•	Nanooptics and photonics

Anomalous nonlinear electromagnetic response of few-layer graphenes
Fil D.V.1,2
1 Institute for Single Crystals of the NAS of Ukraine. 60 Nauky Av., Kharkiv 61001, Ukraine. E-mail: fil@isc.kharkov.ua
2 V.N.Karazin Kharkiv National University, 4 Svobody Sq., Kharkiv 61022, Ukraine.

The results of theoretical study of nonlinear electromagnetic properties of AB-stacked bilayer, ABC-stacked three-layer and ABCA-stacked four-layer graphene (N=2,3,4) are presented and compared with ones of monolayer graphene (N=1). In the bight-binding approximation the low-energy electron spectrum of the N-layer graphene has the form , where is the Fermi velocity in the monolayer graphene, and is the interlayer tunneling amplitude. Non-quadratic dependence of  on the momentum p results in a strong nonlinear response of N=1,3,4 graphenes. The nonlinear response is computed using the quasiclassical approach based on the exact solution of the Boltzmann kinetic equation in the relaxation time approximation, and the quantum approach based on the perturbation solution of the equation of motion for the density matrix.
The quasiclassical approach predicts strong dependence of nonlinear response of graphene on the number of layers N. For N=1 graphene one obtains the effects of saturable absorption and power induced transparency, the response of N=2 graphene does not contain any nonlinear term, the N=3 and N=4 graphenes demonstrate opposite to the N=1 graphene behavior: the absorption increases and the transparency decreases at high intensity of the incident wave radiation. It is shown that the intensity of the third harmonic generation in a few-layer graphene can be smaller as well as larger than of the monolayer graphene depending on the ratio of the Fermi energy to the amplitude .
The quantum calculations show that the quasiclassical approach is applicable if the incident wave frequency is much smaller than the Fermi energy. At larger frequencies monolayer, bilayer and N=3,4 graphenes have similar nonlinear properties. 
The publication is based on the research provided by the grant support of the State Fund For Fundamental Research (project No 33683).