Maximal superconductivity in proximity to the charge density wave quantum critical point in ${\mathrm{Cu}}_x{\mathrm{TiSe}}_2$

Superconductivity emerges in $1T\text{−}{\mathrm{TiSe}}_2$ 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 excitonic fluctuations. We investigated this question as to the nature of doping induced superconductivity in ${\mathrm{Cu}}_x{\mathrm{TiSe}}_2$ by asking about its consistency with the phonon-mediated pairing. We employed the 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 $0\le {\mu }^{*}\le 0.1$. The maximal $T_c$ was found to be pinned to the quantum critical point where 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 ${\mathrm{Cu}}_x{\mathrm{TiSe}}_2$ seems to be consistent with the phonon-mediated pairing. A comparative discussion was made with the pressure induced superconductivity in ${\mathrm{TiSe}}_2$.

Figure 1

(a) Crystal structure of ${\mathrm{Cu}}_x{\mathrm{TiSe}}_2$ illustrating the intercalated Cu atoms between ${\mathrm{TiSe}}_2$ layers. (b) 1st BZ of $1T\text{−}{\mathrm{TiSe}}_2$ corresponding to the crystal structure of (a). (c) Energy dispersion of the hole and electron bands along the $\mathrm{\Gamma }-L_1$ direction. The hole band around the $\mathrm{\Gamma }$ and three electron bands around $L_i$ points are shown in blue and red, respectively. $E_g$ is the indirect gap between the electron and hole bands. BZ is reduced in the CDW phase and the backfolded dispersions appear as shown in the blue- and red- dashed lines.

Figure 2

(a) Electronic band structures of unfolded $3\times 3\times 1$ supercell ${\mathrm{Cu}}_{0.11}{\mathrm{TiSe}}_2$, and (b) VCA ${\mathrm{Li}}_{0.11}{\mathrm{TiSe}}_2$. (c) Electronic DOS for the supercell system with the green area around the Fermi level corresponding to the amount of electrons transferred from the Cu cations to the ${\mathrm{TiSe}}_2$ layers, and (d) DOS for the VCA system with the green area corresponding to the amount of electrons transferred from the virtual Li cations to the ${\mathrm{TiSe}}_2$ layers. The vertical (red dashed) line is a guide to eyes to compare DOS at the Fermi level for a supercell to VCA calculations.

Figure 3

The phonon dispersions and corresponding phonon density of states for ${\mathrm{Cu}}_x{\mathrm{TiSe}}_2$ at $x$ = 0, 0.05, 0.08, and 0.15 are shown, respectively, in (a), (b), (c), and (d). The dopant induced phonon modes appear in between the acoustic and optical modes and are shown by green colored lines.

Figure 4

(a) SC critical temperature $T_c$ for ${\mu }^{*}=0.0$ ($⟐$) and ${\mu }^{*}=0.1$ ($\odot$) together with experimental $T_c$ from Ref. [11] (purple $\times$) and Ref. [17] (green $+$) and (b) logarithmic average of phonon frequency $\langle {\omega }_{\text{ln}}\rangle$ (red $•$) and electron-phonon coupling constant $\lambda$ (blue $*$), as a function of doping concentration $x$ in ${\mathrm{Cu}}_x{\mathrm{TiSe}}_2$.

Figure 5

The Eliashberg function ${\alpha }^{2}F\left(\omega \right)$ calculated using EPW at the dopings indicated in the inset. [(a)–(d)] indicate the phonon modes analyzed in Fig. 6.

Figure 6

The analysis of the phonon modes contributing to superconductivity. The labels (a)–(d) correspond to the modes of Fig. 5. The grey, yellow, and purple balls represent the Ti, Se, and virtual Li atoms, respectively. The green arrows represent the direction of the vibration motion of atoms.