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Two-dimensional (2D) transition-metal dichalcogenide (TMD) MX$_2$ (M = Mo, W;X= S, Se, Te) possess unique properties and novel applications. In this work,we perform first-principles calculations on the van der Waals (vdW) stackedMX$_2$ heterostructures to investigate their electronic, optical and transportproperties systematically. We perform the so-called Anderson's rule to classifythe heterostructures by providing the scheme of the construction of energy banddiagrams for the heterostructure consisting of two semiconductor materials. Formost of the MX$_2$ heterostructures, the conduction band maximum (CBM) andvalence band minimum (VBM) reside in two separate semiconductors, forming typeII band structure, thus the electron-holes pairs are spatially separated. Wealso find strong interlayer coupling at $\Gamma$ point after forming MX$_2$heterostructures, even leading to the indirect band gap. While the bandstructure near $K$ point remain as the independent monolayer. The carriermobilities of MX$_2$ heterostructures depend on three decisive factors, elasticmodulus, effective mass and deformation potential constant, which are discussedand contrasted with those of monolayer MX$_2$, respectively.

Electronic, optical and transport properties of van der Waals Transition-metal Dichalcogenides Heterostructures: A First-principle Study