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We calculate the excitation spectrum of single-layer MoS$_2$ at several hydrogen coverages by using a method based on first-principles Density-Matrix Time-Dependent Density-Functional Theory (TDDFT). Our results show that the fully hydrogenated system is metallic, while in the low-coverage limit the spectrum of single-layer MoS$_2$ includes spin-polarized partially filled localized mid-gap states. These states arise from s-orbitals of H atoms which make a tilted bond with the surface S atoms. The calculated absorption spectrum of the system reveals standard excitonic peaks, which correspond to the bound valence-band hole and conduction-band electron, as well as excitonic peaks that involve the mid-gap charges. As in the case of pristine single-layer MoS$_2$, binding energies of the excitons of the hydrogenated system are found to be relatively large (few tens of meV), making their experimental detection facile and suggesting hydrogenation as a knob for tuning the optical properties of single-layer MoS$_2$. Importantly, we find hydrogenation to suppress visible light photoluminescence, in agreement with experimental observations. As an aside, we contrast the effects of hydrogen coverage to that of the next two elements in the same column of the periodic table (the lightest metals), Li and Na, on the spectral properties of single-layer MoS$_2$ which lead instead to the formation of n-doped non-magnetic semiconductors that do not allow excitonic states.