  The effect of the number of stacking layers and the type of stacking on the
electronic and optical properties of bilayer and trilayer black phosphorus are
investigated by using first principles calcula- tions within the framework of
density functional theory. We find that inclusion of many body effects (i.e.,
electron-electron and electron-hole interactions) modifies strongly both the
electronic and opti- cal properties of black phosphorus. While trilayer black
phosphorus with a particular stacking type is found to be a metal by using
semilocal functionals, it is predicted to have an electronic band gap of 0.82
eV when many-body effects are taken into account within the G0W0 scheme. Though
different stacking types result in similar energetics, the size of the band gap
and the optical response of bilayer and trilayer phosphorene is very sensitive
to the number of layers and the stacking type. Regardless of the number of
layers and the type of stacking, bilayer and trilayer black phosphorus are
direct band gap semiconductors whose band gaps vary within a range of 0.3 eV.
Stacking arrangments different from the ground state structure in both bilayer
and trilayer black phosphorus significantly modify valence bands along the
zigzag direction and results in larger hole effective masses. The optical gap
of bilayer (trilayer) black phosphorus varies by 0.4 (0.6) eV when changing the
stacking type. Due to strong interlayer interaction, some stackings obstruct
the observation of the optical excitation of bound excitons within the
quasi-particle band gap. In other stackings, the binding energy of bound
excitons hardly changes with the type of stacking and is found to be 0.44
(0.30) eV for bilayer (trilayer) phosphorous.
