The electronic properties of graphene
A. H. Castro Neto
Department of Physics, Boston University, 590 Commonwealth Avenue, Boston,
Massachusetts 02215, USA
F. Guinea
Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, E-28049 Madrid, Spain
N. M. R. Peres
Center of Physics and Department of Physics, Universidade do Minho, P-4710-057,
Braga, Portugal
K. S. Novoselov and A. K. Geim
Department of Physics and Astronomy, University of Manchester, Manchester, M13 9PL,
United Kingdom

Published 14 January 2009
This article reviews the basic theoretical aspects of graphene, a one-atom-thick allotrope of carbon,
with unusual two-dimensional Dirac-like electronic excitations. The Dirac electrons can be controlled
by application of external electric and magnetic fields, or by altering sample geometry and/or topology.
The Dirac electrons behave in unusual ways in tunneling, confinement, and the integer quantum Hall
effect. The electronic properties of graphene stacks are discussed and vary with stacking order and
number of layers. Edge
surface states in graphene depend on the edge termination
zigzag or
armchair and affect the physical properties of nanoribbons. Different types of disorder modify the
Dirac equation leading to unusual spectroscopic and transport properties. The effects of
electron-electron and electron-phonon interactions in single layer and multilayer graphene are also
presented.
DOI: 10.1103/RevModPhys.81.109 PACS number
s: 81.05.Uw, 73.20.
r, 03.65.Pm, 82.45.Mp
CONTENTS
I. Introduction 110
II. Elementary Electronic Properties of Graphene 112
A. Single layer: Tight-binding approach 112
1. Cyclotron mass 113
2. Density of states 114
B. Dirac fermions 114
1. Chiral tunneling and Klein paradox 115
2. Confinement and Zitterbewegung 117
C. Bilayer graphene: Tight-binding approach 118
D. Epitaxial graphene 119
E. Graphene stacks 120
1. Electronic structure of bulk graphite 121
F. Surface states in graphene 122
G. Surface states in graphene stacks 124
H. The spectrum of graphene nanoribbons 124
1. Zigzag nanoribbons 125
2. Armchair nanoribbons 126
I. Dirac fermions in a magnetic field 126
J. The anomalous integer quantum Hall effect 128
K. Tight-binding model in a magnetic field 128
L. Landau levels in graphene stacks 130
M. Diamagnetism 130
N. Spin-orbit coupling 131
III. Flexural Phonons, Elasticity, and Crumpling 132
IV. Disorder in Graphene 134
A. Ripples 135
B. Topological lattice defects 136
C. Impurity states 137
D. Localized states near edges, cracks, and voids 137
E. Self-doping 138
F. Vector potential and gauge field disorder 139
1. Gauge field induced by curvature 140
2. Elastic strain 140
3. Random gauge fields 141
G. Coupling to magnetic impurities 141
H. Weak and strong localization 142
I. Transport near the Dirac point 143
J. Boltzmann equation description of dc transport in
doped graphene 144
K. Magnetotransport and universal conductivity 145
1. The full self-consistent Born approximation

FSBA 146
V. Many-Body Effects 148
A. Electron-phonon interactions 148
B. Electron-electron interactions 150
1. Screening in graphene stacks 152
C. Short-range interactions 152
1. Bilayer graphene: Exchange 153
2. Bilayer graphene: Short-range interactions 154
D. Interactions in high magnetic fields 154
VI. Conclusions 154
Acknowledgments 155
References 155
REVIEWS OF MODERN PHYSICS, VOLUME 81, JANUARY–MARCH 2009
0034-6861/2009/81
1/109
54 ©2009 The American Physical Society109

I. INTRODUCTION
Carbon is the materia prima for life and the basis of all
organic chemistry. Because of the flexibility of its bond-
ing, carbon-based systems show an unlimited number of
different structures with an equally large variety of
physical properties. These physical properties are, in
great part, the result of the dimensionality of these
structures. Among systems with only carbon atoms,
graphene—a two-dimensional
2D allotrope of
carbon—plays an important role since it is the basis for
the understanding of the electronic properties in other
allotropes. Graphene is made out of carbon atoms ar-
ranged on a honeycomb structure made out of hexagons

see Fig. 1, and can be thought of as composed of ben-
zene rings stripped out from their hydrogen atoms

Pauling, 1972. Fullerenes
Andreoni, 2000 are mol-
ecules where carbon atoms are arranged spherically, and
hence, from the physical point of view, are zero-
dimensional objects with discrete energy states.
Fullerenes can be obtained from graphene with the in-
troduction of pentagons
that create positive curvature
defects, and hence, fullerenes can be thought as
wrapped-up graphene. Carbon nanotubes
Saito et al.,
1998; Charlier et al., 2007 are obtained by rolling
graphene along a given direction and reconnecting the
carbon bonds. Hence carbon nanotubes have only hexa-
gons and can be thought of as one-dimensional
1D ob-
jects. Graphite, a three dimensional
3D allotrope of
carbon, became widely known after the invention of the
pencil in 1564
Petroski, 1989, and its usefulness as an
instrument for writing comes from the fact that graphite
is made out of stacks of graphene layers that are weakly
coupled by van der Waals forces. Hence, when one
presses a pencil against a sheet of paper, one is actually
producing graphene stacks and, somewhere among
them, there could be individual graphene layers. Al-
though graphene is the mother for all these different
allotropes and has been presumably produced every
time someone writes with a pencil, it was only isolated
440 years after its invention
Novoselov et al., 2004. The
reason is that, first, no one actually expected graphene
to exist in the free state and, second, even with the ben-
efit of hindsight, no experimental tools existed to search
for one-atom-thick flakes among the pencil debris cov-
ering macroscopic areas
Geim and MacDonald, 2007.
Graphene was eventually spotted due to the subtle op-
tical effect it creates on top of a chosen SiO2 substrate

Novoselov et al., 2004 that allows its observation with
an ordinary optical microscope
Abergel et al., 2007;
Blake et al., 2007; Casiraghi et al., 2007. Hence,
graphene is relatively straightforward to make, but not
so easy to find.
The structural flexibility of graphene is reflected in its
electronic properties. The sp2 hybridization between one
s orbital and two p orbitals leads to a trigonal planar
structure with a formation of a  bond between carbon
atoms that are separated by 1.42 Å. The  band is re-
sponsible for the robustness of the lattice structure in all
allotropes. Due to the Pauli principle, these bands have
a filled shell and, hence, form a deep valence band. The
unaffected p orbital, which is perpendicular to the pla-
nar structure, can bind covalently with neighboring car-
bon atoms, leading to the formation of a  band. Since
each p orbital has one extra electron, the  band is half
filled.
Half-filled bands in transition elements have played
an important role in the physics of strongly correlated
systems since, due to their strong tight-binding charac-
ter, the Coulomb energies are large, leading to strong
collective effects, magnetism, and insulating behavior
due to correlation gaps or Mottness
Phillips, 2006. In
fact, Linus Pauling proposed in the 1950s that, on the
basis of the electronic properties of benzene, graphene
should be a resonant valence bond
RVB structure

Pauling, 1972. RVB states have become popular in the
literature of transition-metal oxides, and particularly in
studies of cuprate-oxide superconductors
Maple, 1998.
This point of view should be contrasted with contempo-
raneous band-structure studies of graphene
Wallace,
1947 that found it to be a semimetal with unusual lin-
early dispersing electronic excitations called Dirac elec-
trons. While most current experimental data in
graphene support the band structure point of view, the
role of electron-electron interactions in graphene is a
subject of intense research.
It was P. R. Wallace in 1946 who first wrote on the
band structure of graphene and showed the unusual
semimetallic behavior in this material
Wallace, 1947.
At that time, the thought of a purely 2D structure was
not reality and Wallace’s studies of graphene served him
as a starting point to study graphite, an important mate-
rial for nuclear reactors in the post–World War II era.
During the following years, the study of graphite culmi-
nated with the Slonczewski-Weiss-McClure
SWM band
structure of graphite, which provided a description of
the electronic properties in this material
McClure, 1957;
Slonczewski and Weiss, 1958 and was successful in de-
scribing the experimental data
Boyle and Nozières
1958; McClure, 1958; Spry and Scherer, 1960; Soule et
al., 1964; Williamson et al., 1965; Dillon et al., 1977.
From 1957 to 1968, the assignment of the electron and
hole states within the SWM model were opposite to
FIG. 1.
Color online Graphene
top left is a honeycomb
lattice of carbon atoms. Graphite
top right can be viewed as
a stack of graphene layers. Carbon nanotubes are rolled-up
cylinders of graphene
bottom left. Fullerenes
C60 are mol-
ecules consisting of wrapped graphene by the introduction of
pentagons on the hexagonal lattice. From Castro Neto et al.,
2006a.
110 Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

what is accepted today. In 1968, Schroeder et al.

Schroeder et al., 1968 established the currently ac-
cepted location of electron and hole pockets
McClure,
1971. The SWM model has been revisited in recent
years because of its inability to describe the van der
Waals–like interactions between graphene planes, a
problem that requires the understanding of many-body
effects that go beyond the band-structure description

Rydberg et al., 2003. These issues, however, do not
arise in the context of a single graphene crystal but they
show up when graphene layers are stacked on top of
each other, as in the case, for instance, of the bilayer
graphene. Stacking can change the electronic properties
considerably and the layering structure can be used in
order to control the electronic properties.
One of the most interesting aspects of the graphene
problem is that its low-energy excitations are massless,
chiral, Dirac fermions. In neutral graphene, the chemical
potential crosses exactly the Dirac point. This particular
dispersion, that is only valid at low energies, mimics the
physics of quantum electrodynamics
QED for massless
fermions except for the fact that in graphene the Dirac
fermions move with a speed vF, which is 300 times
smaller than the speed of light c. Hence, many of the
unusual properties of QED can show up in graphene but
at much smaller speeds
Castro Neto et al., 2006a;
Katsnelson et al., 2006; Katsnelson and Novoselov,
2007. Dirac fermions behave in unusual ways when
compared to ordinary electrons if subjected to magnetic
fields, leading to new physical phenomena
Gusynin and
Sharapov, 2005; Peres, Guinea, and Castro Neto, 2006a
such as the anomalous integer quantum Hall effect

IQHE measured experimentally
Novoselov, Geim,
Morozov, et al., 2005a; Zhang et al., 2005. Besides being
qualitatively different from the IQHE observed in Si
and GaAlAs
heterostructures devices
Stone, 1992,
the IQHE in graphene can be observed at room tem-
perature because of the large cyclotron energies for
“relativistic” electrons
Novoselov et al., 2007. In fact,
the anomalous IQHE is the trademark of Dirac fermion
behavior.
Another interesting feature of Dirac fermions is their
insensitivity to external electrostatic potentials due to
the so-called Klein paradox, that is, the fact that Dirac
fermions can be transmitted with probability 1 through a
classically forbidden region
Calogeracos and Dombey,
1999; Itzykson and Zuber, 2006. In fact, Dirac fermions
behave in an unusual way in the presence of confining
potentials, leading to the phenomenon of Zitter-
bewegung, or jittery motion of the wave function
Itzyk-
son and Zuber, 2006. In graphene, these electrostatic
potentials can be easily generated by disorder. Since dis-
order is unavoidable in any material, there has been a
great deal of interest in trying to understand how disor-
der affects the physics of electrons in graphene and its
transport properties. In fact, under certain conditions,
Dirac fermions are immune to localization effects ob-
served in ordinary electrons
Lee and Ramakrishnan,
1985 and it has been established experimentally that
electrons can propagate without scattering over large
distances of the order of micrometers in graphene
No-
voselov et al., 2004. The sources of disorder in graphene
are many and can vary from ordinary effects commonly
found in semiconductors, such as ionized impurities in
the Si substrate, to adatoms and various molecules ad-
sorbed in the graphene surface, to more unusual defects
such as ripples associated with the soft structure of
graphene
Meyer, Geim, Katsnelson, Novoselov, Booth,
et al., 2007a. In fact, graphene is unique in the sense
that it shares properties of soft membranes
Nelson et
al., 2004 and at the same time it behaves in a metallic
way, so that the Dirac fermions propagate on a locally
curved space. Here analogies with problems of quantum
gravity become apparent
Fauser et al., 2007. The soft-
ness of graphene is related with the fact that it has out-
of-plane vibrational modes
phonons that cannot be
found in 3D solids. These flexural modes, responsible
for the bending properties of graphene, also account for
the lack of long range structural order in soft mem-
branes leading to the phenomenon of crumpling
Nelson
et al., 2004. Nevertheless, the presence of a substrate or
scaffolds that hold graphene in place can stabilize a cer-
tain degree of order in graphene but leaves behind the
so-called ripples
which can be viewed as frozen flexural
modes.
It was realized early on that graphene should also
present unusual mesoscopic effects
Peres, Castro Neto,
and Guinea, 2006a; Katsnelson, 2007a. These effects
have their origin in the boundary conditions required for
the wave functions in mesoscopic samples with various
types of edges graphene can have
Nakada et al., 1996;
Wakabayashi et al., 1999; Peres, Guinea, and Castro
Neto, 2006a; Akhmerov and Beenakker, 2008. The
most studied edges, zigzag and armchair, have drastically
different electronic properties. Zigzag edges can sustain
edge
surface states and resonances that are not present
in the armchair case. Moreover, when coupled to con-
ducting leads, the boundary conditions for a graphene
ribbon strongly affect its conductance, and the chiral
Dirac nature of fermions in graphene can be used for
applications where one can control the valley flavor of
the electrons besides its charge, the so-called valleytron-
ics
Rycerz et al., 2007. Furthermore, when supercon-
ducting contacts are attached to graphene, they lead to
the development of supercurrent flow and Andreev pro-
cesses characteristic of the superconducting proximity
effect
Heersche et al., 2007. The fact that Cooper pairs
can propagate so well in graphene attests to the robust
electronic coherence in this material. In fact, quantum
interference phenomena such as weak localization, uni-
versal conductance fluctuations
Morozov et al., 2006,
and the Aharonov-Bohm effect in graphene rings have
already been observed experimentally
Recher et al.,
2007; Russo, 2007. The ballistic electronic propagation
in graphene can be used for field-effect devices such as
p-n
Cheianov and Fal’ko, 2006; Cheianov, Fal’ko, and
Altshuler, 2007; Huard et al., 2007; Lemme et al., 2007;
Tworzydlo et al., 2007; Williams et al., 2007; Fogler,
Glazman, Novikov, et al., 2008; Zhang and Fogler, 2008
and p-n-p
Ossipov et al., 2007 junctions, and as “neu-
111Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

trino” billiards
Berry and Modragon, 1987; Miao et al.,
2007. It has also been suggested that Coulomb interac-
tions are considerably enhanced in smaller geometries,
such as graphene quantum dots
Milton Pereira et al.,
2007, leading to unusual Coulomb blockade effects

Geim and Novoselov, 2007 and perhaps to magnetic
phenomena such as the Kondo effect. The transport
properties of graphene allow for their use in a plethora
of applications ranging from single molecule detection

Schedin et al., 2007; Wehling et al., 2008 to spin injec-
tion
Cho et al., 2007; Hill et al., 2007; Ohishi et al., 2007;
Tombros et al., 2007.
Because of its unusual structural and electronic flex-
ibility, graphene can be tailored chemically and/or struc-
turally in many different ways: deposition of metal at-
oms
Calandra and Mauri, 2007; Uchoa et al., 2008 or
molecules
Schedin et al., 2007; Leenaerts et al., 2008;
Wehling et al., 2008 on top; intercalation as done in
graphite intercalated compounds
Dresselhaus et al.,
1983; Tanuma and Kamimura, 1985; Dresselhaus and
Dresselhaus, 2002; incorporation of nitrogen and/or
boron in its structure
Martins et al., 2007; Peres,
Klironomos, Tsai, et al., 2007 in analogy with what has
been done in nanotubes
Stephan et al., 1994; and using
different substrates that modify the electronic structure

Calizo et al., 2007; Giovannetti et al., 2007; Varchon et
al., 2007; Zhou et al., 2007; Das et al., 2008; Faugeras et
al., 2008. The control of graphene properties can be
extended in new directions allowing for the creation of
graphene-based systems with magnetic and supercon-
ducting properties
Uchoa and Castro Neto, 2007 that
are unique in their 2D properties. Although the
graphene field is still in its infancy, the scientific and
technological possibilities of this new material seem to
be unlimited. The understanding and control of this ma-
terial’s properties can open doors for a new frontier in
electronics. As the current status of the experiment and
potential applications have recently been reviewed

Geim and Novoselov, 2007, in this paper we concen-
trate on the theory and more technical aspects of elec-
tronic properties with this exciting new material.
II. ELEMENTARY ELECTRONIC PROPERTIES OF
GRAPHENE
A. Single layer: Tight-binding approach
Graphene is made out of carbon atoms arranged in
hexagonal structure, as shown in Fig. 2. The structure
can be seen as a triangular lattice with a basis of two
atoms per unit cell. The lattice vectors can be written as
a1 =
a
2

3,3, a2 =
a
2

3,− 3 ,
1
where a1.42 Å is the carbon-carbon distance. The
reciprocal-lattice vectors are given by
b1 =
2
3a

1,3, b2 =
2
3a

1,− 3 .
2
Of particular importance for the physics of graphene are
the two points K and K


at the corners of the graphene
Brillouin zone
BZ. These are named Dirac points for
reasons that will become clear later. Their positions in
momentum space are given by
K =

2
3a
,
2
33a

, K


=

2
3a
,−
2
33a

.
3
The three nearest-neighbor vectors in real space are
given by

1 =
a
2

1,3
2 =
a
2

1,− 3 3 = − a
1,0
4
while the six second-nearest neighbors are located at
1
= ±a1, 2
= ±a2, 3
= ±
a2−a1.
The tight-binding Hamiltonian for electrons in
graphene considering that electrons can hop to both
nearest- and next-nearest-neighbor atoms has the form

we use units such that =1
H = − t


i,j,

a
,i
† b
,j + H.c.
− t





i,j,

a
,i
† a
,j + b
,i
† b
,j + H.c. ,
5
where ai,
ai,
†
 annihilates
creates an electron with
spin 
= ↑ , ↓  on site Ri on sublattice A
an equiva-
lent definition is used for sublattice B, t
2.8 eV is the
nearest-neighbor hopping energy
hopping between dif-
ferent sublattices, and t


is the next nearest-neighbor
hopping energy1
hopping in the same sublattice. The
energy bands derived from this Hamiltonian have the
form
Wallace, 1947
E±
k = ± t3 + f
k − t
f
k ,
1The value of t


is not well known but ab initio calculations

Reich et al., 2002 find 0.02t t


0.2t depending on the tight-
binding parametrization. These calculations also include the
effect of a third-nearest-neighbors hopping, which has a value
of around 0.07 eV. A tight-binding fit to cyclotron resonance
experiments
Deacon et al., 2007 finds t


0.1 eV.
a
a
1
2
b
b
1
2
K
Γ
k
k
x
y
1
2
3
M
δ
δ
δ
A B
K’
FIG. 2.
Color online Honeycomb lattice and its Brillouin
zone. Left: lattice structure of graphene, made out of two in-
terpenetrating triangular lattices
a1 and a2 are the lattice unit
vectors, and i, i=1,2 ,3 are the nearest-neighbor vectors.
Right: corresponding Brillouin zone. The Dirac cones are lo-
cated at the K and K


points.
112 Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

f
k = 2 cos
3kya + 4 cos
3
2
kyacos
3
2
kxa ,
6
where the plus sign applies to the upper
* and the
minus sign the lower
 band. It is clear from Eq.
6
that the spectrum is symmetric around zero energy if t


=0. For finite values of t


, the electron-hole symmetry is
broken and the  and * bands become asymmetric. In
Fig. 3, we show the full band structure of graphene with
both t and t


. In the same figure, we also show a zoom in
of the band structure close to one of the Dirac points
at
the K or K


point in the BZ. This dispersion can be
obtained by expanding the full band structure, Eq.
6,
close to the K
or K


 vector, Eq.
3, as k=K+q, with
q  K
Wallace, 1947,
E±
q  ± vFq +O
q/K
2
 ,
7
where q is the momentum measured relatively to the
Dirac points and vF is the Fermi velocity, given by vF
=3ta /2, with a value vF
1106 m/s. This result was
first obtained by Wallace
1947.
The most striking difference between this result and
the usual case,
q=q2 /
2m, where m is the electron
mass, is that the Fermi velocity in Eq.
7 does not de-
pend on the energy or momentum: in the usual case we
have v=k /m=2E /m and hence the velocity changes
substantially with energy. The expansion of the spectrum
around the Dirac point including t


up to second order
in q /K is given by
E±
q
3t
± vFq − 
9t


a2
4
±
3ta2
8
sin
3
qq
2,
8
where

q = arctan
qx
qy


9
is the angle in momentum space. Hence, the presence of
t


shifts in energy the position of the Dirac point and
breaks electron-hole symmetry. Note that up to order

q /K2 the dispersion depends on the direction in mo-
mentum space and has a threefold symmetry. This is the
so-called trigonal warping of the electronic spectrum

Ando et al., 1998, Dresselhaus and Dresselhaus, 2002.
1. Cyclotron mass
The energy dispersion
7 resembles the energy of ul-
trarelativistic particles; these particles are quantum me-
chanically described by the massless Dirac equation
see
Sec. II.B for more on this analogy. An immediate con-
sequence of this massless Dirac-like dispersion is a cy-
clotron mass that depends on the electronic density as its
square root
Novoselov, Geim, Morozov, et al., 2005;
Zhang et al., 2005. The cyclotron mass is defined, within
the semiclassical approximation
Ashcroft and Mermin,
1976, as
m* =
1
2

A
E
E

E=EF
,
10
with A
E the area in k space enclosed by the orbit and
given by
A
E = q
E2 = 
E2
vF
2 .
11
Using Eq.
11 in Eq.
10, one obtains
m* =
EF
vF
2 =
kF
vF
.
12
The electronic density n is related to the Fermi momen-
tum kF as kF
2 /=n
with contributions from the two
Dirac points K and K


and spin included, which leads to
m* =


vF
n .
13
Fitting Eq.
13 to the experimental data
see Fig. 4
provides an estimation for the Fermi velocity and the
FIG. 3.
Color online Electronic dispersion in the honeycomb
lattice. Left: energy spectrum
in units of t for finite values of
t and t


