Orbital analysis of electronic structure and phonon dispersion in MoS2, MoSe2, WS2, and WSe2monolayers under strain

Abstract

Structures, electronic properties, and phonon dispersions of monolayer $MX$${}_{2}$ ($M=$ Mo, W; $X=$ S, Se) under various types of mechanical strains are investigated with density functional calculations. By analyzing the orbitals near band gaps, we demonstrate that the $X$-$M$-$X$ bond angle, together with the $X$-$X$ distances, plays one of the most important roles in determining band gaps, as it is directly related to the coupling of $d$ orbitals of the $M$ atoms and $p$ orbitals of the $X$ atoms. It is found that the band gaps of monolayer $MX$${}_{2}$ are more sensitive to biaxial strains. This notion can be attributed to the fact that, under biaxial strains, $MX$${}_{2}$ tends to be stretched more significantly along the out-of-plane direction, resulting in greater changes in bond lengths/angles than uniaxial strains. While most theoretical reports suggested systematic reduction of band gaps under mechanical strains, we found that the direct band gaps can be robustly widened by applying compressive biaxial strains. Our results are of immediate importance for the recent experimental advances in applying compressive strains, which was not applicable before. On the other hand, the anisotropy of $X$-$M$-$X$ bond angles induced by uniaxial strains leads to splitting in the phonon dispersion, and is directly relevant to recent Raman spectroscopy results. As most transition-metal dichalcogenides share the same structural characters, we expect that the notions derived here are applicable to other layered transition-metal oxides/chalcogenides and open up new venues to engineer their properties.