PHYSICAL REVIEW B 86, 125450 (2012)
Edge magnetoplasmons and the optical excitations in graphene disks
Weihua Wang,
*
S. Peter Apell, and Jari M. Kinaret
Department of Applied Physics, Chalmers University of Technology, SE-412 96 G¨ oteborg, Sweden
(Received 5 July 2012; revised manuscript received 10 September 2012; published 28 September 2012)
We discuss the edge magnetoplasmon properties in highly doped graphene disks, and the corresponding optical
excitations. Edge magnetoplasmons with nonzero angular momentum (l = 0) have two branches corresponding
to different edge current rotations with respect to the magnetic field. The resonance energies of one branch are
blueshifted and the other redshifted relative to energies at B = 0, with the energy differences linearly proportional
to the magnetic field. Recently, the l = 1 dipole mode has been investigated by two experiments using optical
transmission spectroscopy [Crassee et al., Nano Lett. 12, 2470 (2012); Yan et al., ibid. 12, 3766 (2012)], and
classical cyclotron resonances were found in highly doped graphene samples. These are determined by graphene
magneto-optical conductivities, which behave like a conventional two-dimensional electron system in the high
doping limit.
DOI: 10.1103/PhysRevB.86.125450 PACS number(s): 73.20.Mf, 71.35.Ji, 78.67.Wj
I. INTRODUCTION
Plasmons are fundamental collective oscillations of elec-
trons that have captured the interest of scientists for years,
driving research in theory, experiment, and applications. Dirac
plasmons of graphene,
1
which have also been identified
recently, are predicted to have particularly interesting and
unique properties, e.g., longer plasmon propagating length
and higher field confinement. As an ideal two-dimensional
electron system (2DES), the carrier concentration in graphene
can be controlled through chemical doping or electrostatic
gating.
2–5
Hence, one can tailor the local conductivity by
a bias voltage, and it might provide exciting venues to
achieve wave-guiding in simple graphene flakes without
additional nanostructures, which have been suggested to per-
form transformation optics and cloaking on a one-atom-thick
surface.
6,7
Engineering plasmonic nanostructures in graphene,
such as disks and ribbons, enables rich functionalities to
be obtained, such as modulating the emitter radiation,
8
enhancing the light matter interaction, and realizing plasmon
wave-guiding.
9,10
These kinds of structures have also been
demonstrated theoretically to achieve a very high optical
absorption.
11,12
Quantum effects are also very important in
small-sized graphene nanostructures,
13
and very recently,
atomically localized plasmon was observed in a graphene
flake with a sub-nanoscale localized defect.
14
From microsize
down to nanosize, graphene plasmonics has the potential to
bridge the size gap between electronics and photonics,
15,16
and it may enable new functional optoelectronics devices to be
implemented.
II. MAGNETO-OPTICAL CONDUCTIVITIES
OF GRAPHENE
Active tunability is a key issue in plasmonics. In metal-
based plasmonics, the designed nanostructures can usually
only work at a specific frequency. However, in graphene
plasmonics, the electron concentration can be easily tuned
by electrostatic gating, so the plasmon frequency will be
tuned as well (e.g., the dipolar plasmon frequencies depend
on the chemical potential
11
). Very recently, this idea was
demonstrated experimentally by two groups using a scattering-
type scanning near-field optical microscope.
17,18
In addition to
this electric tuning method, graphene plasmons can also be
tuned through magnetic fields. Graphene magnetoplasmons
have been studied theoretically in infinite graphene sheets,
19–22
semi-infinite sheets,
23,24
and finite structures.
25,26
As a linear
and gapless energy spectrum, graphene has a different Landau
level (LL) distribution compared with a usual 2DES, reading
E
n
=±
√
n¯ hω
B
, (1)
where ω
B
=
√
2v
F
/l
B
, and the ± represent an electronlike
(+) or holelike (−) LL index. Here v
F
≈ 10
6
m/s is the
Fermi velocity in graphene and l
B
=
√
¯ h/eB is the magnetic
length. With this energy dispersion, the conductivities can be
calculated using the Kubo formula,
27
σ
xx
(ω) =
e
2
2π ¯ h
∞
n=0
iω
B
(ω + iτ
−1
)
[n
F
(E
n
) − n
F
(E
n+1
)] + [n
F
(−E
n+1
) − n
F
(−E
n
)]
(ω + iτ
−1
)
2
− f
2
intra
(n)ω
2
B
f
intra
(n)
+ iω
B
(ω + iτ
−1
)
[n
F
(−E
n
) − n
F
(E
n+1
)] + [n
F
(−E
n+1
) − n
F
(E
n
)]
(ω + iτ
−1
)
2
− f
2
inter
(n)ω
2
B
f
inter
(n)
, (2a)
σ
xy
(ω) =
e
2
2π ¯ h
∞
n=0
{[n
F
(E
n
) − n
F
(E
n+1
)] − [n
F
(−E
n+1
) − n
F
(−E
n
)]}
×
ω
2
B
(ω + iτ
−1
)
2
− f
2
intra
(n)ω
2
B
+
ω
2
B
(ω + iτ
−1
)
2
− f
2
inter
(n)ω
2
B
, (2b)
125450-1 1098-0121/2012/86(12)/125450(5) ©2012 American Physical Society