We present a formulation for the nonlinear optical response in gapped graphene, where the low-energy single-particle spectrum is modeled by massive Dirac theory. As a representative example of the formulation presented here, we obtain a closed form formula for the third harmonic generation in gapped graphene. It turns out that the covariant form of the low-energy theory gives rise to peculiar logarithmic singularities in the nonlinear optical spectra. The universal functional dependence of the response function on dimensionless quantities indicates that the optical nonlinearity can be largely enhanced by tuning the gap to smaller values.
The two-sublattice nature of graphene lattice in conjunction with three-fold rotational symmetry, allows for the p-wave hybridization of the impurity state with the Bloch states of carbon atoms. Such an opportunity is not available in normal metals where the wave function is scalar. The p-wave hybridization function V(→k) appears when dealing with vacancies, substitutional adatoms and the hollow site impurities while the s-wave mixing on graphene lattice pertains only to the top site impurities. In this work, we compare the local moment formation in these two cases and find that the local moments formed by p-wave mixing compared to the s-wave one are robust against the changes in the parameters of the model. Furthermore, we investigate the stability of the local moments in the above cases. We find that the quantum fluctuations can destroy the local moments in the case of s-wave hybridization, while the local moments formed by p-wave hybridization survive the quantum fluctuations. Based on these findings, we propose vacancies, substitutional adatoms, and hollow site adatoms as possible candidates to produce stable local moments in graphene.
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