Superconductivity and hybrid soft modes inTiSe2

The interplay between superconductivity and charge-density-wave (CDW) order plays a central role in the layered transition-metal dichalcogenides. $1T\ensuremath{-}\mathrm{Ti}{\mathrm{Se}}₂$ 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 $1T\ensuremath{-}\mathrm{Ti}{\mathrm{Se}}₂$ 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 $1T\ensuremath{-}\mathrm{Ti}{\mathrm{Se}}₂$ under pressure is close to the elusive state of the excitonic insulator.