First-principles calculation of the lattice thermal conductivities of   \alpha-, \beta-, and \gamma-Si$_3$N$_4$
  Lattice thermal conductivities (LTCs) of \alpha-, \beta-, and \gamma-Si$_3$N$_4$ single crystals are investigated from ab initio anharmonic lattice dynamics, within the single-mode relaxation-time approximation of the linearized phonon Boltzmann transport equation. At a temperature of 300 K, a \kappa$_{xx}$ of 70 and a \kappa$_{zz}$ of 98 (in units of Wm$^{-1}$K$^{-1}$) are obtained for \alpha-Si$_3$N$_4$. For \beta-Si$_3$N$_4$, \kappa$_{xx}$ and \kappa$_{zz}$ are found to be 71 and 194, respectively, which are consistent with the reported experimental values of 69 and 180 for individual \beta-Si$_3$N$_4$ grains in a ceramic. The theoretical \kappa$_{xx}$ values of \alpha- and \beta-Si$_3$N$_4$ are comparable, while the \kappa$_{zz}$ value of \beta-Si$_3$N$_4$ is almost twice that of \alpha-Si$_3$N$_4$, which demonstrates the very large anisotropy in the LTC of the \beta phase. It is found that the large anisotropy in the LTC of \beta-Si$_3$N$_4$ was caused by the elongated Brillouin zone along the c* axis, where the acoustic phonons mostly contribute to LTC and have large group velocities even near the Brillouin zone boundary. The LTC of \gamma-Si$_3$N$_4$ is 81, which is as small as that of \alpha-Si$_3$N$_4$, although \gamma-Si$_3$N$_4$ has much larger elastic constants. This means that elastic constants are not always a good indicator of LTC. We show that knowing the detailed distributions of both the group velocities and phonon lifetimes in the Brillouin zones is important for characterizing the LTC of the three phases. 
Kazuyoshi Tatsumi, Atsushi Togo, Isao Tanaka
cond-mat.mtrl-sci