Fig. 4: Properties of MoS₂ in the 1T′ and high-doping 2 × 2 CDW phases.

a Electron band structure obtained for the excess charge of n = 2.62 × 10¹⁴ cm⁻². b Adiabatic phonon dispersion. c Eliashberg spectral function calculated using the adiabatic (A) and nonadiabatic (NA) formula, with the corresponding cumulative EPC constant λ(ω). d Electron band structures for the largest doping values considered in this work; n₁ = 3.0 × 10¹⁴ cm⁻², n₂ = 4.3 × 10¹⁴ cm⁻², n₃ = 4.6 × 10¹⁴ cm⁻². e Corresponding phonon dispersions and f NA Eliashberg spectral functions with cumulative EPC constants λ(ω). g Softening and stabilization of the M* point acoustic phonon. The stabilization of this mode defines the low- and high-doping 2 × 2 CDW phase, while the region where it is unstable corresponds to the polaronic and partial CDW phases.