Stacking-Dependent Interlayer Phonons in 3R and 2H MoS$_{2}$

We have investigated the interlayer shear and breathing phonon modes in MoS$_{2}$ with pure 3R and 2H stacking order by using polarization-dependent ultralow-frequency Raman spectroscopy. We observe up to three shear branches and four breathing branches in MoS$_{2}$ with thickness from 2 to 13 layers. The breathing modes show the same Raman activity behavior for both polytypes, but the 2H breathing frequencies are consistently several wavenumbers higher than the 3R breathing frequencies, signifying that 2H MoS$_{2}$ has slightly stronger interlayer lattice coupling than 3R MoS$_{2}$. In contrast, the shear-mode Raman spectra are strikingly different for 2H and 3R MoS$_{2}$. While the strongest shear mode corresponds to the highest-frequency branch in the 2H structure, it corresponds to the lowest-frequency branch in the 3R structure. Such distinct and complementary Raman spectra of the 3R and 2H polytypes allow us to survey a broad range of shear modes in MoS$_{2}$, from the highest to lowest branch. By combining the linear chain model, group theory, effective bond polarizability model and first-principles calculations, we can account for all the major observations in our experiment.