Role of Interlayer Coupling in the Second Harmonic Generation of Bilayer Transition-metal Dichalcogenides

Little is known about the role of weak interlayer coupling in the second harmonic generation (SHG) effects of two-dimensional van der Waals (vdW) systems. In this article, taking homo-bilayer $MoS_2/MoS_2$ and hetero-bilayer $MoS_2/MoSe_2$ as typical examples, we have systemically investigated their SHG susceptibilities as a function of interlayer hopping strength using first-principles calculations. For the SHG at zero frequency limit of both $MoS_2/MoS_2$ and $MoS_2/MoSe_2$, although the increase of tint can increase the intensities of interlayer optical transitions (IOT), the increased band repulsion around G point can eventually decrease their SHG values; the larger the tint, the smaller the SHG response. For the SHG spectra of $MoS_2/MoSe_2$ in the low photon-energy region, opposite to the $MoS_2/MoS_2$, their peak values are very sensitive to the variable tint, due to the strong $t_{int}$-dependent IOT dominating in the band edge; the larger the tint, the larger the SHG. For the SHG of $MoS_2/MoS_2$ in the high photon-energy region, comparing to the $MoS_2/MoSe_2$, their peak values will decrease in a much more noticeable way as the $t_{int}$ increases, due to the larger reduction of band nesting effect. Our study not only can successfully explain the puzzling experimental observations for the different SHG responses in different bilayer transition-metal dichalcogenides under variable tint, but also may provide a general understanding for designing controllable the SHG effects in the vdW systems.