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Electronic properties of single-layer and multilayer transition metal dichalcogenides $MX_2$ ($M=$ Mo, W and $X=$ S, Se)

Single- and few-layer transition metal dichalcogenides have recently emerged as a new family of layered crystals with great interest, not only from the fundamental point of view, but also because of their potential application in ultrathin devices. Here we review the electronic properties of semiconducting $MX_2$, where $M=$Mo or W and $X=$ S or Se. Based on of density functional theory calculations, which include the effect of spin-orbit interaction, we discuss the band structure of single-layer, bilayer and bulk compounds. The band structure of these compounds is highly sensitive to elastic deformations, and we review how strain engineering can be used to manipulate and tune the electronic and optical properties of those materials. We further discuss the effect of disorder and imperfections in the lattice structure and their effect on the optical and transport properties of $MX_2$. The superconducting transition in these compounds, which has been observed experimentally, is analyzed, as well as the different mechanisms proposed so far to explain the pairing. Finally, we include a discussion on the excitonic effects which are present in these systems.