Effects of uniaxial strain on monolayer transition-metal dichalcogenides revisited

Abstract

Using hybrid density functional calculations including spin-orbit coupling, we compute the strain evolution of the band structure of monolayer 1H-phase transition-metal dichalcogenides, MX$_2$ (M= Mo, W; X= S, Se, Te), emphasizing an accurate reproduction of the quasiparticle band gap (as opposed to the excitonic optical gap). We show that tensile uniaxial strain applied along either the armchair or zigzag directions leads to a pronounced reduction of the fundamental gap, with the conduction-band edge generally exhibiting the stronger strain response. Both the conduction-band electron valleys (CBM) and the valence-band hole valleys (VBM) remain degenerate under uniaxial strain, while simultaneously drifting away from the high-symmetry $K$ point under strain ("valley drift"), such that the band extrema occur at nearby off-symmetry wave vectors. A minimal tight-binding model rationalizes the valley drift and the unequal electron- and hole-valley drift rates in the presence of strain, leading to indirect band gaps. In particular, for MoS$_2$ the indirectness increases with tensile strain, providing a natural explanation for the experimentally observed decrease in photoluminescence intensity under uniaxial deformation. These results provide quantitative guidance for tailoring band structures for optoelectronic and quantum-defect applications.

Figures (7)

• Figure 1: Crystal and reciprocal-space geometry of monolayer 1H-MX2. (a) Top view and (b) side view of the trigonal-prismatic MX2 layer, with transition-metal atoms (blue) and chalcogen atoms (purple). The in-plane lattice vectors and structural descriptors used in this work (including the layer thickness $d$ and the chalcogen--metal--chalcogen bond angle $\phi$) are indicated in (b). (c-e) Brillouin zones (BZs) illustrating the strain-induced distortion of reciprocal space under tensile uniaxial loading: (d) unstrained (equilibrium) hexagonal BZ, compared to the distorted BZ for strain applied along (c) the armchair direction (red) and (e) the zigzag direction (green). The arrows $\mathbf{b}_1$ and $\mathbf{b}_2$ denote the reciprocal-lattice basis vectors. High-symmetry points ($\Gamma$, $K/K'$, $M/M'$, and the $Q/Q'$ points on $\Gamma$--$K$) are labeled for reference. The red dots in (c) and green dots in (e) schematically indicate the strain-induced drift of the electron and hole valleys away from the nominal $K$-corner, as analyzed in Sec. "Valley drift and strain-induced indirectness".
• Figure 2: Electronic band structures of unstrained monolayer MoS2 calculated along $\Gamma$--$Q$--$K$--$M$--$\Gamma$ using different exchange-correlation functionals to highlight the roles of hybrid exchange and spin-orbit coupling (SOC): (a) HSE$\alpha$+SOC, (b) HSE$\alpha$ without SOC, (c) SCAN+SOC, and (d) SCAN without SOC. The valence-band maximum (VBM) is set to zero in all panels; red (blue) curves denote the lowest conduction (highest valence) bands. In (a), the direct gap at $K$ is labeled and the characteristic energy offsets $\Delta_{\Gamma K}$ and $\Delta_{QK}$, as well as the valence-band SOC splitting at $K$ ($\Delta_{\mathrm{SO}}$), are indicated; $K_v$ and $K_c$ mark the VBM and CBM at $K$, respectively.
• Figure 3: HSE$\alpha$+SOC band structures of unstrained monolayer 1H-MX2 for (a) MoS2, (b) MoSe2, (c) MoTe2, (d) WS2, (e) WSe2, and (f) WTe2, plotted along $\Gamma$--$Q$--$K$--$M$--$\Gamma$. Energies are referenced to the VBM (set to zero) in each panel. The fundamental quasiparticle gap $E_g$ and the valence-band SOC splitting at $K$ ($\Delta_{\mathrm{SO}}$) are annotated to facilitate comparison across the series, highlighting the systematic reduction of $E_g$ from S$\rightarrow$Se$\rightarrow$Te and the enhancement of SOC effects with heavier constituents.
• Figure 4: HSE$\alpha$+SOC band structures of monolayer MoS2 under tensile uniaxial strain applied along the armchair direction: (a) 1% to (e) 5%. Energies are referenced to the VBM (set to zero) in each panel, and the dispersions are plotted along the strain-adapted path $M\!\rightarrow\!K\!\rightarrow\!Q\!\rightarrow\!\Gamma\!\rightarrow\!Q'\!\rightarrow\!K'\!\rightarrow\!M'$. With increasing strain, the fundamental gap decreases, and the band-edge valleys exhibit a systematic valley drift away from the nominal $K'$ point. The bottom-right inset (last panel) magnifies the vicinity of the strained valley extrema and explicitly marks the $k$-space displacement of the hole valley (VBM) and electron valley (CBM) from $K'$, quantified as $\Delta k_v$ and $\Delta k_c$, respectively (for 5% strain: $\Delta k_v=0.052~\text{\AA}^{-1}$ and $\Delta k_c=0.064~\text{\AA}^{-1}$ along the indicated direction).
• Figure 5: HSE$\alpha$+SOC band structures of monolayer MoS2 under tensile uniaxial strain applied along the zigzag direction: (a) 1% to (e) 5%. Energies are referenced to the VBM (set to zero) in each panel, and the dispersions are plotted along the strain-adapted path $M\!\rightarrow\!K\!\rightarrow\!Q\!\rightarrow\!\Gamma\!\rightarrow\!Q'\!\rightarrow\!K'\!\rightarrow\!M'$. The fundamental gap decreases with strain, and the band-edge valleys drift continuously away from the nominal $K$ point. The bottom-right inset (last panel) zooms into the vicinity of the valley extrema and highlights the strain-induced hole-valley (VBM) and electron-valley (CBM) displacements from $K$, labeled $\Delta k_v$ and $\Delta k_c$, respectively (for 5% strain: $\Delta k_v=0.065~\text{\AA}^{-1}$ and $\Delta k_c=0.080~\text{\AA}^{-1}$ along the indicated direction), illustrating that the electron and hole valleys drift at different rates under uniaxial loading.