The efficient method of lattice dynamics calculation: Monte Carlo integration with importance sampling

In this study, we aimed to accelerate the thermal conductivity calculations of crystalline nanostructures using anharmonic lattice dynamics. For this we implemented Monte Carlo (MC) integration for relaxation time calculations and achieved a dramatic acceleration of approximately two orders of magnitude. The relaxation times can be calculated by computing the scattering rates for all phonon combinations; however, in this MC integration, we instead calculated the scattering rates of a randomly sampled subset of combinations. Then, we estimated the overall scattering rate. Simple MC integration samples the scattering channels that do not contribute to the total scattering rate, leading to inefficiencies. To address this issue, we implemented an importance sampling method (ISM) for improving sampling efficiency. In this study, we compared the computational speeds of both methods and investigated the differences in accuracy by comparing the results with the exact values obtained from traditional relaxation time calculations. The comparison showed similarity between both methods in terms of speed; however, ISM was faster when the error margin was ∼5%. Furthermore, while simple MC integration risks significantly worse accuracy as the system size increases, the ISM remains relatively robust and reliable.