, with t=2.7 eV and t


=−0.2t. Right: zoom in of the
energy bands close to one of the Dirac points.
FIG. 4.
Color online Cyclotron mass of charge carriers in
graphene as a function of their concentration n. Positive and
negative n correspond to electrons and holes, respectively.
Symbols are the experimental data extracted from the tem-
perature dependence of the SdH oscillations; solid curves are
the best fit by Eq.
13. m0 is the free-electron mass. Adapted
from Novoselov, Geim, Morozov, et al., 2005.
113Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

where the index i=1
i=2 refers to the K
K


 point.
These new fields, ai,n and bi,n, are assumed to vary
slowly over the unit cell. The procedure for deriving
a theory that is valid close to the Dirac point con-
sists in using this representation in the tight-
binding Hamiltonian and expanding the opera-
tors up to a linear order in
. In the derivation, one
uses the fact that

e±iK·
=

e±iK
·
=0. After some
straightforward algebra, we arrive at
Semenoff,
1984
H
− t

dxdyˆ1
†

r


0 3a
1 − i3/4
− 3a
1 + i3/4 0

x + 
0 3a
− i − 3/4
− 3a
i − 3/4 0

yˆ1
r
+ ˆ2
†

r


0 3a
1 + i3/4
− 3a
1 − i3/4 0

x + 
0 3a
i − 3/4
− 3a
− i − 3/4 0

yˆ2
r
= − ivF dxdyˆ1
†

r · ˆ1
r + ˆ2
†

r* · ˆ2
r ,
18
with Pauli matrices =
x ,y, *=
x ,−y, and ˆi
†
=
ai
† ,bi
†

i=1,2. It is clear that the effective Hamil-
tonian
18 is made of two copies of the massless Dirac-
like Hamiltonian, one holding for p around K and the
other for p around K


. Note that, in first quantized lan-
guage, the two-component electron wave function 
r,
close to the K point, obeys the 2D Dirac equation,
− ivF · 
r = E
r .
19
The wave function, in momentum space, for the mo-
mentum around K has the form
±,K
k =
1
2

e−i
k/2
±ei
k/2


20
for HK=vF ·k, where the  signs correspond to the
eigenenergies E= ±vFk, that is, for the * and  bands,
respectively, and
k is given by Eq.
9. The wave func-
tion for the momentum around K


has the form
±,K



k =
1
2

ei
k/2
±e−i
k/2


21
for HK


=vF* ·k. Note that the wave functions at K and
K


are related by time-reversal symmetry: if we set the
origin of coordinates in momentum space in the M point
of the BZ
see Fig. 2, time reversal becomes equivalent
to a reflection along the kx axis, that is,
kx ,ky
→
kx ,−ky. Also note that if the phase
is rotated by
2, the wave function changes sign indicating a phase of

in the literature this is commonly called a Berry’s
phase. This change of phase by  under rotation is char-
acteristic of spinors. In fact, the wave function is a two-
component spinor.
A relevant quantity used to characterize the eigen-
functions is their helicity defined as the projection of the
momentum operator along the
pseudospin direction.
The quantum-mechanical operator for the helicity has
the form
hˆ =
1
2
 ·
p
p
.
22
It is clear from the definition of hˆ that the states K
r
and K



r are also eigenstates of hˆ,
hˆK
r = ±
1
2K
r ,
23
and an equivalent equation for K



r with inverted sign.
Therefore, electrons
holes have a positive
negative
helicity. Equation
23 implies that  has its two eigen-
values either in the direction of
⇑ or against
⇓ the
momentum p. This property says that the states of the
system close to the Dirac point have well defined chiral-
ity or helicity. Note that chirality is not defined in regard
to the real spin of the electron
that has not yet ap-
peared in the problem but to a pseudospin variable as-
sociated with the two components of the wave function.
The helicity values are good quantum numbers as long
as the Hamiltonian
18 is valid. Therefore, the existence
of helicity quantum numbers holds only as an
asymptotic property, which is well defined close to the
Dirac points K and K


. Either at larger energies or due
to the presence of a finite t


, the helicity stops being a
good quantum number.
1. Chiral tunneling and Klein paradox
In this section, we address the scattering of chiral elec-
trons in two dimensions by a square barrier
Katsnelson
et al., 2006; Katsnelson, 2007b. The one-dimensional
scattering of chiral electrons was discussed earlier in the
context on nanotubes
Ando et al., 1998; McEuen et al.,
1999.
We start by noting that by a gauge transformation the
wave function
20 can be written as
115Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

K
k =
1
2

1
±ei
k

.
24
We further assume that the scattering does not mix the
momenta around K and K


points. In Fig. 6, we depict
the scattering process due to the square barrier of width
D.
The wave function in the different regions can be writ-
ten in terms of incident and reflected waves. In region I,
we have
I
r =
1
2

1
sei

ei
kxx+kyy +
r
2

1
sei
−

ei
−kxx+kyy,

25
with =arctan
ky /kx, kx=kF cos , ky=kF sin , and kF
the Fermi momentum. In region II, we have
II
r =
a
2

1
s


ei


ei
qxx+kyy +
b
2

1
s


ei
−


ei
−qxx+kyy,

26
with
=arctan
ky /qx and
qx = 
V0 − E
2/
vF
2
 − ky
2,
27
and finally in region III we have a transmitted wave only,
III
r =
t
2

1
sei

ei
kxx+kyy,
28
with s=sgn
E and s


=sgn
E−V0. The coefficients r, a,
b, and t are determined from the continuity of the wave
function, which implies that the wave function has to
obey the conditions I
x=0,y=II
x=0,y and II
x
=D ,y=III
x=D ,y. Unlike the Schrödinger equation,
we only need to match the wave function but not its
derivative. The transmission through the barrier is ob-
tained from T
= tt* and has the form
T
 =
cos2
cos2 
cos
Dqxcos  cos

2 + sin2
Dqx
1 − ss
sin  sin

2 .
29
This expression does not take into account a contribu-
tion from evanescent waves in region II, which is usually
negligible, unless the chemical potential in region II is at
the Dirac energy
see Sec. IV.A.
Note that T
=T
−, and for values of Dqx satisfy-
ing the relation Dqx=n, with n an integer, the barrier
becomes completely transparent since T
=1, indepen-
dent of the value of . Also, for normal incidence

→0 and
→0 and any value of Dqx, one obtains T
0
=1, and the barrier is again totally transparent. This re-
sult is a manifestation of the Klein paradox
Calogeracos
and Dombey, 1999; Itzykson and Zuber, 2006 and does
not occur for nonrelativistic electrons. In this latter case
and for normal incidence, the transmission is always
smaller than 1. In the limit V0  E, Eq.
29 has the
following asymptotic form:
T

cos2 
1 − cos2
Dqxsin
2

.
30
In Fig. 7, we show the angular dependence of T
 for
two different values of the potential V0; it is clear that
there are several directions for which the transmission is
1. Similar calculations were done for a graphene bilayer

Katsnelson et al., 2006 with the absence of tunneling in
the forward
ky=0 direction its most distinctive behav-
ior.
The simplest example of a potential barrier is a square
potential discussed previously. When intervalley scatter-
ing and the lack of symmetry between sublattices are
neglected, a potential barrier shows no reflection for
electrons incident in the normal direction
Katsnelson et
al., 2006. Even when the barrier separates regions
where the Fermi surface is electronlike on one side and
holelike on the other, a normally incident electron con-
tinues propagating as a hole with 100% efficiency. This
phenomenon is another manifestation of the chirality of
the Dirac electrons within each valley, which prevents
backscattering in general. The transmission and reflec-
tion probabilities of electrons at different angles depend
on the potential profile along the barrier. A slowly vary-
ing barrier is more efficient in reflecting electrons at
nonzero incident angles
Cheianov and Fal’ko, 2006.
Electrons moving through a barrier separating p- and
n-doped graphene, a p-n junction, are transmitted as
D
V0
E
x
Energy
x
y
θ
φ
φ
D
III III
FIG. 6.
Color online Klein tunneling in graphene. Top: sche-
matic of the scattering of Dirac electrons by a square potential.
Bottom: definition of the angles  and
used in the scattering
formalism in regions I, II, and III.
116 Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

holes. The relation between the velocity and the mo-
mentum for a hole is the inverse of that for an electron.
This implies that, if the momentum parallel to the bar-
rier is conserved, the velocity of the quasiparticle is in-
verted. When the incident electrons emerge from a
source, the transmitting holes are focused into an image
of the source. This behavior is the same as that of pho-
tons moving in a medium with negative reflection index

Cheianov, Fal’ko, and Altshuler, 2007. Similar effects
can occur in graphene quantum dots, where the inner
and outer regions contain electrons and holes, respec-
tively
Cserti, Palyi, and Peterfalvi, 2007. Note that the
fact that barriers do not impede the transmission of nor-
mally incident electrons does not preclude the existence
of sharp resonances, due to the confinement of electrons
with a finite parallel momentum. This leads to the pos-
sibility of fabricating quantum dots with potential barri-
ers
Silvestrov and Efetov, 2007. Finally, at half filling,
due to disorder graphene can be divided in electron and
hole charge puddles
Katsnelson et al., 2006; Martin et
al., 2008. Transport is determined by the transmission
across the p-n junctions between these puddles
Che-
ianov, Fal’ko, Altshuler, et al., 2007; Shklovskii, 2007.
There is much progress in the measurement of transport
properties of graphene ribbons with additional top gates
that play the role of tunable potential barriers
Han et
al., 2007; Huard et al., 2007; Lemme et al., 2007; Özyil-
maz et al., 2007; Williams et al., 2007.
A magnetic field and potential fluctuations break both
the inversion symmetry of the lattice and time-reversal
symmetry. The combination of these effects also breaks
the symmetry between the two valleys. The transmission
coefficient becomes valley dependent, and, in general,
electrons from different valleys propagate along differ-
ent paths. This opens the possibility of manipulating the
valley index
Tworzydlo et al., 2007
valleytronics in a
way similar to the control of the spin in mesoscopic de-
vices
spintronics. For large magnetic fields, a p-n junc-
tion separates regions with different quantized Hall con-
ductivities. At the junction, chiral currents can flow at
both edges
Abanin and Levitov, 2007, inducing back-
scattering between the Hall currents at the edges of the
sample.
The scattering of electrons near the Dirac point by
graphene-superconductor junctions differs from An-
dreev scattering process in normal metals
Titov and
Beenakker, 2006. When the distance between the Fermi
energy and the Dirac energy is smaller than the super-
conducting gap, the superconducting interaction hybrid-
izes quasiparticles from one band with quasiholes in the
other. As in the case of scattering at a p-n junction, the
trajectories of the incoming electron and reflected hole

note that hole here is meant as in the BCS theory of
superconductivity are different from those in similar
processes in metals with only one type of carrier
Bhat-
tacharjee and Sengupta, 2006; Maiti and Sengupta,
2007.
2. Confinement and Zitterbewegung
Zitterbewegung, or jittery motion of the wave function
of the Dirac problem, occurs when one tries to confine
the Dirac electrons
Itzykson and Zuber, 2006. Local-
ization of a wave packet leads, due to the Heisenberg
principle, to uncertainty in the momentum. For a Dirac
particle with zero rest mass, uncertainty in the momen-
tum translates into uncertainty in the energy of the par-
ticle as well
this should be contrasted with the nonrela-
tivistic case, where the position-momentum uncertainty
relation is independent of the energy-time uncertainty
relation. Thus, for an ultrarelativistic particle, a par-
ticlelike state can have holelike states in its time evolu-
tion. Consider, for instance, if one tries to construct a
wave packet at some time t=0, and assume, for simplic-
ity, that this packet has a Gaussian shape of width w with
momentum close to K,
0
r =
e−r
2/2w2

w
eiK·r ,
31
where  is spinor composed of positive energy states
associated with +,K of Eq.
20. The eigenfunction of
the Dirac equation can be written in terms of the solu-
tion
20 as

r,t =

d2k

22 a=±1
a,ka,K
ke
−ia
k·r+vFkt,
32
where ±,k are Fourier coefficients. We can rewrite Eq.

31 in terms of Eq.
32 by inverse Fourier transform
and find that
±,k = we
−k2w2/2
±,K
†

k .
33
Note that the relative weight of positive energy states
with respect to negative energy states +/−, given by
Eq.
20 is 1, that is, there are as many positive energy
states as negative energy states in a wave packet. Hence,
these will cause the wave function to be delocalized at
any time t0. Thus, a wave packet of electronlike states
has holelike components, a result that puzzled many re-
-90 -75 -60 -45 -30 -15 0 15 30 45 60 75 90
0
0.2
0.4
0.6
0.8
1
T(
φ
)
V0 = 200 meV
V0 = 285 meV
-90 -75 -60 -45 -30 -15 0 15 30 45 60 75 90
angle φ
0
0.2
0.4
0.6
0.8
1
T(
φ
)
FIG. 7.
Color online Angular behavior of T
 for two dif-
ferent values of V0: V0=200 meV, dashed line; V0=285 meV,
solid line. The remaining parameters are D=110 nm
top, D
=50 nm
bottom E=80 meV, kF=2 /, and =50 nm.
117Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

If V=0 and 3 ,vFk1, one can eliminate the high-
energy states perturbatively and write an effective
Hamiltonian,
HK 

0
vF
2k2
1
+ 33ak*
vF
2

k*2
1
+ 33ak 0 
.
40
The hopping 4 leads to a k-dependent coupling be-
tween the sublattices or a small renormalization of 1.
The same role is played by the inequivalence between
sublattices within a layer.
For 3=0, Eq.
40 gives two parabolic bands, k,±
 ±vF
2k2 / t


, which touch at =0
as shown in Fig. 10.
The spectrum is electron-hole symmetric. There are two
additional bands that start at ±t


. Within this approxi-
mation, the bilayer is metallic, with a constant density of
states. The term 3 changes qualitatively the spectrum at
low energies since it introduces a trigonal distortion, or
warping, of the bands note that this trigonal distortion,
unlike the one introduced by large momentum in Eq.

8, occurs at low energies. The electron-hole symmetry
is preserved but, instead of two bands touching at k=0,
we obtain three sets of Dirac-like linear bands. One
Dirac point is at =0 and k=0, while the three other
Dirac points, also at =0, lie at three equivalent points
with a finite momentum. The stability of points where
bands touch can be understood using topological argu-
ments
Mañes et al., 2007. The winding number of a
closed curve in the plane around a given point is an
integer representing the total number of times that the
curve travels counterclockwise around the point so that
the wave function remains unaltered. The winding num-
ber of the point where the two parabolic bands come
together for 3=0 has winding number +2. The trigonal
warping term 3 splits it into a Dirac point at k=0 and
winding number −1, and three Dirac points at k0 and
winding numbers +1. An in-plane magnetic field, or a
small rotation of one layer with respect to the other,
splits the 3=0 degeneracy into two Dirac points with
winding number +1.
The term V in Eq.
38 breaks the equivalence of the
two layers, or, alternatively, inversion symmetry. In this
case, the dispersion relation becomes
±,k
2 = V2 + vF
2k2 + t


2 /2 ± 4V2vF
2k2 + t2vF
2k2 + t


4 /4,

41
giving rise to the dispersion shown in Fig. 11, and to the
opening of a gap close to, but not directly at, the K
point. For small momenta, and V t, the energy of the
conduction band can be expanded,
k  V − 2VvF
2k2/t


+ vF
4k4/2t


2 V .
42
The dispersion for the valence band can be obtained by
replacing k by − k. The bilayer has a gap at k2
2V2 /vF
2 . Note, therefore, that the gap in the biased
bilayer system depends on the applied bias and hence
can be measured experimentally
McCann, 2006; Mc-
Cann and Fal’ko, 2006; Castro, Novoselov, Morozov, et
al., 2007. The ability to open a gap makes bilayer
graphene interesting for technological applications.
D. Epitaxial graphene
It has been known for a long time that monolayers of
graphene could be grown epitaxially on metal surfaces
using catalytic decomposition of hydrocarbons or carbon
oxide
Shelton et al., 1974; Eizenberg and Blakely, 1979;
Campagnoli and Tosatti, 1989; Oshima and Nagashima,
1997; Sinitsyna and Yaminsky, 2006. When such sur-
faces are heated, oxygen or hydrogen desorbs, and the
carbon atoms form a graphene monolayer. The resulting
graphene structures could reach sizes up to a microme-
ter with few defects, and they were characterized by dif-
ferent surface-science techniques and local scanning
probes
Himpsel et al., 1982. For example, graphene
grown on ruthenium has zigzag edges and also ripples
associated with a
1010 reconstruction
Vázquez de
Parga et al., 2008.
Graphene can also be formed on the surface of SiC.
Upon heating, the silicon from the top layers desorbs,
and a few layers of graphene are left on the surface

Bommel et al., 1975; Forbeaux et al., 1998; Coey et al.,
2002; Berger et al., 2004; Rollings et al., 2006; Hass,
Feng, Millán-Otoya, et al., 2007; de Heer et al., 2007.
The number of layers can be controlled by limiting time
or temperature of the heating treatment. The quality
and the number of layers in the samples depend on the
SiC face used for their growth
de Heer et al., 2007
the
carbon-terminated surface produces few layers but with
a low mobility, whereas the silicon-terminated surface
produces several layers but with higher mobility. Epi-
taxially grown multilayers exhibit SdH oscillations with
FIG. 10.
Color online Band structure of bilayer graphene of
V=0 and 3=0.
FIG. 11.
Color online Band structure of bilayer graphene for
V0 and 3=0.
119Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

a Berry phase shift of 
Berger et al., 2006, which is the
same as the phase shift for Dirac fermions observed in a
single layer as well as for some subbands present in
multilayer graphene and graphite
Luk’yanchuk and Ko-
pelevich, 2004. The carbon layer directly on top of the
substrate is expected to be strongly bonded to it, and it
shows no  bands
Varchon et al., 2007. The next layer
shows a
6363 reconstruction due to the substrate,
and has graphene properties. An alternate route to pro-
duce few layers of graphene is based on synthesis from
nanodiamonds
Affoune et al., 2001.
Angle-resolved photoemission experiments
ARPES
show that epitaxial graphene grown on SiC has linearly
dispersing quasiparticles
Dirac fermions
Zhou,
Gweon, et al., 2006; Bostwick, Ohta, Seyller, et al., 2007;
Ohta et al., 2007, in agreement with the theoretical ex-
pectation. Nevertheless, these experiments show that
the electronic properties can change locally in space, in-
dicating a certain degree of inhomogeneity due to the
growth method
Zhou et al., 2007. Similar inhomogene-
ities due to disorder in the c-axis orientation of graphene
planes are observed in graphite
Zhou, Gweon, and
Lanzara, 2006. Moreover, graphene grown this way is
heavily doped due to the charge transfer from the sub-
strate to the graphene layer
with the chemical potential
well above the Dirac point and therefore all samples
have strong metallic character with large electronic mo-
bilities
Berger et al., 2006; de Heer et al., 2007. There is
also evidence for strong interaction between a substrate
and the graphene layer leading to the appearance of
gaps at the Dirac point
Zhou et al., 2007. Indeed, gaps
can be generated by the breaking of the sublattice sym-
metry and, as in the case of other carbon-based systems
such as polyacethylene
Su et al., 1979, 1980, it can lead
to solitonlike excitations
Jackiw and Rebbi 1976; Hou et
al., 2007. Multilayer graphene grown on SiC have also
been studied with ARPES
Bostwick, Ohta, McChesney,
et al., 2007; Ohta et al., 2007 and the results seem to
agree quite well with band-structure calculations
Mat-
tausch and Pankratov, 2007. Spectroscopy measure-
ments also show the transitions associated with Landau
levels
Sadowski et al., 2006 and weak-localization ef-
fects at low magnetic fields, also expected for Dirac fer-
mions
Wu et al., 2007. Local probes reveal a rich struc-
ture of terraces
Mallet et al., 2007 and interference
patterns due to defects at or below the graphene layers

Rutter et al., 2007.
E. Graphene stacks
In stacks with more than one graphene layer, two
consecutive layers are normally oriented in such a way
that the atoms in one of the two sublattices An of
the honeycomb structure of one layer are directly
above one-half of the atoms in the neighboring layer,
sublattice An±1. The second set of atoms in one layer sits
on top of the
empty center of a hexagon in the other
layer. The shortest distance between carbon atoms in
different layers is dAnAn±1=c=3.4 Å. The next distance is
dAnBn±1=
c2+a2. This is the most common arrangement
of nearest-neighbor layers observed in nature, although
a stacking order in which all atoms in one layer occupy
positions directly above the atoms in the neighboring
layers
hexagonal stacking has been considered theo-
retically
Charlier et al., 1991 and appears in graphite
intercalated compounds
Dresselhaus and Dresselhaus,
2002.
The relative position of two neighboring layers allows
for two different orientations of the third layer. If we
label the positions of the first two atoms as 1 and 2, the
third layer can be of type 1, leading to the sequence 121,
or it can fill a third position different from 1 and 2
see
Fig. 12, labeled 3. There are no more inequivalent po-
sitions where a new layer can be placed, so that thicker
stacks can be described in terms of these three orienta-
tions. In the most common version of bulk graphite, the
stacking order is 1212…
Bernal stacking. Regions with
the stacking 123123…
rhombohedral stacking have
also been observed in different types of graphite
Bacon,
1950; Gasparoux, 1967. Finally, samples with no dis-
cernible stacking order
turbostratic graphite are also
commonly reported.
Beyond two layers, the stack ordering can be arbi-
trarily complex. Simple analytical expressions for the
electronic bands can be obtained for perfect Bernal

1212… and rhombohedral
123123… stacking
Guinea
et al., 2006; Partoens and Peeters, 2006. Even if we con-
sider one interlayer hopping t


