  We present a combined experimental and theoretical study of the surface
vibrational modes of the topological insulator Bi$_2$Te$_3$. Using
high-resolution helium-3 spin-echo spectroscopy we are able to resolve the
acoustic phonon modes of Bi$_2$Te$_3$(111). The low energy region of the
lattice vibrations is mainly dominated by the Rayleigh mode which has been
claimed to be absent in previous experimental studies. The appearance of the
Rayleigh mode is consistent with previous bulk lattice dynamics studies as well
as theoretical predictions of the surface phonon modes. Density functional
perturbation theory calculations including van der Waals corrections are in
excellent agreement with the experimental data. Comparison of the experimental
results with theoretically obtained values for films with a thickness of
several layers further demonstrate, that for an accurate theoretical
description of three-dimensional topological insulators with their layered
structure the inclusion of van der Waals corrections is essential. The presence
of a prominent surface acoustic wave and the contribution of van der Waals
bonding to the lattice dynamics may hold important implications for the
thermoelectric properties of thin-film and nanoscale devices.
