Berry curvature in monolayer MoS${}_2$ with broken mirror symmetry

An ideal 1H phase monolayer MoS${}_2$ has the mirror reflection symmetry but this symmetry is broken in common experimental situations, where the monolayer is placed on a substrate. By using the k$\cdot$p perturbation theory, we investigate the effect of the mirror symmetry breaking on the Berry curvature of the material. We find that the symmetry breaking may modify the Berry curvature considerably and the spin/valley Hall effect due to the modified Berry curvature is in qualitative agreement with a recent experimental result [Science $\mathbf{344}$, 1489 (2014)], which cannot be explained by previous theories that ignore the mirror symmetry breaking.