Simulations (e.g. Zhou et al., Phys. Rev. B 79, 115201 (2009)) show nonlocal effects of the ballistic/diffusive crossover. The local temperature has nonlinear spatial variation not contained in the local Fourier law $\vec{ j}(\vec{ r})=-\kappa\vec{ \nabla}T(\vec{ r})$. The heat current $\vec{ j}(\vec{ r})$ depends not just on the local temperature gradient $\vec{ \nabla}T(\vec{ r})$, but also on temperatures at points $\vec{ r}^{ \ \prime}$ within phonon mean free paths, which can be micrometers long. This paper uses the Peierls-Boltzmann transport theory in non-local form to analyze the spatial variation $\Delta T(\vec{ r})$. The relaxation-time approximation (RTA) is used because full solution is very challenging. Improved methods of extrapolation to obtain the bulk thermal conductivity $\kappa$ are proposed. Callaway invented an approximate method of correcting RTA for the $\vec{ q}$ (phonon wavevector or crystal momentum) conservation of N (normal as opposed to Umklapp) anharmonic collisions This method is generalized to the non-local case where $\kappa(\vec{ k})$ depends on wavevector of the current $\vec{ j}(\vec{ k})$ and temperature gradient $i\vec{ k}\Delta T(\vec{ k})$.
Categories: cond-mat.mtrl-sci