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Chemically and mechanically exfoliated MoS$_2$ single-layer samples havesubstantially different properties. While mechanically exfoliated single-layersare mono-phase (1H polytype with Mo in trigonal prismatic coordination), thechemically exfoliated samples show coexistence of three different phases, 1H,1T (Mo in octahedral coordination) and 1T$^{'}$ (a distorted $2\times 1$1T-superstructure). By using first-principles calculations, we investigate theenergetics and the dynamical stability of the three phases. We show that the 1Hphase is the most stable one, while the metallic 1T phase, strongly unstable,undergoes a phase transition towards a metastable and insulating 1T$^{'}$structure composed of separated zig-zag chains. We calculate electronicstructure, phonon dispersion, Raman frequencies and intensities for the1T$^{'}$ structure. We provide a microscopical description of the J$_1$, J$_2$and J$_3$ Raman features first detected more then $20$ years ago, butunexplained up to now. Finally, we show that H adsorbates, that are naturallypresent at the end of the chemical exfoliation process, stabilize the1T$^{\prime}$ over the 1H one.

Chemically-exfoliated single-layer MoS$_2$ : stability, lattice dynamics and catalytic adsorption from first principles