Analysis of nonlocal phonon thermal conductivity simulations showing the ballistic to diffusive crossover

Simulations [e.g., X. W. 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 (j) over right arrow ((r) over right arrow) = -kappa(del) over right arrow T ((r) over right arrow). The heat current (j) over right arrow ((r) over right arrow) depends not just on the local temperature gradient (del) over right arrow T ((r) over right arrow) but also on temperatures at points (r) over right arrow. within phonon mean free paths, which can be micrometers long. This paper uses the Peierls-Boltzmann transport theory in nonlocal form to analyze the spatial variation Delta T((r) over right arrow). The relaxation-time approximation (RTA) is used because the full solution is very challenging. Improved methods of extrapolation to obtain the bulk thermal conductivity kappa proposed. Callaway invented an approximate method of correcting RTA for the (q) over right arrow (phonon wave vector or crystal momentum) conservation of N (Normal as opposed to Umklapp) anharmonic collisions. This method is generalized to the nonlocal case where kappa((k) over right arrow) depends on the wave vector of the current (j) over right arrow ((k) over right arrow) and temperature gradient i (k) over right arrow Delta T((k) over right arrow).