Quasiparticle band structure and tight-binding model for single- and bilayer black phosphorus

By performing \emph{ab initio} calculations for one- to four-layer black phosphorus within the $GW$ approximation, we obtain a significant difference in the band gap ($\sim$1.5 eV), which is in line with recent experimental data. The results are analyzed in terms of the constructed four-band tight-binding model, which gives accurate descriptions of the mono- and bilayer band structure near the band gap, and reveal an important role of the interlayer hoppings, which are largely responsible for the obtained gap difference.