Analysis of quasiparticles in the mixed state of a d-wave superconductor and NMR in pores with surface relaxation

This thesis is divided into two parts. The first set of problems studied deals with the analysis of quasiparticle excitations of d-wave superconductors in the mixed state. The second part studies correlations between self-diffusion and relaxation of fluids in porous media through nuclear magnetic resonance measurements. The first chapter of this thesis contains an introduction to both topics.; In the second chapter of this thesis, the quasiparticle spectrum of a two-dimensional d-wave superconductor in the mixed state is studied both analytically and numerically using the linearized Bogoliubov-de Gennes equation. The problem of the fate of zero-energy states when a magnetic field is turned on is analyzed and the results are compared to the semiclassical predictions of a constant density of states at low energies.; In the third chapter, we introduce a one-dimensional model which captures the essential physics of the two-dimensional linearized Bogoliubov-de Gennes equation discussed in Chapter 2 for large values of the Fermi-velocity anisotropy ratio alphaD = vF/v Delta. Measurements in high-temperature superconductors find alpha D = 14 in YBCO and alphaD = 20 in Bi2212. This model allows one to access the large-alpha D regime both analytically and numerically and we use it to calculate several features of the quasipartitle band structure in the vortex lattice. In chapter four, we extend this calculation to arbitrary orientations of the vortex lattice with respect to the crystallographic axes.; Finally, in the fifth chapter we turn to the topic of surface relaxation in porous media and study correlations between surface relaxation and diffusion processes in nuclear magnetic resonance measurements in fluid-saturated, isolated pores. In a porous medium, where diffusion is restricted, diffusion and surface relaxation are not independent processes. We consider a single isolated pore and, using an eigenmode expansion of the diffusion equation, we calculate the time-dependent diffusion co-efficient and the fourth cumulant of the distribution of random walkers for several eigenmodes. We consider the implications of these results for the analysis of experimental data and propose an approach to characterize the surface-to-volume ratio and surface relaxivity of the pore simultaneously.