Theory of strain in single-layer transition metal dichalcogenides

Strain engineering has emerged as a powerful tool to modify the optical and electronic properties of two-dimensional crystals. Here we perform a systematic study of strained semiconducting transition metal dichalcogenides. The effect of strain is considered within a full Slater-Koster tight-binding model, which provides us with the band structure in the whole Brillouin zone (BZ). From this, we derive an effective low-energy model valid around the $K$ point of the BZ, which includes terms up to second order in momentum and strain. For a generic profile of strain, we show that the solutions for this model can be expressed in terms of the harmonic oscillator and double quantum well models, for the valence and conduction bands respectively. We further study the shift of the position of the electron and hole band edges due to uniform strain. Finally, we discuss the importance of spin-strain coupling in these 2D semiconducting materials.

Figure 1

A top view schematic of monolayer ${\mathrm{MoS}}_2$ lattice structure. Blue (orange) circles indicate Mo (S) atoms. The nearest-neighbor (${\delta }_i$) and the next-nearest neighbor ($a_i$) vector have been shown in the figure.

Figure 2

Top view of an arc-shaped $M{X}_2$ with $R=4L_y$ in which blue and red lattice points indicates $M$ and $X$ atoms, respectively.

Figure 3

(a) Energy dispersion of unstrained nanoribbon case with $L_y=50a_0$. (b) Energy dispersion a strained nanoribbon for $L_y=50a_0,{\eta }_i=\eta =1$, and $R=0.5L_y$. Noticeably, the first Landau level in the valence band evolves in an electronlike edge mode.

Figure 4

Energy dispersion of arc-shaped case with scalar potential and rotation tensors neglected. The dispersion relation show equidistant parabolic valence subbands, and a set of conduction subbands that present band crossing for the low-energy states and parabolic dispersion above some specific energy. The size of the energy gap and level spacing between the subbands is strongly dependent on the strain strength. We set (a)$L_y=50a_0,R=2.5L_y$ and (b) $L_y=50a_0,R=1.8L_y$.

Figure 5

(a) Potential profile in the conduction band. (b) Five low-energy bands around the $K$ point in the conduction band, as calculated from the single-band model (18) for the parameters $L_y=50a_0$ and $R=1.8L_y$.

Figure 6

Energy dispersion in the presence of gauge fields and deformation potentials. (a) Full tight binding calculation in an unstrained zigzag ribbon with ($L_y=299a_0$). (b) Full tight-binding calculation in a deformed zigzag ribbon with ($L_y=299a_0$, $R=7L_y$). Dashed red line indicates a small shift of conduction band minimum with respect to the maximum of the valence band at the $K$ point. The energy gap around $K$ point is no longer direct originating from the scalar potential associated with this profile of the strain. The interlevel spacing in the conduction and valence bands for the strained system (b) dramatically increases as compared to the unstrained case (a), indicating that the origin of these levels does not depend on the finite-size effects. (c) Band structure calculated from the low-energy model (7) on a ribbon with hard wall boundary condition with ($L_y=299a_0$, $R=7L_y$). Notice spin-orbit coupling has not been considered in this figure.

Figure 7

Energy dispersion of a uniaxially deformed monolayer ${\mathrm{MoS}}_2$ calculated by using the full TB model. Solid (dashed) lines indicate spin up (down) components. The vertical dashed lines mark in addition the position of the conduction band minimum and valence band maximum according to the low-energy model (32) and(33)].