Superconductivity of Transition Metals

1. Introduction.- 1.1 The Distribution of Superconductors Among the Elements of the Periodic Table, Their Alloys and Compounds.- 1.1.1 Superconductivity in Pure Elements.- 1.1.2 Superconductivity in Alloys.- 1.1.3 Superconductivity in Chemical Compounds.- 1.1.4 Outlook.- 1.2 Survey of Theoretical Concepts.- 1.2.1 The Fundamentals of the BCS Theory.- 1.2.2 The Role of the Coulomb Interaction.- 1.2.3 The Effects of Retardation in the Electron-Phonon Interaction.- 1.2.4 The Electron-Phonon Coupling Constant.- 1.2.5 The Peculiarities of the Transition Metals.- 1.2.6 Superconductivity and Magnetism.- 2. The Theory of Strong Coupling Superconductors.- 2.1 Perturbation Theory for Green’s Functions.- 2.1.1 The Gor’kov-Nambu Matrix Formalism.- 2.1.2 Perturbation Theory.- 2.1.3 The Momentum Representation.- 2.1.4 Lower-Order Graphs for ?.- 2.2 Eliashberg Equations for the Electron Green’s Function.- 2.2.1 Formulation of the Equations and Summation over Frequencies.- 2.2.2 Calculation of the Phonon Part ?ph(p,?).- 2.2.3 Calculation of the Coulomb Part ?c (p,?)).- 2.2.4 Basic Equations for a Superconductor.- 2.2.5 Dispersion Relations.- 2.3 The Superconducting Transition Temperature.- 2.3.1 Linearized Eliashberg’s Equations.- 2.3.2 Weak Coupling.- 2.3.3 Intermediate Coupling.- 2.3.4 Strong Coupling.- 2.3.5 Contribution of Low-Frequency Modes.- 2.4 Calculations of the Electron-Phonon Coupling Constant ?.- 2.4.1 Green’s Function of Phonons.- 2.4.2 Expression of ? in Terms of the Phonon Green’s Function.- 2.4.3 The McMillan Representation.- 2.4.4 The Hopfield Representation for a Transition Metal.- 3. Superconductivity and Magnetism.- 3.1 Conditions for the Occurrence of Superconductivity in a Ferromagnetic Metal.- 3.1.1 Four-Dimensional Matrix Formalism.- 3.1.2 Wick’s Theorem.- 3.1.3 Perturbation Theory.- 3.1.4 Matrix Structure of Green’s Functions.- 3.1.5 Superconducting Transition Temperature of a Ferromagnetic Metal.- 3.2 Superconductivity in Dilute Ferromagnetic Alloys.- 3.2.1 The Problem of the Coexistence of Superconductivity and Magnetic Ordering.- 3.2.2 Equations Including Dynamic Effects.- 3.2.3 Quasi-Static Approximation.- 3.3 The Coexistence of Superconductivity and Helical Magnetic Ordering.- 3.3.1 The Anderson-Suhl Hypothesis.- 3.3.2 Equations for the Electron Green’s Function in the Weak-Coupling Approximation.- 3.3.3 Analysis of the Free Energy.- 3.4 Magnetic Ordering in Superconducting Rare-Earth Compounds.- 3.4.1 Superconductivity and Antiferromagnetism.- 3.4.2 Superconductivity and Ferromagnetism.- 3.4.3 Phenomenological Theory of Magnetic Ordering in Superconductors.- 3.4.4 Magnetic Fluctuations and Superconductivity.- 3.4.5 Critical Behaviour of Magnetic Superconductors.- 3.5 Magnetic Impurities in Superconductors.- 3.5.1 The Abri kosov-Gor’kov Theory.- 3.5.2 The Crystalline Field Effects.- 3.6 The Kondo Effect in Superconductors.- 3.6.1 The Equation for the Scattering Amplitude.- 3.6.2 Impurity States Within the Energy Gap.- 3.6.3 The Lowering of the Superconducting Transition Temperature by Magnetic Impurities.- 3.6.4 Comparison with Experiment.- 3.7 Nonmagnetic Localized States in Superconducting Alloys.- 3.7.1 Anderson’s Model.- 3.7.2 The Energy Gap and the Order Parameter.- 3.7.3 Suppression of the Superconducting Transition.- 3.8 Localized Spin Fluctuations in Superconducting Alloys.