1. Nonlinear response of gated graphene in a strong radiation field
Anahit Djotyan and Artak Avetisyan
Yerevan State University
Saratov Fall Meeting September 22-25

2. Electronic structure of graphene
 Massless relativistic
Dirac particles
 Zero gap semiconductor
Graphene grown epitaxially on SiC
has a band gap of about 0.2 eV
[1] S.Y. Zhou et al., Nature Mater. 6,
In bilayer graphene a gap can be tuned by gates [2,3]
[2] E. McCann et al., Solid State Com 143, 110 (2007)
[3] Eduardo V. Castro, K. S. Novoselov, et.al., PRL 99, (2007

3. only two tight-binding parameters
Induced gaps in bilayer graphene
Important to include all the SWMcC parameters
only two tight-binding parameters

4. Interaction of a strong electromagnetic wave with AB-stacked bilayer
Low-energy excitations
vertical interlayer hopping
can be described with 2 by 2 Hamiltonian.
the interaction Hamiltonian between a laser field and bilayer
the laser pulse propagates in the perpendicular direction to graphene plane (XY)
and the electric field of pulse lies in the graphene plane

5. The free solutions in bilayer graphene with an energy gap
Energy spectrum of bilayer graphene
The free solutions in bilayer graphene with an energy gap

6. Expanding the fermionic field operator
over the free wave function of bilayer

7. Laser interaction with bilayer graphene with opened energy gap
The dipole matrix element for the light interaction
with the bilayer depends on electron momentum
In cartesian coordinates, the dipole matrix element

8. In Heisenberg representation operator evolution
The single-particle density matrix in momentum space :
Evolution of the
interband polarization
Heisenberg representation

9. Total Coulomb Hamiltonian

10. The final form for the nonlinear equations
For excitonic effects we apply the Random Phase approximation (RPA) to the many particle system.
We express four field operator averages as products of polarization and population
The final form for the nonlinear equations
in graphene bilayer in the presence of laser pulse:
where
are functions of
and energy gap

12. Larger value of the frequency the red triangle is larger

14. Excitonic absorption in gated graphene monolayer
Fine structure constant
Opened energy gap
Mass of the particle in the gated graphene is defined as
At photon energy
we obtain the exciton peak with

15. Conclusion
We found that changing the energy gap linearly on time, the electron population can be transferred from the top of valence band to the bottom of conduction one after the time t_(final), when the gap comes into resonance with the electromagnetic field.
This is an alternative and more suitable way to bring the system into the resonance in comparison with the method of frequency chirped pulse.
We see that for larger value of the frequency, the red triangle is larger in a figure, and correspondingly larger number of electrons are transferred to the conduction band. It is connected with the fact that for larger value of the gap the bilayer has more flat bands near the Fermi level. Estimations show that the density of electrons in the triangle is about 10¹¹sm-².
For monolayer graphene, the excitonic peak in the light absorption is obtained at the energy about 4.15R* in linear regime, which is in good agreement with experimental results.
The nonlinear optical spectra for monolayer and multilayers of graphene will be investigated on the basis of developed methods.
The high density of excitons can lead to Bose condensation phenomenon.