=1, the two stacking or-
ders show different band structures near =0. A Bernal
stack with N layers, N even, has N /2 electronlike and
N /2 holelike parabolic subbands touching at =0. When
N is odd, an additional subband with linear
Dirac dis-
persion emerges. Rhombohedral systems have only two
subbands that touch at =0. These subbands disperse as
kN, and become surface states localized at the top and
bottom layers when N→. In this limit, the remaining
2N−2 subbands of a rhombohedral stack become Dirac-
like, with the same Fermi velocity as a single graphene
layer. The subband structure of a trilayer with the Ber-
nal stacking includes two touching parabolic bands, and
one with Dirac dispersion, combining the features of bi-
layer and monolayer graphene.
The low-energy bands have different weights on the
two sublattices of each graphene layer. The states at a
FIG. 12.
Color online Sketch of the three inequivalent orien-
tations of graphene layers with respect to each other.
120 Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

site directly coupled to the neighboring planes are
pushed to energies  ± t


. The bands near =0 are lo-
calized mostly at the sites without neighbors in the next
layers. For the Bernal stacking, this feature implies that
the density of states at =0 at sites without nearest
neighbors in the contiguous layers is finite, while it van-
ishes linearly at the other sites. In stacks with rhombo-
hedral stacking, all sites have one neighbor in another
plane, and the density of states vanishes at =0
Guinea
et al., 2006. This result is consistent with the well known
fact that only one of the two sublattices at a graphite
surface can be resolved by scanning tunneling micros-
copy
STM
Tománek et al., 1987.
As in the case of a bilayer, an inhomogeneous charge
distribution can change the electrostatic potential in the
different layers. For more than two layers, this breaking
of the equivalence between layers can take place even in
the absence of an applied electric field. It is interesting
to note that a gap can open in a stack with Bernal or-
dering and four layers if the electronic charge at the two
surface layers is different from that at the two inner
ones. Systems with a higher number of layers do not
show a gap, even in the presence of charge inhomoge-
neity. Four representative examples are shown in Fig. 13.
The band structure analyzed here will be modified by
the inclusion of the trigonal warping term, 3. Experi-
mental studies of graphene stacks have showed that,
with an increasing number of layers, the system becomes
increasingly metallic
concentration of charge carriers at
zero energy gradually increases, and there appear sev-
eral types of electronlike and holelike carries
No-
voselov et al., 2004; Morozov et al., 2005. An inhomo-
geneous charge distribution between layers becomes
very important in this case, leading to 2D electron and
hole systems that occupy only a few graphene layers
near the surface, and can completely dominate transport
properties of graphene stacks
Morozov et al., 2005.
The degeneracies of the bands at =0 can be studied
using topological arguments
Mañes et al., 2007. Multi-
layers with an even number of layers and Bernal stack-
ing have inversion symmetry, leading to degeneracies
with winding number +2, as in the case of a bilayer. The
trigonal lattice symmetry implies that these points can
lead, at most, to four Dirac points. In stacks with an odd
number of layers, these degeneracies can be completely
removed. The winding number of the degeneracies
found in stacks with N layers and orthorhombic ordering
is ±N. The inclusion of trigonal warping terms will lead
to the existence of many weaker degeneracies near
=0.
Furthermore, it is well known that in graphite, the
planes can be rotated relative to each other giving rise to
Moiré patterns that are observed in STM of graphite
surfaces
Rong and Kuiper, 1993. The graphene layers
can be rotated relative to each other due to the weak
coupling between planes that allows for the presence of
many different orientational states that are quasidegen-
erate in energy. For certain angles, the graphene layers
become commensurate with each other leading to a low-
ering of the electronic energy. Such a phenomenon is
quite similar to the commensurate-incommensurate
transitions observed in certain charge-density-wave sys-
tems or adsorption of gases on graphite
Bak, 1982.
This kind of electronic structure dependence on the
relative rotation angle between graphene layers leads to
what is called superlubricity in graphite
Dienwiebel et
al., 2004, namely, the vanishing of the friction between
layers as a function of the rotation angle. In the case of
bilayer graphene, a rotation by a small commensurate
angle leads to the effective decoupling between layers
and recovery of the linear Dirac spectrum of the single
layer, albeit with a modification on the value of the
Fermi velocity
Lopes dos Santos et al., 2007.
1. Electronic structure of bulk graphite
The tight-binding description of graphene described
earlier can be extended to systems with an infinite num-
ber of layers. The coupling between layers leads to hop-
ping terms between  orbitals in different layers, leading
to the so-called Slonczewski-Weiss-McClure model

Slonczewski and Weiss, 1958. This model describes the
band structure of bulk graphite with the Bernal stacking
(b)
(a)
(c)
(d)
FIG. 13.
Color online Electronic bands of graphene multilay-
ers.
a Biased bilayer.
b Trilayer with Bernal stacking.
c
Trilayer with orthorhombic stacking.
d Stack with four layers
where the top and bottom layers are shifted in energy with
respect to the two middle layers by +0.1 eV.
121Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

order in terms of seven parameters: 0, 1, 2, 3, 4, 5,
and . The parameter 0 describes the hopping within
each layer, and it has been considered previously. The
coupling between orbitals in atoms that are nearest
neighbors in successive layers is 1, which we called t

earlier. The parameters 3 and 4 describe the hopping
between orbitals at next nearest neighbors in successive
layers and were discussed in the case of the bilayer. The
couplings between orbitals at next-nearest-neighbor lay-
ers are 2 and 5. Finally,  is an on-site energy that
reflects the inequivalence between the two sublattices in
each graphene layer once the presence of neighboring
layers is taken into account. The values of these param-
eters, and their dependence with pressure, or, equiva-
lently, the interatomic distances, have been extensively
studied
McClure, 1957, Nozières, 1958; Dresselhaus and
Mavroides, 1964; Soule et al., 1964; Dillon et al., 1977;
Brandt et al., 1988. A representative set of values is
shown in Table I.
The unit cell of graphite with Bernal stacking includes
two layers, and two atoms within each layer. The tight-
binding Hamiltonian described previously can be repre-
sented as a 44 matrix. In the continuum limit, the two
inequivalent corners of the BZ can be treated separately,
and the in-plane terms can be described by the Dirac
equation. The next terms of importance for the low-
energy electronic spectrum are the nearest-neighbor
couplings 1 and 3. The influence of the parameter 4
on the low-energy bands is much smaller, as discussed
below. Finally, the fine details of the spectrum of bulk
graphite are determined by , which breaks the
electron-hole symmetry of the bands preserved by 0, 1,
and 3. It is usually assumed to be much smaller than the
other terms.
We label the two atoms from the unit cell in one layer
as 1 and 2, and 3 and 4 correspond to the second layer.
Atoms 2 and 3 are directly on top of each other. Then,
the matrix elements of the Hamiltonian can be written
as
H11
K = 22 cos
2kz/c ,
H12
K = vF
kx + iky ,
H13
K =
34a
2

1 + eikzc
kx + iky ,
H14
K =
33a
2

1 + eikzc
kx − iky ,
H22
K =  + 25 cos
2kz/c ,
H23
K = 1
1 + e
ikzc
 ,
H24
K =
34a
2

1 + eikzc
kx + iky ,
H33
K =  + 25 cos
2kz/c ,
H34
K = vF
kx + iky ,
H44
K = 22 cos
2kz/c ,
43
where c is the lattice constant in the out-of-plane direc-
tion, equal to twice the interlayer spacing. The matrix
elements of HK
can be obtained by replacing kx by −kx

other conventions for the unit cell and the orientation
of the lattice lead to different phases. Recent ARPES
experiments
Ohta et al., 2006; Zhou, Gweon, and Lan-
zara, 2006; Zhou, Gweon, et al., 2006; Bostwick, Ohta,
Seyller, et al., 2007 performed in epitaxially grown
graphene stacks
Berger et al., 2004 confirm the main
features of this model, formulated on the basis of Fermi
surface measurements
McClure, 1957; Soule et al.,
1964. The electronic spectrum of the model can also be
calculated in a magnetic field
de Gennes, 1964; Nakao,
1976, and the results are also consistent with STM on
graphite surfaces
Kobayashi et al., 2005; Matsui et al.,
2005; Niimi et al., 2006; Li and Andrei, 2007, epitaxially
grown graphene stacks
Mallet et al., 2007, and optical
measurements in the infrared range
Li et al., 2006.
F. Surface states in graphene
So far we have discussed the basic bulk properties of
graphene. Nevertheless, graphene has very interesting
surface
edge states that do not occur in other systems.
A semi-infinite graphene sheet with a zigzag edge has a
band of zero-energy states localized at the surface

Fujita et al., 1996; Nakada et al., 1996; Wakabayashi et
al., 1999. In Sec. II.H, we discuss the existence of edge
states using the Dirac equation. Here we discuss the
same problem using the tight-binding Hamiltonian. To
see why these edge states exist, we consider the ribbon
geometry with zigzag edges shown in Fig. 14. The ribbon
width is such that it has N unit cells in the transverse
cross section
y direction. We assume that the ribbon
has infinite length in the longitudinal direction
x direc-
tion.
We rewrite Eq.
5, with t


=0, in terms of the integer
indices m and n, introduced in Fig. 14, and labeling the
unit cells,
TABLE I. Band-structure parameters of graphite
Dressel-
haus and Dresselhaus, 2002.
0 3.16 eV
1 0.39 eV
2 −0.020 eV
3 0.315 eV
4 0.044 eV
5 0.038 eV
 −0.008 eV
122 Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

H = − t

m,n,
a

†

m,nb


m,n + a

†

m,nb


m − 1,n
+ a

†

m,nb


m,n − 1 + H.c. .
44
Given that the ribbon is infinite in the a1 direction, one
can introduce a Fourier decomposition of the operators
leading to
H = − t

dk
2 n,
a

†

k,nb


k,n + eikaa

†

k,nb


k,n
+ a

†

k,nb


k,n − 1 + H.c. ,
45
where c

†

k ,n0= c , ,k ,n and c=a ,b. The one-
particle Hamiltonian can be written as
H1p = − t

dk

n,

1 + eikaa,k,n,
b,k,n,
+ a,k,n,
b,k,n − 1, + H.c. .
46
The solution of the Schrödinger equation H1p ,k ,
=E
,k ,k , can be generally expressed as
,k, =

n

k,na,k,n, + 
k,nb,k,n, ,

47
where the coefficients  and  satisfy the following
equations:
E
,k
k,n = − t
1 + e
ika

k,n + 
k,n − 1 ,
48
E
,k
k,n = − t
1 + e
−ika

k,n + 
k,n + 1 .
49
As the ribbon has a finite width, we have to be careful
with the boundary conditions. Since the ribbon only ex-
ists between n=0 and n=N−1 at the boundary, Eqs.
48
and
49 read
E
,k
k,0 = − t
1 + e
ika

k,0 ,
50
E
,k
k,N − 1 = − t
1 + e
−ika

k,N − 1 .
51
The surface
edge states are solutions of Eqs.
48–
51
with E
,k=0,
0 =
1 + eika
k,n + 
k,n − 1 ,
52
0 =
1 + e−ika
k,n + 
k,n + 1 ,
53
0 = 
k,0 ,
54
0 = 
k,N − 1 .
55
Equations
52 and
53 are easily solved, giving

k,n = − 2 cos
ka/2nei
ka/2n
k,0 ,
56

k,n = − 2 cos
ka/2N−1−n
e−i
ka/2
N−1−n
k,N − 1 .
57
Consider, for simplicity, a semi-infinite system with a
single edge. We must require the convergence condition
−2 cos
ka /21 in Eq.
57 because otherwise the
wave function would diverge in the semi-infinite
graphene sheet. Therefore, the semi-infinite system has
edge states for ka in the region 2 /3ka4 /3, which
corresponds to 1/3 of the possible momenta. Note that
the amplitudes of the edge states are given by

k,n =
2

k
e−n/
k,
58

k,n =
2

k
e−
N−1−n/
k,
59
where the penetration length is given by

k = − 1/ln2 cos
ka/2 .
60
It is easily seen that the penetration length diverges
when ka approaches the limits of the region
2 /3 ,4 /3.
Although the boundary conditions defined by Eqs.

54 and
55 are satisfied for solutions
56 and
57 in
the semi-infinite system, they are not in the ribbon ge-
ometry. In fact, Eqs.
58 and
59 are eigenstates only in
the semi-infinite system. In the graphene ribbon the two
edge states, which come from both sides of the edge, will
overlap with each other. The bonding and antibonding
states formed by the two edge states will then be the
ribbon eigenstates
Wakabayashi et al., 1999
note that
at zero energy there are no other states with which the
edge states could hybridize. As bonding and antibond-
ing states result in a gap in energy, the zero-energy flat
bands of edge states will become slightly dispersive, de-
pending on the ribbon width N. The overlap between
the two edge states is larger as ka approaches 2 /3 and
4 /3. This means that deviations from zero-energy flat-
ness will be stronger near these points.
Edge states in graphene nanoribbons, as in carbon
nanotubes, are predicted to be Luttinger liquids, that is,
interacting one-dimensional electron systems
Castro
Neto et al., 2006b. Hence, clean nanoribbons must have
1D square root singularities in their density of states

Nakada et al., 1996 that can be probed by Raman spec-
troscopy. Disorder may smooth out these singularities,
however. In the presence of a magnetic field, when the
bulk states are gapped, the edge states are responsible
for the transport of spin and charge
Abanin et al., 2006;
Abanin, Lee, and Levitov, 2007; Abanin and Levitov,
2007; Abanin, Novoselov, Zeitler, et al., 2007.
x
y
m− m m+
n+
m+
n+
n
n−
1
1
1
1
2
2
1
a
2
a
A
B
FIG. 14.
Color online Ribbon geometry with zigzag edges.
123Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

G. Surface states in graphene stacks
Single-layer graphene can be considered a zero gap
semiconductor, which leads to the possibility of midgap
states, at =0, as discussed in the previous section. The
most studied such states are those localized near a
graphene zigzag edge
Fujita et al., 1996; Wakayabashi
and Sigrist, 2000. It can be shown analytically
Castro,
Peres, Lopes dos Santos, et al., 2008 that a bilayer zig-
zag edge, like that shown in Fig. 15, analyzed within the
nearest-neighbor tight-binding approximation described
before, has two bands of localized states, one completely
localized in the top layer and indistinguishable from
similar states in single-layer graphene, and another band
that alternates between the two layers. These states, at
=0, have finite amplitudes on one-half of the sites only.
These bands, as in single-layer graphene, occupy one-
third of the BZ of a stripe bounded by zigzag edges.
They become dispersive in a biased bilayer. As graphite
can be described in terms of effective bilayer systems,
one for each value of the perpendicular momentum kz,
bulk graphite with a zigzag termination should show one
surface band per layer.
H. The spectrum of graphene nanoribbons
The spectrum of graphene nanoribbons depends on
the nature of their edges: zigzag or armchair
Brey and
Fertig, 2006a, 2006b; Nakada et al., 1996. In Fig. 16, we
show a honeycomb lattice having zigzag edges along the
x direction and armchair edges along the y direction. If
we choose the ribbon to be infinite in the x direction, we
produce a graphene nanoribbon with zigzag edges; con-
versely, choosing the ribbon to be macroscopically large
along the y but finite in the x direction, we produce a
graphene nanoribbon with armchair edges.
In Fig. 17, we show 14 energy levels, calculated in the
tight-binding approximation, closest to zero energy for a
nanoribbon with zigzag and armchair edges and of width
N=200 unit cells. We show that both are metallic, and
that the zigzag ribbon presents a band of zero-energy
modes that is absent in the armchair case. This band at
zero energy is the surface states living near the edge of
the graphene ribbon. More detailed ab initio calcula-
tions of graphene nanoribbon spectra show that interac-
tion effects can lead to electronic gaps
Son et al., 2006b
and magnetic states close to the graphene edges, inde-
pendent of their nature
Son et al., 2006a; Yang, Cohen,
and Louie, 2007; Yang, Park, Son, et al., 2007.
From the experimental point of view, however,
graphene nanoribbons currently have a high degree of
roughness at the edges. Such edge disorder can change
significantly the properties of edge states
Areshkin and
White, 2007; Gunlycke et al., 2007, leading to Anderson
localization and anomalies in the quantum Hall effect

Castro Neto et al., 2006b; Martin and Blanter, 2007 as
well as Coulomb blockade effects
Sols et al., 2007. Such
effects have already been observed in lithographically
engineered graphene nanoribbons
Han et al., 2007;
x
y
m+1 m+2m- 1 m
n+1
n
2a
1a
B1
A2
B2
A1
FIG. 15. Sketch of a zigzag termination of a graphene bilayer.
As discussed by Castro, Peres, Lopes dos Santos, et al.
2008,
there is a band of surface states completely localized in the
bottom layer, and another surface band which alternates be-
tween the two.
zigzag edge
ar
m
ch
ai
r
ed
g
e
y
x
a
A
B
FIG. 16.
Color online A piece of a honeycomb lattice dis-
playing both zigzag and armchair edges.
0 1 2 3 4 5 6 7
-1
0
1
en
er
gy
/t
0 0.1 0.2 0.3 0.4
-0.2
0
0.2
0 1 2 3 4 5 6 7
momentum ka
-1
0
1
en
er
gy
/t
1.5 2 2.5 3 3.5 4 4.5
momentum ka
-0.2
0
0.2
FIG. 17. Electronic dispersion for graphene nanoribbons. Left:
energy spectrum, as calculated from the tight-binding equa-
tions, for a nanoribbon with armchair
top and zigzag
bot-
tom edges. The width of the nanoribbon is N=200 unit cells.
Only 14 eigenstates are depicted. Right: zoom of the low-
energy states shown on the right.
124 Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

Özyilmaz et al., 2007. Furthermore, the problem of edge
passivation by hydrogen or other elements is not cur-
rently understood experimentally. Passivation can be
modeled in the tight-binding approach by modifications
of the hopping energies
Novikov, 2007 or via addi-
tional phases in the boundary conditions
Kane and
Mele, 1997. Theoretical modeling of edge passivation
indicate that those have a strong effect on the electronic
properties at the edge of graphene nanoribbons
Barone
et al., 2006; Hod et al., 2007.
In what follows, we derive the spectrum for both zig-
zag and armchair edges directly from the Dirac equa-
tion. This was originally done both with and without a
magnetic field
Nakada et al., 1996; Brey and Fertig,
2006a, 2006b.
1. Zigzag nanoribbons
In the geometry of Fig. 16, the unit-cell vectors are
a1=a0
1,0 and a2=a0
1/2 ,3/2, which generate the
unit vectors of the BZ given by b1
=4 /
a03
3/2 ,−1/2 and b2=4 /
a03
0,1. From
these two vectors, we find two inequivalent Dirac points
given by K=
4 /3a0 ,0=
K ,0 and K

=
−4 /3a0 ,0=
−K ,0, with a0=3a. The Dirac Hamil-
tonian around the Dirac point K reads in momentum
space
HK = vF
0 px − ipy
px + ipy 0

,
61
and around the K


as
HK


= vF
0 px + ipy
px − ipy 0

.
62
The wave function, in real space, for the sublattice A is
given by
A
r = e
iK·r
A
r + e
iK


·r
A

r ,
63
and for sublattice B is given by
B
r = e
iK·r
B
r + e
iK


·r
B

r ,
64
where A and B are the components of the spinor wave
function of Hamiltonian
61 and A
and B
have iden-
tical meaning but relative to Eq.
62. Assume that the
edges of the nanoribbons are parallel to the x axis. In
this case, the translational symmetry guarantees that the
spinor wave function can be written as

r = eikxx

A
y
B
y

,
65
and a similar equation for the spinor of Hamiltonian

62. For zigzag edges, the boundary conditions at the
edge of the ribbon
located at y=0 and y=L, where L is
the ribbon width are
A
y = L = 0, B
y = 0 = 0,
66
leading to
0 = eiKxeikxxA
L + e
−iKxeikxxA

L ,
67
0 = eiKxeikxxB
0 + e
−iKxeikxxB

0 .
68
The boundary conditions
67 and
68 are satisfied for
any x by the choice
A
L = A

L = B
0 = B

0 = 0.
69
We need now to find out the form of the envelope func-
tions. The eigenfunction around the point K has the
form

0 kx − y
kx + y 0

A
y
B
y

= ˜

A
y
B
y

,
70
with ˜= /vF and the energy eigenvalue. The eigen-
problem can be written as two linear differential equa-
tions of the form

kx − yB = ˜A,

kx + yA = ˜B.
71
Applying the operator kx+y to the first of Eqs.
71
leads to

− y
2 + kx
2
B = ˜
2
B,
72
with A given by
A =
1
˜

kx − yB.
73
The solution of Eq.
72 has the form
B =Ae
zy + Be−zy,
74
leading to an eigenenergy ˜2=kx
2−z2. The boundary con-
ditions for a zigzag edge require that A
y=L=0 and
B
y=0=0, leading to
B
y = 0 = 0 ⇔ A + B = 0,
A
y = L = 0 ⇔
kx − zAe
zL +
kx + zBe
−zL = 0,

75
which leads to an eigenvalue equation of the form
e−2zL =
kx − z
kx + z
.
76
Equation
76 has real solutions for z, whenever kx is
positive; these solutions correspond to surface waves

edge states existing near the edge of the graphene rib-
bon. In Sec. II.F, we discussed these states from the
point of view of the tight-binding model. In addition to
real solutions for z, Eq.
76 also supports complex ones,
of the form z= ikn, leading to
kx =
kn
tan
knL
.
77
The solutions of Eq.
77 correspond to confined modes
in the graphene ribbon.
125Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

If we apply the same procedure to the Dirac equation
around the Dirac point K


, we obtain a different eigen-
value equation given by
e−2zL =
kx + z
kx − z
.
78
This equation supports real solutions for z if kx is nega-
tive. Therefore, we have edge states for negative values
kx, with momentum around K
. As in the case of K, the
system also supports confined modes, given by
kx = −
kn
tan
knL
.
79
One should note that the eigenvalue equations for K


are obtained from those for K by inversion, kx→−kx.
We finally note that the edge states for zigzag nano-
ribbons are dispersionless
localized in real space when
t


=0. When electron-hole symmetry is broken
t


0,
these states become dispersive with a Fermi velocity ve
 t


a
Castro Neto et al., 2006b.
2. Armchair nanoribbons
We now consider an armchair nanoribbon with arm-
chair edges along the y direction. The boundary condi-
tions at the edges of the ribbon
located at x=0 and x
=L, where L is the width of the ribbon,
A
x = 0 =B
x = 0 =A
x = L =B
x = L = 0.

80
Translational symmetry guarantees that the spinor wave
function of Hamiltonian
61 can be written as

r = eikyy

A
x
B
x

,
81
and a similar equation for the spinor of the Hamiltonian

62. The boundary conditions have the form
0 = eikyyA
0 + e
ikyy
A

0 ,
82
0 = eikyyB
0 + e
ikyy
B

0 ,
83
0 = eiKLeikyyA
L + e
−iKLeikyyA

L ,
84
0 = eiKLeikyyB
L + e
−iKLeikyyB

L ,
85
and are satisfied for any y if



0 + 




0 = 0
86
and
eiKL


L + e−iKL




L = 0,
87
with =A ,B. It is clear that these boundary conditions
mix states from the two Dirac points. Now we must find
the form of the envelope functions obeying the bound-
ary conditions
86 and
87. As before, the functions B
and B
obey the second-order differential equation
72

with y replaced by x, and the functions A and A
are
determined from Eq.
73. The solutions of Eq.
72 have
the form
B =Ae
iknx + Be−iknx,
88
B
= Ce
iknx +De−iknx.
89
Applying the boundary conditions
86 and
87, one ob-
tains
0 =A + B + C +D ,
90
0 =Aei
kn+KL +De−i
kn+KL + Be−i
kn−KL + Cei
kn−KL.