- 3.8.1 The Anomalous Behaviour of a Number of Alloys with Nonmagnetic Impurities.- 3.8.2 Theoretical Study of the Effect of LSF on Tc.- 3.8.3 Alloys Based on Transition Metals.- 3.9 Spin Fluctuations in Pure Transition Metals.- 3.9.1 Itinerant Spin Fluctuations of a Nonferromagnetic Metal (Paramagnons).- 3.9.2 Virtual-Paramagnon Creation and Superconductivity.- 3.9.3 Comparison with Experiment.- 4. Superconductivity in Transition Metals.- 4.1 The Principal Parameters of Superconducting Metals.- 4.1.1 The Superconducting Transition Temperature.- 4.1.2 The Energy Gap.- 4.1.3 Empirical Correlations Between Tc and Physical Properties of Transition Metals.- 4.1.4 The Pressure Dependence of Tc.- 4.1.5 Superconductivity in Thin Films.- 4.2 The Electron Structure of Transition Metals.- 4.2.1 Theoretical Band Structure Calculations.- 4.2.2 The Fermi Surface.- 4.2.3 X-Ray and Optical Spectra.- 4.2.4 Contributions of the Partial Density of States to N(O).- 4.3 Lattice Dynamics of Transition Metals.- 4.4 The Electron-Phonon Interaction and Tc in Transition Metals.- 4.4.1 Analysis of Empirical Values of ?.- 4.4.2 The Contribution of Electron and Lattice Factors to the Quantity x.- 4.5 Calculating the Electron-Phonon Interaction Constant from First Principle.- 4.5.1 Gaspari and Gyorffy’s Representation for .- 4.5.2 Calculation of N(O) for Transition Metals.- 4.5.3 Calculation of the Pressure Dependence of Tc.- 5. Superconductivity in Transition-Metal Alloys.- 5.1 Types of Solid Solutions and Conditions of Their Formation.- 5.1.1 Conditions for the Formation of Solid Solutions.- 5.1.2 Types of Solid Solutions and Phase Diagrams.- 5.2 The Superconducting Transition Temperature of Alloys.- 5.2.1 Types of Concentration Dependences of Tc.- 5.2.2 Binary Alloys.- 5.2.3 The Influence of Pressure on the Tc of Alloys.- 5.3 The Empirical Correlations Between Tc and the Electron and Lattice Properties of Solid Solutions.- 5.3.1 Matthias’ Rule.- 5.3.2 The Rigid-Band Approximation.- 5.3.3 The Relation of Tc to the Lattice Instability.- 5.4 Miedema’s Empirical Method of Describing Concentration Dependences of the Properties of Metals.- 5.5 Theoretical Analysis of the Concentration Dependence of Tc in Alloys.- 5.5.1 The Alloy Model.- 5.5.2 The CPA Method.- 5.5.3 The Superconducting Transition Temperature.- 6. Compounds with A-15 Structure.- 6.1 Physical Properties.- 6.1.1 The Crystal Structure.- 6.1.2 Superconducting Transition Temperature of Pure Compounds and Their Alloys.- 6.1.3 The Electron Concentration Dependence of Electronic Heat Capacity and Magnetic Susceptibility.- 6.1.4 The Temperature Dependence of the Susceptibility, the Knight Shift and X-Ray and Optical Spectra.- 6.1.5 Structural Transformation.- 6.1.6 Acoustic Properties.- 6.1.7 The Phonon Density of States.- 6.1.8 Effect of Pressure on the Superconducting Transition Temperature.- 6.2 Band Structure.- 6.2.1 The Clogston-Jaccarino Model.- 6.2.2 The Noninteracting Chains Model.- 6.2.3 Electron Structure in the Presence of Interchain Interaction.- 6.2.4 Band Structure Calculations from First Principles.- 6.3 Investigation into Theoretical Models.- 6.3.1 The Weger-Labbé-Friedel Model.- 6.3.2 The Cohen-Cody-Halloran Model.- 6.3.3 Gor’kov’s Model I.- 6.3.4 Gor’kov’s Model II.- 6.3.5 The Birman-Lee-Williamson Model.- 6.3.6 Bhatt’s Model.- 6.4 Analysis of the Electron-Phonon Interaction and Tc.