The thermal conductivity is a system property rather than a material property alone, i.e. the mechanisms that govern the thermal conductivity strongly depend on the system (e.g. length scale, phonon confinement, ballistic interface and boundary effects). A specific model, then, capable to account for the different mechanisms should be used. Among the different models to study thermal transport, the most standard one is based on the Fourier’s law, which can be implemented either in the framework of classical or ab-initio molecular dynamics (MD) simulations. MD builds on classical Newton’s equation of motion and, therefore, does not account for nuclear quantum effects that are present below the Debye temperature . Furthermore, ab-inito and MD simulations may feature strong size effects due to the limited size of the computational domain. Fourier’s law based models, thus, do not properly account for the ballistic, wave like, and quantum phenomena of heat transport.
At a length scale, L, of the order of the phonons mean free path, λ, a statistical description of the phonon transport using the Boltzmann transport equation is required (e.g. 50% of thermal conductivity in bulk silicon at RT comes from phonons having mean free paths > 500 nm ). This, however, is very demanding from a computational point of view. Even new developments (e.g. energy based and deviational formalism ) in stochastic Monte Carlo methods, are still too expensive to simulate 3D device problems on standard workstations. Deterministic methods for solving the phonon Boltzmann transport equation, on the other hand, suffer from ray effect and / or false scattering . These effects strongly limit their accuracy in the high Knudsen number regime (i.e. λ<< L).
Here, we present a set of improvements on the lattice Boltzmann method (LBM), in order to extend its applicability far into the ballistic regime. First of all, we get rid of the numerical anisotropy by allowing an adaptive time step. This allows to go beyond the grey approximation (i.e. average phonon properties), and to introduce several phonon species with different group velocities on one grid. Secondly, we go beyond the next neighbour hopping for the LBM stencil. This allows to increase the number of discretized directions (e.g. to reduce the ray effect) without introducing false scattering. The new numerical algorithms are applied to standard 2D transient and stationary phonon heat transport cases, and are benchmarked against analytical and numerical solutions given by the discrete ordinate method and MC simulations.
 O.N. Bedoya-Martínez, J.-L. Barrat, D. Rodney, Phys. Rev. B 89 (2014).
 A.S. Henry, G. Chen, J. Comput. Theor. Nanosci. (2008) 1–12.
 J.-P.M. Péraud, N.G. Hadjiconstantinou, Phys. Rev. B 84 (2011) 107.
 A. Nabovati, D.P. Sellan, C.H. Amon, J. Comput. Phys. 230 (2011) 5864–5876.