Effects of strain on band structure and effective masses in MoS${}_2$

We use hybrid density functional theory to explore the band structure and effective masses of MoS${}_2$, and the effects of strain on the electronic properties. Strain allows engineering the magnitude as well as the nature (direct versus indirect) of the band gap. Deformation potentials that quantify these changes are reported. The calculations also allow us to investigate the transition in band structure from bulk to monolayer, and the nature and degeneracy of conduction-band valleys. Investigations of strain effects on effective masses reveal that small uniaxial stresses can lead to large changes in the hole effective mass.

Figure 1

(a) Side view of the bulk unit cell of MoS${}_2$. (b) A monolayer of MoS${}_2$. (c) The hexagonal Brillouin zone. High symmetry points and lines are indicated. The conduction-band valleys at ${\Lambda }_{\min }$ and $K$ are schematically depicted, with ellipsoids representing constant-energy surfaces.

Figure 2

Band structure of bulk MoS${}_2$ under uniaxial stress along the $c$ axis, plotted along high-symmetry lines in the in-plane Brillouin zone. The band structure of unstrained bulk and an unstrained monolayer are also shown. For clarity we only show the highest two valence bands and lowest two conduction bands. The labels underneath the segments of the Brillouin zone indicate the names of the high-symmetry lines [see Fig. 1c].

Figure 3

Band gaps of MoS${}_2$ as a function of strain for (a) uniaxial stress along $c$ and (b) biaxial (in-plane) stress. The various gaps are indicated by the locations in the Brillouin zone where the VB and CB extrema occur: indirect $\Gamma$-to-${\Lambda }_{\min }$ and $\Gamma$-to-$K$ gaps, and direct $K$-to-$K$ gap. The vertical dashed lines indicate unstrained MoS${}_2$.

Figure 4

Effective masses (in units of electron mass $m_0$) for (a) holes and (b) electrons as a function of strain under uniaxial stress conditions. The masses are labeled by the band extremum and (in square brackets) the direction along which the mass is calculated [see Fig. 1c]. The vertical dashed line indicates unstrained MoS${}_2$.

Figure 5

Effective masses (in units of electron mass $m_0$) for (a) holes and (b) electrons as a function of strain under biaxial stress conditions. The masses are labeled by the band extremum and (in square brackets) the direction along which the mass is calculated [see Fig. 1c]. The vertical dashed line indicates unstrained MoS${}_2$.