  We investigate the relative thermodynamic stability of the $3\times3$,
$5\times5$, $7\times7$, $9\times9$ and infinitely large structures related to
the dimers-adatoms-stacking faults family of Si$(111)$ surface reconstructions
by means of first-principles calculations. Upon accounting for the vibrational
contribution to the surface free energy, we find that the $5\times5$ structure
is more stable than the $7\times7$ at low temperatures. While a phase
transition is anticipated to occur at around room temperature, the
$7\times7\rightarrow5\times5$ transformation upon cooling is hindered by the
limited mobility of Si atoms. The results not only flag a crucial role of
vibrational entropy in the formation of the $7\times7$ structure at elevated
temperatures, but also point for its metastable nature below room temperature.
