Spectral characteristics of the sum-frequency generation from atomically thin hexagonal crystals lacking center-of-inversion symmetry

We calculate the second-order nonlinear optical susceptibility ${\chi }^{\left(2\right)}(-{\omega }_{\sigma };{\omega }_1,{\omega }_2)$ for the sum-frequency generation (SFG) from two-dimensional (2D) hexagonal crystals with broken space inversion symmetry, such as graphene, grown epitaxially on a SiC substrate, and boronitrene (a single atomic layer of hexagonal boron nitride (h-BN)). We show that in the presence of two normally incident pump beams at frequencies ${\omega }_1$ and ${\omega }_2$, a remarkably large output signal at frequency ${\omega }_{\sigma }={\omega }_1+{\omega }_2$ is generated in those materials in the two-photon resonant regime where the frequencies ${\omega }_1$ and ${\omega }_2$ are matched to provide a resonance for the radiated photon energy $\hslash {\omega }_{\sigma }$ with the bandgap energy E_g of the materials. In addition, it is found that the absolute magnitude of the ${\chi }^{\left(2\right)}$ is resonantly enhanced when the photon energy of one of the input beams is tuned to the fundamental absorption edge. These resonant behaviors lead to very large values of $|{\chi }^{\left(2\right)}|$, reaching the order of 10⁻⁴ esu for the graphene/SiC overlayer system and 10⁻⁶ esu for monolayer h-BN. Furthermore, even outside the resonances, the magnitude of $|{\chi }^{\left(2\right)}|$ is fairly large ($\sim {10}^{-5}$ esu for the overlying graphene and 10⁻⁸ esu for boronitrene), which opens up new opportunities for practical exploitation of the SFG effect in the examined 2D crystals. (paper)