  Nanostructured semiconducting materials are promising candidates for
thermoelectrics due to their potential to suppress phonon transport while
preserving electrical properties. Modeling phonon-boundary scattering in
complex geometries is crucial for predicting materials with high conversion
efficiency. However, the simultaneous presence of ballistic and diffusive
phonons challenges the development of models that are both accurate and
computationally tractable. Using the recently developed first-principles
Boltzmann transport equation (BTE) approach, we investigate diffusive phonons
in nanomaterials with wide mean-free-path (MFP) distributions. First, we derive
the short MFP limit of the suppression function, showing that it does not
necessarily recover the value predicted by standard diffusive transport,
challenging previous assumptions. Second, we identify a Robin type boundary
condition describing diffuse surfaces within Fourier's law, extending the
validity of diffusive heat transport in terms of Knudsen numbers. Finally, we
use this result to develop a hybrid Fourier/BTE approach to model realistic
materials, obtaining excellent agreement with experiments. These results
provide insight on thermal transport in materials that are within experimental
reach and open opportunities for large-scale screening of nanostructured
thermoelectric materials.
