Solution of the Phonon Boltzmann Transport Equation Employing Rigorous Implementation of Phonon Conservation Rules

A finite volume scheme is developed to solve the phonon Boltzmann transport equation in an energy form accounting for phonon dispersion and polarization. The physical space and the first Brillouin zone are discretized into finite volumes and the phonon BTE is integrated over them. Second-order accurate differencing schemes are used for the discretization. The scattering term employs a rigorous implementation of phonon momentum and energy conservation laws in determining the rate of normal and Umklapp processes. The method is applied to a variety of bulk silicon and silicon thin-film conduction problems and shown to perform satisfactorily. NOMENCLATURE a Lattice constant, m A Amplitude of the wave, m b Reciprocal wave vector, 1/m CP Propagating mode specific heat D Density of states e" Phonon energy density, J/m-sr G Number of atoms per unit volume K Phonon wave vector, 1/m k Thermal conductivity, W/mK kB Boltzmann’s constant, J/K M Atomic mass, kg N Phonon distribution function r Position vector, m t Time, s T Temperature, K v Speed of sound, m/s vg Group velocity of phonons, m/s ħ Reduced Plank’s constant, Js η Number of polarizations Ω Solid angle, sr τ Relaxation time, s ω Angular frequency of phonons, 1/s γ Gruneisen parameter ρ Density, kg/m θ Polar angle, rad φ Azimuthal angle, rad INTRODUCTION In recent years, the aggressive scaling trends of modern microelectronic devices have resulted in increased power dissipation and thermal failures. Accordingly, there has been increasing interest in modeling sub-micron thermal transport in transistors, and therefore in semi-conductors and dielectrics. When the mean free path of phonons is much longer than the domain length scale, Fourier conduction no longer obtains. The phonon Boltzmann transport equation (BTE) has been used to make thermal predictions at the sub-micron scale [1-7]. However, nearly all implementations presented thus far make substantial simplifications, both to the polarization and dispersion behavior of phonons, and to the representation of phonon-phonon scattering. It is unclear what the influence of these approximations is when phonon confinement effects are important or when phonon groups are strongly out of equilibrium with each other, as in modern ultra-scaled MOSFETs. Among the earliest numerical simulations of the phonon BTE were those published by Majumdar and co-workers [1][2] who employed a discrete ordinates-technique. Frequencydependence was considered but the scattering terms employed a relaxation time approximation with the frequency-dependent relaxation times being derived to fit bulk thermal conductivity. Ju et al. [3][4] and Sverdrup [5] employed a two-fluid BTE model for heat conduction. The two-fluid model [8] divides phonons into reservoir and propagation modes. The reservoir mode phonons are regarded as capacitive because of their small group velocities, while the propagation mode phonons account for the entire heat transport. The propagation mode employs a single phonon group velocity and a single effective relaxation time is used for energy exchange between reservoir and propagating modes. Under this model, the bulk thermal conductivity is given in terms of the propagating mode 2 1 3 P P P k C v τ = (1) The relaxation time τP in [5] is extracted from the bulk thermal conductivity using the heat capacity and group velocity of longitudinal acoustic phonons, which are considered to be in the propagation mode. In contrast, the transverse acoustic and optical phonons are lumped together in the reservoir mode, and