  We determined the lattice dynamics of metastable, surface-stabilized
$\alpha$-phase FeSi$_2$ nanoislands epitaxially grown on the Si(111) surface
with average heights and widths ranging from 1.5 to 20 nm and 18 to 72 nm,
respectively. The crystallographic orientation, surface morphology and local
crystal structure of the nanoislands were investigated by reflection
high-energy electron diffraction, atomic force microscopy and X-ray absorption
spectroscopy. The Fe-partial phonon density of states (PDOS), obtained by
nuclear inelastic scattering, exhibits a pronounced damping and broadening of
the spectral features with decreasing average island height. First-principles
calculations of the polarization-projected Si- and Fe-partial phonon
dispersions and PDOS enable the disentanglement of the contribution of the
$xy$- and $z$-polarized phonons to the experimental PDOS. Modeling of the
experimental data with the theoretical results unveils an enhanced damping of
the $z$-polarized phonons for islands with average sizes below 10 nm. This
phenomenon is attributed to the fact that the low-energy $z$-polarized phonons
couple to the low-energy surface/interface vibrational modes. The thermodynamic
and elastic properties obtained from the experimental data show a pronounced
size-dependent behavior.
