Superconducting Transitions from the Pseudo-Gap State: D-Wave Symmetry, Lattice, and Low Dimensional Effects

We investigate the behavior of the superconducting transition temperature within a previously developed BCS-Bose Einstein crossover picture. This picture, based on a diagrammatic decoupling scheme of Kadanoff and Martin, further extended by Patton, can be used to derive a simple form for the superconducting transition temperature in the presence of a pseudogap. We extend previous work which addressed the case of s-wave pairing in jellium, to explore the solutions for $T_c$ as a function of variable coupling in more physically relevant situations. We thereby ascertain the effects of reduced dimensionality, periodic lattices and a d-wave pairing interaction. Implications for the cuprate superconductors are discussed.