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Superconductivity emerges in $1T$-TiSe$_2$ when its charge density wave (CDW)order is suppressed by Cu intercalation or pressure. Since the CDW state isthought to be an excitonic insulator, an interesting question is whether thesuperconductivity is also mediated by the excitonic fluctuations. Weinvestigated this question as to the nature of doping induced superconductivityin Cu$_x$TiSe$_2$ by asking if it is consistent with the phonon-mediatedpairing. We employed the {\it ab initio} density functional theory and densityfunctional perturbation theory to compute the electron-phonon couplingEliashberg function from which to calculate the superconducting (SC) criticaltemperature $T_c$. The calculated $T_c $ as a function of the dopingconcentration $x$ exhibits a dome shape with the maximum $T_c$ of $2-6$ K at $x\approx 0.05$ for the Coulomb pseudopotential $0 \leq \mu^* \leq 0.1$. Themaximal $T_c$ was found to be pinned to the quantum critical point at which theCDW is completely suppressed and the corresponding phonon mode becomes soft.Underlying physics is that the reduced phonon frequency enhances theelectron-phonon coupling constant $\lambda$ which overcompensates the frequencydecrease to produce a net increase of $T_c$. The doping inducedsuperconductivity in Cu$_x$TiSe$_2$ seems to be consistent with thephonon-mediated pairing. Comparative discussion was made with the pressureinduced superconductivity in TiSe$_2$.

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