Role of Phonon Dispersion in Lattice Thermal Conductivity Modeling role of phonon dispersion in the prediction of the thermal conductivity of germanium

between temperatures of 2 K and 1000 K is investigated using the Holland approach. If no dispersion is assumed, a large, nonphysical discontinuity is found in the transverse pho- non relaxation time over the entire temperature range. However, this effect is masked in the final prediction of the thermal conductivity by the use of fitting parameters. As the treatment of the dispersion is refined, the magnitude of the discontinuity is reduced. At the same time, discrepancies between the high temperature predictions and experimental data become apparent, indicating that the assumed heat transfer mechanisms (i.e., the relax- ation time models) are not sufficient to account for the expected thermal transport. Mo- lecular dynamics simulations may be the most suitable tool available for addressing this issue. @DOI: 10.1115/1.1723469# The thermal conductivity of dielectric crystals ~where phonons dominate the thermal transport! has been theoretically investi- gated since the works of Debye @1# and Peierls @2#. The exact or even numerical prediction of the thermal conductivity is a formi- dable task. This is due to the complexity of the Boltzmann trans- port equation ~BTE! , which is the basis for most solution tech- niques. Analytical thermal conductivity models have been developed based on the single mode relaxation time ~SMRT! ap- proximation in the BTE with different degrees of complexity @3,4#. In SMRT models, a single relaxation time is associated with each phonon mode, and describes how that mode would respond if excited while all others remain at equilibrium. The temperature and frequency dependencies of the relaxation times are assumed, and the predictions must be fit to the experimental data. Callaway @3# developed an SMRT model that can successfully predict the low temperature thermal conductivity of germanium. This approach uses the Debye approximation, which assumes that there is no phonon dispersion and that the longitudinal and trans- verse polarizations behave identically. Holland @4# extended the work of Callaway by separating the contributions of longitudinal acoustic ~LA! and transverse acoustic ~TA! phonons, including some phonon dispersion, and using different forms of the relax- ation times. Better high temperature agreement was found for ger- manium than with the Callaway model. The Holland model has been refined to include further detail on the phonon dispersion and relaxation times @5-7#. The added complexities lead to more fitted parameters. One could argue that the resulting better fits with the experimental data are due to this increase in the number of fitted parameters, and not to an improvement of the actual physical model. This study examines the impact of refining the phonon disper- sion model beyond the Debye approximation for germanium using an SMRT approach. Consideration is given to the distinction be- tween the group and phase velocities, their frequency dependen- cies, and the rigorous treatment of impurity scattering. In the con- text of the improved dispersion model, some issues with the Holland relaxation time model are addressed. 2 Thermal Conductivity Model By solving the BTE and using the Fourier law of conduction, the thermal conductivity of germanium, assuming an isotropic crystal, and neglecting the contributions of optical phonons, has been predicted to be of the form @4#