Callaway approximation to the Boltzmann equation fails to predict phonon hydrodynamics

The Callaway approximation to the linearized Peierls-Boltzmann equation (LPBE) is often used to describe the steady-state and the transient phonon dynamics in ultrahigh thermal conductivity (κ) materials, particularly under cryogenic conditions, due to its low computational cost relative to the full LPBE solution. However, the predictive capability of the Callaway approximation to the LPBE is not well-established, since a direct quantitative comparison with the full solution of LPBE from first-principles does not exist in the literature under these conditions. Here, we present a quantitative comparison of the steady-state and transient thermal transport predictions from the Callaway approximation and the full LPBE solution for 20 different semiconductor materials below 200 K. We find that the Callaway approximation successfully predicts the conventional Fourier-diffusive and the quasiballistic heat transfer regimes at temperatures where momentum-dissipative Umklapp processes (U-processes) are comparable or only slightly weaker than the momentum-conserving Normal processes (N-processes). As the temperature is further lowered, the U-processes become exceptionally weaker than the N-processes-a necessary condition for the unconventional hydrodynamic heat flow regime, and the Callaway model fails to closely approximate the full LPBE solution. Using quantitative comparisons of the predicted steady-stateκand temporal temperature dynamics in a pump-probe experiment called the transient grating, we uncover a strong correlation between the failure of the Callaway approximation to the LPBE and the onset of phonon hydrodynamics, as the temperature is lowered. Our work provides guidelines to determine the applicability of the Callaway approximation to the LPBE to describe phonon dynamics at low temperatures.