Scaling laws of cumulative thermal conductivity for short and long phonon mean free paths

Cumulative thermal conductivity (CTC), an accumulation function of lattice thermal conductivity with respect to the phonon mean free path (PMFP), is a useful single-crystal property to gain insight into how much nanostructuring can potentially reduce thermal conductivity. While the details of the CTC profile depend on each material, we have identified that the profile has universal features in the short and long PMFP regimes with each characteristic length scale. In each PMFP regime, by scaling the PMFP with the characteristic length derived using phenomenological models, CTC calculated based on first principles for various materials collapse on a master curve. We also find an empirical relation between the short and long PMFP characteristic length scales, which allows us to roughly estimate the onset/offset PMFP of CTC (i.e., PMFP when CTC are 10%/90% of the total thermal conductivity) only with the knowledge of bulk thermal conductivity and averaged group velocity. The finding provides a facile way to estimate the range of PMFP with noticeable contribution to lattice thermal conductivity, which is useful for designing nanostructured materials with low thermal conductivity, particularly in developing thermoelectric materials.