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Recently,thermal rectification and thermal non-equilibrium phenomena in nanoscale heat transfer have received considerable attention,and in this talk,two examples of our work will be presented: thermal rectification in asymmetric graphene nanoribbons; and thermal non-equilibrium among electrons,optical phonons,and acoustic phonons in Raman measurements of thermal conductivity of graphene.We show that thermal rectification(TR)in asymmetric graphene nanoribbons(GNRs)is originated from phonon confinement in the lateral dimension,which is a fundamentally new mechanism different from that in macroscopic heterojunctions.Our molecular dynamics simulations reveal that,though TR is significant in nanosized asymmetric GNRs,it diminishes at larger width.By solving the heat diffusion equation,we prove that TR is indeed absent in both the total heat transfer rate and local heat flux for bulk-size asymmetric single materials,regardless of the device geometry or the anisotropy of the thermal conductivity.We have performed phonon spectra analysis and shown that phonon lateral confinement can enable three possible mechanisms for TR: phonon spectra overlap,inseparable dependence of thermal conductivity on temperature and space,and phonon edge localization,which are essentially related to each other in a complicated manner.We show that other asymmetric nanostructures,such as asymmetric nanowires,thin films,and quantum dots,of a single material are potentially high-performance thermal rectifiers.We have also predicted the thermal non-equilibrium among electrons,optical phonons,and acoustic phonons in Raman spectroscopy,and investigated its impact on the thermal conductivity measurement of two-dimensional materials.Raman spectroscopy has been widely used to measure thermal conductivity(κ)of 2D materials such as graphene.In this method the temperature measured from Raman peaks is used in a Fourier model to derive the thermal conductivity κ,and a basic assumption is that different modes of phonons are in local thermal equilibrium.However,this assumption was not validated yet.In this work,we give comprehensive and predictive simulations of the intrinsic electron-phonon coupling processes and the resulting non-equilibrium in graphene,using first principles density functional theory(DFT)calculations.We calculated the electron cooling rate due to phonon scattering as a function of their corresponding temperatures(Te,Tph),and our results clearly illustrate that optical phonons dominate the hot electron relaxation process for Te >Tph ≥ 300 K.We then used these results in conjunction with the phonon scattering rates computed using perturbation theory to develop a multi-temperature model,and resolved the spatial temperature distributions of the energy carriers.Our results show that electrons,optical phonons,and acoustic phonons are in strong non-equilibrium,with the ZA phonons showing strongest non-equilibrium mainly due to their weak coupling to other carriers in suspended graphene.We estimate that neglecting this non-equilibrium leads to under prediction of thermal conductivity in experiments by a factor of 1.8-2.6 at room temperature.Such under-estimation is also expected in Raman measurements of thermal conductivity of other 2D materials.