New understandings of third harmonic generation coefficient of bulk silicon: Implementation and application of the full ab initio band structure sum-over-states method

We make a full ab initio band structure analysis of interband and intraband contributions for the third-order nonlinear optical susceptibilities of bulk silicon by implementing the Aversa and Sipe sum-over-states formulism. The band structure and momentum matrix elements were calculated by using the highly accurate all-electron full potential linearized augmented plane wave method within the local density approximation. The convergence tests including the scissor correction with different k-points meshes and empty states were performed. Both real and imaginary parts of susceptibility were directly calculated and checked by the Kramers–Kronig relation. The converged results are compared with other theoretical and experimental ones and in agreement with the recent ab initio real-time-based calculation. The nonlinear optical coefficient comes from three parts: the pure interband contribution (P_inter), the modulation of interband terms by intraband terms (P_mod), and the intraband contribution (J_intra). For each part, the origin of enhanced peaks is explored by tracing the sum-over-states process. The interband contribution is found to be dramatically modulated by the intraband contribution.