Probing the uniaxial strains in MoS$_2$ using polarized Raman spectroscopy: A first-principles study

Characterization of strain in two-dimensional (2D) crystals is important for understanding their properties and performance. Using first-principles calculations, we study effects of uniaxial strain on the Raman-active modes in monolayer MoS$_2$. We show that the in-plane $E'$ mode at 384 cm$^{-1}$ can serve as a fingerprint for the uniaxial strain in this 2D material. Specifically, under a uniaxial strain, the doubly degenerate $E'$ mode splits into two non-degenerate modes: one is $E_{\parallel}'$ mode in which atoms vibrate in parallel to the strain direction, and the other is $E_\perp'$ mode in which atoms vibrate perpendicular to the strain direction. The frequency of the $E_{\parallel}'$ mode blue-shifts for a compressive strain, but red-shifts for a tensile strain. In addition, due to the strain-induced anisotropy in the MoS$_2$ lattice, the polarized Raman spectra of the $E_{\parallel}'$ and $E_{\perp}'$ modes exhibit distinct angular dependence, allowing for a precise determination of the direction of the uniaxial strain. Thus, the polarized Raman spectroscopy offers an efficient non-destructive way to characterize the uniaxial strains in monolayer MoS$_2$.