Monte Carlo simulations of phonon transport in silicon

In this paper, the development of a computational procedure to simulate thermal transport in small semiconductor structures was described. On a microscopic scale, heat transport can be described mathematically using a Boltzmann equation for phonons. Direct numerical solution of this equation is difficult, without extensive approximation, because of the quantity and complexity of the anharmonic phonon-phonon interactions. Therefore, Monte Carlo simulation approach was developed, analogous to that used for electron transport modelling, which models phonon trajectories and phonon scattering events. The simulation domain is subdivided into cells, and a discretized phonon distribution is monitored in every cell. Anharmonic three-phonon processes (of both ´absorption´ and ´emission´ type) are simulated for acoustic phonon modes in silicon. For phonon-phonon absorption, a ´partner´ phonon must be selected from within the same real space cell to participate in the interaction, in a similar manner to the algorithms used in Monte Carlo simulations of electron-electron scattering. An important difference between electron and phonon transport simulations is the necessity, in the latter case, of simulating Umklapp processes, since these are essential in defining the thermal conductivity. Whereas in previous derivations of analytical approximations for phonon lifetimes and thermal conductivities, it has been very difficult to determine the relative contribution of normal and Umklapp processes in phonon-phonon interactions, in principle, this information can be extracted directly from a Monte Carlo simulation.