Exploiting non-local analysis of lattice thermal conductivity

Small Q phonons have very slow relaxation rates $1/\tau_Q$. This causes the heat current $j(x)$ to depend on the temperature gradient $dT(x^\prime)/dx^\prime$ at long distances $|x-x^\prime|$. In a homogeneous crystal, the Fourier-space representation $j(q)=-\kappa(q) dT/dx(q)$ is helpful; I use this to analyze simulations\footnote{X. W. Zhou {\it et al.}, Phys. Rev. B \textbf{79}, 115201 (2009).}\footnote{Z. Liang {\it et al.}, J. Appl. Phys. \textbf{118}, 125104 (2015).} of GaN thermal conductivity. The Peierls-Boltzmann equation in relaxation time approximation gives a formula for $\kappa(q)$. Using a Debye model, explicit results $\kappa_p(q)$ are found for models where $1/\tau_Q\propto Q^p$. Numerics often gives exponents $p$ to be 2, 3, or 4. When $p=2$, $\kappa_2(q)\sim\kappa_0-C\sqrt{q}$. This shows that simulations on samples of size $L$ should be extrapolated by plotting $\kappa(L)$ {\it versus} $1/\sqrt{L}$. For exponent $p\ge 3$, $\kappa(q)$ diverges as $q\rightarrow 0$, which means that $\kappa(L)$ diverges as $L\rightarrow\infty$. An improved analysis is described, which uses Callaway's version of the relaxation time approximation, treating $N$ and $U$ processes separately.