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Polarization analysis of first- and second-order Raman scattering from MoTe$_2$ single crystal

We report on Raman experiments performed on a single crystal MoTe$_2$ sample. The system belongs to the wide family of Transition Metal Dichalcogenides which includes several of the most interesting two dimensional materials for both basic and applied physics. Measurements were performed in the standard basal plane configuration, by placing the $ab$ plane of the crystal perpendicular to the wave vector $k_i$ of the incident beam to explore the in plane vibrational modes, and in the edge plane configuration with $k_i$ perpendicular to the crystal $c$ axis, thus mainly exciting out-of-plane modes. For both configurations we performed a polarization-dependent Raman study and we were able to provide a complete assignment of the observed first- and second-order Raman peaks fully exploiting the polarization selection rules. Present findings are in complete agreement with previous first-order Raman data whereas a thorough analysis of the second-order Raman bands, either in basal- or edge-plane configurations, provides new information and a precise assignment of these spectral structures. In particular, we have observed Raman active modes of the $M$ point of the Brillouin zone previously predicted by ab-initio calculations and ascribed to either combination or overtone but never previously measured.