Unified Description of the Optical Phonon Modes in $N$-Layer MoTe$_2$

$N$-layer transition metal dichalcogenides (denoted MX$_2$) provide a unique platform to investigate the evolution of the physical properties between the bulk (3D) and monolayer (quasi-2D) limits. Here, we present a unified analysis of the optical phonon modes in $N$-layer $2H$-MX$_2$ [1]. The $2H$-phase (or hexagonal phase) is the most common polytype for semiconducting MX$_2$ (such as MoS$_2$). Using Raman spectroscopy, we have measured the manifold of low-frequency (rigid layer), mid-frequency (involving intralayer displacement of the chalcogen atoms only), and high-frequency (involving intralayer displacements of all atoms) Raman-active modes in $N=1$ to $12$ layer $2H$-molybdenenum ditelluride (MoTe$_2$). For each monolayer mode, the $N$-dependent phonon frequencies give rise to fan diagrams that are quantitatively fit to a force constant model. This analysis allows us to deduce the frequencies of $all$ the bulk (including silent) optical phonon modes. [1] G. Froehlicher, E. Lorchat, F. Fernique, C. Joshi, A. Monlina-S\'anchez, L. Wirtz, and S. Berciaud, Unified Description of the Optical Phonon Modes in $N$-Layer MoTe$_2$, Nano Letters, \textbf{15} (10), pp 6481-6489 (2015)