91
The boundary conditions are satisfied with the choice
A = −D, B = C = 0,
92
which leads to sin
kn+KL=0. Therefore, the allowed
values of kn are given by
kn =
n
L
−
4
3a0
,
93
and the eigenenergies are given by
˜
2 = ky
2 + kn
2 .
94
No surface states exist in this case.
I. Dirac fermions in a magnetic field
We now consider the problem of a uniform magnetic
field B applied perpendicular to the graphene plane.2
We use the Landau gauge: A=B
−y ,0. Note that the
magnetic field introduces a new length scale in the prob-
lem,

B =
c
eB
,
95
which is the magnetic length. The only other scale in the
problem is the Fermi-Dirac velocity. Dimensional analy-
sis shows that the only quantity with dimensions of en-
ergy we can make is vF /
B. In fact, this determines the
cyclotron frequency of the Dirac fermions,
c = 2
vF

B

96

the 2 factor comes from the quantization of the prob-
lem, see below. Equations
95 and
96 show that the
cyclotron energy scales like B, in contrast with the non-
relativistic problem where the cyclotron energy is linear
in B. This implies that the energy scale associated with
the Dirac fermions is rather different from the one
found in the ordinary 2D electron gas. For instance, for
fields of the order of B10 T, the cyclotron energy in
the 2D electron gas is of the order of 10 K. In contrast,
2The problem of transverse magnetic and electric fields can
also be solved exactly. See Lukose et al.
2007 and Peres and
Castro
2007.
126 Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

for the Dirac fermion problem and the same fields, the
cyclotron energy is of the order of 1000 K, that is, two
orders of magnitude larger. This has strong implications
for the observation of the quantum Hall effect at room
temperature
Novoselov et al., 2007. Furthermore, for
B=10 T the Zeeman energy is relatively small, gBB
5 K, and can be disregarded.
We now consider the Dirac equation in more detail.
Using the minimal coupling in Eq.
19
i.e., replacing
−i by −i +eA /c, we find
vF
·
− i  + eA/c
r = E
r ,
97
in the Landau gauge the generic solution for the wave
function has the form 
x ,y=eikx
y, and the Dirac
equation reads
vF
0 y − k + Bey/c
− y − k + Bey/c 0


y = E
y ,

98
which can be rewritten as
c
0 O
O† 0


 = E
 ,
99
or equivalently

O+ +O†− =
2E/c ,
100
where ±=x± iy, and we have defined the dimension-
less length scale
 =
y

B
−
Bk
101
and 1D harmonic-oscillator operators
O =
1
2



+  ,
O† =
1
2

− 

+  ,
102
which obey canonical commutation relations O ,O†=1.
The number operator is simply N=O†O.
First, we note that Eq.
100 allows for a solution with
zero energy,

O+ +O†−0 = 0,
103
and since the Hilbert space generated by 
is of dimen-
sion 2, and the spectrum generated by O† is bounded
from below, we just need to ensure that
O0 = 0,

−
0 = 0,
104
in order for Eq.
103 to be fulfilled. The obvious zero-
mode solution is
0
 = 0
   ⇓  ,
105
where ⇓ indicates the state localized on sublattice A,
and ⇑ indicates the state localized on sublattice B. Fur-
thermore,
O0
 = 0
106
is the ground states of the 1D harmonic oscillator. All
the solutions can now be constructed from the zero
mode,
N,±
 = N−1
   ⇑  ± N
   ⇓  = 
N−1

±N


,

107
and their energy is given by
McClure, 1956
E±
N = ± cN ,
108
where N=0,1 ,2 , . . . is a positive integer and N
 is the
solution of the 1D harmonic oscillator explicitly, N

=2−N/2
N!−1/2 exp−2 /2HN
, where HN
 is a Her-
mite polynomial. The Landau levels at the opposite
Dirac point K


have exactly the same spectrum and
hence each Landau level is doubly degenerate. Of par-
ticular importance for the Dirac problem discussed here
is the existence of a zero-energy state N=0, which is
responsible for the anomalies observed in the quantum
Hall effect. This particular Landau level structure has
been observed by many different experimental probes,
from Shubnikov–de Haas oscillations in single layer
graphene
see Fig. 18
Novoselov, Geim, Morozov, et
al., 2005; Zhang et al., 2005, to infrared spectroscopy
FIG. 18.
Color online SdH oscillations observed in longitu-
dinal resistivity xx of graphene as a function of the charge
carrier concentration n. Each peak corresponds to the popula-
tion of one Landau level. Note that the sequence is not inter-
rupted when passing through the Dirac point, between elec-
trons and holes. The period of oscillations is n=4B /0,
where B is the applied field and 0 is the flux quantum
No-
voselov, Geim, Morozov, et al., 2005.
127Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

Jiang, Henriksen, Tung, et al., 2007, and to scanning
tunneling spectroscopy
Li and Andrei, 2007
STS on a
graphite surface.
J. The anomalous integer quantum Hall effect
In the presence of disorder, Landau levels get broad-
ened and mobility edges appear
Laughlin, 1981. Note
that there will be a Landau level at zero energy that
separates states with hole character
0 from states
with electron character
0. The components of the
resistivity and conductivity tensors are given by
xx =
xx
xx
2 + xy
2 ,
xy =
xy
xx
2 + xy
2 ,
109
where xx
xx is the longitudinal component and xy

xy is the Hall component of the conductivity
resistiv-
ity. When the chemical potential is inside a region of
localized states, the longitudinal conductivity vanishes,
xx=0, and hence xx=0, xy=1/xy. On the other hand,
when the chemical potential is in a region of delocalized
states, when the chemical potential is crossing a Landau
level, we have xx0 and xy varies continuously
Sheng
et al., 2006, 2007.
The value of xy in the region of localized states can
be obtained from Laughlin’s gauge invariance argument

Laughlin, 1981: one imagines making a graphene rib-
bon such as shown in Fig. 19 with a magnetic field B
normal through its surface and a current I circling its
loop. Due to the Lorentz force, the magnetic field pro-
duces a Hall voltage VH perpendicular to the field and
current. The circulating current generates a magnetic
flux  that threads the loop. The current is given by
I = c
E

,
110
where E is the total energy of the system. The localized
states do not respond to changes in , only the delocal-
ized ones. When the flux is changed by a flux quantum
=0=hc /e, the extended states remain the same by
gauge invariance. If the chemical potential is in the re-
gion of localized states, all the extended states below the
chemical potential will be filled both before and after
the change of flux by 0. However, during the change of
flux, an integer number of states enter the cylinder at
one edge and leave at the opposite edge.
The question is: How many occupied states are trans-
ferred between edges? We consider a naive and, as
shown further, incorrect calculation in order to show the
importance of the zero mode in this problem. Each Lan-
dau level contributes with one state times its degeneracy
g. In the case of graphene, we have g=4 since there are
two spin states and two Dirac cones. Hence, we expect
that when the flux changes by one flux quantum, the
change in energy would be Einc= ±4NeVH, where N is
an integer. The plus sign applies to electron states

charge +e and the minus sign to hole states
charge −e.
Hence, we conclude that Iinc= ±4
e2 /hVH and hence
xy,inc=I /VH= ±4Ne2 /h, which is the naive expectation.
The problem with this result is that when the chemical
potential is exactly at half filling, that is, at the Dirac
point, it would predict a Hall plateau at N=0 with
xy,inc=0, which is not possible since there is an N=0
Landau level, with extended states at this energy. The
solution for this paradox is rather simple: because of the
presence of the zero mode that is shared by the two
Dirac points, there are exactly 2
2N+1 occupied
states that are transferred from one edge to another.
Hence, the change in energy is E= ±2
2N+1eVH for a
change of flux of =hc /e. Therefore, the Hall conduc-
tivity is
Schakel, 1991; Gusynin and Sharapov, 2005;
Herbut, 2007; Peres, Guinea, and Castro Neto, 2006a,
2006b
xy =
I
VH
=
c
VH
E

= ± 2
2N + 1
e2
h
,
111
without any Hall plateau at N=0. This result has been
observed experimentally
Novoselov, Geim, Morozov, et
al., 2005; Zhang et al., 2005 as shown in Fig. 20.
K. Tight-binding model in a magnetic field
In the tight-binding approximation, the hopping inte-
grals are replaced by a Peierls substitution,
eieR
R

A·drtR,R


= ei
2/0R
R

A·drtR,R


,
112
where tR,R


represents the hopping integral between the
sites R and R


, with no field present. The tight-binding
Hamiltonian for a single graphene layer, in a constant
magnetic field perpendicular to the plane, is conve-
niently written as
H = − t

m,n,
ei
/0n
1+z/2a

†

m,nb


m,n
+ e−i
/0na

†

m,nb

„m − 1,n −
1 − z/2…
+ ei
/0n
z−1/2a

†

m,nb


m,n − z + H.c. ,

113
FIG. 19.
Color online Geometry of Laughlin’s thought ex-
periment with a graphene ribbon: a magnetic field B is applied
normal to the surface of the ribbon; a current I circles the loop,
generating a Hall voltage VH and a magnetic flux .
128 Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

holding for a graphene stripe with a zigzag
z=1 and
armchair
z=−1 edges oriented along the x direction.
Fourier transforming along the x direction gives
H = − t

k,n,
ei
/0n
1+z/2a

†

k,nb


k,n
+ e−i
/0neikaa

†

k,nb

„k,n −
1 − z/2…
+ ei
/0n
z−1/2a

†

k,nb


k,n − z + H.c. .
We now consider the case of zigzag edges. The eigen-
problem can be rewritten in terms of Harper’s equations

Harper, 1955, and for zigzag edges we obtain
Rammal,
1985
E
,k
k,n = − te
ika/22 cos



0
n −
ka
2


k,n
+ 
k,n − 1

,
114
E
,k
k,n = − te
−ika/22 cos



0
n −
ka
2


k,n
+ 
k,n + 1

,
115
where the coefficients 
k ,n and 
k ,n show up in
Hamiltonian’s eigenfunction 
k written in terms of
lattice-position-state states as

k =

n,

k,na ;k,n, + 
k,nb ;k,n, .

116
Equations
114 and
115 hold in the bulk. Considering
that the zigzag ribbon has N unit cells along its width,
from n=0 to n=N−1, the boundary conditions at the
edges are obtained from Eqs.
114 and
115, and read
E
,k
k,0 = − te
ika/22 cos

ka
2


k,0 ,
117
E
,k
k,N − 1 = − 2te
−ika/2 cos



0

N − 1 −
ka
2


k,N − 1 .
118
Similar equations hold for a graphene ribbon with arm-
chair edges.
In Fig. 21, we show 14 energy levels, around zero en-
ergy, for both the zigzag and armchair cases. The forma-
tion of the Landau levels is signaled by the presence of
flat energy bands, following the bulk energy spectrum.
From Fig. 21, it is straightforward to obtain the value of
the Hall conductivity in the quantum Hall effect regime.
We assume that the chemical potential is in between two
Landau levels at positive energies, shown by the dashed
line in Fig. 21. The Landau level structure shows two
zero-energy modes; one of them is electronlike
hole-
like, since close to the edge of the sample its energy is
shifted upwards
downwards. The other Landau levels
are doubly degenerate. The determination of the values
for the Hall conductivity is done by counting how many
energy levels
of electronlike nature are below the
chemical potential. This counting produces the value
2N+1, with N=0,1 ,2 , . . .
for the case of Fig. 21 one has
FIG. 20.
Color online Quantum Hall effect in graphene as a
function of charge-carrier concentration. The peak at n=0
shows that in high magnetic fields there appears a Landau level
at zero energy where no states exist in zero field. The field
draws electronic states for this level from both conduction and
valence bands. The dashed lines indicate plateaus in xy de-
scribed by Eq.
111. Adapted from Novoselov, Geim, Moro-
zov, et al., 2005.
0 1 2 3 4 5 6
7
-1
-0.5
0
0.5
1
e
n
e
r
g
y
/
t
zigzag: N=200, φ/φ
0
=1/701
0 1 2 3 4 5 6
7
-1
-0.5
0
0.5
1
armchair: N=200, φ/φ
0
=1/701
2 3 4 5 6
momentum ka
-0.4
-0.2
0
0.2
0.4
e
n
e
r
g
y
/
t
0.5 1 1.5 2
momentum ka
-0.4
-0.2
0
0.2
0.4
µ
µ
FIG. 21.
Color online Fourteen energy levels of tight-binding
electrons in graphene in the presence of a magnetic flux 
=0 /701, for a finite stripe with N=200 unit cells. The bottom
panels are zoom-in images of the top ones. The dashed line
represents the chemical potential .
129Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

N=2. From this counting, the Hall conductivity is given,
including an extra factor of 2 accounting for the spin
degree of freedom, by
xy = ± 2
e2
h

2N + 1 = ± 4
e2
h

N +
1
2

.
119
Equation
119 leads to the anomalous integer quantum
Hall effect discussed previously, which is the hallmark of
single-layer graphene.
L. Landau levels in graphene stacks
The dependence of the Landau level energies on the
Landau index N roughly follows the dispersion relation
of the bands, as shown in Fig. 22. Note that, in a trilayer
with Bernal stacking, two sets of levels have a N de-
pendence, while the energies of other two depend lin-
early on N. In an infinite rhombohedral stack, the Lan-
dau levels remain discrete and quasi-2D
Guinea et al.,
2006. Note that, even in an infinite stack with the Ber-
nal structure, the Landau level closest to E=0 forms a
band that does not overlap with the other Landau levels,
leading to the possibility of a 3D integer quantum Hall
effect
Luk’yanchuk and Kopelevich, 2004; Kopelevich
et al., 2006; Bernevig et al., 2007.
The optical transitions between Landau levels can
also be calculated. The selection rules are the same as
for a graphene single layer, and only transitions between
subbands with Landau level indices M and N such that
N= M±1 are allowed. The resulting transitions, with
their respective spectral strengths, are shown in Fig. 23.
The transitions are grouped into subbands, which give
rise to a continuum when the number of layers tends to
infinity. In Bernal stacks with an odd number of layers,
the transitions associated with Dirac subbands with lin-
ear dispersion have the largest spectral strength, and
they give a significant contribution to the total absorp-
tion even if the number of layers is large, NL30–40

Sadowski et al., 2006.
M. Diamagnetism
Back in 1952, Mrozowski
Mrozowski, 1952 studied
diamagnetism of polycrystalline graphite and other
condensed-matter molecular-ring systems. It was con-
cluded that in such ring systems diamagnetism has two
contributions:
i diamagnetism from the filled bands of
electrons, and
ii Landau diamagnetism of free elec-
trons and holes. For graphite the second source of dia-
magnetism is by far the largest of the two.
McClure
1956 computed diamagnetism of a 2D hon-
eycomb lattice using the theory introduced by Wallace

1947, a calculation he later generalized to three-
dimensional graphite
McClure, 1960. For the honey-
comb plane, the magnetic susceptibility
in units of
emu/g is
= −
0.0014
T
0
2 sech2


2kBT

,
120
where  is the Fermi energy, T is the temperature, and
kB is the Boltzmann constant. For graphite, the magnetic
susceptibility is anisotropic and the difference between
(b)
(a)
(c)
(d)
FIG. 22.
Color online Landau levels of graphene stacks
shown in Fig. 13. The applied magnetic field is 1 T.
FIG. 23.
Color online Relative spectral strength of the low energy optical transitions between Landau levels in graphene stacks
with Bernal ordering and an odd number of layers. The applied magnetic field is 1 T. Left: 3 layers. Middle: 11 layers. Right: 51
layers. The large circles are the transitions in a single layer.
130 Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

the susceptibility parallel to the principal axis and that
perpendicular to the principal axis is −21.5
10−6 emu/g. The susceptibility perpendicular to the
principal axis is equal to about the free-atom suscepti-
bility, −0.510−6 emu/g. In the 2D model, the suscepti-
bility turns out to be large due to the presence of fast
moving electrons, with a velocity of the order of vF
106 m/s, which in turn is a consequence of the large
value of the hopping parameter 0. In fact, the suscepti-
bility turns out to be proportional to the square of 0.
Later Sharma et al. extended the calculation of McClure
for graphite by including the effect of trigonal warping
and showed that the low-temperature diamagnetism in-
creases
Sharma et al., 1974.
Safran and DiSalvo
1979, interested in the suscepti-
bility of graphite intercalation compounds, recalculated,
in a tight-binding model, the susceptibility perpendicular
to a graphite plane using Fukuyama’s theory
Fuku-
yama, 1971, which includes interband transitions. The
results agree with those by McClure
1956. Later, Saf-
ran computed the susceptibility of a graphene bilayer
showing that this system is diamagnetic at small values
of the Fermi energy, but there appears a paramagnetic
peak when the Fermi energy is of the order of the inter-
layer coupling
Safran, 1984.
The magnetic susceptibility of other carbon-based ma-
terials, such as carbon nanotubes and C60 molecular sol-
ids, was measured
Heremans et al., 1994 showing a dia-
magnetic response at finite magnetic fields different
from that of graphite. Studying the magnetic response of
nanographite ribbons with both zigzag and armchair
edges was done by Wakabayashi et al. using a numerical
differentiation of the free energy
Wakabayashi et al.,
1999. From these two systems, the zigzag edge state is
of particular interest since the system supports edge
states even in the presence of a magnetic field, leading to
very high density of states near the edge of the ribbon.
The high-temperature response of these nanoribbons
was found to be diamagnetic, whereas the low-
temperature susceptibility is paramagnetic.
The Dirac-like nature of the electronic quasiparticles
in graphene led Ghosal et al.
2007 to consider in gen-
eral the problem of the diamagnetism of nodal fermions,
and Nakamura to study the orbital magnetism of Dirac
fermions in weak magnetic fields
Nakamura, 2007. Ko-
shino and Ando considered the diamagnetism of disor-
dered graphene in the self-consistent Born approxima-
tion, with a disorder potential of the form U
r=1ui
r
−R
Koshino and Ando, 2007. At the neutrality point
and zero temperature, the susceptibility of disordered
graphene is given by

0 = −
gvgs
32
e20
22W
!0
,
121
where gs=gv=2 is the spin and valley degeneracies, W is
a dimensionless parameter for the disorder strength, de-
fined as W=niui
2 /40
2, ni is the impurity concentration,
and !0= c exp−1/
2W, with c a parameter defining a
cutoff function used in the theory. At finite Fermi energy
F and zero temperature, the magnetic susceptibility is
given by

F = −
gvgs
3
e20
2 W
 F
.
122
In the clean limit, the susceptibility is given by
Mc-
Clure, 1956; Safran and DiSalvo, 1979; Koshino and
Ando, 2007

F = −
gvgs
6
e20
2

F .
123
N. Spin-orbit coupling
Spin-orbit coupling describes a process in which an
electron changes simultaneously its spin and angular
momentum or, in general, moves from one orbital wave
function to another. The mixing of the spin and the or-
bital motion is a relativistic effect, which can be derived
from Dirac’s model of the electron. The mixing is large
in heavy ions, where the average velocity of the elec-
trons is higher. Carbon is a light atom, and the spin orbit
interaction is expected to be weak.
The symmetries of the spin-orbit interaction, however,
allow the formation of a gap at the Dirac points in clean
graphene. The spin-orbit interaction leads to a spin-
dependent shift of the orbitals, which is of a different
sign for the two sublattices, acting as an effective mass
within each Dirac point
Dresselhaus and Dresselhaus,
1965; Kane and Mele, 2005; Wang and Chakraborty,
2007a. The appearance of this gap leads to a nontrivial
spin Hall conductance, similar to other models that
study the parity anomaly in relativistic field theory in 2
+1 dimensions
Haldane, 1988. When the inversion
symmetry of the honeycomb lattice is broken, either be-
cause the graphene layer is curved or because an exter-
nal electric field is applied
Rashba interaction, addi-
tional terms, which modulate the nearest-neighbor
hopping, are induced
Ando, 2000. The intrinsic and
extrinsic spin-orbit interactions can be written as

Dresselhaus and Dresselhaus, 1965; Kane and Mele,
2005
HSO;int  so d2rˆ†
rsˆzˆz"ˆzˆ
r ,
HSO;ext  R d2rˆ†
r
− sˆxˆy + sˆyˆx"ˆzˆ
r ,
124
where ˆ and "ˆ are Pauli matrices that describe the sub-
lattice and valley degrees of freedom and sˆ are Pauli
matrices acting on actual spin space. so is the spin-orbit
coupling and R is the Rashba coupling.
The magnitude of the spin-orbit coupling in graphene
can be inferred from the known spin-orbit coupling in
the carbon atom. This coupling allows for transitions be-
tween the pz, px, and py orbitals. An electric field also
induces transitions between the pz and s orbitals. These
intra-atomic processes mix the  and  bands in
131Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

graphene. Using second-order perturbation theory, one
obtains a coupling between the low-energy states in the
 band. Tight-binding
Huertas-Hernando et al., 2006;
Zarea and Sandler, 2007 and band-structure calcula-
tions
Min et al., 2006; Yao et al., 2007 give estimates for
the intrinsic and extrinsic spin-orbit interactions in the
range 0.01−0.2 K, and hence much smaller than the
other energy scales in the problem
kinetic, interaction,
and disorder.
III. FLEXURAL PHONONS, ELASTICITY, AND
CRUMPLING
Graphite, in the Bernal stacking configuration, is a
layered crystalline solid with four atoms per unit cell. Its
basic structure is essentially a repetition of the bilayer
structure discussed earlier. The coupling between layers,
as discussed, is weak and, therefore, graphene has al-
ways been the starting point for the discussion of
phonons in graphite
Wirtz and Rubio, 2004. Graphene
has two atoms per unit cell, and if we consider graphene
as strictly 2D it should have two acoustic modes with
dispersion ac
k
k as k→0 and two optical modes
with dispersion op
k
const, as k→0 solely due to
the in-plane translation and stretching of the graphene
lattice. Nevertheless, graphene exists in the 3D space
and hence the atoms can oscillate out-of-plane leading
to two out-of-plane phonons
one acoustic and another
optical called flexural modes. The acoustic flexural
mode has dispersion flex
k
k2 as k→0, which repre-
sents the translation of the whole graphene plane
essen-
tially a one-atom-thick membrane in the perpendicular
direction
free-particle motion. The optical flexural
mode represents the out-of-phase, out-of-plane oscilla-
tion of the neighboring atoms. In first approximation, if
we neglect the coupling between graphene planes,
graphite has essentially the same phonon modes, al-
though they are degenerate. The coupling between
planes has two main effects:
i it lifts the degeneracy of
the phonon modes, and
ii it leads to a strong suppres-
sion of the energy of the flexural modes. Theoretically,
flexural modes should become ordinary acoustic and op-
tical modes in a fully covalent 3D solid, but in practice
the flexural modes survive due to the fact that the planes
are coupled by weak van der Waals–like forces. These
modes have been measured experimentally in graphite

Wirtz and Rubio, 2004. Graphene can also be obtained
as a suspended membrane, that is, without a substrate,
supported only by a scaffold or bridging micrometer-
scale gaps
Bunch et al., 2007; Meyer, Geim, Katsnelson,
Novoselov, Booth, et al., 2007; Meyer, Geim, Katsnelson,
Novoselov, Obergfell, et al., 2007. Figure 24 shows a
suspended graphene sheet and an atomic resolution im-
age of its crystal lattice.
Because the flexural modes disperse like k2, they
dominate the behavior of structural fluctuations in
graphene at low energies
low temperatures and long
wavelengths. It is instructive to understand how these
modes appear from the point of view of elasticity theory

Chaikin and Lubensky, 1995; Nelson et al., 2004. Con-
sider, for instance, a graphene sheet in 3D and param-
etrize the position of the sheet relative of a fixed coor-
dinate frame in terms of the in-plane vector r and the
height variable h
r so that a position in the graphene is
given by the vector R= „r ,h
r…. The unit vector normal
to the surface is given by
N =
z − h
1 +
h2
,
125
where =
x ,y is the 2D gradient operator and z is the
unit vector in the third direction. In a flat graphene con-
figuration, all the normal vectors are aligned and there-
fore  ·N=0. Deviations from the flat configuration re-
quire misalignment of the normal vectors and cost
elastic energy. This elastic energy can be written as
E0 =
#
2

d2r
 ·N2 
#
2

d2r
2h2,
126
where # is the bending stiffness of graphene, and the
expression in terms of h
r is valid for smooth distor-
tions of the graphene sheet 
h21. The energy
126
is valid in the absence of a surface tension or a substrate
that breaks the rotational and translational symmetry of
the problem, respectively. In the presence of tension,
(b)
(a)
FIG. 24.
Color online Suspended graphene sheet.
a Bright-
field transmission-electron-microscope image of a graphene
membrane. Its central part
homogeneous and featureless re-
gion is monolayer graphene. Adapted from Meyer, Geim,
Katsnelson, Novoselov, Booth, et al., 2007.
b Despite only
one atom thick, graphene remains a perfect crystal at this
atomic resolution. The image is obtained in a scanning trans-
mission electron microscope. The visible periodicity is given by
the lattice of benzene rings. Adapted from Booth et al., 2008.
132 Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

there is an energy cost for solid rotations of the
graphene sheet
h0 and hence a new term has to be
added to the elastic energy,
ET =

2

d2r
h2,
127
where  is the interfacial stiffness. A substrate, de-
scribed by a height variable s
r, pins the graphene sheet
through van der Waals and other electrostatic potentials
so that the graphene configuration tries to follow the
substrate h
rs
r. Deviations from this configuration
cost extra elastic energy that can be approximated by a
harmonic potential
Swain and Andelman, 1999,
ES =
v
2

d2rs
r − h
r2,
128
where v characterizes the strength of the interaction po-
tential between substrate and graphene.
First, consider the free floating graphene problem

126 that we can rewrite in momentum space as
E0 =
#
2 k
k4h−khk.
129
We now canonically quantize the problem by introduc-
ing a momentum operator Pk that has the following
commutator with hk:
hk,Pk


 = ik,k


,
130
and we write the Hamiltonian as
H =

k

P−kPk
2
+
#k4
2
h−khk ,
131
where  is graphene’s 2D mass density. From the
Heisenberg equations of motion for the operators, it is
trivial to find that hk oscillates harmonically with a fre-
quency given by
flex
k = 
#


1/2
k2,
132
which is the long-wavelength dispersion of flexural
modes. In the presence of tension, it is easy to see that
the dispersion is modified to

k = k
#

k2 +


,
133
indicating that the dispersion of flexural modes becomes
linear in k, as k→0, under tension. That is what happens
in graphite, where the interaction between layers breaks
the rotational symmetry of the graphene layers.
Equation
132 also allows us to relate the bending
energy of graphene with the Young modulus Y of graph-
ite. The fundamental resonance frequency of a macro-
scopic graphite sample of thickness t is given by
Bunch
et al., 2007
$
k =

Y


1/2
tk2,
134
where = / t is the 3D mass density. Assuming that Eq.

