Thermal conductivity modeling using the Debye approximation can significantly reduce the complexity involved in the evaluation of thermal conductivity integral. Nevertheless, there are several compelling reasons (e.g., inconsistency between the estimations of the density of states used for heat capacity and thermal conductivity) that motivate more detailed modeling of the phonon dispersion in silicon or other diamond-like dielectrics. This manuscript presents closed-form expressions for the dispersion of the longitudinal and transverse modes in diamond-like dielectrics, which are then used to estimate the isotopic scattering rate, phonon spectrum and specific heat. The combinations of the above parameters are then used to predict the thermal conductivity of the bulk silicon and germanium with orders of magnitude variation in the mass fluctuation of isotopes and for the temperature range 2 to 1000 K. While this approach offers an elegant and consistent account of the phonon dispersion in diamond-like materials, however, provides no additional insight into the relative contributions of the longitudinal and transverse phonons to the heat transport.
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