Density functional theory calculations not only allow to predict the vibrational and optical properties of solids but also to understand and disentangle the mechanisms playing a key role in the generation of coherent optical phonons. Recent experiments performed on a ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ nanoscale thin film have shown that a terahertz (THz) pulse launches at least a coherent $A_{1g}^1$ phonon as the transient transmittance measured using an isotropic detection scheme displays oscillations with a frequency matching the frequency of the $A_{1g}^1$ mode measured in Raman experiments. Such an observation can be explained by invoking either a sum-frequency process or cubic or quartic phonon-phonon couplings as considered for ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, a parent compound of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$. By resorting to group theory and calculating energy surfaces from first principles, the main phonon-phonon couplings can be identified. Furthermore, a minimal model can be built to compute the dynamics of the Raman active modes coupled to the infrared active mode driven by the experimental THz pulse. Our model firmly establishes that cubic phonon-phonon interactions are relevant as the agreement between the computed and experimental transmittance is noteworthy.

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• Anharmonic lattice dynamics
• Lattice dynamics
• Ab initio calculations
• Density functional theory
• Ultrafast femtosecond pump probe