134 works down to the single plane level, that is, when
t is the distance between planes, we find
# = Yt3,
135
which provides a simple relationship between the bend-
ing stiffness and the Young modulus. Given that Y
1012 N/m and t3.4 Å we find #1 eV. This result is
in good agreement with ab initio calculations of the
bending rigidity
Lenosky et al., 1992; Tu and Ou-Yang,
2002 and experiments in graphene resonators
Bunch et
al., 2007.
The problem of structural order of a “free-floating”
graphene sheet can be fully understood from the exis-
tence of the flexural modes. Consider, for instance, the
number of flexural modes per unit of area at a certain
temperature T,
Nph =
d2k

22
nk =
1
2

0

dk
k
e#/k
2
− 1
,
136
where nk is the Bose-Einstein occupation number

=1/kBT. For T0, the above integral is logarithmically
divergent in the infrared
k→0 indicating a divergent
number of phonons in the thermodynamic limit. For a
system with finite size L, the smallest possible wave vec-
tor is of the order of kmin2 /L. Using kmin as a lower
cutoff in the integral
136, we find
Nph =

LT
2 ln
1
1 − e−LT
2 /L2
,
137
where
LT =
2
kBT

#


1/4

138
is the thermal wavelength of flexural modes. Note that
when LLT, the number of flexural phonons in Eq.

137 diverges logarithmically with the size of the sys-
tem,
Nph 
2
LT
2 ln
L
LT

,
139
indicating that the system cannot be structurally ordered
at any finite temperature. This is nothing but the crum-
pling instability of soft membranes
Chaikin and Luben-
sky, 1995; Nelson et al., 2004. For LLT, one finds that
Nph goes to zero exponentially with the size of the sys-
tem indicating that systems with finite size can be flat at
sufficiently low temperatures. Note that for #1 eV, 
2200 kg/m3, t=3.4 Å
7.510−7 kg/m2, and T
300 K, we find LT1 Å, indicating that free-floating
graphene should always crumple at room temperature
due to thermal fluctuations associated with flexural
phonons. Nevertheless, the previous discussion involves
only the harmonic
quadratic part of the problem. Non-
linear effects such as large bending deformations
Peliti
133Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

and Leibler, 1985, the coupling between flexural and
in-plane modes or phonon-phonon interactions
Le
Doussal and Radzihovsky, 1992; Bonini et al., 2007, and
the presence of topological defects
Nelson and Peliti,
1987 can lead to strong renormalizations of the bending
rigidity, driving the system toward a flat phase at low
temperatures
Chaikin and Lubensky, 1995. This situa-
tion has been confirmed in numerical simulations of free
graphene sheets
Adebpour et al., 2007; Fasolino et al.,
2007.
The situation is rather different if the system is under
tension or in the presence of a substrate or scaffold that
can hold the graphene sheet. In fact, static rippling of
graphene flakes suspended on scaffolds has been ob-
served for single layer as well as bilayers
Meyer, Geim,
Katsnelson, Novoselov, Booth, et al., 2007; Meyer, Geim,
Katsnelson, Novoselov, Obergfell, et al., 2007. In this
case the dispersion, in accordance with Eq.
133, is at
least linear in k, and the integral in Eq.
136 converges
in the infrared
k→0, indicating that the number of
flexural phonons is finite and graphene does not
crumple. We should note that these thermal fluctuations
are dynamic and hence average to zero over time, there-
fore the graphene sheet is expected to be flat under
these circumstances. Obviously, in the presence of a sub-
strate or scaffold described by Eq.
128, static deforma-
tions of the graphene sheet are allowed. Also, hydrocar-
bon molecules that are often present on top of free
hanging graphene membranes could quench flexural
fluctuations, making them static.
Finally, one should note that in the presence of a me-
tallic gate the electron-electron interactions lead to the
coupling of the phonon modes to the electronic excita-
tions in the gate. This coupling could be partially re-
sponsible for the damping of the phonon modes due to
dissipative effects
Seoanez et al., 2007 as observed in
graphene resonators
Bunch et al., 2007.
IV. DISORDER IN GRAPHENE
Graphene is a remarkable material from an electronic
point of view. Because of the robustness and specificity
of the sigma bonding, it is very hard for alien atoms to
replace the carbon atoms in the honeycomb lattice. This
is one of the reasons why the electron mean free path in
graphene can be so long, reaching up to 1 m in the
existing samples. Nevertheless, graphene is not immune
to disorder and its electronic properties are controlled
by extrinsic as well as intrinsic effects that are unique to
this system. Among the intrinsic sources of disorder,
highlight are surface ripples and topological defects. Ex-
trinsic disorder can come about in many different forms:
adatoms, vacancies, charges on top of graphene or in the
substrate, and extended defects such as cracks and
edges.
It is easy to see that from the point of view of single
electron physics that is, terms that can be added to Eq.

5, there are two main terms to which disorder couples.
The first one is a local change in the single site energy,
Hdd =
i
Vi
ai
†ai + bi
†bi ,
140
where Vi is the strength of the disorder potential on site
Ri, which is diagonal in the sublattice indices and hence,
from the point of view of the Dirac Hamiltonian
18,
can be written as
Hdd = d
2r

a=1,2
Va
rˆa
†

rˆa
r ,
141
which acts as a chemical potential shift for the Dirac
fermions, it that is, shifts locally the Dirac point.
Because the density of states vanishes in single-layer
graphene, and as a consequence the lack of electrostatic
screening, charge potentials may be rather important in
determining the spectroscopic and transport properties

Ando, 2006b; Adam et al., 2007; Nomura and Mac-
Donald, 2007. Of particular importance is the Coulomb
impurity problem, where
Va
r =
e2
0
1
r
,
142
with 0 the dielectric constant of the medium. The solu-
tion of the Dirac equation for the Coulomb potential in
2D has been studied analytically
DiVincenzo and Mele,
1984; Biswas et al., 2007; Novikov, 2007a; Pereira, Nils-
son, and Castro Neto, 2007; Shytov et al., 2007. Its so-
lution has many of the features of the 3D relativistic
hydrogen atom problem
Baym, 1969. As in the case of
the 3D problem, the nature of the eigenfunctions de-
pends strongly on graphene’s dimensionless coupling
constant,
g =
Ze2
0vF
.
143
Note that the coupling constant can be varied by either
changing the charge of the impurity Z or modifying the
dielectric environment and changing 0. For ggc=
1
2 ,
the solutions of this problem are given in terms of Cou-
lomb wave functions with logarithmic phase shifts. The
local density of states
LDOS is affected close to the
impurity due the electron-hole asymmetry generated by
the Coulomb potential. The local charge density decays
like 1/r3 plus fast oscillations of the order of the lattice
spacing in the continuum limit this gives rise to a Dirac
delta function for the density
Kolezhuk et al., 2006. As
in 3D QED, the 2D problem becomes unstable for g
gc=
1
2 leading to supercritical behavior and the so-
called fall of electron to the center
Landau and Lifshitz,
1981. In this case, the LDOS is strongly affected by the
presence of the Coulomb impurity with the appearance
of bound states outside the band and scattering reso-
nances within the band
Pereira, Nilsson, and Castro
Neto, 2007 and the local electronic density decays
monotonically like 1/r2 at large distances.
Schedin et al.
2007 argued that without a high
vacuum environment, these Coulomb effects can be
strongly suppressed by large effective dielectric con-
stants due to the presence of a nanometer thin layer of
134 Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

absorbed water
Sabio et al., 2007. In fact, experiments
in ultrahigh vacuum conditions
Chen, Jang, Fuhrer, et
al., 2008 display scattering features in the transport that
can be associated with charge impurities. Screening ef-
fects that affect the strength and range of the Coulomb
interaction are rather nontrivial in graphene
Fogler,
Novikov, and Shklovskii, 2007; Shklovskii, 2007 and,
therefore, important for the interpretation of transport
data
Bardarson et al., 2007; Nomura et al., 2007; San-
Jose et al., 2007; Lewenkopf et al., 2008.
Another type of disorder is the one that changes the
distance or angles between the pz orbitals. In this case,
the hopping energies between different sites are modi-
fied, leading to a new term to the original Hamiltonian

5,
Hod =
i,j
tij

ab

ai
†bj + H.c. + tij

aa

ai
†aj + bi
†bj ,

144
or in Fourier space,
Hod =
k,k


ak
†bk



i,
ab
ti

abei
k−k
·Ri−i


aa·k
+ H.c.
+
ak
†ak


+ bk
†bk




i,
aa
ti

aaei
k−k
·Ri−i


ab·k
,
145
where tij

ab

tij

aa
 is the change of the hopping energy
between orbitals on lattice sites Ri and Rj on the same

different sublattices
we have written Rj=Ri+
, where



ab is the nearest-neighbor vector and 
aa is the next-
nearest-neighbor vector. Following the procedure of
Sec. II.B, we project out the Fourier components of the
operators close to the K and K


points of the BZ using
Eq.
17. If we assume that tij is smooth over the lattice
spacing scale, that is, it does not have a Fourier compo-
nent with momentum K−K



so the two Dirac cones are
not coupled by disorder, we can rewrite Eq.
145 in
real space as
Hod = d
2rA
ra1†
rb1
r + H.c.
+ 
ra1
†

ra1
r + b1
†

rb1
r ,
146
with a similar expression for cone 2 but with A replaced
by A*, where
A
r =




ab
t
ab
re−i


ab·K,
147

r =




aa
t
aa
re−i


aa·K.
148
Note that whereas 
r=*
r, because of the inversion
symmetry of the two triangular sublattices that make up
the honeycomb lattice, A is complex because of a lack of
inversion symmetry for nearest-neighbor hopping.
Hence,
A
r = Ax
r + iAy
r .
149
In terms of the Dirac Hamiltonian
18, we can rewrite
Eq.
146 as
Hod = d
2rˆ1
†

r · A

rˆ1
r + 
rˆ1†
rˆ1
r ,

150
where A
=
Ax ,Ay. This result shows that changes in the
hopping amplitude lead to the appearance of vector A

and scalar  potentials in the Dirac Hamiltonian. The
presence of a vector potential in the problem indicates
that an effective magnetic field B
=
c /evFA
should
also be present, naively implying a broken time-reversal
symmetry, although the original problem was time-
reversal invariant. This broken time-reversal symmetry
is not real since Eq.
150 is the Hamiltonian around
only one of the Dirac cones. The second Dirac cone is
related to the first by time-reversal symmetry, indicating
that the effective magnetic field is reversed in the second
cone. Therefore, there is no global broken symmetry but
a compensation between the two cones.
A. Ripples
Graphene is a one-atom-thick system, the extreme
case of a soft membrane. Hence, like soft membranes, it
is subject to distortions of its structure due to either
thermal fluctuations
as we discussed in Sec. III or in-
teraction with a substrate, scaffold, and absorbands

Swain and Andelman, 1999. In the first case, the fluc-
tuations are time dependent
although with time scales
much longer than the electronic ones, while in the sec-
ond case the distortions act as quenched disorder. In
both cases, disorder occurs because of the modification
of the distance and relative angle between the carbon
atoms due to the bending of the graphene sheet. This
type of off-diagonal disorder does not exist in ordinary
3D solids, or even in quasi-1D or quasi-2D systems,
where atomic chains and atomic planes, respectively, are
embedded in a 3D crystalline structure. In fact,
graphene is also different from other soft membranes
because it is
semimetallic, while previously studied
membranes were insulators.
The problem of bending graphitic systems and its ef-
fect on the hybridization of the  orbitals has been stud-
ied in the context of classical minimal surfaces
Lenosky
et al., 1992 and applied to fullerenes and carbon nano-
tubes
Tersoff, 1992; Kane and Mele, 1997; Zhong-can et
al., 1997; Xin et al., 2000; Tu and Ou-Yang, 2002. In
order to understand the effect of bending on graphene,
consider the situation shown in Fig. 25. The bending of
the graphene sheet has three main effects: the decrease
of the distance between carbon atoms, a rotation of
the pZ orbitals compression or dilation of the lattice
are energetically costly due to the large spring con-
stant of graphene 57 eV/Å2
Xin et al., 2000, and a
rehybridization between  and  orbitals
Eun-Ah Kim
135Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

and Castro Neto, 2008. Bending by a radius R de-
creases the distance between the orbitals from
to d
=2R sin
/
2R
−
3 /24R2 for R
. The decrease in
the distance between the orbitals increases the overlap
between the two lobes of the pZ orbital
Harrison, 1980:
VppaVppa
0
1+
2 /
12R2, where a=, , and Vppa
0 is the
overlap for a flat graphene sheet. The rotation of the pZ
orbitals can be understood within the Slater-Koster for-
malism, namely, the rotation can be decomposed into a
pz−pz
 bond plus a px−px
 bond hybridization
with energies Vpp and Vpp, respectively
Harrison,
1980: V

=Vpp cos2

−Vpp sin2

Vpp−
Vpp
+Vpp
/
2R2, leading to a decrease in the overlap.
Furthermore, the rotation leads to rehybridization be-
tween  and  orbitals leading to a further shift in en-
ergy of the order of
Eun-Ah Kim and Castro Neto,
2008 


Vsp
2 +Vpp
2
 /


− a.
In the presence of a substrate, as discussed in Sec. III,
elasticity theory predicts that graphene can be expected
to follow the substrate in a smooth way. Indeed, by mini-
mizing the elastic energy
126–
128 with respect to the
height h, we get
Swain and Andelman, 1999
#
4h
r − 2h
r + vh
r = vs
r ,
151
which can be solved by Fourier transform,
h
k =
s
k
1 +

tk
2 +

ck
4 ,
152
where

t = 

v

1/2
,

c = 
#
v

1/4
.
153
Equation
152 gives the height configuration in terms of
the substrate profile, and
t and
c provide the length
scales for elastic distortion of graphene on a substrate.
Hence, disorder in the substrate translates into disorder
in the graphene sheet
albeit restricted by elastic con-
straints. This picture has been confirmed by STM mea-
surements on graphene
Ishigami et al., 2007; Stolyarova
et al., 2007 in which strong correlations were found be-
tween the roughness of the substrate and the graphene
topography. Ab initio band-structure calculations also
give support to this scenario
Dharma-Wardana, 2007.
The connection between the ripples and the electronic
problem comes from the relation between the height
field h
r and the local curvature of the graphene sheet
R,
2
R
r
 
2h
r ,
154
hence we see that due to bending the electrons are sub-
ject to a potential that depends on the structure of a
graphene sheet
Eun-Ah Kim and Castro Neto, 2008,
V
r  V0 − 2h
r2,
155
where 
10 eV Å2 is the constant that depends on
microscopic details. The conclusion from Eq.
155 is
that Dirac fermions are scattered by ripples of the
graphene sheet through a potential that is proportional
to the square of the local curvature. The coupling be-
tween geometry and electron propagation is unique to
graphene, and results in additional scattering and resis-
tivity due to ripples
Katsnelson and Geim, 2008.
B. Topological lattice defects
Structural defects of the honeycomb lattice like pen-
tagons, heptagons, and their combinations such as
Stone-Wales defect
a combination of two pentagon-
heptagon pairs are also possible in graphene and can
lead to scattering
Cortijo and Vozmediano, 2007a,
2007b. These defects induce long-range deformations,
which modify the electron trajectories.
We consider first a disclination. This defect is equiva-
lent to the deletion or inclusion of a wedge in the lattice.
The simplest one in the honeycomb lattice is the absence
of a 60° wedge. The resulting edges can be glued in such
a way that all sites remain threefold-coordinated. The
honeycomb lattice is recovered everywhere, except at
the apex of the wedge, where a fivefold ring, a pentagon,
is formed. One can imagine a situation in which the
nearest-neighbor hoppings are unchanged. Nevertheless,
the existence of a pentagon implies that the two sublat-
tices of the honeycomb structure can no longer be de-
fined. A trajectory around the pentagon after a closed
circuit has to change the sublattice index.
The boundary conditions imposed at the edges of a
disclination are shown in Fig. 26, identifying sites from
different sublattices. In addition, the wave functions at
the K and K


points are exchanged when moving around
the pentagon.
Far away from the defect, a slow rotation of the spino-
rial wave function components can be described by a
gauge field that acts on the valley and sublattice indices

González et al., 1992, 1993b. This gauge field is techni-
cally non-Abelian, although a transformation can be de-
fined that makes the resulting Dirac equation equivalent
to one with an effective Abelian gauge field
González et
al., 1993b. The final continuum equation gives a reason-
able description of the electronic spectrum of fullerenes
l
+ +
−−
θ
α
β
RR
z z
x
z z1 2
d
FIG. 25. Bending the surface of graphene by a radius R and its
effect on the pz orbitals.
136 Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

Plain text is unavailable for this page.

case, the number of midgap states near the edge is
roughly proportional to the difference in sites between
the two sublattices near this boundary.
Along a zigzag edge there is one localized state per
three lattice units. This implies that a precursor structure
for localized states at the Dirac energy can be found in
ribbons or constrictions of small lengths
Muñoz-Rojas
et al., 2006, which modifies the electronic structure and
transport properties.
Localized solutions can also be found near other de-
fects that contain broken bonds or vacancies. These
states do not allow an analytical solution, although, as
discussed above, the continuum Dirac equation is com-
patible with many boundary conditions, and it should
describe well localized states that vary slowly over dis-
tances comparable to the lattice spacing. The existence
of these states can be investigated by analyzing the scal-
ing of the spectrum near a defect as a function of the
size of the system L
Vozmediano et al., 2005. A num-
ber of small voids and elongated cracks show states
whose energy scales as L−2, while the energy of ex-
tended states scales as L−1. A state with energy scaling
L−2 is compatible with continuum states for which the
modulus of the wave function decays as r−2 as a function
of the distance from the defect.
E. Self-doping
The band-structure calculations discussed in the pre-
vious sections show that the electronic structure of a
single graphene plane is not strictly symmetrical in en-
ergy
Reich et al., 2002. The absence of electron-hole
symmetry shifts the energy of the states localized near
impurities above or below the Fermi level, leading to a
transfer of charge from or to the clean regions. Hence,
the combination of localized defects and the lack of per-
fect electron-hole symmetry around the Dirac points
leads to the possibility of self-doping, in addition to the
usual scattering processes.
Extended lattice defects, like edges, grain boundaries,
or microcracks, are likely to induce a number of elec-
tronic states proportional to their length L /a, where a is
of the order of the lattice constant. Hence, a distribution
of extended defects of length L at a distance equal to L
itself gives rise to a concentration of L /a carriers per
carbon in regions of size of the order of
L /a2. The
resulting system can be considered a metal with a low
density of carriers, ncarrier
a /L per unit cell, and an elas-
tic mean free path lelas
L. Then, we obtain
F
vF
aL
,
1
"elas

vF
L
,
158
and, therefore,
"elas−1 F when a /L1. Hence, the
existence of extended defects leads to the possibility of
self-doping but maintaining most of the sample in the
clean limit. In this regime, coherent oscillations of trans-
port properties are expected, although the observed
electronic properties may correspond to a shifted Fermi
energy with respect to the nominally neutral defect-free
system.
One can describe the effects that break electron-hole
symmetry near the Dirac points in terms of a finite next-
nearest-neighbor hopping between  orbitals t


in Eq.

