We have used an atomistic {it ab initio} approach with no adjustable parameters to compute the lattice thermal conductivity of Si$_{0.5}$Ge$_{0.5}$ with a low concentration of embedded Si or Ge nanoparticles of diameters up to 4.4 nm. Through exact G
reens function calculation of the nanoparticle scattering rates, we find that embedding Ge nanoparticles in $text{Si}_{0.5}text{Ge}_{0.5}$ provides 20% lower thermal conductivities than embedding Si nanoparticles. This contrasts with the Born approximation which predicts an equal amount of reduction for the two cases, irrespective of the sign of the mass difference. Despite these differences, we find that the Born approximation still performs remarkably well, and it permits investigation of larger nanoparticle sizes, up to 60 nm in diameter, not feasible with the exact approach.
We theoretically compute the thermal conductivity of SiGe alloy nanowires as a function of nanowire diameter, alloy concentration, and temperature, obtaining a satisfactory quantitative agreement with experimental results. Our results account for the
weaker diameter dependence of the thermal conductivity recently observed in Si$_{1-x}$Ge$_x$ nanowires ($x<0.1$), as compared to pure Si nanowires. We also present calculations in the full range of alloy concentrations, $0 leq x leq 1$, which may serve as a basis for comparison with future experiments on high alloy concentration nanowires.
