Ballistic effects on thermal conductivity in 1D and 2D configurations from single and multiple localized sub mean free path heat sources: a numerical investigation

International audienceThe effective thermal conductivity of materials decreases when their dimensions are comparable to orlower than the mean free path of heat carriers (phonons in dielectrics) and when the influence of the thermal boundaryconductance increases. Distinguishing the two phenomena is not experimentally straightforward, while the phononBoltzmann transport equation (BTE) in the Relaxation Time Approximation (RTA) can be used to investigate thisnumerically. We solve a particular form of the BTE under the RTA, the Equation of Phonon Radiative Transfer, withthe Discrete Ordinates Method (DOM) in one-dimensional and two-dimensional configurations for gray and non-graymedia. The simulations give access to local effective temperature and conductive heat flux fields. We consider inparticular the cross-plane and the in-plane thermal conductivities through 1D and 2D calculations, respectively.Diffuse, specular and diffuse-specular reflections of the heat carriers on the boundaries are implemented. The casesof extended and localized thermal sources are studied in 2D. Reduction factors associated to the effective thermalconductivities are obtained for the different cases and compared to the predictions based on the Fuchs-Sondheimer-Casimir-Ziman theory [1–4]. Constant temperature and adiabatic boundary conditions are also distinguished and theireffects on the thermal transport are highlighted. Collective effects due to multiple localized heat sources are alsoconsidered. The competition between different characteristic length-scales (heat source size, distance between heatsources, sample size) is investigated. Finally, we discuss the obtained results in light of the recent thermo-optical andelectrical experiments involving 2D heat spreading from sub-mean free path heat sources [5,6].References:[1] Fuchs K.., The conductivity of thin metallic films according to the electron theory of metals, Math. Proc. Camb.Philos. Soc. 34, 100 (1938)[2] Sondheimer E.H., The mean free path of electrons in metals, Adv. Phys. 1, 1 (1952)[3] Casimir H.B.G., Note on the conduction of heat in crystals, Physica 5, 495 (1938)[4] Ziman J.M., Electrons and Phonons (Clarendon Press), 1960[5] Hoogeboom-Pot K., et al., A new regime of nanoscale thermal transport: Collective diffusion increases dissipationefficiency, Proc. Natl. Acad. Sci. 112, 4846 (2015)[6] Hu Y., et al., Spectral mapping of thermal conductivity through nanoscale ballistic transport, Nat. Nanotechnol., 10 ,701 (2015