Abstract
Three phonon scattering is the primary mechanism by which phonon transport is impeded in insulating and semiconducting bulk materials. Accurate computational modeling of this scattering mechanism is required for high fidelity simulations of thermal transport across the ballistic (quantum mechanics) to Fourier (continuum mechanics) range of behavior. Traditional Monte Carlo simulations of phonon transport use a scaling factor such that each scattering event is considered representative of a large number of phonons, often on the order of 104 physical phonons per simulated event. The ability to account for every phonon scattering event is desirable to enhance model fidelity. A physics-based model using time dependent perturbation theory (Fermi’s Golden Rule) is implemented to compute three phonon scattering rates for each permissible phonon interaction subject to selection rules. The strength of the interaction is based on use of a Gruneisen-like parameter. Both Type I and Type II scattering rates are computed for the allowable interactions that conserve energy and momentum (up to the addition of a reciprocal lattice vector) on a given discretization of momentum space. All of the phonons in the computational domain are represented and phonon populations are updated in momentum space and real space based on the computed number of phonons involved in given scattering events. The computational algorithm is tested in an adiabatic single cell of silicon of dimension 100 × 100 × 100 nm at a nominal temperature of 500 Kelvin containing approximately 108 fully anisotropic phonons. The results indicate that phonon populations return to equilibrium if artificially displaced from that condition. Two approaches are introduced to model the relaxation time of phonon states: the single mode relaxation time (SMRT) which is consistent with the underlying assumptions for previously reported theoretical estimates, and the multi model relaxation time (MMRT) which is more consistent with in-situ conditions. The trends meet physical expectations and are comparable to other literature results. In addition, an estimate of error associated with the relaxation times is presented using the statistical nature of the model. The three phonon scattering model presented provides a high fidelity representation of this physical process that improves the computational prediction of anisotropic phonon transport in the statistical phonon transport model.
