Coupling between Phonon-Phonon and Phonon-Impurity Scattering: A Critical Revisit of the Spectral Matthiessen's Rule

The spectral Matthiessen’s rule is commonly used to calculate the total phonon scattering rate when multiple scattering mechanisms exist. Here we predict the spectral phonon relaxation time $\tau$ of defective bulk silicon using normal mode analysis based on molecular dynamics, and show that the spectral Matthiessen’s rule is not accurate due to the neglect of the coupling between anharmonic phonon-phonon scattering ${\tau }_a^{-1}$ and phonon-impurity scattering ${\tau }_i^{-1}$. As a result, the spectral Matthiessen’s rule underestimates the total phonon scattering rate, and hence overestimates the thermal conductivity $\kappa$ of mass-doped and Ge-doped silicon by about 20-40%. We have also directly estimated this coupling scattering rate, so called coupled five-phonon scattering ${\tau }_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{e}}^{-1}$, and achieved good agreement between ${\tau }_a^{-1}+{\tau }_i^{-1}+{\tau }_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{e}}^{-1}$ and the total scattering rate ${\tau }_{tot}^{-1}$.