Giant valley drifts in uniaxially strained monolayer ${\mathrm{MoS}}_2$

Using first-principles calculations, we study the electronic structure of monolayer ${\mathrm{MoS}}_2$ under uniaxial strain. We show that the energy valleys drift far off the corners of the Brillouin zone (K points), about 12 times the amount observed in graphene. Therefore, it is essential to take this effect into consideration for a correct identification of the band gap. The system remains a direct band gap semiconductor up to 4% uniaxial strain, while the size of the band gap decreases from 1.73 to 1.54 eV. We also demonstrate that the splitting of the valence bands due to inversion symmetry breaking and spin-orbit coupling is not sensitive to strain.

Figure 1

(a) Unit cell of ${\mathrm{MoS}}_2$ without strain. Brillouin zone under uniaxial strain along the (b) $\theta =0^{\circ }$, (c) $\theta ={15}^{\circ }$, and (d) $\theta ={30}^{\circ }$ directions. The red arrow in (a) defines the direction of the applied strain.

Figure 2

(a) Band structures of monolayer ${\mathrm{MoS}}_2$ without strain and under uniaxial strain ($\theta =0^{\circ }$). (b) Zoom of the valence bands near the K point. (c) Brillouin zone.

Figure 3

Color map of the band structure (a) without strain as well as under uniaxial strain along the (b) $\theta =0^{\circ }$, (c) $\theta ={15}^{\circ }$, and (d) $\theta ={30}^{\circ }$ directions. Zero energy (dark red) corresponds to the valence band maximum. The lower left and upper right corners of the color maps are the points $(-1,-1)\times 2\pi /a_0$ and $(1,1)\times 2\pi /a_0$ of the reciprocal space, where $a_0$ is the lattice constant of the unstrained system.

Figure 4

Color map of the band structure near the K${}_2$ point (a) without strain as well as under uniaxial strain along the (b) $\theta =0^{\circ }$, (c) $\theta ={15}^{\circ }$, and (d) $\theta ={30}^{\circ }$ directions. Zero energy (dark red) corresponds to the valence band maximum. The lower left and upper right corners of the color maps are the points $(-0.06,-0.06)\times 2\pi /a_0+{\mathrm{K}}_2$ and $(0.06,0.06)\times 2\pi /a_0+{\mathrm{K}}_2$ of the reciprocal space.

Figure 5

Energy difference between the highest valence and lowest conduction bands near the K${}_2$ point (a) without strain as well as under uniaxial strain along the (b) $\theta =0^{\circ }$, (c) $\theta ={15}^{\circ }$, and (d) $\theta ={30}^{\circ }$ directions. The maps cover the same area as in Fig. 4.