The role of phonons in antiferromagnetic spin-fluctuation mediated superconductors within FLEX approximations.

The physics of phonon mediated conventional superconductors is well explained by the Bardeen-
Cooper-Schrieer (BCS) theory. However, the role of phonons in unconventional superconduc-
tors such as the cuprates is still not fully understood. It has been suggested that the electron-
phonon coupling due to forward-scattering or buckling mode generally helps cooper pairing,
and therefore enhances the transition temperature T_c. Here, we consider a tight-binding model typical for cuprates and treat the Coulomb interaction U within Fluctuation-Exchange (FLEX) approximations together with a momentum dependent electron-phonon coupling in a fully self-consistent Migdal-Eliashberg formalism. In the simulations, the T_c were estimated for various U and electron-phonon coupling constant λ for three types of phonon modes. The in-plane breathing mode mostly suppresses T_c while out-of-plane buckling mode and forward scattering phonons can enhance T_c; however, depending on the values of U and λ, the T_c is not always enhanced or suppressed in either case. Finally, the single-particle spectral function with/without phonons will also be shown.