1. Tunable excitons in gated graphene systems
Anahit Djotyan1, Artak Avetisyan1, and
Konstantinos Moulopoulos2
1Yerevan State University
2University of Cyprus
Saratov Fall Meeting September 27-30

2. MOTIVATION
Graphene is a unique bridge between condensed matter physics and relativistic quantum field theory and due to these properties is of great interest for nonlinear optical applications.
The coherent optical response of multilayer graphene systems to an intense laser radiation field may reveal many particle correlation effects. Excitons are expected to modify strongly the optical response. In monolayer graphene, since there is no energy gap, the Coulomb problem has no true bound states, but resonances [1]. A signature of the presence of the excitonic resonances was observed in its optical properties [2]. The optical response of graphene with an opened energy gap between the conduction and valence bands is dominated by bound excitons [3].
We developed a theoretical method for investigation of nonlinear optical properties and excitonic effects in gated monolayer and bilayer graphene systems. To describe the band structure of graphene systems we use a tight binding approach. In order to take into account the Coulomb interaction, we use second quantized Hamiltonian.
1. N. M. R. Peres, R. M. Ribeiro, and A. H. Castro Neto, arXiv: v2
2. Kin Fai Mak et al., Phys. Rev. Lett. 101, (2008).
3. C.-H. Park and S. G. Louie, Nano Lett. 10, 426 (2010).

3. Electronic structure of graphene monolayer and bilayer
Monolayer of graphene grown epitaxially on SiC has a band gap of about 0.2 eV
[1] S.Y. Zhou et al., Nature Mater. 6,
A perpendicular electric field applied to bilayer of graphene may open an energy gap between the conduction and valence bands , which is tunable by the gate voltage between zero and midinfrared energies.
[2] E. McCann et al., Solid State Com 143, 110 (2007)
[3] Eduardo V. Castro, K. S. Novoselov, et.al., PRL 99, (2007

4. Energy spectrum of tuned monolayer graphene
In the case of gapless monolayer of graphene:
In the presence of the gap induced by the interaction of the charge carriers with the substrate
we introduce the mass of the electron ( hole) by the expression
is the hopping parameter between A and B atoms in the same plane

5. Energy spectrum of bilayer graphene in perpendicular electric field
U is the gap induced by electric field
We introduce the mass of the electron ( hole) by the expression
is vertical interlayer hopping parameter,
The free solutions in bilayer graphene with an energy gap

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

7. Interaction of a strong electromagnetic wave with AB-stacked bilayer
Low-energy excitations
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

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

9. 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 we have found the expression

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

11. The Coulomb Hamiltonian

12. In order to investigate the 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 nonlinear equations in graphene bilayer in the presence of laser pulse
are functions of
and energy gap

13. Excitonic peak in absorption spectrum of gated graphene monolayer
Effective fine structure constant
Opened energy gap
Mass of the particle in the gated graphene
At photon energy
we obtain the excitonic peak with

14. Excitonic peaks in absorption spectrum in gated graphene bilayer: the dependence of the binding energy on the parameter
Exciton binding energy:
detuning

15. The binding energy of exciton in gated graphene bilayer
The excitonic peaks:
for
detuning
Opened energy gap

16. The dependence of excitonic binding energy in bilayer graphene on the mass of the electron ( on the hopping parameter )
The mass of the electron
The band parameters
can be changed by changing doping concentration of bilayer*
*Science 313, 951 (2006);
Taisuke Ohta, et al.

17. Conclusion
We consider the excitonic states in graphene multilayers with opened energy gap. To take into account the Coulomb interaction, we use Hartree- Fock approximation that leads to closed set of equations for the single-particle density matrix.
We have found the compact expression for Coulomb Hamiltonian in gated graphene systems.
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 theoretical results*.
The absorption spectra of gated bilayer is investigated on the basis of developed methods and using the expression for Coulomb Hamiltonian for different values of parameters and .
We found that due to relative flatness of the bottom (top) of conduction (valence) band in bilayer graphene systems in the presence of perpendicular electric field, the density of coherently created particle-hole pairs becomes quite large, which can make possible Bose-Einstein condensation of electron-hole pairs.
* S. H. Guo, X. L. Yang, F. T. Chan et al., Phys. Rev. A 43, 1197 (1991). *.