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.

Chapter 4 introduces the single-electron spectral function and the basic models for describing the measurement spectra of angle-resolved photoemission. It is shown that the photoemission density is proportional to the single-particle spectral function under the "sudden" approximation in the three-step model. This offers an irreplaceable tool for probing the momentum dependence of the energy gap as well as other single-particle properties of quasiparticle excitations. The Luttinger theorem, which relates the Fermi momentum with the filling factor of electrons, the particle-hole mixing, and the effect of quasiparticle scattering on the transport properties are also discussed.

Chapter 11 discusses the properties of Raman response functions in d-wave superconductors. It is shown that the vertex function of high-symmetric A1g mode is strongly modified by the screening effect of the fluctuating charges to the long-range Coulomb interaction. Unlike in an s-wave superconductor, the Raman spectral functions behave quite differently in different symmetric channels in a d-wave superconductor. Particularly, the B2g-mode shows a characteristic peak at a higher frequency than that of the B2g mode. The behavior of the Raman response function in a two-band system, such as the electron-doped cuprate superconductors, is also discussed.

Chapter 5 discusses the Andreev reflection on a metal-superconducting interface and its effect on the tunneling conductance in d-wave superconductors. A zero-energy surface bound state is shown to exist in a d-wave pairing system. The tunneling conductance of superconducting quasiparticles in an applied bias is also derived from the tunneling Hamiltonian. The interlayer tunneling is suppressed by the anisotropic hopping integral, which leads to a modified power law with a relatively high exponent in the bias dependence of the c-axis conductance of high-Tc cuprates.

Chapter 9 calculates the superfluid density as a function of temperature in d-wave superconductors. The in-plane superfluid density varies linearly at low temperatures. The c-axis superfluid density, on the other hand, is predicted to behave as T5 in copper oxides at low temperatures if the gap function possesses the dx2-y2-wave symmetry. This peculiar T5-dependence of the superfluid density was confirmed by experiments. Both the in-plane and c-axis superfluid densities, however, vary quadratically with temperature if the impurity scattering becomes important. The superfluid response in a two-gap system is discussed and applied to explain the anomalous temperature dependence of the superfluid density in electron-doped high-Tc materials. Both the nonlinear and nonlocal effects on the superfluid density of d-wave superconductors are also discussed.

Chapter 10 studies the properties of optical conductivity in d-wave superconductors. The optical sum rule is first derived and discussed in connection with the missing low frequency spectral weight in underdoped cuprate superconductors. In the dirty limit, as the coherence length is significantly larger than the mean free path, the light absorption is determined by the optical conductivity in the local limit. Unlike in an s-wave superconductor, there is no absorption edge in a d-wave superconductor. In the limit the elastic scattering rate is significantly lower than the gap amplitude, the conductance is universal, independent on the strength of impurity scattering potential. The microwave conductivity is shown to vary cubically with temperature in cuprate superconductors, resulting from the hot spot nature of the scattering rate. The thermal conductivity of d-wave superconductors with its universal behavior in the zero temperature limit is also discussed and compared with the measurement data.

Using our strict definition of Atomic Scale Analytical Tomography (ASAT), we explore the current landscape of materials characterization tools and discuss how electron microscopy, field ion microscopy, and atom probe tomography are each approaching ASAT. State-of-the-art electron microscopy can achieve sub-angstrom spatial resolution imaging in 2-D and small volumes in 3-D but lacks single-atom chemical sensitivity, especially in 3-D. Field Ion Microscopy can achieve 3-D imaging on small volumes but not for all materials. Atom probe tomography can achieve single-atom elemental quantification in 3-D but lacks the spatial resolution necessary for ASAT. The chapter concludes with a comparison of the different techniques and discusses how different techniques may be complementary.