Linear and nonlinear optical response based on the GW-Bethe-Salpeter and Kadanoff-Baym approaches for two-dimensional layered semiconductor materials

The family of 2D layered semiconductors, including transition metal chalcogenides (TMCs) of the form MX (M=Ga, In; X=S, Se, Te) and their Janus structures M$_2$XX' (X$\ne$X') exhibit exceptional nonlinear optical properties. The energetically most favorable crystal ordering for nonlinear response is the AB layer stacking, which breaks central inversion symmetry for an arbitrary number of layers, resulting in non-zero off-diagonal elements of the $\chi^{(2)}$ tensor for arbitrary thickness of the materials. Experimentally, the second harmonic generation (SHG) in GaSe is the strongest among all the 2D layered crystals. We perform first-principles calculations of bandstructures and linear and nonlinear optical responses of monolayer and bulk TMC crystals and their Janus structures based on $GW$-Bethe-Salpeter and Kadanoff-Baym approaches in and out of equilibrium, respectively, while taking band gap renormalization and excitonic effects into account. We provide detailed analysis of the linear and nonlinear optical selection rules by means of group and representation theory. In particular, we derive general formulas for the nonlinear optical response based on exciton states in semiconductor materials. We show that by choosing elements with larger mass and by breaking the symmetry of TMC monolayers using Janus structures it is possible to increase the nonlinear $\chi^{(2)}$ and $\chi^{(3)}$ response in 2D semiconductor materials.