The maximal superconductivity in proximity to charge density wave quantum critical point in Cu$_x$TiSe$_2$

Abstract

Superconductivity emerges in $1T$-TiSe$_2$ when its charge density wave (CDW) order is suppressed by Cu intercalation or pressure. 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$_2$ by asking if it is consistent with the phonon-mediated pairing. We employed the {\it ab initio} density functional theory and density functional perturbation theory to compute the electron-phonon coupling Eliashberg function from which to calculate the superconducting (SC) critical temperature $T_c$. The calculated $T_c $ as a function of the doping concentration $x$ exhibits a dome shape with the maximum $T_c$ of $2-6$ K at $x \approx 0.05$ for the Coulomb pseudopotential $0 \leq \mu^* \leq 0.1$. 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. Comparative discussion was made with the pressure induced superconductivity in TiSe$_2$.