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

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

Figure 1

(a) Top and (b) side view of an ideal $1H$ phase monolayer ${\mathrm{MoS}}_2$, which consists of Mo atoms (larger dot) and S atoms (smaller dot) and has mirror symmetry with respect to the middle sublayer plane. (c) Schematic band structure near the $K$ point of an ideal monolayer ${\mathrm{MoS}}_2$ with mirror reflection symmetry. Only the two lowest conduction bands and the two highest valence bands are shown. Band-gap energy is $\mathrm{\Delta }=E_{\text{c},\downarrow }\left(0\right)-E_{\text{v},\downarrow }\left(0\right)\approx 1.82\text{eV}$ and spin-splitting energies are ${\mathrm{\Delta }}_{\text{c}}\approx 3\text{meV}$ and ${\mathrm{\Delta }}_{\text{v}}\approx 148\text{meV}$ for conduction and valence bands, respectively. About $70\text{meV}$ above $E_{\text{c},\downarrow }\left(0\right)$, $E_{\text{c},\uparrow }\left(\mathit{q}\right)$, and $E_{\text{c},\downarrow }\left(\mathit{q}\right)$ become degenerate.

Figure 2

(a) Spin-momentum coupling turns up when the mirror symmetry is broken. (b) The spin expectation value of the lower conduction band (left) without mirror symmetry breaking and (right) with mirror symmetry breaking.

Figure 3

Calculated values of the Berry curvature near the $K$ point (a) with mirror symmetry ($\beta =0\text{meV}\AA$) and (b) without mirror symmetry ($\beta =20\text{meV}\AA$). Note the scale difference in vertical axes of (a) and (b). (c) Berry curvature (left) and (d) orbital magnetic moment (right) at the $K$ point as a function of $\beta$.

Figure 4

(a) Valley Hall conductivity as a function of the photocarrier density $n_{\text{c},\downarrow }^K$ for various values of $\beta$. (b) Spin/valley Hall conductivity in a monolayer ${\mathrm{MoS}}_2$.