  Which sizes of nanoparticles embedded in a crystalline solid yield the lowest
thermal conductivity? Nanoparticles have long been demonstrated to reduce the
thermal conductivity of crystals by scattering phonons, but most previous works
assumed the nanoparticles to have a single size. Here, we use optimization
methods to show that the best nanoparticle size distribution to scatter the
broad thermal phonon spectrum is not a similarly broad distribution but rather
several discrete peaks at well-chosen nanoparticle radii. For SiGe, the best
size distribution yields a thermal conductivity below that of amorphous
silicon. Further, we demonstrate that a simplified distribution yields nearly
the same low thermal conductivity and can be readily fabricated. Our work
provides important insights into how to manipulate the full spectrum of phonons
and will guide the design of more efficient thermoelectric materials.
