Monolayer MoS${}_2$: Trigonal warping, the $\Gamma$ valley, and spin-orbit coupling effects

We use a combined ab initio calculations and $\mathbf{k}·\mathbf{p}$ theory based approach to derive a low-energy effective Hamiltonian for monolayer ${\mathrm{MoS}}_2$ at the $K$ point of the Brillouin zone. It captures the features which are present in first-principles calculations but not explained by the theory of Xiao et al. [Phys Rev Lett 108, 196802 (2012)], namely the trigonal warping of the valence and conduction bands, the electron-hole symmetry breaking, and the spin splitting of the conduction band. We also consider other points in the Brillouin zone which might be important for transport properties. Our findings lead to a more quantitative understanding of the properties of this material in the ballistic limit.

Figure 1

(a) Spin-resolved band structure of $Mo{S}_2$ from DFT LSDA calculations. (b) Contour plot showing the isoenergy contours of the valence band (for zero SOC) from DFT calculations at the $K$ point of the BZ (symbols) and as obtained from Eq. (2) (solid lines). (c) The same as in (b) for the conduction band. $a_0$ is the lattice constant. The energy difference between the two innermost contours is 0.02 eV, between every other contours is 0.04 eV.

Figure 2

Top view of the $Mo{S}_2$ lattice. Mo atoms are indicated by gray (solid line) circles, S atoms by yellow (dotted line) circles. The lattice vectors ${\mathbf{a}}_1=\frac{a_0}{2}(1,\sqrt{3})$ and ${\mathbf{a}}_2=\frac{a_0}{2}(1,-\sqrt{3})$ are also shown ($a_0=3.129$ Å  is the lattice constant).

Figure 3

Conduction band of $Mo{S}_2$ from DFT calculations using the HSE06 functional (red, solid line) and the LDA (green, dashed line).