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: $h\text{−}\mathrm{BN}$, $\mathrm{Mo}{\mathrm{S}}_2$, 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.

• Received 17 July 2017

DOI:https://doi.org/10.1103/PhysRevB.96.155410