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The spectral Matthiessen's rule is commonly used to calculate the totalphonon scattering rate when multiple scattering mechanisms exist. Here wepredict the spectral phonon relaxation time $\tau$ of defective bulk siliconusing normal mode analysis based on molecular dynamics, and show that thespectral Matthiessen's rule is not accurate due to the neglect of the couplingbetween anharmonic phonon-phonon scattering $\tau_a^{-1}$ and phonon-impurityscattering $\tau_i^{-1}$. As a result, the spectral Matthiessen's ruleunderestimates the total phonon scattering rate, and hence overestimates thethermal conductivity $\kappa$ of mass-doped and Ge-doped silicon by about20-40%. We have also directly estimated this coupling scattering rate, socalled coupled five-phonon scattering $\tau_{\rm couple}^{-1}$, and achievedgood agreement between $\tau_a^{-1}+\tau_i^{-1}+\tau_{\rm couple}^{-1}$ and thetotal scattering rate $\tau_{tot}^{-1}$.

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