of the Present Book
21
References
22
Chapter 2.
Free-Electron Model for a Metal
2.1.
Conduction Electrons in a Metal; The Hamiltonian . . . . . . . . . . . .
23
2.2.
Free Electrons; The Fermi Energy
. . . . . . . . . . . . . . . . . . . . . . . .
26
2.3.
Density of States
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2.4.
Heat Capacity of Degenerate Electrons 1; Qualitative Discussions
. .
33
2.5.
Heat Capacity of Degenerate Electrons 2; Quantitative Calculations . .
34
2.6.
Ohm’s Law, Electrical Conductivity, and Matthiessen’s Rule
. . . . . .
38
2.7.
Motion of a Charged Particle in Electromagnetic Fields
. . . . . . . . . .
40
Chapter 3.
Lattice Vibrations: Phonons
3.1.
Crystal Lattices
45
3.2.
Lattice Vibrations; Einstein’s Theory of Heat Capacity
46
3.3.
Oscillations of Particles on a String; Normal Modes
49
3.4.
Transverse Oscillations of a Stretched String
54
3.5.
Debye’s Theory of Heat Capacity
58
References
65
Chapter 4.
Liquid Helium: Bose–Einstein Condensation
4.1.
Liquid Helium
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
4.2.
Free Bosons; Bose–Einstein Condensation . . . . . . . . . . . . . . . . . .
68
4.3.
Bosons in Condensed Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
xi
xii
CONTENTS
Chapter 5. Bloch Electrons; Band Structures
5.1.
The Bloch Theorem
75
5.2.
The Kronig–Penney Model
79
5.3.
Independent-Electron Approximation; Fermi Liquid Model
81
5.4.
The Fermi Surface
83
5.5.
Electronic Heat Capacity; The Density of States
88
5.6.
de-Haas–van-Alphen Oscillations; Onsager’s Formula
90
5.7.
The Hall Effect; Electrons and Holes
93
5.8.
Newtonian Equations of Motion for a Bloch Electron
95
5.9.
Bloch Electron Dynamics
100
5.10.
Cyclotron Resonance
103
References
106
Chapter 6. Second Quantization; Equation-of-Motion Method
6.1.
Creation and Annihilation Operators for Bosons
109
6.2.
Physical Observables for a System of Bosons
113
6.3.
Creation and Annihilation Operators for Fermions
114
6.4.
Second Quantization in the Momentum (Position) Space
115
6.5.
Reduction to a One-Body Problem
117
6.6.
One-Body Density Operator; Density Matrix
120
6.7.
Energy Eigenvalue Problem
122
6.8.
Quantum Statistical Derivation of the Fermi Liquid Model
124
Reference
125
Chapter 7.
Interparticle Interaction; Perturbation Methods
7.1.
Electron–Ion Interaction; The Debye Screening
127
7.2.
Electron–Electron Interaction
129
7.3.
More about the Heat Capacity; Lattice Dynamics
130
7.4.
Electron–Phonon Interaction; The Fröhlich Hamiltonian
135
7.5.
Perturbation Theory 1; The Dirac Picture
138
7.6.
Scattering Problem; Fermi’s Golden Rule
141
7.7.
Perturbation Theory 2; Second Intermediate Picture
144
7.8.
Electron–Impurity System; The Boltzmann Equation
145
7.9.
Derivation of the Boltzmann Equation
147
7.10.
Phonon-Exchange Attraction
150
References
154
Chapter 8. Superconductors at 0 K
8.1.
Introduction
155
8.2.
The Generalized BCS Hamiltonian
156
8.3.
The Cooper Problem 1; Ground Cooper Pairs
161
8.4.
The Cooper Problem 2; Excited Cooper Pairs
164
8.5.
The Ground State
167
8.6.
Discussion
172
8.7.
Concluding Remarks
178
CONTENTS
xiii
References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
179
Chapter 9.
Bose–Einstein Condensation of Pairons
9.1.
Pairons Move as Bosons
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
181
9.2.
Free Bosons Moving in Two Dimensions with ∈ = cp
. . . . . . . . . .
184
9.3.
Free Bosons Moving in Three Dimensions with ∈ = cp . . . . . . . . . .
187
9.4. B–E Condensation of Pairons
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
189
9.5.
Discussion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
193
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
199
Chapter 10. Superconductors below T
c
10.1.
Introduction
201
10.2.
Energy Gaps for Quasi-Electrons at 0 K
202
10.3.
Energy Gap Equations at 0 K
204
10.4.
Energy Gap Equations; Temperature-Dependent Gaps
206
10.5.
Energy Gaps for Pairons
208
10.6.
Quantum Tunneling Experiments 1; S–I–S Systems
211
10.7.
Quantum Tunneling Experiments 2; S
1
–I–S
2
and S–I–N
219
10.8.
Density of the Supercondensate
222
10.9.
Heat Capacity
225
10.10.
Discussion
227
References
232
Chapter 11. Supercurrents, Flux Quantization, and Josephson Effects
11.1.
Ring Supercurrent; Flux Quantization 1
233
11.2.
Josephson Tunneling; Supercurrent Interference
236
11.3.
Phase of the Quasi-Wave Function
239
11.4.
London’s Equation and Penetration Depth; Flux Quantization 2
241
11.5.
Ginzburg–Landau Wave Function; More about the Supercurrent
245
11.6.
Quasi-Wave Function: Its Evolution in Time
247
11.7.
Basic Equations Governing a Josephson Junction Current
250
11.8.
AC Josephson Effect; Shapiro Steps
253
11.9.
Discussion
255
References
260
Chapter 12. Compound Superconductors
12.1.
Introduction
263
12.2.
Type II Superconductors; The Mixed State
263
12.3.
Optical Phonons
268
12.4.
Discussion
270
References
270
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