5. Consider, for instance, the electronic structure of a
ribbon of width L terminated by zigzag edges, which, as
discussed, lead to surface states for t


=0. The transla-
tional symmetry along the axis of the ribbon allows us to
define bands in terms of a wave vector parallel to this
axis. On the other hand, the localized surface bands, ex-
tending from k

=
2 /3 to k

=−
2 /3, acquire a dis-
persion of the order of t


. Hence, if the Fermi energy
remains unchanged at the position of the Dirac points

Dirac=−3t
, this band will be filled, and the ribbon will
no longer be charge neutral. In order to restore charge
neutrality, the Fermi level needs to be shifted by an
amount of the order of t


. As a consequence, some ex-
tended states near the Dirac points are filled, leading to
the phenomenon of self-doping. The local charge is a
function of distance to the edges, setting the Fermi en-
ergy so that the ribbon is globally neutral. Note that the
charge transferred to the surface states is localized
mostly near the edges of the system.
The charge transfer is suppressed by electrostatic ef-
fects, as large deviations from charge neutrality have an
associated energy cost
Peres, Guinea, and Castro Neto,
2006a. In order to study these charging effects, we add
to the free-electron Hamiltonian
5 the Coulomb en-
ergy of interaction between electrons,
HI =
i,j
Ui,jninj,
159
where ni= 
ai,
† ai,+bi,
† bi, is the number operator at
site Ri, and
Ui,j =
e2
0Ri −Rj

160
is the Coulomb interaction between electrons. We ex-
pect, on physics grounds, that an electrostatic potential
builds up at the edges, shifting the position of the sur-
face states, and reducing the charge transferred to or
FIG. 27.
Color online Sketch of a rough graphene surface.
The full line gives the boundary beyond which the lattice can
be considered undistorted. The number of midgap states
changes depending on the difference in the number of re-
moved sites for two sublattices.
138 Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

from them. The potential at the edge induced by a con-
stant doping  per carbon atom is 
e2 /aW /a
e2 /a is
the Coulomb energy per carbon, and W the width of the
ribbon
W /a is the number of atoms involved. The
charge transfer is stopped when the potential shifts the
localized states to the Fermi energy, that is, when t



e2 /a
W /a. The resulting self-doping is therefore
 
t


a2
e2W
,
161
which vanishes when W→.
We treat the Hamiltonian
159 within the Hartree ap-
proximation
that is, we replace HI by HM.F.= iVini,
where Vi= jUi,j
nj, and solve the problem self-
consistently for
ni. Numerical results for graphene rib-
bons of length L=803a and different widths are shown
in Fig. 28
t


/ t=0.2 and e2 /a=0.5t. The largest width
studied is 0.1 m, and the total number of carbon at-
oms in the ribbon is 105. Note that as W increases, the
self-doping decreases indicating that, for a perfect
graphene plane
W→, the self-doping effect disap-
pears. For realistic parameters, we find that the amount
of self-doping is 10−4−10−5 electrons per unit cell for
sizes 0.1–1 m.
F. Vector potential and gauge field disorder
As discussed in Sec. IV, lattice distortions modify the
Dirac equation that describes the low-energy band struc-
ture of graphene. We consider here deformations that
change slowly on the lattice scale, so that they do not
mix the two inequivalent valleys. As shown earlier, per-
turbations that hybridize the two sublattices lead to
terms that change the Dirac Hamiltonian from vF ·k to
vF ·k+ ·A. Hence, the vector A can be thought of as
if induced by an effective gauge field A. In order to
preserve time-reversal symmetry, this gauge field must
have opposite signs at the two Dirac cones AK=−AK


.
A simple example is a distortion that changes the hop-
ping between all bonds along a given axis of the lattice.
We assume that the sites at the ends of those bonds
define the unit cell, as shown in Fig. 29. If the distortion
is constant, its only effect is to displace the Dirac points
away from the BZ corners. The two inequivalent points
are displaced in opposite directions. This uniform distor-
tion is the equivalent of a constant gauge field, which
does not change the electronic spectrum. The situation
changes if one considers a boundary that separates two
domains where the magnitude of the distortion is differ-
ent. The shift of Dirac points leads to a deflection of the
electronic trajectories that cross the boundary, also
shown in Fig. 29. The modulation of the gauge field
leads to an effective magnetic field, which is of opposite
sign for the two valleys.
We have shown in Sec. IV.B how topological lattice
defects, such as disclinations and dislocations, can be de-
scribed by an effective gauge field. Those defects can
only exist in graphene sheets that are intrinsically
curved, and the gauge field only depends on the topol-
ogy of the lattice. Changes in the nearest-neighbor hop-
ping also lead to effective gauge fields. We next consider
two physical processes that induce effective gauge fields:

i changes in the hopping induced by hybridization be-
tween  and  bands, which arise in curved sheets, and
0 100 200 300 400 500
Position
-0.04
-0.02
0
0.02
0.04
0.06
El
ec
tro
sta
tic
po
te
nt
ia
l
0 10 200.4
0.45
0.5
0.5
0.5005
Ch
ar
ge
de
ns
it y

0 200 400 600 800 1000 1200 1400
Width
0
0.0001
0.0002
0.0003
0.0004
0.0005
D
op
in
g
FIG. 28.
Color online Displaced electronic charge close to a
graphene zigzag edge. Top: self-consistent analysis of the dis-
placed charge density
in units of number of electrons per car-
bon shown as a continuous line, and the corresponding elec-
trostatic potential
in units of t shown as a dashed line, for a
graphene ribbon with periodic boundary conditions along the
zigzag edge
with a length of L=960a and with a circumfer-
ence of size W=803a. Inset: The charge density near the
edge. Due to the presence of the edge, there is a displaced
charge in the bulk
bottom panel that is shown as a function of
width W. Note that the displaced charge vanishes in the bulk
limit
W→, in agreement with Eq.
161. Adapted from
Peres, Guinea, and Castro Neto, 2006a.
K
K’
K
K’
FIG. 29.
Color online Gauge field induced by a simple elastic
strain. Top: the hopping along the horizontal bonds is assumed
to be changed on the right-hand side of the graphene lattice,
defining a straight boundary between the unperturbed and per-
turbed regions
dashed line. Bottom: the modified hopping
acts like a constant gauge field, which displaces the Dirac
cones in opposite directions at the K and K


points of the
Brillouin zone. The conservation of energy and momentum
parallel to the boundary leads to a deflection of electrons by
the boundary.
139Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

ii changes in the hopping due to modulation in the
bond length, which is associated with elastic strain. The
strength of these fields depends on parameters that de-
scribe the value of the - hybridization, and the depen-
dence of hopping on the bond length.
A comparison of the relative strengths of the gauge
fields induced by intrinsic curvature, - hybridization

extrinsic curvature, and elastic strains, arising from a
ripple of typical height and size, is given in Table II.
1. Gauge field induced by curvature
As discussed in Sec. IV.A, when the  orbitals are not
parallel, the hybridization between them depends on
their relative orientation. The angle
i determines the
relative orientation of neighboring orbitals at some po-
sition ri in the graphene sheet. The value of
i depends
on the local curvature of the layer. The relative angle of
rotation of two pz orbitals at position ri and rj can be
written as cos

i−
j=Ni ·Nj, where Ni is the unit vector
perpendicular to the surface, defined in Eq.
125. If rj
=ri+uij, we can write
Ni ·Nj  1 +Ni · 
uij · Ni +
1
2Ni · 
uij · 
2Ni ,

162
where we assume smoothly varying N
r. We use Eq.

125 in terms of the height field h
r N
rz
−h
r−
h2z /2 to rewrite Eq.
162 as
Ni ·Nj  1 −
1
2 
uij ·   h
ri
2.
163
Hence, bending of the graphene sheet leads to a modi-
fication of the hopping amplitude between different sites
of the form
tij  −
tij
0
2

uij ·   h
ri
2,
164
where tij
0 is the bare hopping energy. A similar effect
leads to changes in the electronic states of carbon nano-
tubes
Kane and Mele, 1997. Using the results of Sec.
IV, namely, Eq.
147, we can now show that a vector
potential is generated for nearest-neighbor hopping
u
=
ab
Eun-Ah Kim and Castro Neto, 2008,
Ax
h = −
3Eaba
2
8

x
2h2 −
y
2h2 ,
Ay
h =
3Eaba
2
4

x
2h + y
2hxhyh ,
165
where the coupling constant Eab depends on microscopic
details
Eun-Ah Kim and Castro Neto, 2008. The flux of
effective magnetic field through a ripple of lateral di-
mension l and height h is given by
 
Eaba
2h2
vFl
3 ,
166
where the radius of curvature is R−1hl−2. For a ripple
with l20 nm, h1 nm, taking Eab /vF10 Å−1, we find
10−30.
2. Elastic strain
The elastic free energy for graphene can be written in
terms of the in-plane displacement u
r=
ux ,uy as
Fu =
1
2 
d2r


B − G


i=1,2
uii

2
+ 2G

i,j=1,2
uij
2

,

167
where B is the bulk modulus, G is the shear modulus,
and
uij =
1
2

ui
xj
+
uj
xj


168
is the strain tensor
x1=x and x2=y.
There are many types of static deformation of the
honeycomb lattice that can affect the propagation of
Dirac fermions. The simplest one is due to changes in
the area of the unit cell due to either dilation or contrac-
tion. Changes in the unit-cell area lead to local changes
in the density of electrons and, therefore, local changes
in the chemical potential in the system. In this case, their
effect is similar to the one found in Eq.
148, and we
have
dp
r = g
uxx + uyy ,
169
and their effect is diagonal in the sublattice index.
The nearest-neighbor hopping depends on the length
of the carbon bond. Hence, elastic strains that modify
the relative orientation of atoms also lead to an effective
gauge field, which acts on each K point separately, as
first discussed in relation to carbon nanotubes
Suzuura
and Ando, 2002b; Mañes, 2007. Consider two carbon
atoms located in two different sublattices in the same
unit cell at Ri. The change in the local bond length can
be written as
TABLE II. Estimates of the effective magnetic length, and
effective magnetic fields generated by the deformations. The
intrinsic curvature entry also refers to the contribution from
topological defects.
lB
B
T
h=1 nm, l=10 nm, a=0.1 nm
Intrinsic curvature l

l
h

0.06
Extrinsic curvature l
t
E
l3
ah2
0.006
Elastic strains l
1

al
h2
6
140 Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

ui =



ab
a
· uA
Ri − uB
Ri + 
ab .
170
The local displacements of the atoms in the unit cell can
be related to u
r by
Ando, 2006a




ab · u = #
−1

uA − uB ,
171
where # is a dimensionless quantity that depends on mi-
croscopic details. Changes in the bond length lead to
changes in the hopping amplitude,
tij  tij
0 +
tij
a
ui,
172
and we can write
t
ab
r  
u
r
a
,
173
where
 =
t
ab
 ln
a
.
174
Substituting Eq.
170 into Eq.
173 and the final result
into Eq.
147, one finds
Ando, 2006a
Ax
s =
3
4#
uxx − uyy ,
Ay
s =
3
2#uxy.
175
We assume that the strains induced by a ripple of dimen-
sion l and height h scale as uij
h / l2. Then, using
 /vFa−11 Å−1, we find that the total flux through a
ripple is
 
h2
al
.
176
For ripples such that h1 nm and l20 nm, this esti-
mate gives 10−10, in reasonable agreement with the
estimates by Morozov et al.
2006.
The strain tensor must satisfy some additional con-
straints, as it is derived from a displacement vector field.
These constraints are called Saint-Venant compatibility
conditions
Landau and Lifshitz, 1959,
Wijkl =
uij
xkxl
+
ukl
xixj
−
uil
xjxk
−
ujk
xixl
= 0.
177
An elastic deformation changes the distances in the crys-
tal lattice and can be considered as a change in the met-
ric,
gij = ij + uij.
178
The compatibility equations
177 are equivalent to the
condition that the curvature tensor derived from Eq.

178 is zero. Hence, a purely elastic deformation cannot
induce intrinsic curvature in the sheet, which only arises
from topological defects. The effective fields associated
with elastic strains can be large
Morozov et al., 2006,
leading to significant changes in the electronic wave
functions. An analysis of the resulting state, and the pos-
sible instabilities that may occur, can be found in Guinea
et al.
2008.
3. Random gauge fields
The preceding discussion suggests that the effective
fields associated with lattice defects can modify signifi-
cantly the electronic properties. This is the case when
the fields do not change appreciably on scales compa-
rable to the
effective magnetic length. The general
problem of random gauge fields for Dirac fermions has
been extensively analyzed before the current interest in
graphene, as the topic is also relevant for the IQHE

Ludwig et al., 1994 and d-wave superconductivity

Nersesyan et al., 1994. The one-electron nature of this
two-dimensional problem makes it possible, at the Dirac
energy, to map it onto models of interacting electrons in
one dimension, where many exact results can be ob-
tained
Castillo et al., 1997. The low-energy density of
states 
 acquires an anomalous exponent 


1−, where 0. The density of states is enhanced
near the Dirac energy, reflecting the tendency of disor-
der to close gaps. For sufficiently large values of the
random gauge field, a phase transition is also possible

Chamon et al., 1996; Horovitz and Doussal, 2002.
Perturbation theory shows that random gauge fields
are a marginal perturbation at the Dirac point, leading
to logarithmic divergences. These divergences tend to
have the opposite sign with respect to those induced by
the Coulomb interaction
see Sec. V.B. As a result, a
renormalization-group
RG analysis of interacting elec-
trons in a random gauge field suggests the possibility of
nontrivial phases
Stauber et al., 2005; Aleiner and Efe-
tov, 2006; Altland, 2006; Dell’Anna, 2006; Foster and
Ludwig, 2006a, 2006b; Nomura et al., 2007; Khvesh-
chenko, 2008, where interactions and disorder cancel
each other.
G. Coupling to magnetic impurities
Magnetic impurities in graphene can be introduced
chemically by deposition and intercalation
Calandra
and Mauri, 2007; Uchoa et al., 2008, or self-generated
by the introduction of defects
Kumazaki and
Hirashima, 2006, 2007. The energy dependence of the
density of states in graphene leads to changes in the
formation of a Kondo resonance between a magnetic
impurity and the graphene electrons. The vanishing of
the density of states at the Dirac energy implies that a
Kondo singlet in the ground state is not formed unless
the exchange coupling exceeds a critical value, of the
order of the electron bandwidth, a problem already
studied in connection with magnetic impurities in
d-wave superconductors
Cassanello and Fradkin, 1996,
1997; Polkovnikov et al., 2001; Polkovnikov, 2002; Fritz
et al., 2006. For weak exchange couplings, the magnetic
impurity remains unscreened. An external gate changes
the chemical potential, allowing for a tuning of the
Kondo resonance
Sengupta and Baskaran, 2008. The
situation changes significantly if the scalar potential in-
141Castro Neto et al.: The electronic properties of graphene
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duced by the magnetic impurity is taken into account.
This potential that can be comparable to the bandwidth
allows the formation of midgap states and changes the
phase shift associated with spin scattering
Hentschel
and Guinea, 2007. These phase shifts have a weak loga-
rithmic dependence on the chemical potential, and a
Kondo resonance can exist, even close to the Dirac en-
ergy.
The RKKY interaction between magnetic impurities
is also modified in graphene. At finite fillings, the ab-
sence of intravalley backscattering leads to a reduction
of the Friedel oscillations, which decay as sin
2kFr / r3

Ando, 2006b; Cheianov and Fal’ko, 2006; Wunsch et al.,
2006. This effect leads to an RKKY interaction at finite
fillings, which oscillate and decay as r−3. When interval-
ley scattering is included, the interaction reverts to the
usual dependence on distance in two dimensions r−2

Cheianov and Fal’ko, 2006. At half filling extended de-
fects lead to an RKKY interaction with an r−3 depen-
dence
Vozmediano et al., 2005; Dugaev et al., 2006.
This behavior is changed when the impurity potential is
localized on atomic scales
Brey et al., 2007; Saremi,
2007, or for highly symmetrical couplings
Saremi,
2007.
H. Weak and strong localization
In sufficiently clean systems, where the Fermi wave-
length is much shorter than the mean free path kFl1,
electronic transport can be described in classical terms,
assuming that electrons follow well-defined trajectories.
At low temperatures, when electrons remain coherent
over long distances, quantum effects lead to interference
corrections to the classical expressions for the conduc-
tivity, the weak-localization correction
Bergman, 1984;
Chakravarty and Schmid, 1986. These corrections are
usually due to the positive interference between two
paths along closed loops, traversed in opposite direc-
tions. As a result, the probability that the electron goes
back to the origin is enhanced, so that quantum correc-
tions decrease the conductivity. These interferences are
suppressed for paths longer than the dephasing length l

determined by interactions between the electron and en-
vironment. Interference effects can also be suppressed
by magnetic fields that break down time-reversal sym-
metry and add a random relative phase to the process
discussed above. Hence, in most metals, the conductivity
increases when a small magnetic field is applied
nega-
tive magnetoresistance.
Graphene is special in this respect, due to the chirality
of its electrons. The motion along a closed path induces
a change in the relative weight of the two components of
the wave function, leading to a new phase, which con-
tributes to the interference processes. If the electron
traverses a path without being scattered from one valley
to the other, this
Berry phase changes the sign of the
amplitude of one path with respect to the time-reversed
path. As a consequence, the two paths interfere destruc-
tively, leading to a suppression of backscattering
Suz-
uura and Ando, 2002a. Similar processes take place in
materials with strong spin-orbit coupling, as the spin di-
rection changes along the path of the electron
Berg-
man, 1984; Chakravarty and Schmid, 1986. Hence, if
scattering between valleys in graphene can be neglected,
one expects a positive magnetoresistance, i.e., weak an-
tilocalization. In general, intravalley and intervalley
elastic scattering can be described in terms of two differ-
ent scattering times "intra and "inter, so that if "intra
"inter, one expects weak antilocalization processes,
while if "inter"intra, ordinary weak localization will take
place. Experimentally, localization effects are always
strongly suppressed close to the Dirac point but can be
partially or, in rare cases, completely recovered at high
carrier concentrations, depending on a particular single-
layer sample
Morozov et al., 2006; Tikhonenko et al.,
2008. Multilayer samples exhibit an additional positive
magnetoresistance in higher magnetic fields, which can
be attribued to classical changes in the current distribu-
tion due to a vertical gradient of concentration
Moro-
zov et al., 2006 and antilocalization effects
Wu et al.,
2007.
The propagation of an electron in the absence of in-
tervalley scattering can be affected by the effective
gauge fields induced by lattice defects and curvature.
These fields can suppress the interference corrections to
the conductivity
Morozov et al., 2006; Morpurgo and
Guinea, 2006. In addition, the description in terms of
free Dirac electrons is only valid near the neutrality
point. The Fermi energy acquires a trigonal distortion
away from the Dirac point, and backward scattering
within each valley is no longer completely suppressed

McCann et al., 2006, leading to further suppression of
antilocalization effects at high dopings. Finally, the gra-
dient of external potentials induces a small asymmetry
between the two sublattices
Morpurgo and Guinea,
2006. This effect will also contribute to reduce antilocal-
ization, without giving rise to localization effects.
The above analysis has to be modified for a graphene
bilayer. Although the description of the electronic states
requires a two-component spinor, the total phase around
a closed loop is 2, and backscattering is not suppressed

Kechedzhi et al., 2007. This result is consistent with
experimental observations, which show the existence of
weak localization effects in a bilayer
Gorbachev et al.,
2007.
When the Fermi energy is at the Dirac point, a replica
analysis shows that the conductivity approaches a uni-
versal value of the order of e2 /h
Fradkin, 1986a, 1986b.
This result is valid when intervalley scattering is ne-
glected
Ostrovsky et al., 2006, 2007; Ryu et al., 2007.
Localization is induced when these terms are included

Aleiner and Efetov, 2006; Altland, 2006, as also con-
firmed by numerical calculations
Louis et al., 2007. In-
teraction effects tend to suppress the effects of disorder.
The same result, namely, a conductance of the order of
e2 /h, is obtained for disordered graphene bilayers where
a self-consistent calculation leads to universal conductiv-
ity at the neutrality point
Nilsson, Castro Neto, Guinea,
et al., 2006; Katsnelson, 2007c; Nilsson et al., 2008. In a
biased graphene bilayer, the presence of impurities leads
142 Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

to the appearance of impurity tails in the density of
states due to the creation of midgap states, which are
sensitive to the applied electric field that opens the gap
between the conduction and valence bands
Nilsson and
Castro Neto, 2007.
We point out that most calculations of transport prop-
erties assume self-averaging, that is, one can exchange a
problem with a lack of translational invariance by an
effective medium system with damping. This procedure
only works when the disorder is weak and the system is
in the metallic phase. Close to the localized phase this
procedure breaks down, the system divides itself into
regions of different chemical potential and one has to
think about transport in real space, usually described in
terms of percolation
Cheianov, Fal’ko, Altshuler, et al.,
2007; Shklovskii, 2007. Single electron transistor
SET
measurements of graphene show that this seems to be
the situation in graphene at half filling
Martin et al.,
2008.
Finally, we point out that graphene stacks suffer from
another source of disorder, namely, c axis disorder,
which is due to either impurities between layers or rota-
tion of graphene planes relative to each other. In either
case, the in-plane and out-of-plane transport is directly
affected. This kind of disorder has been observed ex-
perimentally by different techniques
Bar et al., 2007;
Hass, Varchon, Millán-Otoya, et al., 2008. In the case of
the bilayer, the rotation of planes changes substantially
the spectrum restoring the Dirac fermion description

Lopes dos Santos et al., 2007. The transport properties
in the out-of-plane direction are determined by the in-
terlayer current operator jˆn,n+1= it
cA,n,s
† cA,n+1,s
−cA,n+1,s
† cA,n,s, where n is a layer index and A is a ge-
neric index that defines the sites coupled by the inter-
layer hopping t. If we only consider hopping between
nearest-neighbor sites in consecutive layers, these sites
belong to one of the two sublattices in each layer.
In a multilayer with Bernal stacking, these connected
sites are the ones where the density of states vanishes at
zero energy, as discussed above. Hence, even in a clean
system, the number of conducting channels in the direc-
tion perpendicular to the layers vanishes at zero energy

