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The quasiparticle band-gap renormalization(BGR) induced by the doped carriers is an important and well-known feature in two-dimensional (2D) semiconductors including transition metal dichalcogenides (TMDs) and it is of both theoretical and practical interest. To get a quantitative understanding of this effect, here we calculate the quasiparticle band-gap renormalization of the electron-doped monolayer MoS$_2$, a prototypical member of TMDs. The many-body electron-electron interaction-induced renormalization of the self-energy is found within random phase approximation (RPA) and to account for the quasi-2D character of the Coulomb interaction in this system a Keldysh-type interaction with nonlocal dielectric constant is used. Considering the renormalization of both the valence and the conduction bands, our calculations reveal a large and nonlinear band-gap renormalization upon adding free carriers to the conduction band. We find a 410 meV reduction of the band-gap for the monolayer MoS$_2$ on SiO$_2$ substrate at the free carrier density $n=4.9\times 10^{12} \rm{cm^{-2}}$ which is in excellent agreement with available experimental results. We also discuss the role of exchange and correlation parts of the self-energy on the overall band-gap renormalization of the system. The strong dependence of the band-gap renormalization on the surrounding dielectric environment is also demonstrated in this work, and a much larger shrinkage of the band-gap is predicted for the freestanding monolayer MoS$_2$.
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