Heat transport in silicon from first-principles calculations

Abstract

Using harmonic and anharmonic force constants extracted from density-functional calculations within a supercell, we have developed a relatively simple but general method to compute thermodynamic and thermal properties of any crystal. First, from the harmonic, cubic, and quartic force constants we construct a force field for molecular dynamics (MD). It is exact in the limit of small atomic displacements and thus does not suffer from inaccuracies inherent in semi-empirical potentials such as Stillinger-Weber's. By using the Green-Kubo (GK) formula and molecular dynamics simulations, we extract the bulk thermal conductivity. This method is accurate at high temperatures where three-phonon processes need to be included to higher orders, but may suffer from size scaling issues. Next, we use perturbation theory (Fermi Golden rule) to extract the phonon lifetimes and compute the thermal conductivity κ from the relaxation time approximation. This method is valid at most temperatures, but will overestimate κ at very high temperatures, where higher order processes neglected in our calculations, also contribute. As a test, these methods are applied to bulk crystalline silicon, and the results are compared and differences discussed in more detail. The presented methodology paves the way for a systematic approach to model heat transport in solids using multiscale modeling, in which the relaxation time due to anharmonic 3-phonon processes is calculated quantitatively, in addition to the usual harmonic properties such as phonon frequencies and group velocities. It also allows the construction of accurate bulk interatomic potentials database.

Thermal transport in phononic crystals and the observation of coherent phonon scattering at room temperature

“…It is worth noting here that this method is quite accurate for the acoustic phonon branches but to a lesser degree for the optical branches. However, the latter are estimated to contribute at most about 3.5% of the overall k in bulk Si before any reduction 16,17 due to patterning. While our treatment clearly overestimates the reduction in k due to the PnC patterning and exaggerates the dispersion effects of the PnCs on the acoustic branches, it is worth noting the reversal of the trend in k PnC /k Slab versus f (and n), consistent with our expectations based on Fig.…”

“…Since it is reasonable to expect a phonon to remain coherent over its MFP, one suggestion is to use the MFP to discern the scattering mechanism. We thus propose a model that incorporates frequency-dependent phonon MFPs in Si 16,17 and introduce the concept of a threshold-MFP (TMFP) that is essentially a measure of the phonon minimum coherence length. Phonons possessing a MFP above the designated TMFP value are able to sample the periodicity of the lattice and undergo coherent boundary scattering by the PnC, thus exhibiting a modified Si dispersion due to the PnC lattice, while phonons possessing a MFP below it do not 'see' the periodicity and thus undergo only incoherent boundary scattering and follow the unmodified Si dispersion.…”

“…According to ref. 16, which uses a density functional theory approach to study heat transport in Si, it is clear that there exist certain heavily-populated phonon modes that dominate over other less-populated ones 16 . More specifically, ref.…”

“…More specifically, ref. 16 identifies several distinct MFP bands that result collectively in a nonlinear increase in the accumulation function; a range beyond 10 mm, and two bands starting around 7.9 and 4.3 mm that are of particular interest. Examining our reported TMFP-SS values, it is seen that the TMFP-SS for the 1 Â 1, 2 Â 2 and 3 Â 3 supercells are about or o4.3 mm, while both the SC and 4 Â 4 supercells are found to be much larger than that value (approaching 7 mm).…”

“…In particular, Hao et al 12 , used Monte Carlo simulations combined with frequency-dependent phonon MFPs 16 to study phonon transport in a 2D square-lattice PnC, and concluded that phonon size effects caused by the periodic arrangement of the holes can be remarkable even when the hole size and spacing are much greater than the average phonon MFP. Tang et al 6 suggested the possibility of a 'necking effect' for phonons with MFPs longer than the smallest dimension of the lattice.…”