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The electronic structure and elastic property of monolayer and bilayer transition metal dichalcogenides MX$_2$ (M=Mo,W;X=O,S,Se,Te): A comparative first-principles study

First-principle calculations with different exchange-correlation functionals, including LDA, PBE and vdW-DF functional in form of optB88-vdW, have been performed to investigate the electronic and elastic properties of two dimensional transition metal dichalcogenides(TMDCs) with the formula of MX$_2$(M=Mo,W; X=O,S,Se,Te) in both monolayer and bilayer structures. The calculated band structures show a direct band gap for monolayer TMDCs at the K point except for MoO$_2$ and WO$_2$. When the monolayers are stacked into bilayer, the reduced indirect band gaps are found except for bilayer WTe$_2$, in which direct gap is still present at the K point. The calculated in-plane Young moduli are comparable to graphene, which promises the possible application of TMDCs in future flexible and stretchable electronic devices. We also evaluated the performance of different functionals including LDA, PBE, and optB88-vdW in describing elastic moduli of TMDCs and found that LDA seems to be the most qualified method. Moreover, our calculations suggest that the Young moduli for bilayers are insensitive to stacking orders and the mechanical coupling between monolayers seems to be negligible.