Heat conduction in crystalline semiconductor materials occurs by lattice vibrations that result in the propagation of quanta of energy called phonons. The Boltzmann Transport Equation (BTE), which describes the evolution of the number density (or energy) distribution for phonons as a result of transport (or drift) and inter-phonon collisions (or scattering), can predict heat conduction in crystalline solids over several orders of magnitude in length-scale, making it an attractive choice for calculation of heat transport in practical devices. Modeling non-equilibrium heat conduction using the BTE presents two main challenges. First, the physics of the phonons needs to be accurately incorporated in the numerical solution of the BTE. Second, the BTE, being a seven-dimensional equation, is a difficult equation to solve numerically.
In this work, the BTE has been solved using three different methods. First, a Monte-Carlo (MC) algorithm was developed and implemented for solution of the BTE for two- and three-dimensional geometries. Although not the primary focus of this study, a method to calculate three-phonon scattering time-scales for both acoustic and optical modes for silicon was developed. These time-scales represent the most important inputs for the solution of the BTE. The calculated time-scales were compared against results obtained using Molecular Dynamics (MD) calculations available in the literature, and were found to be in good agreement.
With this data, MC calculations were performed to predict thermal conductivity of silicon thin films and a good agreement with experimental results was obtained. Studies to elucidate the role of various phonon polarizations (modes) revealed that Transverse Acoustic (TA) phonons are the primary carriers of energy at low temperatures. At high temperatures (T > 200 K), Longitudinal Acoustic (LA) phonons carry more energy than TA phonons. Optical phonons, when included, were found to carry about 10-25% of the energy at room temperature for silicon thin films.
Although the MC method is very attractive for modeling sub-micron heat conduction, it suffers from being prohibitively slow. Its use is further limited by the fact that it is susceptible to statistical errors. This prompted the development of deterministic solutions of the BTE.
Deterministic solution of the BTE was first obtained using the Discrete Ordinate Method (DOM) based on the SN approximation. Since, DOM is susceptible to the so called “ray effects,” the Control Angle DOM (CADOM) method was also implemented. In an effort to reduce the cost of simulation, the so-called “Ballistic-Diffusive” approximation to the BTE was formulated and numerical procedure to solve this equation was developed. The resulting method, referred to as the Hybrid SN-PN method, along with DOM and CADOM, was first compared against benchmark MC method for the solution of the “gray” BTE (i.e., BTE without phonon dispersion and polarization) for a two-dimensional transient heat conduction problem at various Knudsen numbers. Subsequently, the deterministic methods were explored for large-scale two- and three-dimensional geometries. It was found that the Hybrid SN-PN method is accurate at all Knudsen numbers. From an efficiency standpoint, the hybrid method was found to be superior to direct solution of the BTE using DOM or CADOM both for steady state as well as for unsteady non-equilibrium heat conduction calculations with the computational gains increasing with increase in problem size.
Finally, dispersion and polarization (or non-gray) effects were incorporated in the numerical solution of the BTE. The non-gray BTE was solved using the CADOM technique and thermal conductivity of silicon thin films was predicted. The predicted thermal conductivity was found to be in good agreement with the Monte-Carlo solutions and experimental data. The numerical solution of the non-gray BTE was then demonstrated for complex two- and three-dimensional geometries.
This dissertation highlights the role of the Boltzmann Transport Equation as a viable computational tool to model non-equilibrium heat conduction in complex three-dimensional geometries, and represents the first step towards modeling thermal transport in real-life semiconductor devices.