A multi $k$-point nonadiabatic molecular dynamics for periodic systems

With the rapid development of ultra-fast experimental techniques used for carrier dynamics in solid-state systems, a microscopic understanding of the related phenomena, particularly a first-principle calculation is highly desirable. Non-adiabatic molecular dynamics (NAMD) offers a real-time direct simulation of the carrier transfer or carrier thermalization. However, when applied to a periodic supercell, due to the $\Gamma$-point phonon movement during the molecular dynamics, there is no supercell electronic $k$-point crossing during the NAMD simulation. This often leads to a significant underestimation of the transition rate due to significant energy gaps in the single supercell $k$-point band structure. In this work, based on the surface hopping scheme used for NAMD, we propose a practical method to enable the cross-$k$ transition for a periodic system. We demonstrate our formalism by showing that the hot electron thermalization process by the multi $k$-point NAMD in a small supercell is equivalent to such simulation in a large supercell with single $\Gamma$ $k$-point. The hot carrier thermalization process in the bulk silicon is also carried out and compared with the recent ultra-fast experiments with excellent agreements.