  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$_2$.
