### 0510 TStauber

```Plasmonics in double-layer
graphene
Tobias Stauber and Guillermo Gómez-Santos
Graphene Nanophotonics
Benasque, 5th March 2013
Overview
Optical properties double-layer graphene
Effect of temperature and inhomogeneous dielectric
background on Plasmons
Near-field amplification
Perfect transmission
Optical properties of twisted bilayer graphene
(Work in progress with L. Brey, P. San Jose, E. Prada)
Drude weight
Plasmons excitations
Plasmons in double-layer
graphene
Double-layer graphene
Coulomb drag, field effect tunneling transistor,
and optical modulator.
S. Kim, et. al., Phys. Rev.
B 83, 161401(R) (2011).
L. A. Ponomarenko et. al., Nature Physics 7, 958 (2011).
L. Britnell et. al., Science 335 (6071) 947-950 (2012)
Ming Liu et al., Nano
Lett. 12, 1482 (2012).
Johan Christensen et
al, ACS Nano 2011
Double-layer graphene
Linear response in
matrix form:
Define loss function:
 j1    11
   
 j 2    21
 12   A1ext 
 22
  ext 
A 
 2 
  11
 ( q ,  )   Im Tr 
  21
 12 
 22


Previous approaches
Often, the dielectric
function is discussed:
 1  v 0  10
 ( q ,  )  det  0 11 0
 v 
1
21

The loss function is given by:
 ( q ,  )   Im


0
0 
1  v 22  2 
v 12  2
0
0
1
 (q,  )
Problems:
•This function changes sign, because it is
not based on a true response function .
• The absolute value gives incorrect weight
for Landau damping regime.
Results for the loss function at
finite temperature
Plasmons at finite temperature
The plasmon dispersion is red-shifted for intermediate
temperatures and blue-shifted for high temperatures.
T  TF / 4
T  TF
TS and G. Gómez-Santos, New J. Phys. 14, 105018 (2012).
Plasmons at zero doping
There are plasmons at zero doping at T=300K:
kT 
kT
v
2 ln 2   T  0 . 035 eV
TS and G. Gómez-Santos, New J. Phys. 14, 105018 (2012).
Inhomogeneous dielectric medium
An inhomogeneous
dielectric medium can
shift relative weight
of in-phase and out-ofphase plasmons.
Topological insulators have
high-dielectric buffer layer:
TS and G. Gómez-Santos, New J. Phys. 14, 105018 (2012).
Acoustic plasmon mode
A substrate with large dielectric constant turns
plasmonic mode into acoustic mode:
va 
2 g dk
2
1
F
vF
Graphene on
top of Pt(111):
v a  1 . 15 v F
TS and G. Gómez-Santos, New J. Phys. 14, 105018 (2012).
Near-field amplification
Near-field amplification
Exponential
amplification for
R=0.
T  
1
2
e
2 qd
Analogy to Pendry´s perfect lens
Numerical results
Longitudinal
polarization:
Transverse
polarization:
by A. Gutiérrez
TS and G. Gómez-Santos, Phys. Rev. B 85, 075410 (2012).
Numerical results
For different densities: order of
layers determines amplification:
n1>n2
n1<n2
Retardation effects
Strong light-matter coupling
Plasmon Dispersion:
1  2
2  1 3  2
The presence of doped graphene at
the interfaces leads strong lightmatter coupling for ω<αωF:
q  q
 
r   1   
  F
• Quenched Fabry-Pérot resonances
• Extraordinary transmission in tunnel region
G. Gómez-Santos and TS, Europhys. Lett. 99, 27006 (2012).




Fabry-Pérot resonances
Quenched Fabry-Pérot
resonances:
Response shows Fano lineshape: Particle-in-a-box
states leak out and interact with continuum.
Im    
(Q  / 2   s )
2
( s  s * )  (  / 2 )
2
2
s* 
 /F
d
 Q  23
Quantum-Dot model
Quasi-localized states between two
doped graphene layers
Extraordinary transmission
Extraordinary transmission
in tunnel region:
Transmission
between light cones:
Finite relaxation time
Non-linear absorption sets in for
angles beyond total reflections:
Different layer distances
Different relaxation times
Optical properties of
Twisted bilayer
Atomic structure
For small angles, the formation of
periodic Moiré superlattices is seen.
P. Moon and M. Koshino, arXive:1302.5218 (2013).
Electronic structure
The electronic structure changes for small twist angles.
Renormalization of
the Fermi velocity:

t

v  vF 1  9
vF K

 K  2 K sin(  / 2 )




J. M. B. Lopes dos Santos et al., Phys. Rev. Lett. 99, 256802 (2007).
Optical conductivity
The optical conductivity is characterized by a van Hove
singularity independent of the angle.
Drude weight
Drude weight follows the shell structure of the DOS.
Drude weight
For small angles, a substructure appears in the Drude
weight not present in the DOS:
Plasmonic excitations
For small
chemical
potential:
Interband
plasmons
Plasmonic excitations
For large
chemical
potential:
Intraband
plasmons
Summary
Concluding remarks
•
There is spectral transfer of in-phase and
out-of-phase mode, near-field
amplification and perfect transmission
in double-layer graphene.
•
Plasmonic spectrum of twisted bilayer
graphene stronly depends on doping.