The spatial extension and "shape" of phonon modes has been used previously to distinguish vibrational characters. Here, we use the participation ratio for such analysis (Figure g, details in Supplemental Note 10). 9,10 A participation ratio close to unity is indicative of a high spatial extension of a mode and typically found in simple crystalline materials like c-Si that have textbook phonon-gas transport, and so a participation ratio of 1 can be called the phonon limit.