  Nanocrystalline silicon and silicon-germanium alloys are promising
thermoelectric materials that have achieved substantially improved figure of
merits compared to their bulk counterparts. This enhancement is typically
attributed to a reduction in lattice thermal conductivity by phonon scattering
at grain boundaries. However, further improvements are difficult to achieve
because grain boundary scattering is poorly understood, with recent
experimental observations suggesting that the phonon transmissivity may depend
on phonon frequency rather than being constant as in the commonly used gray
model. Here, we examine the impact of frequency-dependent grain boundary
scattering in nanocrystalline silicon and silicon-germanium alloys in a
realistic 3D geometry using frequency-dependent variance-reduced Monte Carlo
simulations. We find that the grain boundary may not be as effective as
predicted by the gray model in scattering certain phonons, with a substantial
amount of heat being carried by low frequency phonons with mean free paths
longer than the grain size. Our result will help guide the design of more
efficient thermoelectrics.
