Superlattices have been used to design thermoelectric materials with ultra-low thermal conductivities. Indeed, the thermoelectric figure of merit ZT varies as the inverse of the material thermal conductivity. However, the design of a thermoelectric material with ZT superior to the alloy limit usually fails with the superlattices because of two major drawbacks: First, a lattice mismatch can occur between the different layers of a superlattice as in a Si/Ge superlattice. This leads to the formation of defects and dislocations, which reduces the electrical conductivity and therefore avoids the increase of ZT compared to the alloy limit. On the other hand, the superlattices only affect heat transfer in one direction. To cancel heat conduction in the three spatial directions, we propose atomic-scale three-dimensional (3D) phononic crystals. Because the lattice constant of our phononic crystal is of the order of some nanometers, we obtain phonon confinement in the THz range and a nanomaterial with a very low thermal conductivity. This is not possible with the usual phononic crystals, which show band gaps in the sub-MHz range owing to their large lattice constant of the order of 1 mm. A period of our atomic-scale 3D phononic crystal is composed of a given number of diamond-like silicon cells forming a supercell. A periodic Si/Ge heterostructure is obtained since we substitute at each supercell center the Si atoms in a smaller number of cells by Ge atoms. The Ge atoms in the cells located at each supercell center form a box-like nanoparticle with a size that can be varied to obtain different atomic configurations of our nanomaterial. We also propose another design for our phononic crystal where we introduce a small number of diamond-like silicon cells at the center of a periodic supercell of diamond-like germanium cells. In this second design, we form box-like nanoparticles of Si atoms in a germanium matrix instead of boxlike nanoparticles of Ge atoms in a silicon matrix in the first design. With the dispersion curves computed by lattice dynamics and a general equation, we obtain the thermal conductivities of several atomic configurations of our phononic crystal. Compared to a bulk material, the thermal conductivity can be reduced by at least one order of magnitude in our phononic crystal. This reduction is only due to the phonon group velocities, and we expect a further decrease owing to the diminution of the phonon mean free path in our phononic crystal.

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