Momentum dependence and losses in
graphene plasmons
Graphene Nanophotonics
Benasque, 2013, Mar 03 -- Mar 08
Outline
 Damping mechanisms
 Plasmons in ribbons
 Experimental results
Mid-infrared plasmons in scaled graphene nanostructures
H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. G., P. Avouris, F. Xia
arXiv:1209.1984, Nature Photonics, in press
L. Martín-Moreno, A. Nikitin, F. García-Vidal, M. M. Fogler
Damping mechanisms
 Higher order processes, multiple
electron-hole pairs
 Lack of momentum conservation
 Decay into other excitations:
phonons, …
Elastic scattering
j  
e 2 vF k F
   
   i
  k j

2 k


k
k+q
k’

 vF



elastic mean free path
k

 p q   i  2 e 2vF k F q
2
Finite systems

 vF


D
D
Inhomogeneous electric fields
T. Low, M. M. Fogler, F. G, unpublished
Edges
Local excitations
p(q) 4 q 4
L-1
Inhomogeneous electric fields
 
F r , t  0 e
q y
Non local conductivity, clean system
 2  kF

q vF  
Im q,    vF2 q
 0
  q vF

e2 
 q, 
Im q, 
2
 q
1ms, EF=0.4eV
E
t
p 

E
1
A
  
 d q F q 
2
2
 d q F q  q, 
1 F
A 2
2
2

vF3 q 3p
2

vF2 q 2p
e2kF

1
D2
Surface polar modes
H e sp 
M sp
2

1
M sp ak q ak bq  bq

A pq

 e 2 e 2 q z 2

F
 env q
0
sp 

1
1


F 


2      env  0   env 
2


 pl2 q 
 sp2 *
   env 1 


2
2
2
2
   i     i  sp    sp   sp * 
4
 sp2 * 
 sp F 2

Si O2 has polar modes
which induce long range
electrostatic potentials.
The dielectric constant of
the system is modified.
Optical phonons
q
k’
k
k+q
Optical phonons at G, ph0.2 eV
Coupling through changes in bond lengths
Weak dispersion
H e  ph 
t 

 a  u x  iu y
a t
 23
t a
0
1/
u x  iu y  1 t

j u

0  vF  a
2

 t 

 
D     F   ph 
    a  M  ph
1
Experiments
Y. Yan, T. Low, W. Zhu,Y. Wu, M. Freitag, X. Li, F. G., P. Avouris, and F. Xia, arXiv:1209.1984, Nature Phys., in press
Nanoribbons, antidots , and
nanodisks defined by
electron beam litography.
The samples lie on CVD
graphene on SiO 2 and
diamond like carbon (DLC)
substrates.
Plasmon dispersion
graphene on diamond like carbon (DLC)
q

W  W0
W0  28nm
dead layer
asymmetric lineshape
Plasmon dispersion
graphene on SiO2
sp1 806cm1
sp 2 1168cm1
op 1598cm1
Plasmon damping
G
1
G01  a  Gpl1 ph
W
Plasmon damping
Comparison between
graphene on SiO 2 and
graphene on DLC
Doping dependence
Conclusions. Open questions
 Plasmon dispersion can be accurately measured in
nanoribbons.
 Plasmon linewidth can be explained by simple
mechanisms
 The main decay channel at high frequencies is decay
into optical phonons and electron-hole pairs
 The role of other decay channels is in reasonable
agreement with simple estimates
 Coulomb blockade, interplay between plasmons and
dc transport