Temperature dependent excitonic effects in the optical properties of single-layer MoS$_2$

Temperature influences the performance of two-dimensional materials in optoelectronic devices. Indeed, the optical characterization of these materials is usually realized at room temperature. Nevertheless most {\it ab-initio} studies are yet performed without including any temperature effect. As a consequence, important features are thus overlooked, such as the relative intensity of the excitonic peaks and their broadening, directly related to the temperature and to the non-radiative exciton relaxation time. We present {\it ab-initio} calculations of the optical response of single-layer MoS$_2$, a prototype 2D material, as a function of temperature using density functional theory and many-body perturbation theory. We compute the electron-phonon interaction using the full spinorial wave functions, i.e., fully taking into account effects of spin-orbit interaction. We find that bound excitons ($A$ and $B$ peaks) and resonant excitons ($C$ peak) exhibit different behavior with temperature, displaying different non-radiative linewidths. We conclude that the inhomogeneous broadening of the absorption spectra is mainly due to electron-phonon scattering mechanisms. Our calculations explain the shortcomings of previous (zero-temperature) theoretical spectra and match well with the experimental spectra acquired at room temperature. Moreover, we disentangle the contributions of acoustic and optical phonon modes to the quasi-particles and exciton linewidths. Our model also allows to identify which phonon modes couple to each exciton state, useful for the interpretation of resonant Raman scattering experiments.