<i>Ab initio</i> calculations for electronic and optical properties of Er_W defects in single-layer tungsten disulfide

Ab-initio calculations for the electronic and optical properties of single-layer (SL) tungsten disulfide (WS$_2$) in presence substitutional Erbium defects (Er$_{\rm W}$) are presented, where W atom is replaced by an Er atom. Defects usually play important role tailoring semiconductors. We show that neutral lead to localized defect states (LDS) band structure due f-orbital Er, which turn give rise sharp transitions in-plane out-of-plane absorption spectra, $\alpha_{\parallel}$ $\alpha_{\perp}$. identify at 3 $\mu$m, 1.5 1.2 920 nm, 780 660 550 nm originate from Er$_{\rm W}$ states. In order provide a clear description we use group theory derive selection rules between LDS both