Extending the theory of phonon-mediated superconductivity in quasi-2D

I present results from an extended Migdal-Eliashberg theory of electron-phonon interactions and superconductivity. The history of the electron-phonon problem is introduced, and then study of the intermediate parameter regime is justified from the energy scales in the cuprate superconductors. The Holstein model is detailed, and limiting cases are examined to demonstrate the need for an extended theory of superconductivity. Results of the extended approximation are shown, including spectral functions and phase diagrams. These are discussed with reference to Hohenbergs theorem, the Bardeen-Cooper-Schrieffer theory and Coulomb repulsion.