Theoretical calculations of nonlinear refraction and absorption coefficients of doped graphene

In this study, we present the first theoretical predictions concerning the nonlinear refractive and absorptive properties of the doped graphene in which the Fermi energy E-F of charge carriers (noninteracting massless Dirac fermions) is controlled by an external gate voltage. We base our study on the original perturbation theory technique developed by Genkin and Mednis (1968 Sov. Phys. JETP 27 609) for calculating the nonlinear-optical (NLO) response coefficients of bulk crystalline semiconductors with partially filled bands. Using a simple tight-binding model for the pi-electron energy bands of graphene, we obtain analytic expressions for the nonlinear refractive index n(2) (omega) and the nonlinear absorption coefficient alpha(2) (omega) of the doped graphene at photon energies above twice the value of the Fermi energy (h omega > 2E(F)). We show that in this spectral region, both the nonlinear refraction ant the nonlinear absorption are determined predominantly by the combined processes which simultaneously involve intraband and interband motion of pi-electrons. Our calculations indicate that extremely large negative values of n(2) (of the order of -10(-6) cm(2)W(-1)) can be achieved in the graphene at a relatively low doping level (of about 10(12) cm(-2)) provided that the excitation frequency slightly exceeds the threshold frequency corresponding to the onset of interband transitions. With a further increase of the radiation frequency, the n(2) (omega) becomes positive and begins to decrease in its absolute magnitude. The peculiar frequency dispersion of n(2) and a negative sign of the alpha(2) (absorption bleaching), as predicted by our theory, suggest that the doped graphene is a prospective NLO material to be used in practical optical switching applications.