Extensions of BCS theory for non-conventional superconductors

Abstract

The Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity affords a very good microscopic understanding of metallic superconductors, where the electron-electron interaction is known to be phonon-mediated. However, it does not adequately describe non-conventional, in particular high-Tc, superconductivity. In this dissertation, BCS theory has been extended in an attempt to provide a framework within which the mechanisms responsible for exotic superconductivity can be understood. The BCS gap equation is solved without the usual restriction of small interaction width for the intermediating bosons, since the electron-electron interaction responsible for superconductive pairing need not necessarily be phonon mediated. Bulk high-Tc superconductors are usually polycrystalline materials, composed of weakly coupled grains which may as a first approximation be treated as small isolated superconducting systems. The finite size of these individual grains needs to be considered in an effective description of high temperature superconductivity. Nuclei that exhibit a. BCS-type pairing transition also fall into the category of small superconducting systems. For these finite systems, the effect of the thermodynamic and quantum fluctuations needs to be included. In particular, the effect of fluctuations on the macroscopic order parameter of the system, which drops smoothly to zero in the thermodynamic limit to indicate a phase transition, is examined. Furthermore, the expectation value of the pairing potential, rather than the conventional BCS energy gap, is proposed and motivated as the more appropriate order parameter for pairing systems. Finally, canonical number projection is also performed to consider the finite number of superconducting particles in small superconducting systems.