Superconductivity and hybrid soft modes in $\mathrm{Ti}{\mathrm{Se}}_2$

The interplay between superconductivity and charge-density-wave (CDW) order plays a central role in the layered transition-metal dichalcogenides. $1T-\mathrm{Ti}{\mathrm{Se}}_2$ forms a prime example, featuring superconducting domes on intercalation as well as under applied pressure. Here, we present high energy-resolution inelastic x-ray scattering measurements of the CDW soft phonon mode in intercalated ${\mathrm{Cu}}_x\mathrm{Ti}{\mathrm{Se}}_2$ and pressurized $1T-\mathrm{Ti}{\mathrm{Se}}_2$ along with detailed ab-initio calculations for the lattice dynamical properties and phonon-mediated superconductivity. We find that the intercalation-induced superconductivity can be explained by a solely phonon-mediated pairing mechanism, while this is not possible for the superconducting phase under pressure. We argue that a hybridization of phonon and exciton modes in the pairing mechanism is necessary to explain the full observed temperature-pressure-intercalation phase diagram. These results indicate that $1T-\mathrm{Ti}{\mathrm{Se}}_2$ under pressure is close to the elusive state of the excitonic insulator.

Figure 1

(a) Phonon dispersion lines along various high symmetry directions of $1T-\mathrm{Ti}{\mathrm{Se}}_2$ for $\sigma =0.1\mathrm{eV}$ calculated in the $2\times 2\times 2$ supercell corresponding to the distorted structure in the CDW phase. The red ellipsoid marks the soft modes carrying strong EPC folded back from the three $L_{\mathrm{tri}}$ points of the trigonal high temperature structure. (b) Energies of CDW soft mode(s) for $\sigma =0.1–0.2\mathrm{eV}$. Calculations for $\sigma \le 0.124\mathrm{eV}$ were performed in the $2\times 2\times 2$ supercell corresponding to the structure in the CDW phase. Here, we plot the energies of the three ${\mathrm{\Gamma }}_{\mathrm{CDW}}$ modes carrying the strong EPC [see (a)]. Error bars correspond to the calculated values of the electronic contribution to the phonon linewidth $\gamma$. The line is a guide to the eye. Squares and ellipsoid mark the soft mode energies denoted similarly in (a) and (c). (c) Phonon dispersion lines along various high symmetry directions of $1T-\mathrm{Ti}{\mathrm{Se}}_2$ for different smearing parameters $\sigma =0.15–0.5\mathrm{eV}$ in the trigonal structure. The vertical (blue) arrow marks the hardening of the CDW soft mode on going from $\sigma =0.15\mathrm{eV}$ to $0.5\mathrm{eV}$. Red squares denote the phonon energies of the soft mode also shown in (b).

Figure 2

(a), (b) Calculated Eliashberg function ${\alpha }^{2}F\left(\omega \right)$ and (c) EPC constant $\lambda \left(\omega \right)$ of $1T-\mathrm{Ti}{\mathrm{Se}}_2$ for different smearing parameters $\sigma =0.1–0.5\mathrm{eV}$ from DFPT. (d) Superconducting transition temperatures $T_c$ for $1T-\mathrm{Ti}{\mathrm{Se}}_2$ (dots, circles) as function of the electronic smearing $\sigma$ estimated from results shown in (a)–(c) using effective electron-electron interactions of ${\mu }^{*}=0.1$ (dots) and $0.15$ (circles) (see text). The CDW distorted phase is stable for $\sigma <0.15\mathrm{eV}$ (blue shaded area). Lines are guides to the eye.