Nilsson, Castro Neto, Guinea, and Peres, 2006; Nilsson
et al., 2008. This situation is reminiscent of the in-plane
transport properties of a single-layer graphene. Similar
to the latter case, a self-consistent Born approximation
for a small concentration of impurities leads to a finite
conductivity, which becomes independent of the number
of impurities.
I. Transport near the Dirac point
In clean graphene, the number of channels available
for electron transport decreases as the chemical poten-
tial approaches the Dirac energy. As a result, the con-
ductance through a clean graphene ribbon is at most
4e2 /h, where the factor of 4 stands for the spin and val-
ley degeneracy. In addition, only one out of every three
possible clean graphene ribbons has a conduction chan-
nel at the Dirac energy. The other two-thirds are semi-
conducting, with a gap of the order of vF /W, where W is
the width. This result is a consequence of the additional
periodicity introduced by the wave functions at the K
and K


points of the Brillouin zone, irrespective of the
boundary conditions.
A wide graphene ribbon allows for many channels,
which can be approximately classified by the momentum
perpendicular to the axis of the ribbon, ky. At the Dirac
energy, transport through these channels is inhibited by
the existence of a gap ky=vFky. Transport through
these channels is suppressed by a factor of the order of
e−kyL, where L is the length of the ribbon. The number
of transverse channels increases as W /a, where W is the
width of the ribbon and a is a length of the order of the
lattice spacing. The allowed values of ky are
ny /W,
where ny is an integer. Hence, for a ribbon such that
WL, there are many channels that satisfy kyL1.
Transport through these channels is not strongly inhib-
ited, and their contribution dominates when the Fermi
energy lies near the Dirac point. The conductance aris-
ing from these channels is given by
Katsnelson, 2006b;
Tworzydlo et al., 2006
G 
e2
h
W
2

dkye
−kyL

e2
h
W
L
.
179
The transmission at normal incidence ky=0 is 1, in
agreement with the absence of backscattering in
graphene, for any barrier that does not induce interval-
ley scattering
Katsnelson et al., 2006. The transmission
of a given channel scales as T
ky=1/cosh2
kyL /2.
Equation
179 shows that the contribution from all
transverse channels leads to a conductance that scales,
similar to a function of the length and width of the sys-
tem, as the conductivity of a diffusive metal. Moreover,
the value of the effective conductivity is of the order of
e2 /h. It can also be shown that the shot noise depends on
current in the same way as in a diffusive metal. A de-
tailed analysis of possible boundary conditions at the
contacts and their influence on evanescent waves can be
found in Robinson and Schomerus
2007 and Schom-
erus
2007. The calculations leading to Eq.
179 can be
extended to a graphene bilayer. The conductance is,
again, a summation of terms arising from evanescent
waves between the two contacts, and it has the depen-
dence on sample dimensions of a 2D conductivity of the
order of e2 /h
Snyman and Beenakker, 2007, although
there is a prefactor twice as big as the one in single-layer
graphene.
The calculation of the conductance of clean graphene
in terms of transmission coefficients, using the Landauer
method, leads to an effective conductivity that is equal
to the value obtained for bulk graphene using diagram-
matic methods, the Kubo formula
Peres, Guinea, and
Castro Neto, 2006b, in the limit of zero impurity con-
centration and zero doping. Moreover, this correspon-
dence remains valid for the case of a bilayer without and
with trigonal warping effects
Koshino and Ando, 2006;
Cserti, Csordés, and Dévid, 2007.
143Castro Neto et al.: The electronic properties of graphene
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Disorder at the Dirac energy changes the conductance
of graphene ribbons in two opposite directions
Louis et
al., 2007:
i a sufficiently strong disorder, with short-
range
intervalley contributions, leads to a localized re-
gime, where the conductance depends exponentially on
the ribbon length, and
ii at the Dirac energy, disorder
allows midgap states that can enhance the conductance
mediated by evanescent waves discussed above. A fluc-
tuating electrostatic potential also reduces the effective
gap for the transverse channels, further enhancing the
conductance. The resonant tunneling regime mediated
by midgap states was suggested by analytical calcula-
tions
Titov, 2007. The enhancement of the conductance
by potential fluctuations can also be studied semianalyti-
cally. In the absence of intervalley scattering, it leads to
an effective conductivity that grows with ribbon length

San-Jose et al., 2007. In fact, analytical and numerical
studies
Bardarson et al., 2007; Nomura et al., 2007; San-
Jose et al., 2007; Lewenkopf et al., 2008, show that the
conductivity obeys a universal scaling with the lattice
size L,

L =
2e2
h
A ln
L/ + B ,
180
where  is a length scale associated with a range of in-
teractions and A and B are numbers of the order of unit
A0.17 and B0.23 for a graphene lattice in the shape
of a square of size L
Lewenkopf et al., 2008. Note,
therefore, that the conductivity is always of the order of
e2 /h and has a weak dependence on size.
J. Boltzmann equation description of dc transport in doped
graphene
It was shown experimentally that the dc conductivity
of graphene depends linearly on the gate potential
No-
voselov et al., 2004; Novoselov, Geim, Morozov, et al.,
2005; Novoselov, Jiang, Schedin, et al., 2005, except very
close to the neutrality point
see Fig. 30. Since the gate
potential depends linearly on the electronic density n,
one has a conductivity 
n. As shown by Shon and
Ando
1998, if the scatterers are short ranged, the dc
conductivity should be independent of the electronic
density, at odds with the experimental result. It has been
shown
Ando, 2006b; Nomura and MacDonald, 2006,
2007 that, by considering a scattering mechanism based
on screened charged impurities, it is possible to obtain
from a Boltzmann equation approach a conductivity
varying linearly with the density, in agreement with the
experimental result
Ando, 2006b; Novikov, 2007b;
Peres, Lopes dos Santos, and Stauber, 2007; Trushin and
Schliemann, 2007; Katsnelson and Geim, 2008.
The Boltzmann equation has the form
Ziman, 1972
− vk · rf
k − e
E + vkH · kf
k = − 
fk
t

scatt
.

181
The solution of the Boltzmann equation in its general
form is difficult and one needs, therefore, to rely upon
some approximation. The first step in the usual approxi-
mation scheme is to write the distribution as f
k
= f0
k+g
k, where f0
k is the steady-state distribu-
tion function and g
k is assumed small. Inserting this
ansatz in into Eq.
181 and keeping only terms that are
linear in the external fields, one obtains the linearized
Boltzmann equation
Ziman, 1972, which reads
−
f0
k
 k
vk · −
k − 
T

rT + eE −
1
e
r
= −

fk
t

scatt
+ vk · rgk + e
vkH · kgk.
182
The second approximation has to do with the form of
the scattering term. The simplest approach is to intro-
duce a relaxation-time approximation,
−

fk
t

scatt
→
gk
"k
,
183
where "k is the relaxation time, assumed to be
momentum-dependent. This momentum dependence is
determined phenomenologically in such way that the
dependence of the conductivity upon the electronic den-
sity agrees with experimental data. The Boltzmann
equation is certainly not valid at the Dirac point, but
since many experiments are performed at finite carrier
density, controlled by an external gate voltage, we ex-
pect the Boltzmann equation to give reliable results if an
appropriate form for "k is used
Adam et al., 2007.
FIG. 30.
Color online Changes in conductivity  of graphene
with varying gate voltage Vg and carrier concentration n. Here
 is proportional to n. Note that samples with higher mobility

1 m2/V s normally show a sublinear dependence, presum-
ably indicating the presence of different types of scatterers.
Inset: Scanning-electron micrograph of one of experimental
devices
in false colors matching those seen in visible optics.
The scale of the micrograph is given by the width of the Hall
bar, which is 1 m. Adapted from Novoselov, Geim, Morozov,
et al., 2005.
144 Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

We compute the Boltzmann relaxation time "k for two
different scattering potentials:
i a Dirac delta function
potential and
ii an unscreened Coulomb potential. The
relaxation time "k is defined as
1
"k
= ni d

k


dk



22
S
k,k



1 − cos
 ,
184
where ni is impurity concentration per unit of area, and
the transition rate S
k ,k


 is given, in the Born approxi-
mation, by
S
k,k


 = 2Hk


,k
2 1
vF

k


− k ,
185
where vFk is the dispersion of Dirac fermions in
graphene and Hk


,k is defined as
Hk


,k = drk


*

rUS
rk
r ,
186
with US
r the scattering potential and k
r is the elec-
tronic spinor wave function of a clean graphene sheet. If
the potential is short ranged
Shon and Ando, 1998 of
the form US=v0
r, the Boltzmann relaxation time is
given by
"k =
4vF
niv0
2
1
k
.
187
On the other hand, if the potential is the Coulomb po-
tential, given by US
r=eQ /4 0 r for charged impuri-
ties of charge Q, the relaxation time is given by
"k =
vF
u0
2k ,
188
where u0
2=niQ2e2 /16 0
2

2. As argued below, the phenom-
enology of Dirac fermions implies that the scattering in
graphene must be of the form
188.
Within the relaxation time approximation, the solu-
tion of the linearized Boltzmann equation when an elec-
tric field is applied to the sample is
gk = −
f0
k
 k
e"kvk · E ,
189
and the electric current reads
spin and valley indexes
included
J =
4
A k
evkgk.
190
Since at low temperatures the following relation
−f0
k / k→
−vFk holds, one can see that assum-
ing Eq.
188 where k is measured relatively to the Dirac
point, the electronic conductivity turns out to be
xx = 2
e2
h

2
u0
2 = 2
e2
h
vF
2
u0
2 n ,
191
where u0 is the strength of the scattering potential
with
dimensions of energy. The electronic conductivity de-
pends linearly on the electron density, in agreement with
the experimental data. We stress that the Coulomb po-
tential is one possible mechanism of producing a scatter-
ing rate of the form
188, but we do not exclude that
other mechanisms may exist see, for instance, Katsnel-
son and Geim
2008.
K. Magnetotransport and universal conductivity
The description of the magnetotransport properties of
electrons in a disordered honeycomb lattice is complex
because of the interference effects associated with the
Hofstadter problem
Gumbs and Fekete, 1997. We sim-
plify our problem by describing electrons in the honey-
comb lattice as Dirac fermions in the continuum ap-
proximation, introduced in Sec. II.B. Furthermore, we
focus only on the problem of short-range scattering in
the unitary limit since in this regime many analytical
results are obtained
Kumazaki and Hirashima, 2006;
Pereira et al., 2006; Peres, Guinea, and Castro Neto,
2006a; Skrypnyk and Loktev, 2006, 2007; Mariani et al.,
2007; Pereira, Lopes dos Santos, and Castro Neto, 2008.
The problem of magnetotransport in the presence of
Coulomb impurities, as discussed, is still an open re-
search problem. A similar approach was considered by
Abrikosov in the quantum magnetoresistance study of
nonstoichiometric chalcogenides
Abrikosov, 1998. In
the case of graphene, the effective Hamiltonian describ-
ing Dirac fermions in a magnetic field
including disor-
der can be written as H=H0+Hi, where H0 is given by
Eq.
5 and Hi is the impurity potential given by
Peres,
Guinea, and Castro Neto, 2006a
Hi = V
j=1
Ni

r − rjI .
192
The formulation of the problem in second quantization
requires the solution of H0, which was done in Sec. II.I.
The field operators, close to the K point, are defined as

the spin index is omitted for simplicity

r =

k
eikx
L

0
0
y

ck,−1
+

n,k,
eikx
2L

n
y − klB
2

n+1
y − klB
2


ck,n,,
193
where ck,n, destroys an electron in band = ±1, with
energy level n and guiding center klB
2 ; ck,−1 destroys an
electron in the zero Landau level; the cyclotron fre-
quency is given by Eq.
96. The sum over n=0,1 ,2 , . . . is
cut off at n0 given by E
1,n0=W, where W is of the
order of the electronic bandwidth. In this representa-
tion, H0 becomes diagonal, leading to Green’s functions
of the form
in the Matsubara representation
G0
k,n, ;i =
1
i − E
,n
,
194
where
145Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

E
,n = cn
195
are the Landau levels for this problem
= ±1 labels the
two bands. Note that G0
k ,n , ; i is effectively k in-
dependent, and E
 ,−1=0 is the zero-energy Landau
level. When expressed in the Landau basis, the scatter-
ing Hamiltonian
192 connects Landau levels of nega-
tive and positive energy.
1. The full self-consistent Born approximation (FSBA)
In order to describe the effect of impurity scattering
on the magnetoresistance of graphene, the Green’s func-
tion for Landau levels in the presence of disorder needs
to be computed. In the context of the 2D electron gas,
an equivalent study was performed by Ohta and Ando

Ohta, 1971a, 1971b; Ando, 1974a, 1974b, 1974c, 1975;
Ando and Uemura, 1974 using the averaging procedure
over impurity positions of Duke
1968. Below, the aver-
aging procedure over impurity positions is performed in
the standard way, namely, having determined the
Green’s function for a given impurity configuration

r1 , . . . ,rNi, the position averaged Green’s function is de-
termined from

G
p,n, ;i ;r1, . . . ,rNi  G
p,n, ;i
= L−2Ni

j=1
Ni

drj
G
p,n, ;i ;r1, . . . ,rNi .

196
In the presence of Landau levels, the average over im-
purity positions involves the wave functions of the one-
dimensional harmonic oscillator. After lengthy algebra,
the Green’s function in the presence of vacancies, in the
FSBA, can be written as
G
p,n, ; + 0+ =  − E
n, − %1

−1,
197
G
p,− 1; + 0+ =  − %2

−1,
198
where
%1
 = − niZ

−1,
199
%2
 = − nigcG
p,− 1; + 0
+
/2 + Z
−1,
200
Z
 = gcG
p,− 1; + 0
+
/2
+ gc
n,
G
p,n, ; + 0+/2,
201
and gc=Ac /2lB
2 is the degeneracy of a Landau level per
unit cell. One should note that the Green’s functions do
not depend upon p explicitly. The self-consistent solu-
tion of Eqs.
197–
201 gives the density of states, the
electron self-energy, and the change of Landau level en-
ergy position due to disorder.
The effect of disorder on the density of states of Dirac
fermions in a magnetic field is shown in Fig. 31. For
reference, we note that E
1,1=0.14 eV for B=14 T,
and E
1,1=0.1 eV for B=6 T. From Fig. 31 we see
that, for a given ni, the effect of broadening due to im-
purities is less effective as the magnetic field increases. It
is also clear that the position of Landau levels is renor-
malized relatively to the nondisordered case. The renor-
malization of the Landau level position can be deter-
mined from poles of Eqs.
197 and
198,
 − E
,n − Re %
 = 0.
202
Due to the importance of scattering at low energies, the
solution to Eq.
202 does not represent exact eigen-
states of the system since the imaginary part of the self-
energy is nonvanishing. However, these energies do de-
termine the form of the density of states, as discussed
below.
In Fig. 32, the graphical solution to Eq.
202 is given
for two different energies E
−1,n, with n=1,2. It is
clear that the renormalization is important for the first
Landau level. This result is due to the increase in scat-
tering at low energies, which is already present in the
case of zero magnetic field. The values of  satisfying
Eq.
202 show up in the density of states as the energy
values where the oscillations due to the Landau level
quantization have a maximum. In Fig. 31, the position of
the renormalized Landau levels is shown in the top
panel
marked by two arrows, corresponding to the
bare energies E
−1,n, with n=1,2. The importance of
this renormalization decreases with the decrease in the
number of impurities. This is clear in Fig. 31, where a
0
0.002
0.004
0.006
0.008
D
O
S
pe
ru
.c
.(
1/
eV
)
ni = 0.001
ni = 0.005
ni = 0.0009
ωc = 0.14 eV, B = 12 T
-3 -2 -1 0 1 2 3
ω / ωc
0
0.002
0.004
0.006
0.008
D
O
S
pe
ru
.c
.(
1/
eV
) ni = 0.001
ni = 0.0008
ni = 0.0006
ωc = 0.1 eV, B = 6 T
FIG. 31.
Color online Density of states of Dirac fermions in
a magnetic field. Top: electronic density of states
DOS 

as a function of  /c
c=0.14 eV in a magnetic field B
=12 T for different impurity concentrations ni. Bottom: 
 as
a function of  /c
c=0.1 eV is the cyclotron frequency in a
magnetic field B=6 T. The solid line shows the DOS in the
absence of disorder. The position of the Landau levels in the
absence of disorder is shown as vertical lines. The two arrows
in the top panel show the position of the renormalized Landau
levels
see Fig. 32 given by the solution of Eq.
202. Adapted
from Peres, Guinea, and Castro Neto, 2006a.
146 Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

'0
q  0 −
22
a20

q,0 ,
209
where 0 is the bare phonon frequency, and the
electron-phonon polarization function is given by

q, =

s,s


=±1

d2k

22
fEs
k + q − fEs



k
0 − Es
k + q + Es



k + i(
,

210
where Es
q is the Dirac fermion dispersion
s= +1 for
the upper band and s=−1 for the lower band, and fE
is the Fermi-Dirac distribution function. For Raman
spectroscopy, the response of interest is at q=0, where
only the interband processes such that ss


=−1
that is,
processes between the lower and upper cones contrib-
ute. The electron-phonon polarization function can be
calculated using the linearized Dirac fermion dispersion

7 and the low-energy density of states
15,

0,0 =
63
vF
2
0
vF&
dEE
f− E − fE


1
0 + 2E + i(
−
1
 − 2E + i(

,
211
where we have introduced the cutoff momentum &

1/a so that the integral converges in the ultraviolet.
At zero temperature T=0 we have fE=

−E and
we assume electron doping 0, so that f−E=1
for
the case of hole doping, 0, obtained by electron-hole
symmetry. The integration in Eq.
211 gives

0,0 =
63
vF
2 vF& − 
+
0
4

ln

0/2 + 
0/2 − 

+ i

0/2 −  ,

212
where the cutoff-dependent term is a contribution com-
ing from the occupied states in the lower  band and
hence is independent of the chemical potential value.
This contribution can be fully incorporated into the bare
value of 0 in Eq.
209. Hence the relative shift in the
phonon frequency can be written as
0
0
 −

4

−

0
+ ln

0/2 + 
0/2 − 

+ i

0/2 −  ,

213
where
 =
363


2
8Ma20

214
is the dimensionless electron-phonon coupling. Note
that Eq.
213 has a real and an imaginary part. The real
part represents the actual shift in frequency, while the
imaginary part gives the damping of the phonon mode
due to pair production
see Fig. 35. There is a clear
change in behavior depending on whether  is larger or
smaller than 0 /2. For 0 /2, there is a decrease in
the phonon frequency implying that the lattice is soften-
ing, while for 0 /2, the lattice hardens. The interpre-
tation for this effect is also given in Fig. 35. On the one
hand, if the frequency of the phonon is larger than twice
the chemical potential, real electron-hole pairs are pro-
duced, leading to stronger screening of the electron-ion
interaction, and hence to a softer phonon mode. At the
same time, the phonons become damped and decay. On
the other hand, if the frequency of the phonon is smaller
than twice the chemical potential, the production of
electron-hole pairs is halted by the Pauli principle and
only virtual excitations can be generated, leading to po-
larization and lattice hardening. In this case, there is no
damping and the phonon is long lived. This result has
been observed experimentally by Raman spectroscopy

Pisana et al., 2007; Yan et al., 2007. Electron-phonon
coupling has also been investigated theoretically for a
finite magnetic field
Ando, 2007a; Goerbig et al., 2007.
In this case, resonant coupling occurs due to the large
degeneracy of the Landau levels, and different Raman
transitions are expected as compared with the zero-field
case. The coupling of electrons to flexural modes on a
free-standing graphene sheet was discussed by Mariani
and von Oppen
2008.
0 0.5 1 1.5 2
Μ
Ω0
-1.5
-1
-0.5
0
0.5
∆
Ω
0



Λ
Ω
0

(b)
(a)
FIG. 35. Kohn anomaly in graphene. Top: the continuous line
is the relative phonon frequency shift as a function of  /0,
and the dashed line is the damping of the phonon due to
electron-hole pair creation. Bottom:
a electron-hole process
that leads to phonon softening
02, and
b electron-hole
process that leads to phonon hardening
02.
149Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

B. Electron-electron interactions
Of all disciplines of condensed-matter physics, the
study of electron-electron interactions is probably the
most complex since it involves understanding the behav-
ior of a macroscopic number of variables. Hence, the
problem of interacting systems is a field in constant mo-
tion and we shall not try to give here a comprehensive
survey of the problem for graphene. Instead, we focus
on a small number of topics that are of current discus-
sion in the literature.
Since graphene is a truly 2D system, it is informative
to compare it with the more standard 2DEG that has
been studied extensively in the past 25 years since the
development of heterostructures and the discovery of
the quantum Hall effect for a review, see Stone
1992.
At the simplest level, metallic systems have two main
kinds of excitations: electron-hole pairs and collective
modes such as plasmons.
Electron-hole pairs are incoherent excitations of the
Fermi sea and a direct result of Pauli’s exclusion prin-
ciple: an electron inside the Fermi sea with momentum k
is excited outside the Fermi sea to a new state with mo-
mentum k+q, leaving a hole behind. The energy associ-
ated with such an excitation is = k+q− k, and for states
close to the Fermi surface
kkF their energy scales
linearly with the excitation momentum qvFq. In a
system with nonrelativistic dispersion such as normal
metals and semiconductors, the electron-hole continuum
is made out of intraband transitions only and exists even
at zero energy since it is always possible to produce
electron-hole pairs with arbitrarily low energy close to
the Fermi surface, as shown in Fig. 36
a. Besides that,
the 2DEG can also sustain collective excitations such as
plasmons that have dispersion plasmon
q
q, and exist
outside the electron-hole continuum at sufficiently long
wavelengths
Shung, 1986a.
In systems with relativisticlike dispersion, such as
graphene, these excitations change considerably, espe-
cially when the Fermi energy is at the Dirac point. In
this case, the Fermi surface shrinks to a point and hence
intraband excitations disappear and only interband tran-
sitions between the lower and upper cones can exist see
Fig. 36
b. Therefore, neutral graphene has no electron-
hole excitations at low energy, instead each electron-
hole pair costs energy and hence the electron hole occu-
pies the upper part of the energy versus momentum
diagram. In this case, plasmons are suppressed and no
coherent collective excitations can exist. If the chemical
potential is moved away from the Dirac point, then in-
traband excitations are restored and the electron-hole
continuum of graphene shares features of the 2DEG
and undoped graphene. The full electron-hole con-
tinuum of doped graphene is shown in Fig. 36
c, and in
this case plasmon modes are allowed. As the chemical
potential is moved away from the Dirac point, graphene
resembles more and more the 2DEG.
These features in the elementary excitations of
graphene reflect its screening properties as well. In fact,
the polarization and dielectric functions of undoped
graphene are different from the ones of the 2DEG

Lindhard function. In the random-phase approxima-
tion
RPA, the polarization function can be calculated
analytically
Shung, 1986a; González et al., 1993a, 1994,
)
q, =
q2
4vF
2q2 − 2
,
215
and hence, for vFq, the polarization function is
imaginary, indicating the damping of electron-hole pairs.
Note that the static polarization function
=0 vanishes
linearly with q, indicating the lack of screening in the
system. This polarization function has also been calcu-
lated in the presence of a finite chemical potential

Shung, 1986a, 1986b; Ando, 2006b; Wunsch et al., 2006;
Hwang and Das Sarma, 2007.
Undoped, clean graphene is a semimetal, with a van-
ishing density of states at the Fermi level. As a result,
the linear Thomas-Fermi screening length diverges, and
the long-range Coulomb interaction is not screened. At
finite electron density n, the Thomas-Fermi screening
length reads
TF 
1
4
1
kF
=
1
4
1

n
,
216
where
 =
e2
0vF

217
is the dimensionless coupling constant in the problem
the analog of Eq.
143 in the Coulomb impurity prob-
lem. Going beyond the linear Thomas-Fermi regime, it
FIG. 36.
Color online Electron-hole continuum and collec-
tive modes of
a a 2DEG,
b undoped graphene, and
c
doped graphene.
150 Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

has been shown that the Coulomb law is modified

Katsnelson, 2006a; Fogler, Novikov, and Shklovskii,
2007; Zhang and Fogler, 2008.
The Dirac Hamiltonian in the presence of interactions
can be written as
H  − ivF d2rˆ†
r · ˆ
r
+
e2
2 0

d2rd2r


1
r − r



ˆ
rˆ
r


 ,
218
where
ˆ
r = ˆ†
rˆ
r
219
is the electronic density. One can observe that the Cou-
lomb interaction, unlike in QED, is assumed to be in-
stantaneous since vF /c1/300 and hence retardation ef-
fects are very small. Moreover, photons propagate in 3D
space whereas electrons are confined to the 2D
graphene sheet. Hence, the Coulomb interaction breaks
the Lorentz invariance of the problem and makes the
many-body situation different from the one in QED

