Thermodynamics and Dynamics of Vortex Lattices in Type-Ii Superconductors.

This thesis presents a theoretical study of vortex lattice melting in high-temperature superconductors. Most of the work is devoted to understanding the thermodynamics of the vortex system, but an approach to the dynamical properties is also outlined. The thermodynamics are carried out in an approximation in which only the fluctuations within the lowest Landau level are included ("LLL approximation"). This approximation is appropriate for superconductors in high magnetic fields, near their upper critical field H _{c2}(T). After an introductory Chapter, we outline in Chapter II a new derivation of the classic 1957 mean-field solution of Abrikosov, using a basis of extended LLL states. The state of the minimum free energy is a triangular lattice of vortices; the square lattice is shown to be unstable to shear. In Chapter III, we present a numerical application of the LLL approximation to vortex melting in two-dimensional superconductors and calculate the temperature dependence of the shear modulus of the vortex lattice. Our results indicate that the melting is first order, with a shear modulus discontinuity close to a value predicted in dislocation unbinding theories. In Chapter IV we present numerical evidence that the vortex solid in a two-dimensional superconductor is a density wave state--a state with quasi-long range positional order, but no off-diagonal long range order. In Chapter V we demonstrate a possibility of two qualitatively different fluctuation regimes within the vortex solid state of a three-dimensional layered superconductor, which can be termed a '2D' solid and a '3D' solid. In the thermodynamic phase diagram these states are separated by a smooth crossover. In Chapter VI we calculate the fluctuation magnetization of a high-T_{c} superconductor YBCO, in good agreement with experiment. The melting of the vortex solid is shown to be a first-order phase transition with a discontinuous magnetization and entropy. The vortex liquid is a non-superconducting, normal phase with short -range correlations in all directions. Finally, in Chapter VII, we outline an extension of the LLL formalism to describe dynamical transport phenomena, using a time-dependent Ginzburg-Landau formalism.