Figure 3

Low temperature IXS in $\mathrm{C}{\mathrm{u}}_x\mathrm{Ti}{\mathrm{Se}}_2$: (a) Elastic intensity measured in the $x=0.06$ sample close to the CDW ordering wavevector ${\mathit{q}}_{\mathrm{CDW}}$ at three different temperatures. This inset shows the temperature dependence of the intensity at ${\mathit{q}}_{\mathrm{CDW}}$. (b) Observed energies of the soft mode close to ${\mathit{q}}_{\mathrm{CDW}}$ in the $x=0.09$ sample as function of temperature. (c)–(e) Raw IXS data taken close to the CDW superlattice peak position at low temperatures in intercalated samples with $x=0,0.06$, and $0.09$, respectively. Solid (red) lines are fits consisting of DHO functions for the phonons (thick dashed lines), a pseudo-voigt function for the resolution limited elastic line (thin dashed line) and the estimated experimental background (dotted straight line). Note that the comparably strong elastic lines in (d), (e) result from increased incoherent scattering in the intercalated, i.e., intrinsically dirty samples and not from CDW order. (f) Observed dispersions for different Cu concentrations $x$ at low temperatures. For clarity, we show all observed phonon modes only for $x=0.09\left(\mathrm{dots}\right)$ and only the soft mode energies for $x=0$ (open circles) and $0.06$ (open squares). Solid lines are DFPT calculations with $\sigma =0.15\mathrm{eV}$. (g) Ratio of the soft mode's line width $\mathrm{\Gamma }$ and its energy $\omega$ along the A-L at low temperatures. The thick line is the result of DFPT with $\sigma =0.15\mathrm{eV}$. Dashed lines are guides to the eye.

Figure 4

(a), (b) Raw IXS data taken (a) close to and (b) at the CDW superlattice peak position for $T_{\mathrm{CDW}}\left(p=1.75\mathrm{GPa}\right)=150\mathrm{K}$ (same symbol/line code as in Figs. 3–3. (c) Dispersion of the soft phonon mode for two sets of $\left(T_{\mathrm{CDW}},p\right)$ values. Thin lines denote DFPT calculations with $\sigma =0.15\mathrm{eV}$ but scaled with a factor of 1.1. The thick (red) line marks the dispersion of the soft mode character across the anti-crossing at $h\approx 0.3–0.35$.

Figure 5

(a)–(d) Raw IXS data taken at $T=5\mathrm{K}$ and increasing pressures $2.5\mathrm{GPa}\le p\le 9.6\mathrm{GPa}$ [same symbol/line code as in Figs. 3–3]. Data at $p<p_c=5.1\mathrm{GPa}$ were taken slightly away from ${\mathit{q}}_{\mathrm{CDW}}$. At higher pressures IXS scans were performed at ${\mathit{q}}_{\mathrm{CDW}}$. (e) Ratio of the soft mode's line width $\mathrm{\Gamma }$ and its energy $\omega$ as shown in the fits in (a)–(d).

Figure 6

(a) Sketch of the excitation energies appearing in the theoretical model without (dashed lines) and including mode hybridization (solid lines) (see text). (b) Summary of the low temperature excitations observed in IXS (see text). Solid lines are guides to the eye of the measured energies including an estimate for the actual soft mode energy at the CDW ordering wavevector for $P<P_c$ based on the observed difference of $1.5\mathrm{meV}$ between IXS at $\mathit{q}=\left(-1.45,3,0.5\right)$ and energies observed in Raman scattering [55]. Dotted lines indicate the expected behavior of the soft mode approaching a quantum critical point at $P=P_c$. (c) The superconducting transition temperature as a function of pressure according to Eq. (8). This figure uses the parameter values $tv=2tc=3.0\mathrm{meV}, E_c^0-E_{\nu }^0=20\mathrm{meV}, {\varepsilon }_0=17\mathrm{meV}$,$E_{\mathrm{exc}}=5.0\mathrm{meV}$,$g=10\mathrm{meV}$, and $\gamma =17.8\mathrm{meV}$, as well as $\eta =0.18$,$\sigma =0.75$, and $p^{*}/p_c=0.6$. (d) The phase diagram emerging from the theoretical model (see text).