Monte Carlo simulated heat transport in semiconductor nanostructures

Abstract

Nanoscale energy transport is a topic of considerable interest as heat transport at these scales can no longer be accurately predicted by diffusion theory. An alternative approach is to use the Boltzmann transport equation, but this equation is challenging to solve in the case of phonon transport, and its exact resolution is currently one of the open research subjects in mathematics. A program has been developed to study nanoscale heat transport by solving the Boltzmann transport equation using two variations of a phonon Monte-Carlo method. The first variation is primarily derived from the works of Mazumber and Majumber (2001). The second variation follows the process of Peraud and Hadjiconstantinou (2012). While both variations follow methodology from existing works, the implementation details are unique. The simulation procedures differ from existing methods by incorporating a ‘system evolution’ algorithm that allows temperatures throughout the system to be periodically updated while simulating phonons one-by-one. The resulting software can rapidly simulate heat transport in relatively complex geometries. The Monte Carlo portion of the software is implemented using parallelized C++ code. Simulating phonons one-by-one makes the parallelization scheme natural and straightforward, although more sophisticated parallelization schemes may result in further computational speedup. The user input is a self-documenting JSON file generated via a Python script. The software is used to study thermal transport through various silicon and germanium nanostructures. Benchmark simulation testing shows that the temperature profiles produced by the simulations largely agree with analytical results and results from the literature, as does the predicted thermal conductivity. However, the thermal conductivity is quite sensitive to the relaxation rates that are used. While both variations of the phonon Monte Carlo method presented in this study strike a good balance between accuracy and efficiency and retain an intuitive connection to the problem physics, a noticeable difference in computational efficiency and precision is observed. With the exceptions of low-temperature ranges and possibly systems with extreme temperature differences, the second variation should be preferred when considering computational performance and precision.