Quantum and classical mode softening near the charge-density-wave/superconductor transition of Cu$_{x}$TiSe$_{2}$: Raman spectroscopic studies*

We report temperature- and x-dependent Raman studies of the charge density wave (CDW) amplitude modes in Cu$_{x}$TiSe$_{2}$, which allow us to study the temperature- and x-dependence of the soft mode in this system. Among the key results: we find that the A$_{1g}$ amplitude mode exhibits identical power law scaling with the reduced temperature, p=T/T$_{CDW}$, and the reduced Cu content, p=x/x$_{c}$, i.e., $\omega _{o}\sim $(1-p)$^{0.15}$, suggesting that mode softening is independent of the control parameter used to approach the CDW transition; we provide evidence that x-dependent mode softening originates from the expansion of the lattice, which leads to a x-dependent reduction of the electron-phonon coupling constant; and we infer from our x-dependent mode softening results the presence of a quantum critical point, x$_{c}$(T=0)$\sim $0.07, within the superconducting phase of Cu$_{x}$TiSe$_{2}$.