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Single layers of hexagonal two-dimensional nanostructures such as graphene,silicene, and germanene exhibit large carrier Fermi velocities and,consequently, large light-matter coupling strength making these materialspromising elements for nano-opto-electronics. Although these materials arecentrosymmetric, the spatial dispersion turns out to be quite large allowingthe second-order nonlinear response of such materials to be comparable to thenon-centrosymmetric 2D ones. The second-order response of massless Diracfermions has been extensively studied, however a general approach correct overthe full Brillouin zone is lacking so far. To complete this gap, in the currentpaper we develop a general quantum-mechanical theory of the in-planesecond-order nonlinear response beyond the Dirac cone approximation andapplicable to the full Brillouin zone of the hexagonal tight-bindingnanostructures. We present explicit calculation of the nonlinear susceptibilitytensor of 2D hexagonal nanostructures applicable to arbitrary three-wave mixingprocesses.

Microscopic quantum description of second-order nonlinearities in 2D hexagonal nanostructures beyond the Dirac cone approximation