Phonon-mediated superconductivity in Cu$x$TiSe$2$ and the maximal $T_c $ in proximity to charge density wave quantum critical point

We present the superconducting properties of Cu intercalated $1T$-TiSe$_2$ by utilizing the {\it ab initio} density functional theory and density functional perturbation theory calculations. The superconductivity emerges as the charge density wave (CDW) order is suppressed by the Cu intercalation. Since the CDW state is thought to be an excitonic insulator, an interesting question is whether the superconductivity is also mediated by the excitonic fluctuations. We investigated this question as to the nature of doping induced superconductivity in Cu$_x$TiSe$_2$. The maximal $T_c$ was found to be pinned to the quantum critical point at which the CDW is completely suppressed and the corresponding phonon mode becomes soft. Underlying physics is that the reduced phonon frequency enhances the electron-phonon coupling constant $\lambda$ which overcompensates the frequency decrease to produce a net increase of $T_c$. The doping induced superconductivity in Cu$_x$TiSe$_2$ seems to be consistent with the phonon-mediated pairing.