Search Results

After a brief review of the normal state of a “textbook” metal, the origin of the effective electron-electron attraction believed to exist in superconducting metals is discussed and the calculation of Cooper leading to the instability of the normal Fermi sea is presented. A derivation of the BCS theory is presented within a particle-number-conserving formalism; results are given for both zero and nonzero temperature, but the properties of the normal component in the superconducting phase are not discussed in any detail. The microscopic basis of the two-fluid model of superconductivity and of the Ginzburg-Landau phenomenology is discussed. Generalizations of the BCS theory are made for the case of both “non-pair-breaking” and “pair-breaking” perturbations. Finally, the microscopic basis of the Josephson effect is presented. Appendices cover inter alia, Landau Fermi-liquid theory, and the phonon-induced inter-electron attraction.

Critical phenomena occur very close to a second-order phase transition. For example, a ferromagnet near its Curie point behaves quite similarly to a liquid near its critical point, and a superconducting transition is not very different from a second-order ferroelectric one. The simplest view of phenomena near a critical point, attributable in its general form to Landau, is of a universal character and therefore attributes certain common characteristics to all phase transitions. Although the Landau theory does not agree quantitatively with general experimental observations very close to a critical point, it provides much qualitative insight and also points to the origin of the breakdown of the simple theory as the critical point is approached. This chapter explores the Landau theory of critical phenomena and other modern theories of critical phenomena, experimental observation of static critical phenomena in ferroelectrics and antiferrodistortive transitions, dynamic scaling and soft modes, experimental observation of critical dynamics, displacement transitions, and order-disorder transitions.

The present chapter explains the mean-field approximation, the Landau theory, the infinite-range model, and the Bethe approximation, and shows that all these (mean-field) theories are essentially equivalent to each other. The Landau theory is a phenomenological approach that uses the concept of symmetry and the order parameter, a measure of the breaking of that symmetry, as fundamental collective degrees of freedom. Also described are the Landau theory of tricritical behaviour, correlation functions, the limit of applicability of the mean-field theory, known as the Ginzburg criterion, and dynamic critical phenomena. Mean-field theories yield the exact critical exponents for dimensions larger than the upper critical dimension, and their solutions provide a reasonable starting point for more advanced methods including the renormalization group.

This chapter provides a scheme of nonlocal electromagnetism and develops a scheme of superconductivity. In essence, nonlocality is meant as a description of constitutive properties through appropriate spatial gradients. This is allowed by generalizing the expressions of the energy and the entropy fluxes while keeping the possible dependence on the history. Superconductivity is developed by starting from an improvement of the London theory and the thermodynamic analysis is performed. The Ginzburg-Landau theory is improved through an evolution model.

This chapter aims to develop the fundamental equations of thermodynamics which govern equilibrium between reacting species or phases. It begins with a review of the basic thermodynamic state variables of a homogeneous phase, followed by a discussion of the basic concepts of phase equilibrium in pure species as well as in multi-component systems. The ideal and non-ideal behaviour of mixtures, which facilitates the development of free energy formalisms associated with the mixture, is discussed in the subsequent sections. An introduction to phase diagram calculation techniques for binary and briefly for multi-component systems is presented. Finally, the concept of metastability and equilibrium suppression is introduced, and the significance and applications of metastable phase diagrams are outlined.

This chapter describes a semi-phenomenological theory due to Landau that accounts for a wide range of experimental results. The theory also predicts a new form of acoustic propagation, termed zero sound.

This chapter introduces a quantum field theory for interacting boson systems. It develops a mean-field theory to study the superfluid phase. A path integral formulation is then developed to re-derive the superfuid phase, which results in a low energy effective non-linear sigma model. A renormalization group approach is introduced to study the zero temperature quantum phase transition between superfluid and Mott insulator phase, and finite temperature phase transition between superfluid and normal phase. The physics and the importance of symmetry breaking in phase transitions and in protecting gapless excitations are discussed. The phenomenon of superfluidity and superconductivity is also discussed, where the coupling to U(1) gauge field is introduced.

This chapter reviews theoretical developments since the 1960s. These theories are of two basic types: microscopic theories, in which the ground-state properties are calculated from the mass of the ³He atom and the interatomic pair, and phenomenological theories that are alternatives to, or extensions of, Landau's theory.

