k.p theory for two-dimensional transition metal dichalcogenide semiconductors

We present $\mathbf{k}\cdot \mathbf{p}$ Hamiltonians parametrised by {\it ab initio} density functional theory calculations to describe the dispersion of the valence and conduction bands at their extrema (the $K$, $Q$, $\mathrm{\Gamma }$, and $M$ points of the hexagonal Brillouin zone) in atomic crystals of semiconducting monolayer transition metal dichalcogenides. We review the parametrisation of the essential parts of the $\mathbf{k}\cdot \mathbf{p}$ Hamiltonians for MoS${}_2$, MoSe${}_2$, WS${}_2$, and WSe${}_2$, including the spin-splitting and spin-polarisation of the bands, and we discuss the vibrational properties of these materials. We then use $\mathbf{k}\cdot \mathbf{p}$ theory to analyse optical transitions in two-dimensional transition metal dichalcogenides over a broad spectral range that covers the Van Hove singularities in the band structure (the $M$ points). We also discuss the visualisation of scanning tunnelling microscopy maps.