Chemistry Reference
In-Depth Information
Chapter 8
Phonons and Electron-Phonon Interaction
in Graphene and Nanotube
Tsuneya Ando
Department of Physics, Tokyo Institute of Technology,
2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan
ando@phys.titech.ac.jp
A brief review is given on long-wavelength acoustic phonons, long-
wavelength optical phonons, and zone-boundary phonons in graphene
and carbon nanotubes together with effects of their interaction with
electrons from a theoretical point of view.
1. Introduction
Monolayer graphene was fabricated using the so-called scotch-tape tech-
nique
1
and the magnetotransport was measured including the integer quan-
tum Hall effect.
2,3
Since then the graphene became the subject of ex-
tensive theoretical and experimental study.
4,5
The carbon nanotube is
graphene rolled into a cylindrical form, discovered and synthesized ear-
lier than graphene.
6
The purpose of this paper is to give a brief review on
phonons and effects of electron-phonon interaction in graphene and nano-
tubes.
2. Monolayer Graphene and Nanotube
In a monolayer graphene the conduction and valence bands consisting of
π
orbitals cross at K and K' points of the Brillouin zone, where the Fermi
level is located.
7,8
Electronic states near a K point are described by the
k
·
p
equation equivalent to Weyl's equation or a Dirac equation with vanishing
rest mass.
6,9-14
In the vicinity of the K point, in particular, we have
F
(
r
)=
F
A
(
r
)
F
B
(
r
)
k
)
F
(
r
)=
γ
(
σ·
ε
F
(
r
)
,
,
(1)
135
