Transcript Презентация

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
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:1002.0464v2
2. Kin Fai Mak et al., Phys. Rev. Lett. 101, 196405 (2008).
3. C.-H. Park and S. G. Louie, Nano Lett. 10, 426 (2010).
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, 770 2007.
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, 16802 (2007
Energy spectrum of tuned monolayer graphene
In the case of gapless monolayer of graphene:
3
Ee / h    0
2
m0  0 ,
E  m02vF4  p 2 v 2F
3 0 a
6
vF 
 10 m/s
2
In the presence of the gap U induced by the interaction of the
charge carriers with the substrate
E U p v
2
2 2
F
we introduce the mass of the electron ( hole) by the expression
0
mvF2  U 2 4
is the hopping parameter between A and B atoms in
the same plane
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
1
is vertical interlayer hopping parameter,
m   1 / 2vF2
v F  106 m / s
v3  0.1vF
The free solutions in bilayer graphene with an energy gap
Induced gaps in bilayer graphene
Important to include
all the SWMcC parameters
 0 , 1 ,  2 , 3 , 4 , 5

only two
tight-binding parameters
 0  0,  1  0
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
Expanding the fermionic field operator
over the free wave function of bilayer
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
In Heisenberg representation operator evolution
The single-particle density matrix in momentum
space :
Evolution of the interband polarization
In Heisenberg representation
The Coulomb Hamiltonian
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
U
Ti ( Pk q , nk q , , U )
are functions of
 ( p,  )
and energy gap
U
Excitonic peak in absorption spectrum of gated
graphene monolayer
Effective fine
structure constant
e2

 0.175
vF 
R  me4 / 2
2
 2  2meV
Opened energy gap
U  500meV  261.1R*
Mass of the particle in
the gated graphene
  (  - U ) / R*
At photon energy
  257R*
2mvF2  U
E  U 2 / 4  p 2 vF2
we obtain the excitonic
peak with
Ebind  4.15R*
Excitonic peaks in absorption spectrum in gated graphene bilayer:
the dependence of the binding energy on the parameter v3
v3  0.1vF
 1  400 meV
Exciton binding energy:
EB  U -   / R*
m   1 / 2vF2
detuning
(  - U ) / R*
The binding energy of exciton in gated
graphene bilayer
The excitonic
peaks:
Ebind  2.15R*
for
v3  0.1vF
Ebind  1.75R* for v3  0
detuning
(  - U ) / R*
Opened energy gap U  100R
*
The dependence of excitonic binding energy in bilayer graphene on the
mass of the electron ( on the hopping parameter  1 )
U  250R *
The mass of the electron
m   1 / 2vF2
The band parameters  1
can be changed by changing
doping concentration of
bilayer*
  (  - U ) / R*
*Science 313, 951 (2006);
Taisuke Ohta, et al.
Conclusion
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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 v3 and  1.
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).