Unified description of the optical phonon modes in $N$-layer MoTe$_2$

$N$-layer transition metal dichalcogenides provide a unique platform to investigate the evolution of the physical properties between the bulk (three dimensional) and monolayer (quasi two-dimensional) limits. Here, using high-resolution micro-Raman spectroscopy, we report a unified experimental description of the $\Gamma$-point optical phonons in $N$-layer $2H$-molybdenum ditelluride (MoTe$_2$). We observe a series of $N$-dependent low-frequency interlayer shear and breathing modes (below $40~\rm cm^{-1}$, denoted LSM and LBM) and well-defined Davydov splittings of the mid-frequency modes (in the range $100-200~\rm cm^{-1}$, denoted iX and oX), which solely involve displacements of the chalcogen atoms. In contrast, the high-frequency modes (in the range $200-300~\rm cm^{-1}$, denoted iMX and oMX), arising from displacements of both the metal and chalcogen atoms, exhibit considerably reduced splittings. The manifold of phonon modes associated with the in-plane and out-of-plane displacements are quantitatively described by a force constant model, including interactions up to the second nearest neighbor and surface effects as fitting parameters. The splittings for the iX and oX modes observed in $N$-layer crystals are directly correlated to the corresponding bulk Davydov splittings between the $E_{2u}/E_{1g}$ and $B_{1u}/A_{1g}$ modes, respectively, and provide a measurement of the frequencies of the bulk silent $E_{2u}$ and $B_{1u}$ optical phonon modes. Our analysis could readily be generalized to other layered crystals.