Optimal Boson energy for superconductivity in the Holstein model

We examine the superconducting solution in the Holstein model, where the conduction electrons couple to the dispersionless Boson fields, using the Migdal-Eliashberg theory and Dynamical Mean Field Theory. Although different in numerical values, both methods imply the existence of an optimal Boson energy for superconductivity at a given electron-Boson coupling. This non-monotonous behavior can be understood as an interplay between the polaron and superconducting physics, as the electron-Boson coupling is the origin of the superconductor, but at the same time traps the conduction electrons making the system more insulating. Our calculation provides a simple explanation on the recent experiment on sulfur hydride (H$_2$S), where an optimal pressure for the superconductivity was observed. The validities of both methods are discussed.