  The quasiparticle band-gap renormalization induced by the doped carriers is
an important and well-known feature in two-dimensional 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 the random phase approximation and to account for the quasi-2D
character of the Coulomb interaction in this system a Keldysh-type interaction
with a 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$_2$.
