Quantum and Classical Mode Softening Near the Charge-Density-Wave–Superconductor Transition of ${\mathrm{Cu}}_x{\mathrm{TiSe}}_2$

Temperature- and $x$-dependent Raman scattering studies of the charge-density-wave (CDW) amplitude modes in ${\mathrm{Cu}}_x{\mathrm{TiSe}}_2$ show that the amplitude mode frequency ${\omega }_0$ exhibits identical power-law scaling with the reduced temperature $T/T_{\mathrm{CDW}}$ and the reduced Cu content $x/x_c$, i.e., ${\omega }_0\sim (1-p{)}^{0.15}$ for $p=T/T_{\mathrm{CDW}}$ or $x/x_c$, suggesting that mode softening is independent of the control parameter used to approach the CDW transition. We provide evidence that $x$-dependent mode softening in ${\mathrm{Cu}}_x{\mathrm{TiSe}}_2$ is associated with the reduction of the electron-phonon coupling constant, and that $x$-dependent “quantum“ ($T\sim 0$) mode softening suggests the presence of a quantum critical point within the superconductor phase of ${\mathrm{Cu}}_x{\mathrm{TiSe}}_2$.

Figure 1

(a) Doping ($x$) dependence of the Raman spectrum of ${\mathrm{Cu}}_x{\mathrm{TiSe}}_2$ at $T=6\mathrm{K}$, illustrating the $E_g$- and $A_{1g}$-symmetry CDW amplitude modes (arrows) as a function of $x$. (b)–(d) Temperature dependence of the $A_{1g}$-symmetry CDW mode spectra at (b) $x=0$, (c) $x=0.02$, and (d) $x=0.03$ in ${\mathrm{Cu}}_x{\mathrm{TiSe}}_2$. The spectra have been offset for clarity, and the vertical scales in (c) and (d) are respectively $3\times$ and $4\times$ the vertical scale in (b). Also shown are example Lorentzian fits to the $A_{1g}$ amplitude mode (dashed line) and nearby optical modes (dotted line).

Figure 2

(a) Summary of the $T=6\mathrm{K}$ $A_{1g}$-symmetry CDW mode frequency (${\omega }_0$) vs $x$ (filled squares) and the $T=6\mathrm{K}$ $A_{1g}$-symmetry CDW mode linewidth (FWHM) vs $x$ (filled circles) for ${\mathrm{Cu}}_x{\mathrm{TiSe}}_2$. The solid line is a fit to the doping dependence of the $T=6\mathrm{K}$ frequency data using ${\omega }_0/{\omega }_0(0)=(1-x/x_c{)}^{\beta }$ with $\beta \sim 0.15$ and $x_c\sim 0.07$, and the dashed line is a fit to $\Gamma \sim (x_c-x{)}^{-2}$ with $x_c\sim 0.07$. (b) Summary of the $x=0$ $A_{1g}$-symmetry CDW mode frequency (${\omega }_0$) vs temperature (filled squares) and the $x=0$ $A_{1g}$-symmetry CDW mode linewidth (FWHM) vs temperature (filled circles) for ${\mathrm{TiSe}}_2$. The solid line is a fit to the frequency data with ${\omega }_0/{\omega }_0(0)=(1-T/T_{\mathrm{CDW}}{)}^{\beta }$ with $\beta \sim 0.15$. The dashed line illustrates the contribution to the linewidth expected from two-phonon damping. The estimated errors from the Lorentzian fits are $\le 1\%$ for the frequency data and $\le 8\%$ for the linewidth data.

Figure 3

Plots of ${\omega }_0/{\omega }_0(0)$ vs $x/x_c$ for $T=60\mathrm{K}$ ($x_c=0.06$ [5]) (filled squares), and ${\omega }_0/{\omega }_0(0)$ vs $T/T_{\mathrm{CDW}}$ for $x=0$ ($T_{\mathrm{CDW}}=218\mathrm{K}$ [5]) (filled circles), $x=0.02$ ($T_{\mathrm{CDW}}=180\mathrm{K}$ [5]) (open triangles), and (iv) $x=0.03$ ($T_{\mathrm{CDW}}=140\mathrm{K}$ [5]) (open diamonds). The solid line shows that all these data sets collapse onto the same curve given by ${\omega }_0/{\omega }_0(0)=(1-p{)}^{\beta }$, where $\beta =0.15$ and $p=x/x_c$ or $T/T_{\mathrm{CDW}}$. The gray lines provide an estimate of the uncertainty in $\beta$ by comparing fits with $\beta =0.2$ (bottom) and $\beta =0.10$ (top). The symbol sizes represent estimated errors ($\le 1\%$).

Figure 4

Summary of the $A_{1g}$ amplitude mode frequency ${\omega }_0$) vs $x$ for different temperatures. The solid lines are fits using ${\omega }_0/{\omega }_0(0)=(1-x/x_c(T){)}^{\beta }$, $w/\beta =0.15–0.16$. (inset) Estimated values of $T(x_c)$ from the fits (filled circles); $T_{\mathrm{CDW}}$ data from Ref. 5 (open circles) are shown for comparison. The inset also shows a contour plot of the temperature and $x$-dependent linewidth $\Gamma$ data, which ranges from light red ($\Gamma \sim 3{\mathrm{cm}}^{-1}$) to dark red ($\Gamma \sim 17{\mathrm{cm}}^{-1}$). The solid line denotes the superconductor (SC) phase boundary line. The symbol sizes represent estimated errors ($\le 1\%$).