This chapter begins with a discussion of the mechanisms for creating aperiodicity. It then discusses the Landau theory of phase transitions, semi-microscopic models, composites, electronic instabilities, and the numerical modeling of aperiodic crystals.

This chapter introduces the theory of superfluid ³He. It begins with a summary of the BCS theory of superconductivity, which is the basis for the development of the most complex theory of superfluid ³He. This is followed by the Ginzburg–Landau theory that is only valid for superfluids at temperatures near their transition temperature, Tc. A discussion of spin-triplet pairing leads to the identification of the B phase with the Balian–Werthamer state and the A phase with the Anderson–Morel state.

This chapter examines pattern-forming phenomena in thin layers of granular materials subjected to low-frequency periodic vertical vibration above the acceleration of gravity. Compared to driven granular gases discussed in Chapter 4, dense layers of granular materials under sufficiently strong excitation exhibit fluid-like motion. The most spectacular manifestation of the fluid-like behavior of granular layers is the occurrence of surface gravity waves which are quite similar to the corresponding patterns in ordinary fluids. To understand the nature of these collective phenomena, many theoretical and computational approaches have been developed. The most straightforward approach is to use molecular dynamics simulations which are feasible for sufficiently thin layers of grains. On the other hand, since the scale of observed pattern typically is much greater than the size of the individual grain, a variety of continuum approaches, ranging from phenomenological Ginzburg-Landau type theories to granular hydrodynamics, are discussed.

This chapter describes how the phenomenological understanding of superconductors was vastly enhanced by the Ginzburg–Landau theory. Its success rested on the fact that it could quantitatively yield the two characteristic lengths λ and ξ and address the question of boundary energy between superconducting and normal phases in terms of the Ginzburg–Landau parameter κ. for κ <1/√2, the sign of the boundary is positive and these are type I superconductors (including pure elements) where superconductivity is abruptly lost at H_c. When κ ≥1/√2, the boundary energy is negative and such materials (including impure metals and alloys) are type II superconductors. Field penetration now begins at a lower critical field H_c1 (<H_c) and continues until an upper critical field H_c2 (>H_c), where the bulk of the superconductivity is lost. Between the two fields, the sample is in a mixed state. Basic features of type II superconductors are discussed.

Interacting many-particle systems may undergo phase transitions and exhibit critical phenomena in the limit of infinite system size, while the precursors of these phenomena are studied in the theory of finite-size scaling. After surveying the basic notions of phases, phase diagrams, and phase transitions, this chapter focuses on critical behaviour at a second-order phase transition. The Landau-Ginzburg theory and the concept of scaling prepare readers for an elementary introduction to the concepts of the renormalization group, followed by an introduction into the field of quantum phase transitions where quantum fluctuations take over the role of thermal fluctuations.

Chapter 13 addresses Bose condensation in superfluids (and superconductors), which involves the field operator ψ‎ having a c-number component (<ψ(x,t)>≠0), challenging number conservation. The nonlinear Gross-Pitaevskii equation is derived for this condensate wave function<ψ>=ψ−ψ˜, facilitating identification of the coherence length and the core region of vortex motion. The noncondensate Green’s function G˜1(1,1′)=−i<(ψ˜(1)ψ˜+(1′))+> and the nonvanishing anomalous correlation function F˜∗(2,1′)=−i<(ψ˜+(2)ψ˜+(1′))+> describe the dynamics and elementary excitations of the non-condensate states and are discussed in conjunction with Landau’s criterion for viscosity. Associated concepts of off-diagonal long-range order and the interpretation of <ψ> as a superfluid order parameter are also introduced. Anderson’s Bose-condensed state, as a phase-coherent wave packet superposition of number states, resolves issues of number conservation. Superconductivity involves bound Cooper pairs of electrons capable of Bose condensation and superfluid behavior. Correspondingly, the two-particle Green’s function has a term involving a product of anomalous bound-Cooper-pair condensate wave functions of the type F(1,2)=−i<(ψ(1)ψ(2))+>≠0, such that G2(1,2;1′,2′)=F(1,2)F+(1′,2′)+G˜2(1,2;1′,2′). Here, G˜2 describes the dynamics/excitations of the non-superfluid-condensate states, while nonvanishing F,F+ represent a phase-coherent wave packet superposition of Cooper-pair number states and off-diagonal long range order. Employing this form of G₂ in the G₁-equation couples the condensed state with the non-condensate excitations. Taken jointly with the dynamical equation for F(1,2), this leads to the Gorkov equations, encompassing the Bardeen–Cooper–Schrieffer (BCS) energy gap, critical temperature, and Bogoliubov-de Gennes eigenfunction Bogoliubons. Superconductor thermodynamics and critical magnetic field are discussed. For a weak magnetic field, the Gorkov-equations lead to Ginzburg–Landau theory and a nonlinear Schrödinger-like equation for the pair wave function and the associated supercurrent, along with identification of the Cooper pair density. Furthermore, Chapter 13 addresses the apparent lack of gauge invariance of London theory with an elegant variational analysis involving re-gauging the potentials, yielding a manifestly gauge invariant generalization of the London equation. Consistency with the equation of continuity implies the existence of Anderson’s acoustic normal mode, which is supplanted by the plasmon for Coulomb interaction. Type II superconductors and the penetration (and interaction) of quantized magnetic flux lines are also discussed. Finally, Chapter 13 addresses Josephson tunneling between superconductors.

