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

Ab-initio calculations for electronic and optical properties of Er$_{\rm W}$ defects in tungsten disulfide