  Doped free carriers can substantially renormalize electronic self-energy and
quasiparticle band gaps of two-dimensional (2D) materials. However, it is still
challenging to quantitatively calculate this many-electron effect, particularly
at the low doping density that is most relevant to realistic experiments and
devices. Here we develop a first-principles-based effective-mass model within
the GW approximation and show a dramatic band gap renormalization of a few
hundred meV for typical 2D semiconductors. Moreover, we reveal the roles of
different many-electron interactions: The Coulomb-hole contribution is dominant
for low doping densities while the screened-exchange contribution is dominant
for high doping densities. Three prototypical 2D materials are studied by this
method, h-BN, MoS2, and black phosphorus, covering insulators to
semiconductors. Especially, anisotropic black phosphorus exhibits a
surprisingly large band gap renormalization because of its smaller
density-of-state that enhances the screened-exchange interactions. Our work
demonstrates an efficient way to accurately calculate band gap renormalization
and provides quantitative understanding of doping-dependent many-electron
physics of general 2D semiconductors.
