Strain theory of the two-dimensional transition metal dichalcogenides

We develop a theory of the electronic structure of the two-dimensional monolayer transition-metal dichalcogenides in the 2H structure, with the formula $\mathrm{2H}\text{−}{MX}_2$ under a general strain condition. We focus on the low-energy Hamiltonian around the $K/K^{\prime }$ valley points, which control many electronic properties of the material. The strain Hamiltonian is derived from a combination of symmetry considerations and an effective $d$-orbital tight-binding model, adopting a simplifying approach, where the chalcogen atoms are only implicitly considered, which makes the derivations much simpler. Unlike previous treatments in the literature, our formulation properly describes the electronic structure in the neighborhood of the valley points, including the well-known valley point drift under strain. The strain Hamiltonian is validated from comparison with the density-functional theory calculations, and the total energy is shown to be consistent with the theory of elasticity. The Hamiltonian parameters, including the strain dependence of the spin-orbit coupling strength, are given for several insulating $M{X}_2$ compounds, where $M=\mathrm{Mo}$ or W and $X=S$, Se, or Te, obtained by fitting with the density-functional calculations, as well as for the metallic counterparts ${\mathrm{NbS}}_2$ and ${\mathrm{NbSe}}_2$. These strain models are not only useful for a fundamental understanding of the electronic structure under strain but also essential for the design of electronic device applications, where strain may already be present or can be tuned by external means.