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Characterization of strain in two-dimensional (2D) crystals is important forunderstanding their properties and performance. Using first-principlescalculations, we study the effects of uniaxial strain on the Raman-active modesin monolayer MoS$_2$. We show that the in-plane $E'$ mode at 384 cm$^{-1}$ andthe out-of-plane $A_1'$ mode at 403 cm$^{-1}$ can serve as fingerprints for theuniaxial strain in this 2D material. Specifically, under a uniaxial strain, thedoubly degenerate $E'$ mode splits into two non-degenerate modes: one is$E_{\parallel}'$ mode in which atoms vibrate in parallel to the straindirection, and the other is $E_\perp'$ mode in which atoms vibrateperpendicular 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, thepolarized Raman spectra of the $E_{\parallel}'$ and $E_{\perp}'$ modes exhibitdistinct angular dependence for specific laser polarization setups, allowingfor a precise determination of the direction of the uniaxial strain withrespect to the crystallographic orientation. Furthermore, we find that thepolarized Raman intensity of the $A_1'$ mode also shows evident dependence onthe applied strain, providing additional effective clues for determining thedirection of the strain even without knowledge of the crystallographicorientation. Thus, polarized Raman spectroscopy offers an efficientnon-destructive way to characterize the uniaxial strains in monolayer MoS$_2$.

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