Chapter 6 studies the Josephson tunneling effect in a superconductor-insulator-superconductor junction. The d-wave energy gap gives rise to a geometry dependent phase factor in the tunneling current. This leads to a unique phase-sensitive tool for experimentally detecting the d-wave pairing symmetry through a corner-sharing or tri-crystal junction. It is this kind of measurement that yields the strongest evidence for identifying the pairing symmetry in cuprate superconductors. The paramagnetic Meissner effect is discussed at the end of the chapter.

Chapter 3 derives the gap equation and determines the critical transition temperature as well as the zero-temperature energy gap as a function of coupling constant for d-wave superconductors. The energy dependence of the density of states and its effects on the temperature dependence of the gap function, entropy and other thermodynamic quantities are also discussed. Low energy nodal excitations lead to characteristic power-law behaviors in the specific heat or other thermodynamic response functions of d-wave superconductors at low temperatures, in contrast to the activated behaviors in s-wave superconductors. The probability density current and charge density current operators of d-wave quasiparticles, together with the gap operators in the continuum limit, are derived and discussed with the BdG framework.

Starting from a brief introduction to the Meissner effect and other defining properties of superconductivity, Chapter 1 recapitulates the phenomenological theories, including the two-fluid model and the Ginzburg-Landau theory, and the groundbreaking microscopic theory of Bardeen-Cooper-Schrieffer for describing this macroscopic quantum phenomenon. The Cooper pairing and other basic concepts of superconductivity, such as the gap function, off-diagonal long-range order, quasiparticle excitations, coherence length, penetration depth, type-I and type-II superconductors, and phase fluctuations are also introduced, followed by a summary on the classification and experimental identification for the pairing symmetry of high-Tc superconductors.

Chapter 14 introduces the theory of d-wave superconductors in the mixed state. It starts with a detailed derivation for the Caroli-de Gennes-Matricon vortex core states and then discusses the properties of low-lying excitations under the semi-classical approximation. The universal scaling laws for several different thermodynamic quantities are derived and compared with experimental observations for high-Tc cuprates.

Chapter 13 studies the dynamic spin response function measured by neutron scattering experiments. In particular, the magnetic resonance states revealed by the neutron scattering measurements for high-Tc cuprates in the superconducting state are discussed. It is argued that this spin resonance mode may arise either from a spin exciton excitation induced by an attractive residual spin interaction in the particle-hole channel or from a collective ?-resonance mode in the particle-particle channel which emerges in the neutron scattering spectrum thanks to the particle-hole mixing in the superconducting state.

Chapter 12 studies the property of magnetic response functions of electrons probed by nuclear magnetic resonance (NMR) experiments. The Knight shift is shown to be proportional to the real part of the local magnetic susceptibility. The spin-lattice relaxation, on the other hand, provides an effective measure of the imaginary part of the susceptibility averaged by the interaction form factor over the whole Brillouin zone. The effect of impurity scattering, particularly the impurity induced resonance states, on the NMR spectra is discussed and compared with experimental results.

Chapter 8 studies the many-impurity scattering effects in d-wave superconductors, particularly in the unitary or Born scattering limit. The impurity corrections to self-energy, density of states, superconducting critical temperature, entropy and specific heat are derived and compared with measurement data for high-Tc superconductors.

Chapter 2 starts with a brief review on the phase diagram of high-Tc cuprates, particularly on the phases of Mott insulators and pseudogaps. A number of microscopic models of high-Tc superconductors, including the three-band Hubbard model and its effective low-energy models in the strong coupling limit, namely the t-J model or its equivalent single-band Hubbard model, are then introduced. The models for describing the interlayer hopping and the system with Zn or Ni impurities in the copper oxides are also discussed. The Friedel sum rule is shown to be severely modified in the strong coupling limit, which reveals the perplexing but inherent nature of Zn as a unitary scattering potential of non-magnetic impurity.

Chapter 7 studies the single-impurity scattering effect on the density of states and other physical quantities. A low-energy resonance state induced by a unitary scattering potential is shown to exist in a d-wave superconductor, but is absent in an s-wave superconductor. The tunneling spectrum associated with a Zn impurity and the phenomenological theory of quasiparticle interference are discussed and compared with the experimental results. The in-gap resonance or bound states induced by a magnetic impurity and the Kondo effect in a d-wave superconductor are also discussed.