Strain-tunable orbital, spin-orbit, and optical properties of monolayer transition-metal dichalcogenides

When considering transition-metal dichalcogenides (TMDCs) in van der Waals heterostructures for periodic ab initio calculations, usually, lattice mismatch is present, and the TMDC needs to be strained. In this study we provide a systematic assessment of biaxial strain effects on the orbital, spin-orbit, and optical properties of the monolayer TMDCs using ab initio calculations. We complement our analysis with a minimal tight-binding Hamiltonian that captures the low-energy bands of the TMDCs around the $K$ and $K^{\prime }$ valleys. We find characteristic trends of the orbital and spin-orbit parameters as a function of the biaxial strain. Specifically, the orbital gap decreases linearly, while the valence (conduction) band spin splitting increases (decreases) nonlinearly in magnitude when the lattice constant increases. Furthermore, employing the Bethe-Salpeter equation and the extracted parameters, we show the evolution of several exciton peaks, with biaxial strain, on different dielectric surroundings, which are particularly useful for interpreting experiments studying strain-tunable optical spectra of TMDCs.

Figure 1

Calculated orbital decomposed band structure of ${\mathrm{MoS}}_2$ as a representative example of a TMDC. SOC is not included, and the different colors correspond to different orbitals or atoms.

Figure 2

Geometry of a TMDC monolayer with general structure ${MX}_2$, where $M$ is the transition metal (Mo, W) and $X$ is the chalcogen atom (S, Se). (a) Side and (b) top views of the geometry, with labels for the lattice constant $a$, distance $d_{\mathrm{MX}}$ (between the transition metal and the chalcogen atom), and $d_{\mathrm{XX}}$ (between the two chalcogen atoms).

Figure 3

(a) Calculated band structure of ${\mathrm{MoS}}_2$ including SOC. The color corresponds to the $s_z$ expectation value. (b) and (c) Calculated low-energy CB and VB around the $K$ point (symbols) with a fit to the model Hamiltonian (solid line).

Figure 4

Summary of the fit parameters for ${\mathrm{MoS}}_2$ as a function of the lattice constant. (a) The gap parameter $\mathrm{\Delta }$ and the total energy $E_{\mathrm{tot}}$. The black data (SOC on $M,X$) correspond to calculations where SOC is included for both atoms $M$ and $X$, while for the red data (SOC on $M$), we turned off SOC on the $X$ atoms. Dashed vertical lines indicate the equilibrium lattice constant. (b) The Fermi velocity $v_{\mathrm{F}}$. (c) The distances $d_{\mathrm{XX}}$ and $d_{\mathrm{MX}}$. (d) and (e) The SOC parameters ${\lambda }_{\mathrm{c}}$ and ${\lambda }_{\mathrm{v}}$. The difference (blue curve) is between the black and the red curves.

Figure 5

(a) Band structure of ${\mathrm{MoS}}_2$ at zero strain, highlighting the important optical transitions mostly affected by strain. The transitions are identified by the reciprocal space point $(K,Q,\mathrm{\Gamma })$ and by the energy band (subindices v and c stand for the valence and conduction bands, respectively). Evolution of the transition energies depicted in (a) as a function of strain for (b) ${\mathrm{MoS}}_2$, (c) ${\mathrm{MoSe}}_2$, (d) ${\mathrm{WS}}_2$, and (e) ${\mathrm{WSe}}_2$. The shaded regions indicate indirect band gap regimes ($K-Q$ for negative strain and $\mathrm{\Gamma }-K$ for positive strain).

Figure 6

Sketch of the energy bands that contribute to the formation of A and B excitons in (a) Mo-based and (b) W-based TMDCs. Evolution of the total exciton energy as a function of the biaxial strain for (c) ${\mathrm{MoS}}_2$, (d) ${\mathrm{MoSe}}_2$, (e) ${\mathrm{WS}}_2$, and (f) ${\mathrm{WSe}}_2$. The thin solid (dashed) lines are the single-particle energies for the A (B) optical transitions. The shaded regions indicate indirect band gap regimes ($K-Q$ for negative strain and $\mathrm{\Gamma }-K$ for positive strain).

Figure 7

Evolution of the energy difference between distinct excitonic levels as a function of the biaxial strain for (a) ${\mathrm{MoS}}_2$, (b) ${\mathrm{MoSe}}_2$, (c) ${\mathrm{WS}}_2$, and (d) ${\mathrm{WSe}}_2$. The shaded regions have the same meaning as in Fig. 6.