  Controlling the thermal conductivity of semiconductors is of practical
interest in optimizing the performance of thermoelectric and phononic devices.
The insertion of inclusions of nanometer size in a semiconductor is an
effective means of achieving such control; it has been proposed that the
thermal conductivity of silicon could be reduced to 1 W/m/K using this approach
and that a minimum in the heat conductivity would be reached for some optimal
size of the inclusions. Yet the practical verification of this design rule has
been limited. In this work, we address this question by studying the thermal
properties of silicon metalattices that consist of a periodic distribution of
spherical inclusions with radii from 7 to 30 nm, embedded into silicon.
Experimental measurements confirm that the thermal conductivity of silicon
metalattices is as low as 1 W/m/K for silica inclusions, and that this value
can be further reduced to 0.16 W/m/K for silicon metalattices with empty pores.
A detailed model of ballistic phonon transport suggests that this thermal
conductivity is close to the lowest achievable by tuning the radius and spacing
of the periodic inhomogeneities. This study is a significant step in
elucidating the scaling laws that dictate ballistic heat transport at the
nanoscale in silicon and other semiconductors.
