First-principles Raman spectra of MoS$_2$, WS$_2$ and their heterostructures*

MoS$_2$ and WS$_2$ are graphene-like layered structures that are considered as alternative and complement to graphene. Raman spectroscopy is a very powerful tool to study them. Despite the extensive experimental Raman study on MoS$_2$ and WS$_2$, it remains unclear how Raman intensities and especially intensity ratio of Raman modes E$_{2g}$ and A$_{1g}$ depend on the thickness. To clarify such issues, we carried out density functional theory calculations for both MoS$_2$ and WS$_2$ to simulate their Raman spectra and reveal the intrinsic thickness dependence of Raman intensities and intensity ratio. More importantly, we quantitatively analyzed the laser polarization effect on the intensity ratio and revealed its high sensitivity to laser polarization, which could explain the large discrepancy between measured intensity ratios by different groups. We also studied $\textit{ab initio}$ Raman spectra of MoS$_2$/WS$_2$ heterostructures up to four layers in every possible combinations and stacking orders. Each configuration is found to possess a unique Raman spectrum in both frequency and intensity that can be explained by changes in dielectric screening and interlayer interactions. Our findings serve as guidelines for the experimental identification of heterostructure configurations.