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The phonon Boltzmann transport equation (BTE) has been widely utilized tostudy thermal transport in solids. While for a number of materials the exactsolution to the BTE has been obtained for a uniform heat flow, problems arisingin micro/nanoscale heat transport have been analyzed within the relaxation timeapproximation (RTA). Since the RTA breaks down at temperatures low compared tothe Debye temperature, this approximation prevents the study of an importantclass of high Debye temperature materials such as diamond, graphite, grapheneand some other 2D materials. We present a full scattering matrix formalism thatgoes beyond the RTA approximation and obtain a Green's function solution forthe linearized BTE, which leads to an explicit expression for the phonondistribution and temperature field produced by an arbitrary spatio-temporaldistribution of heat sources in an unbounded medium. The presented formalism iscapable of describing a wide range of phenomena, from heat dissipation bynanoscale hot spots to the propagation of second sound waves. We providenumerical results for graphene for a spatially sinusoidal heating profile anddiscuss the importance of using the full scattering matrix compared to the RTA.

Green's Functions of the Boltzmann Transport Equation with the Full  Scattering Matrix for Phonon Nanoscale Transport beyond the Relaxation Time  Approximation