Examining the Callaway model for lattice thermal conductivity

The Callaway model [J. Callaway, Phys. Rev. 113, 1046 (1959)], regarded as an improvement over the relaxation time approximation (RTA) for the phonon Boltzmann transport equation (BTE), is widely used in studying lattice thermal conductivity (kappa). However, its accuracy needs to be systematically examined. By solving BTE accurately using an iterative method along with the first principles calculation of phonon scatterings, we conduct such an examination of the Callaway model as well as a modified version proposed by Allen [Phys. Rev. B 88, 144302 (2013)] for Si, diamond, and wurtzite AlN. At room temperature, the RTA underestimates kappa by 5%, 32%, 11%, and 12% for Si, diamond, and in-plane and cross-plane AlN, respectively. The deviation of the original Callaway model from the accurate kappa is -1%, 25%, 1%, and -12%, respectively, while the deviation of Allen's modified model is 7%, 44%, 13%, and -8%, respectively. The room temperature anisotropy of AlN is 5%, and the anisotropy predicted by RTA, the Callaway model, and Allen's modified version is 7%, 19%, and 29%, respectively. We conclude that neither the original Callaway model nor Allen's modified version can generally guarantee an improvement over RTA. In these three systems, we also find that the relaxation times for umklapp processes scale as 1/omega(3) at low frequencies for both transverse acoustic (TA) and longitudinal acoustic (LA) modes, and those for normal processes scale as 1/omega and 1/omega(2) for TA and LA modes, respectively.