Coupling between shear and tensile strains in layered two-dimensional crystals

We report a theoretical study revealing unavoidable coupling between shear and tensile strains in several layered two-dimensional crystals. It is shown that the coupling can explain a recent Raman experiment exhibiting an anomalous splitting in the low frequency interlayer shear modes of bilayer MoS$_2$ under uniaxial strain. We have found that the splitting comes from the strain-induced interlayer sliding. Our calculation shows that the direction of the induced sliding is related to the strain-induced polarization, piezoelectricity, that is calculated using electronic Berry phase, thus connecting piezoelectricity of a layered material with its elastic effect. We will present the results of our calculations for shear-tensile strain coupling of graphene, $h$-BN, and MoS$_2$ respectively and demonstrate that the Raman measurement can determine the off-diagonal elements of compliance tensor of the layered materials.