  The ability of graphene to support long-lived, electrically tunable plasmons
that interact strongly with light, combined with its highly nonlinear optical
response, has generated great expectations for application of the
atomically-thin material to nanophotonic devices. These expectations are mainly
reinforced by classical analyses performed using the response derived from
extended graphene, neglecting finite-size and nonlocal effects that become
important when the carbon layer is structured on the nanometer scale in actual
device designs. Here we show that finite-size effects produce large
contributions that increase the nonlinear response of nanostructured graphene
to significantly higher levels than those predicted by classical theories. We
base our analysis on a quantum-mechanical description of graphene using
tight-binding electronic states combined with the random-phase approximation.
While classical and quantum descriptions agree well for the linear response
when either the plasmon energy is below the Fermi energy or the size of the
structure exceeds a few tens of nanometers, this is not always the case for the
nonlinear response, and in particular, third-order Kerr-type nonlinearities are
generally underestimated by the classical theory. Our results reveal the
complex quantum nature of the optical response in nanostructured graphene,
while further supporting the exceptional potential of this material for
nonlinear nanophotonic devices.
