  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 materials
promising elements for nano-opto-electronics. Although these materials are
centrosymmetric, the spatial dispersion turns out to be quite large allowing
the second-order nonlinear response of such materials to be comparable to the
non-centrosymmetric 2D ones. The second-order response of massless Dirac
fermions has been extensively studied, however a general approach correct over
the full Brillouin zone is lacking so far. To complete this gap, in the current
paper we develop a general quantum-mechanical theory of the in-plane
second-order nonlinear response beyond the Dirac cone approximation and
applicable to the full Brillouin zone of the hexagonal tight-binding
nanostructures. We present explicit calculation of the nonlinear susceptibility
tensor of 2D hexagonal nanostructures applicable to arbitrary three-wave mixing
processes.
