Efficient simulation of multidimensional phonon transport using energy-based variance-reduced Monte Carlo formulations

We present a Monte Carlo method for obtaining solutions of the Boltzmann equation to describe phonon transport in micro-and nanoscale devices. The proposed method can resolve arbitrarily small signals (e. g., temperature differences) at small constant cost and thus represents a considerable improvement compared to traditional Monte Carlo methods, whose cost increases quadratically with decreasing signal. This is achieved via a control-variate variance-reduction formulation in which the stochastic particle description solves only for the deviation from a nearby equilibrium, while the latter is described analytically. We also show that simulation of an energy-based Boltzmann equation results in an algorithm that lends itself naturally to exact energy conservation, thereby considerably improving the simulation fidelity. Simulations using the proposed method are used to investigate the effect of porosity on the effective thermal conductivity of silicon. We also present simulations of a recently developed thermal conductivity spectroscopy process. The latter simulations demonstrate how the computational gains introduced by the proposed method enable the simulation of otherwise intractable multiscale phenomena.