Baym and Chin, 1976. Furthermore, the problem de-
pends on two parameters: vF and e2 / 0. Under a dimen-
sional scaling, r→r , t→t ,→−1, both parameters
remain invariant. In RG language, the Coulomb interac-
tion is a marginal variable, whose strength relative to the
kinetic energy does not change upon a change in scale. If
the units are chosen such that vF is dimensionless, the
value of e2 / 0 will also be rendered dimensionless. This
is the case in theories considered renormalizable in
quantum field theory.
The Fermi velocity in graphene is comparable to that
in half-filled metals. In solids with lattice constant a, the
total kinetic energy per site 1/ma2, where m is the bare
mass of the electron, is of the same order of magnitude
as the electrostatic energy e2 / 0a. The Fermi velocity for
fillings of the order of unity is vF1/ma. Hence,
e2 / 0vF1. This estimate is also valid in graphene.
Hence, unlike in QED, where QED=1/137, the cou-
pling constant in graphene is 1.
Despite the fact that the coupling constant is of the
order of unity, a perturbative RG analysis can be ap-
plied. RG techniques allow us to identify stable fixed
points of the model, which may be attractive over a
broader range than the one where a perturbative treat-
ment can be rigorously justified. Alternatively, an RG
scheme can be reformulated as the process of piecewise
integration of high-energy excitations
Shankar, 1994.
This procedure leads to changes in the effective low-
energy couplings. The scheme is valid if the energy of
the renormalized modes is much larger than the scales of
interest.
The Hartree-Fock correction due to Coulomb interac-
tions between electrons
given by Fig. 37 gives a loga-
rithmic correction to the electron self-energy
González
et al., 1994,
%HF
k =

4
k ln

&
k

,
220
where & is a momentum cutoff which sets the range of
validity of the Dirac equation. This result remains true
even to higher order in perturbation theory
Mish-
chenko, 2007 and is also obtained in large N expansions

Rosenstein et al., 1989, 1991; Son, 2007
where N is the
number of flavors of Dirac fermions, with the only
modification the prefactor in Eq.
220. This result im-
plies that the Fermi velocity is renormalized toward
higher values. As a consequence, the density of states
near the Dirac energy is reduced, in agreement with the
general trend of repulsive interactions to induce or in-
crease gaps.
This result can be understood from the RG point of
view by studying the effect of reducing the cutoff from &
to &−d& and its effect on the effective coupling. It can
be shown that  obeys
González et al., 1994
&

&
= −

2
4
.
221
Therefore, the Coulomb interaction becomes marginally
irrelevant. These features are confirmed by a full relativ-
istic calculation, although the Fermi velocity cannot sur-
pass the velocity of light
González et al., 1994. This
result indicates that strongly correlated electronic
phases, such as ferromagnetism
Peres et al., 2005 and
Wigner crystals
Dahal et al., 2006, are suppressed in
clean graphene.
A calculation of higher-order self-energy terms leads
to a wave-function renormalization, and to a finite qua-
siparticle lifetime, which grows linearly with quasiparti-
cle energy
González et al., 1994, 1996. The wave-
function renormalization implies that the quasiparticle
weight tends to zero as its energy is reduced. A strong-
coupling expansion is also possible, assuming that the
number of electronic flavors justifies an RPA expansion,
keeping only electron-hole bubble diagrams
González
et al., 1999. This analysis confirms that the Coulomb
interaction is renormalized toward lower values.
The enhancement in the Fermi velocities leads to a
widening of the electronic spectrum. This is consistent
with measurements of the gaps in narrow single-wall
nanotubes, which show deviations from the scaling with
R−1, where R is the radius, expected from the Dirac
equation
Kane and Mele, 2004. The linear dependence
of the inverse quasiparticle lifetime with energy is con-
sistent with photoemission experiments in graphite, for
energies larger with respect to the interlayer interactions

Xu et al., 1996; Zhou, Gweon, and Lanzara, 2006; Zhou,
Gweon, et al., 2006; Bostwick, Ohta, Seyller, et al., 2007;
k
k+q
v
q
FIG. 37.
Color online Hartree-Fock self-energy diagram that
leads to a logarithmic renormalization of the Fermi velocity.
151Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

Sugawara et al., 2007. Note that in graphite, band-
structure effects modify the lifetimes at low energies

Spataru et al., 2001. The vanishing of the quasiparticle
peak at low energies can lead to an energy-dependent
renormalization of the interlayer hopping
Vozmediano
et al., 2002, 2003. Other thermodynamic properties of
undoped and doped graphene can also be calculated

Barlas et al., 2007; Vafek, 2007.
Nonperturbative calculations of the long-range inter-
action effects in undoped graphene show that a transi-
tion to a gapped phase is also possible, when the number
of electronic flavors is large
Khveshchenko, 2001;
Luk’yanchuk and Kopelevich, 2004; Khveshchenko and
Shively, 2006. The broken symmetry phase is similar to
the excitonic transition found in materials where it be-
comes favorable to create electron-hole pairs that then
form bound excitons
excitonic transition.
Undoped graphene cannot have well-defined plas-
mons, as their energies fall within the electron-hole con-
tinuum, and therefore have a significant Landau damp-
ing. At finite temperatures, however, thermally excited
quasiparticles screen the Coulomb interaction, and an
acoustic collective charge excitation can exist
Vafek,
2006.
Doped graphene shows a finite density of states at the
Fermi level, and the long-range Coulomb interaction is
screened. Accordingly, there are collective plasma inter-
actions near q→0, which disperse as pq, since the
system is 2D
Shung, 1986a, 1986b; Campagnoli and To-
satti, 1989. The fact that the electronic states are de-
scribed by the massless Dirac equation implies that P
n1/4, where n is the carrier density. The static dielectric
constant has a continuous derivative at 2kF, unlike in the
case of the 2D electron gas
Ando, 2006b; Wunsch et al.,
2006; Sarma et al., 2007. This fact is associated with the
suppressed backward scattering in graphene. The sim-
plicity of the band structure of graphene allows analyti-
cal calculation of the energy and momentum depen-
dence of the dielectric function
Wunsch et al., 2006;
Sarma et al., 2007. The screening of the long-range Cou-
lomb interaction implies that the low-energy quasiparti-
cles show a quadratic dependence on energy with re-
spect to the Fermi energy
Hwang et al., 2007.
One way to probe the strength of the electron-
electron interactions is via the electronic compressibility.
Measurements of the compressibility using a single-
electron transistor
SET show little sign of interactions
in the system, being well fitted by the noninteracting
result that, contrary to the two-dimensional electron gas

2DEG
Eisenstein et al., 1994; Giuliani and Vignale,
2005, is positively divergent
Polini et al., 2007; Martin
et al., 2008. Bilayer graphene, on the other hand, shares
characteristics of the single layer and the 2DEG with a
nonmonotonic dependence of the compressibility on the
carrier density
Kusminskiy et al., 2008. In fact, bilayer
graphene very close to half filling has been predicted to
be unstable toward Wigner crystallization
Dahal et al.,
2007, just like the 2DEG. Furthermore, according to
Hartree-Fock calculations, clean bilayer graphene is un-
stable toward ferromagnetism
Nilsson et al., 2006a.
1. Screening in graphene stacks
The electron-electron interaction leads to the screen-
ing of external potentials. In a doped stack, the charge
tends to accumulate near the surfaces, and its distribu-
tion is determined by the dielectric function of the stack
in the out-of-plane direction. The same polarizability de-
scribes the screening of an external field perpendicular
to the layers, similar to the one induced by a gate in
electrically doped systems
Novoselov et al., 2004. The
self-consistent distribution of charge in a biased
graphene bilayer has been studied by McCann
2006.
From the observed charge distribution and self-
consistent calculations, an estimate of the band-structure
parameters and their relation with the induced gap can
be obtained
Castro, Novoselov, Morozov, et al., 2007.
In the absence of interlayer hopping, the polarizability
of a set of stacks of 2D electron gases can be written as
a sum of the screening by the individual layers. Using
the accepted values for the effective masses and carrier
densities of graphene, this scheme gives a first approxi-
mation to screening in graphite
Visscher and Falicov,
1971. The screening length in the out-of-plane direction
is of about two graphene layers
Morozov et al., 2005.
This model is easily generalizable to a stack of semimet-
als described by the 2D Dirac equation
González et al.,
2001. At half filling, the screening length in all direc-
tions diverges, and the screening effects are weak.
Interlayer hopping modifies this picture significantly.
The hopping leads to coherence
Guinea, 2007. The
out-of-plane electronic dispersion is similar to that of a
one-dimensional conductor. The out-of-plane polariz-
ability of a multilayer contains intraband and interband
contributions. The subbands in a system with Bernal
stacking have a parabolic dispersion, when only the
nearest-neighbor hopping terms are included. This band
structure leads to an interband susceptibility described
by a sum of terms like those in Eq.
228, which diverges
at half filling. In an infinite system, this divergence is
more pronounced at k


= /c, that is, for a wave vector
equal to twice the distance between layers. This effect
enhances Friedel-like oscillations in the charge distribu-
tion in the out-of-plane direction, which can lead to the
changes in the sign of the charge in neighboring layers

Guinea, 2007. Away from half filling, a graphene bi-
layer behaves, from the point of view of screening, in a
way very similar to the 2DEG
Wang and Chakraborty,
2007b.
C. Short-range interactions
In this section, we discuss the effect of short-range
Coulomb interactions on the physics of graphene. The
simplest carbon system with a hexagonal shape is the
benzene molecule. The value of the Hubbard interaction
among  electrons was, for this system, computed long
ago by Parr et al.
1950, yielding U=16.93 eV. For com-
parison purposes, in polyacetylene the value for the
Hubbard interaction is U
10 eV and the hopping en-
ergy is t2.5 eV
Baeriswyl et al., 1986. These two ex-
152 Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

amples show that the value of the on-site Coulomb in-
teraction is fairly large for  electrons. As a first guess
for graphene, one can take U to be of the same order as
for polyacethylene, with the hopping integral t
2.8 eV.
Of course in pure graphene the electron-electron inter-
action is not screened, since the density of states is zero
at the Dirac point, and one should work out the effect of
Coulomb interactions by considering the bare Coulomb
potential. On the other hand, as shown before, defects
induce a finite density of states at the Dirac point, which
could lead to an effective screening of the long-range
Coulomb interaction. We assume that the bare Coulomb
interaction is screened in graphene and that Coulomb
interactions are represented by the Hubbard interaction.
This means that we must add to the Hamiltonian
5 a
term of the form
HU =U
Ri
a↑
†

Ria↑
Ria↓
†

Ria↓
Ri
+ b↑
†

Rib↑
Rib↓
†

Rib↓
Ri .
222
The simplest question one can ask is whether this system
shows a tendency toward some kind of magnetic order
driven by the interaction U. Within the simplest
Hartree-Fock approximation
Peres et al., 2004, the in-
stability line toward ferromagnetism is given by
UF
 =
2


,
223
which is merely the Stoner criterion. Similar results are
obtained in more sophisticated calculations
Herbut,
2006. At half filling the value for the density of states is

0=0 and the critical value for UF is arbitrarily large.
Therefore, we do not expect a ferromagnetic ground
state at the neutrality point of one electron per carbon
atom. For other electronic densities, 
 becomes finite
producing a finite value for UF. We note that the inclu-
sion of t


does not change these findings, since the den-
sity of states remains zero at the neutrality point.
The critical interaction strength toward an antiferro-
magnetic ground state is given by
Peres et al., 2004
UAF
 =
2

1/N

k,0
1/E+
k
,
224
where E+
k is given in Eq.
6. This result gives a finite
UAF at the neutrality point
Sorella and Tosatti, 1992;
Martelo et al., 1997,
UAF
0 = 2.23t .
225
Quantum Monte Carlo calculations
Sorella and Tosatti,
1992; Paiva et al., 2005, however, raise its value to
UAF
0
5t .
226
Taking for graphene the same value for U as in poly-
acetylene and t=2.8 eV, one obtains U / t
3.6, which
puts the system far from the transition toward an anti-
ferromagnet ground state. Yet another possibility is that
the system may be in a sort of a quantum spin liquid

Lee and Lee, 2005 as originally proposed by Pauling

1972 in 1956 since mean-field calculations give a criti-
cal value for U to be of the order of U / t
1.7. Whether
this type of ground state really exists and whether quan-
tum fluctuations push this value of U toward larger val-
ues is not known.
1. Bilayer graphene: Exchange
The exchange interaction can be large in an unbiased
graphene bilayer with a small concentration of carriers.
It was shown that the exchange contribution to the elec-
tronic energy of a single graphene layer does not lead to
a ferromagnetic instability
Peres et al., 2005. The rea-
son for this is a significant contribution from the inter-
band exchange, which is a term usually neglected in
doped semiconductors. This contribution depends on
the overlap of the conduction and valence wave func-
tions, and it is modified in a bilayer. The interband ex-
change energy is reduced in a bilayer
Nilsson et al.,
2006b, and a positive contribution that depends loga-
rithmically on the bandwidth in graphene is absent in its
bilayer. As a result, the exchange energy becomes nega-
tive, and scales as n3/2, where n is the carrier density,
similar to the 2DEG. The quadratic dispersion at low
energies implies that the kinetic energy scales as n2,
again as in the 2DEG. This expansion leads to
E = Ekin + Eexc 
vF
2n2
8t


−
e2n3/2
27 0
.
227
Writing n↑=
n+s /2, n↓=
n−s /2, where s is the magne-
tization, Eq.
227 predicts a second-order transition to a
ferromagnetic state for n=4e4t2 /813vF
4
0. Higher-order
corrections to Eq.
227 lead to a first-order transition at
slightly higher densities
Nilsson et al., 2006b. For a ra-
tio 1 /00.1, this analysis implies that a graphene bi-
layer should be ferromagnetic for carrier densities such
that n41010 cm−2.
A bilayer is also the unit cell of Bernal graphite, and
the exchange instability can also be studied in an infinite
system. Taking into account nearest-neighbor interlayer
hopping only, bulk graphite should also show an ex-
change instability at low doping. In fact, there is experi-
mental evidence for a ferromagnetic instability in
strongly disordered graphite
Esquinazi et al., 2002,
2003; Kopelevich and Esquinazi, 2007.
The analysis described above can be extended to the
biased bilayer, where a gap separates the conduction and
valence bands
Stauber et al., 2007. The analysis of this
case is somewhat different, as the Fermi surface at low
doping is a ring, and the exchange interaction can
change its bounds. The presence of a gap further re-
duces the mixing of the valence and conduction bands,
leading to an enhancement of the exchange instability.
At all doping levels, where the Fermi surface is ring
shaped, the biased bilayer is unstable toward ferromag-
netism.
153Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

2. Bilayer graphene: Short-range interactions
The band structure of a graphene bilayer, at half fill-
ing, leads to logarithmic divergences in different re-
sponse functions at q=0. The two parabolic bands that
are tangent at k=0 lead to a susceptibility given by that

q
,

q
&
d2k
1
 −
vF
2/tk2
ln

&

t/vF
2 
,

228
where &t2 /vF
2 is a high momentum cutoff. These
logarithmic divergences are similar to the ones that
show up when the Fermi surface of a 2D metal is near a
saddle point in the dispersion relation
González et al.,
1996. A full treatment of these divergences requires a
RG approach
Shankar, 1994. Within a simpler mean-
field treatment, however, it is easy to note that the diver-
gence of the bilayer susceptibility gives rise to an insta-
bility toward an antiferromagnetic phase, where the
carbon atoms that are not connected to the neighboring
layers acquire a finite magnetization, while the magneti-
zation of atoms with neighbors in the contiguous layers
remains zero. A scheme of the expected ordered state is
shown in Fig. 38.
D. Interactions in high magnetic fields
The formation of Landau levels enhances the effect of
interactions due to the quenching of the kinetic energy.
This effect is most pronounced at low fillings, when only
the lowest levels are occupied. New phases may appear
at low temperatures. We consider here phases different
from the fractional quantum Hall effect, which has not
been observed in graphene so far. The existence of new
phases can be inferred from the splitting of the valley or
spin degeneracy of the Landau levels, which can be ob-
served in spectroscopy measurements
Sadowski et al.,
2006; Henriksen, Tung, Jiang, et al., 2007, or in the ap-
pearance of new quantum Hall plateaus
Zhang et al.,
2006; Abanin, Novoselov, Zeitler, et al., 2007; Giesbers et
al., 2007; Goswami et al., 2007; Jiang, Zhang, Stormer, et
al., 2007.
Interactions can lead to new phases when their effect
overcomes that of disorder. An analysis of the competi-
tion between disorder and interactions has been found
by Nomura and MacDonald
2007. The energy splitting
of the different broken symmetry phases, in a clean sys-
tem, is determined by lattice effects, so that it is reduced
by factors of order a / lB, where a is a length of the order
of the lattice spacing and lB is the magnetic length
Ali-
cea and Fisher, 2006; Goerbig et al., 2006, 2007; Wang et
al., 2008. The combination of disorder and a magnetic
field may also lift the degeneracy between the two val-
leys, favoring valley-polarized phases
Abanin, Lee, and
Levitov, 2007.
In addition to phases with enhanced ferromagnetism
or with broken valley symmetry, interactions at high
magnetic fields can lead to excitonic instabilities
Gusy-
nin et al., 2006 and Wigner crystal phases
Zhang and
Joglekar, 2007. When only the n=0 state is occupied,
the Landau levels have all their weight in a given sub-
lattice. Then, the breaking of valley degeneracy can be
associated with a charge-density wave, which opens a
gap
Fuchs and Lederer, 2007. It is interesting to note
that in these phases new collective excitations are pos-
sible
Doretto and Morais Smith, 2007.
Interactions modify the edge states in the quantum
Hall regime. A novel phase can appear when the n=0 is
the last filled level. The Zeeman splitting shifts the elec-
tronlike and holelike chiral states, which disperse in op-
posite directions near the boundary of the sample. The
resulting level crossing between an electronlike level
with spin antiparallel to the field, and a holelike level
with spin parallel to the field, may lead to Luttinger-
liquid features in the edge states
Fertig and Brey, 2006;
Abanin, Novoselov, Zeitler, et al., 2007.
VI. CONCLUSIONS
Graphene is a unique system in many ways. It is truly
2D, has unusual electronic excitations described in terms
of Dirac fermions that move in a curved space, is an
interesting mix of a semiconductor
zero density of
states and a metal
gaplessness, and has properties of
soft matter. The electrons in graphene seem to be almost
insensitive to disorder and electron-electron interactions
and have very long mean free paths. Hence, graphene’s
properties are different from what is found in usual met-
als and semiconductors. Graphene has also a robust but
flexible structure with unusual phonon modes that do
not exist in ordinary 3D solids. In some sense, graphene
brings together issues in quantum gravity and particle
physics, and also from soft and hard condensed matter.
Interestingly enough, these properties can be easily
modified with the application of electric and magnetic
fields, addition of layers, control of its geometry, and
chemical doping. Moreover, graphene can be directly
and relatively easily probed by various scanning probe
techniques from mesoscopic down to atomic scales, be-
cause it is not buried inside a 3D structure. This makes
graphene one of the most versatile systems in
condensed-matter research.
Besides the unusual basic properties, graphene has
the potential for a large number of applications
Geim
and Novoselov, 2007, from chemical sensors
Chen, Lin,
Rooks, et al., 2007; Schedin et al., 2007 to transistors

Nilsson et al., 2006b; Oostinga et al., 2007. Graphene
can be chemically and/or structurally modified in order
to change its functionality and henceforth its potential
applications. Moreover, graphene can be easily obtained
FIG. 38.
Color online Sketch of the expected magnetization
for a graphene bilayer at half filling.
154 Castro Neto et al.: The electronic properties of graphene
Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

from graphite, a material that is abundant on the Earth’s
surface. This particular characteristic makes graphene
one of the most readily available materials for basic re-
search since it frees economically challenged research
institutions in developing countries from the depen-
dence of expensive sample-growing techniques.
Many of graphene’s properties are currently subject of
intense research and debate. Understanding the nature
of the disorder and how it affects the transport proper-
ties
a problem of fundamental importance for applica-
tions, the effect of phonons on electronic transport, the
nature of electron-electron interactions, and how they
modify its physical properties are research areas that are
still in their infancy. In this review, we have only touched
the surface.
Whereas many papers have been written on mono-
layer graphene in the past few years, only a small frac-
tion actually deal with multilayers. The majority of the
theoretical and experimental efforts have concentrated
on the single layer, perhaps because of its simplicity and
the natural attraction that a one atom thick material,
which can be produced by simple methods in almost any
laboratory, creates. Nevertheless, few-layer graphene is
equally interesting and unusual with a technological po-
tential, perhaps larger than the single layer. Indeed, the
theoretical understanding and experimental exploration
of multilayers is far behind the single layer. This is a
fertile and open field of research for the future.
Finally, we have focused entirely on pure carbon
graphene where the band structure is dominated by the
Dirac description. Nevertheless, chemical modification
of graphene can lead to entirely new physics. Depending
on the nature of chemical dopants and how they are
introduced into the graphene lattice
adsorption, substi-
tution, or intercalation, there can be many results.
Small concentrations of adsorbed alkali metal can be
used to change the chemical potential while adsorbed
transition elements can lead to strong hybridization ef-
fects that affect the electronic structure. In fact, the in-
troducion of d- and f-electron atoms in the graphene
lattice may produce a significant enhancement of the
electron-electron interactions. Hence, it is easy to envi-
sion a plethora of many-body effects that can be induced
by doping and have to be studied in the context of Dirac
electrons: Kondo effect, ferromagnetism, antiferromag-
netism, and charge- and spin-density waves. The study
of chemically induced many-body effects in graphene
would add a new chapter to the short but fascinating
history of this material. Only time will tell, but the po-
tential for more amazement is lurking on the horizon.
ACKNOWLEDGMENTS
We have benefited immensely from discussions with
many colleagues and friends in the last few years, but we
would like to especially thank Boris Altshuler, Eva An-
drei, Alexander Balatsky, Carlo Beenakker, Sankar Das
Sarma, Walt de Heer, Millie Dresselhaus, Vladimir
Falko, Andrea Ferrari, Herb Fertig, Eduardo Fradkin,
Ernie Hill, Mihail Katsnelson, Eun-Ah Kim, Philip Kim,
Valery Kotov, Alessandra Lanzara, Leonid Levitov, Al-
lan MacDonald, Sergey Morozov, Johan Nilsson, Vitor
Pereira, Philip Phillips, Ramamurti Shankar, João Lopes
dos Santos, Shan-Wen Tsai, Bruno Uchoa, and Maria
Vozmediano. N.M.R.P. acknowledges financial support
from POCI 2010 via project PTDC/FIS/64404/2006. F.G.
was supported by MEC
Spain Grant No. FIS2005-
05478-C02-01 and EU Contract No. 12881
NEST.
A.H.C.N was supported through NSF Grant No. DMR-
0343790. K.S.N. and A.K.G. were supported by EPSRC

UK and the Royal Society.
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