  Ultrafast lattice dynamics of few quintuple layers of topological insulator
(TI) Bi$_2$Te$_3$ is studied with time-resolved optical pump-probe
spectroscopy. Both optical and acoustic phonons are photogenerated and
detected. Here, in order to get new insights on the out-of-equilibrium
electron-phonon coupling and phonons dynamics in confined TI, different
nanostructures have been investigated (single or polycrystalline QLs assemblies
and nano-crystallized islands). Contrary to previous literature claims, we show
that even for nanostructures containing only 10 quintuple layers (QLs), the
symmetric A1g(I) coherent optical phonon is efficiently photogenerated and no
restriction due to the structural confinement appears. We also observe that
whatever the arrangement of the nanostructures, the A1g(I) optical phonon
features are similar (lifetime). We also report the observation of confined
coherent acoustic phonons propagating from QLs to QLs whose spectrum is, this
time, very sensitive to the atomic arrangement. In the case of the single
crystalline ultrathin film, the time of flight analysis of these acoustic
phonons provides direct estimate of the elastic properties of these
nanostructures as well as some estimates of Van der Waals interactions between
QLs.
