Stability Constraints and Local Criteria for the Bounds on $T_c$ of Conventional Superconductors

In the regime of weak electron-phonon coupling, the scale of and bounds on $T_c$ are set by the real phonon frequencies of a material. In the strong coupling limit of Eliashberg theory, the phonon frequencies no longer limit the transition temperature because $T_c$ can grow without bound as $\sim \sqrt {\lambda}\omega_{ph}$, where $\lambda$ is the electron-phonon coupling parameter. However, structural stability of the system puts bounds on $T_c$ even in the strongly coupled regime due to the softening of phonons. In this case, $T_c$ is also bounded by an averaged un-renormalized phonon frequency defined for a non-metallic precursor material. Additional features of this model will also be discussed.