Extensive research about superlattices with a very low thermal conductivity was performed to design thermoelectric materials. Indeed, the thermoelectric figure of merit ZT varies with the inverse of the thermal conductivity but is directly proportional to the power factor. Unfortunately, as nanowires, superlattices reduce heat transfer in only one main direction. Moreover, they often show dislocations owing to lattice mismatches. Therefore, fabrication of nanomaterials with a ZT larger than the alloy limit usually fails with the superlattices. Self-assembly is a major epitaxial technology to fabricate ultradense arrays of germaniums quantum dots (QD) in a silicon matrix for many promising electronic and photonic applications as quantum computing. We theoretically demonstrate that high-density three-dimensional (3-D) periodic arrays of small self-assembled Ge nanoparticles (i.e. the QDs), with a size of some nanometers, in Si can show a very low thermal conductivity in the three spatial directions. This property can be considered to design thermoelectric devices, which are compatible with the complementary metal-oxide-semiconductor (CMOS) technologies. To obtain a computationally manageable model of these nanomaterials, we simulate their thermal behavior with atomic-scale 3-D phononic crystals. A phononic-crystal period (supercell) consists of diamond-like Si cells. At each supercell center, we substitute Si atoms by Ge atoms in a given number of cells to form a box-like Ge nanoparticle. The phononic-crystal dispersion curves, which are computed by classical lattice dynamics, are flat compared to those of bulk Si. In an example phononic crystal, the thermal conductivity can be reduced below the value of only 0.95 W/mK or by a factor of at least 165 compared to bulk silicon at 300 K. Close to the melting point of silicon, we obtain a larger decrease of the thermal conductivity below the value of 0.5 W/mK, which is twice smaller than the classical Einstein Limit of single crystalline Si. In this paper, we use an incoherent-scattering approach for the nanoparticles. Therefore, we expect an even larger decrease of the phononic-crystal thermal conductivity when multiple-scattering effects, as multiple reflections and diffusions of the phonons between the Ge nanoparticles, will be considered in a more realistic model. As a consequence of our simulations, a large ZT could be achieved in 3-D ultradense self-assembled Ge nanoparticle arrays in Si. Indeed, these nanomaterials with a very small thermal conductivity are crystalline semiconductors with a power factor that can be optimized by doping using CMOS-compatible technologies, which is not possible with other recently-proposed nanomaterials.

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