Phase transitions as a collective response of an ensemble, with appearance of unique stable properties spontaneously, is critical to a variety of devices: electronic, magnetic, optical, and their coupled forms. This chapter starts with a discussion of broken symmetry and its manifestation in the property changes in thermodynamic phase transition and the Landau mean-field articulation. It then follows it with an exploration of different phenomena and their use in devices. The first is ferroelectricity—spontaneous electric polarization—and its use in ferroelectric memories. Electron correlation effects are explored, and then conductivity transition from electron-electron and electron-phonon coupling and its use in novel memory and device forms. This is followed by development of an understanding of spin correlations and interactions and magnetism—spontaneous magnetic polarization. The use and manipulation of the magnetic phase transition in disk drives, magnetic and spin-torque memory as well as their stability is explored. Finally, as a fourth example, amorphous-crystalline structural transition in optical, electronic, and optoelectronic form are analyzed. This latter’s application include disk drives and resistive memories in the form of phase-change as well as those with electochemical transport.

This chapter begins with an analysis of Ginzburg–Landau theory. The discussion in Chapter 32 has brought out the idea that superconductivity is some kind of macroscopic quantum state. Ginzburg and Landau built this idea into the Landau second-order phase transition theory by assuming the existence of a macroscopic ‘wave function’, which they took as the order parameter associated with superconductivity. The chapter then discusses boundaries and boundary conditions, the upper critical field and the phase diagram of a type II superconductor, the Josephson effects, magnetic field effects, and the behaviour of a single planar Josephson junction in a magnetic field. A sample problem is also provided at the end of the chapter.

The balance equations resulting from the nonlocal kinetic equation are derived. They show besides the Landau-like quasiparticle contributions explicit two-particle correlated parts which can be interpreted as molecular contributions. It looks like as if two particles form a short-living molecule. All observables like density, momentum and energy are found as a conserving system of balance equations where the correlated parts are in agreement with the forms obtained when calculating the reduced density matrix with the extended quasiparticle functional. Therefore the nonlocal kinetic equation for the quasiparticle distribution forms a consistent theory. The entropy is shown to consist also of a quasiparticle part and a correlated part. The explicit entropy gain is proved to complete the H-theorem even for nonlocal collision events. The limit of Landau theory is explored when neglecting the delay time. The rearrangement energy is found to mediate between the spectral quasiparticle energy and the Landau variational quasiparticle energy.

Phase transitions in the solid state are classified as being continuous or discontinuous. According to Landau theory, continuous phase transitions require that there is a group-subgroup relation between the space groups of the involved crystal structures. They are driven by certain vibrational modes (soft modes) that must satisfy certain symmetry conditions. A phase transition which involves a symmetry reduction can often result in a topotactic texture consisting of domains. These are twin domains if there is a translationengleiche group-subgroup relation between the space groups of the phases, and antiphase domains if it is a klassengleiche relation. The index of the symmetry reduction determines how many kinds of domains may appear. Reconstructive phase transitions cannot proceed via a hypothetical intermediate phase whose symmetry is an assumed common subgroup of the two involved space groups.