- 6.4.1 Experimental Data on the Electron-Phonon Coupling Constant ?.- 6.4.2 Analysis of ? and Tc.- 6.4.3 Pairing of Electrons via Acoustic Plasmons.- 6.5 The Theory of Structural Transition in A-15 Compounds.- 6.5.1 The Character of the Phase Transition.- 6.5.2 The Noolandi and Sham Theory [6.280].- 6.5.3 Effect of Pressure on Structural Phase Transition.- 6.5.4 Structural Transition as a Result of a Peierls Instability in Chains.- 6.5.5 A Comparison of Different Theories of Martensitic Transformations.- 7. Other Compounds Based on Transition Metals.- 7.1 Intermetallics.- 7.1.1 Laves Phases.- 7.1.2 Compounds with CsCl (B-2) Structure.- 7.1.3 ? Phases and Compounds with ?-Mn (A-12) Structure.- 7.1.4 C-40 Structure Compounds.- 7.2 Compounds with NaCl Structure.- 7.2.1 Crystal Chemistry.- 7.2.2 Superconductivity.- 7.2.3 Band Structure.- 7.2.4 Phonon Spectra.- 7.2.5 Electron-Phonon Interactions.- 7.3 Compounds of Palladium with Hydrogen and Deuterium.- 7.3.1 The Superconductivity of PdH and PdD and the Reverse Isotopic Effect.- 7.3.2 The Mechanisms of Superconductivity in Pd Hydrides.- 7.3.3 Other Systems Based on Palladium.- 7.4 Sesquicarbides and Ternary Borides.- 7.4.1 Crystal Chemistry.- 7.4.2 Superconductivity.- 7.5 Chalcogenides.- 7.5.1 Sulphides.- 7.5.2 Chevrel Phases.- 7.5.3 Oxides.- 8. High Temperature Superconductors and Lattice Instability of Compounds.- 8.1 Comparison of Superconductivity in Various Crystal Structures.- 8.1.1 Symmetry.- 8.1.2 Crystallochemistry.- 8.1.3 Stoichiometry.- 8.1.4 Alloying.- 8.1.5 Electron Concentration.- 8.2 Types of Lattice Instability.- 8.2.1 Structural Transition.- 8.2.2 “Virtual” Structural Transition.- 8.2.3 Softening of Long-Wave Lattice Moduli with Decreasi ng Temperature.- 8.2.4 Softening of Short-Wave Phonons.- 8.2.5 The Coexistence of Different Structural Phases in the Vicinity of a Given Composition.- 8.3 Phase Transition in 0n Structures from the Point of View of Symmetry Theory.- 8.3.1 Characteristics of the Phase Transitions.- 8.3.2 Construction of the Mechanical Representation and Analysis of Phonon Modes at the Point ?.- 8.3.3 Seeking the Group of the New Phase.- 8.3.4 Discussion of Results.- 8.4 Correlation Between High Tc and Lattice Instability.- 9. Radiation Effects on Superconductors.- 9.1 Radiation Resistance of Superconducting Materials.- 9.1.1 Compounds with A-15 Structure.- 9.1.2 Disordered Alloys of Transition Metals.- 9.1.3 Laves Phases.- 9.1.4 Chevrel Phases.- 9.1.5 Carbides and Nitrides of Transition Metals.- 9.2 Structural Investigations of Radiation Defects.- 9.2.1 Lattice Parameter Measurements.- 9.2.2 Determination of the Degree of Long-Range Order.- 9.2.3 Investigation by Small-Angle Scattering of Neutrons.- 9.2.4 Electron-Microscopic Investigations.- 9.2.5 Channelling of ? Particles.- 9.2.6 The Mössbauer Effect.- 9.3 Physical Properties of Irradiated Superconducting Materials.- 9.3.1 Magnetic Susceptibility.- 9.3.2 Heat Capacity.- 9.3.3 The Velocity of Sound.- 9.3.4 Inelastic and Elastic Scattering of Neutrons.- 9.3.5 Resistivity.- 9.3.6 The Temperature Coefficient of the Upper Critical Field.- 9.4 Theoretical Models for the Degradation of Superconducti ng Properties.- 9.4.1 Variations of the Electron-Phonon Interaction.- 9.4.2 Smearing of the Peak in the Density of Electron States.- 9.4.3 Radiation Defects in the Cluster Approximation.- References.