' 2005 Nature Publishing Group

Two-dimensional gas of massless Dirac fermions in
graphene
K. S. Novoselov
1
, A. K. Geim
1
, S. V. Morozov
2
, D. Jiang
1
, M. I. Katsnelson
3
, I. V. Grigorieva
1
, S. V. Dubonos
2
& A. A. Firsov
2
Quantum electrodynamics (resulting from the merger of quantum
mechanics and relativity theory) has provided a clear understand-
ing of phenomena ranging from particle physics to cosmology and
from astrophysics to quantum chemistry
1–3
. The ideas underlying
quantum electrodynamics also influence the theory of condensed
matter
4,5
, but quantum relativistic effects are usually minute in
the known experimental systems that can be described accurately
by the non-relativistic Schro¨dinger equation. Here we report an
experimental study of a condensed-matter system (graphene, a
single atomic layer of carbon
6,7
) in which electron transport is
essentially governed by Dirac’s (relativistic) equation. The charge
carriers in graphene mimic relativistic particles with zero rest
mass and have an effective ‘speed of light’ c
*
< 10
6
ms
21
. Our
study reveals a variety of unusual phenomena that are character-
istic of two-dimensional Dirac fermions. In particular we have
observed the following: first, graphene’s conductivity never falls
below a minimum value corresponding to the quantum unit of
conductance, even when concentrations of charge carriers tend to
zero; second, the integer quantum Hall effect in graphene is
anomalous in that it occurs at half-integer filling factors; and
third, the cyclotron mass m
c
of massless carriers in graphene is
described by E 5 m
c
c
*
2
. This two-dimensional system is not only
interesting in itself but also allows access to the subtle and rich
physics of quantum electrodynamics in a bench-top experiment.
Graphene is a monolayer of carbon atoms packed into a dense
honeycomb crystal structure that can be viewed as an individual
atomic plane extracted from graphite, as unrolled single-wall carbon
nanotubes or as a giant flat fullerene molecule. This material has not
been studied experimentally before and, until recently
6,7
, was pre-
sumed not to exist in the free state. To obtain graphene samples we
used the original procedures described in ref. 6, which involve the
micromechanical cleavage of graphite followed by the identification
and selection of monolayers by using a combination of optical
microscopy, scanning electron microscopy and atomic-force
microscopy. The selected graphene films were further processed
into multi-terminal devices such as that shown in Fig. 1, by following
standard microfabrication procedures
7
. Despite being only one atom
thick and unprotected from the environment, our graphene devices
remain stable under ambient conditions and exhibit high mobility of
charge carriers. Below we focus on the physics of ‘ideal’ (single-layer)
grapheme, which has a different electronic structure and exhibits
properties qualitatively different from those characteristic of either
ultrathin graphite films (which are semimetals whose material
properties were studied recently
7–10
) or even of other devices con-
sisting of just two layers of graphene (see below).
Figure 1 shows the electric field effect
7–9
in graphene. Its conduc-
tivity j increases linearly with increasing gate voltage V
g
for both
polarities, and the Hall effect changes its sign at V
g
< 0. This
behaviour shows that substantial concentrations of electrons
(holes) are induced by positive (negative) gate voltages. Away from
the transition region V
g
< 0, Hall coefficient R
H
¼ 1/ne varies as
1/V
g
, where n is the concentration of electrons or holes and e is the
electron charge. The linear dependence 1/R
H
/ V
g
yields n ¼ aV
g
with a < 7.3 £ 10
10
cm
22
V
21
, in agreement with the theoretical
estimate n/V
g
< 7.2 £ 10
10
cm
22
V
21
for the surface charge density
induced by the field effect (see the caption to Fig. 1). The agreement
indicates that all the induced carriers are mobile and that there are no
trapped charges in graphene. From the linear dependence j(V
g
)we
found carriermobilities m ¼ j/ne, which reached 15,000 cm
2
V
21
s
21
for both electrons and holes, were independent of temperature T
between 10 and 100K and were probably still limited by defects in
parent graphite.
To characterize graphene further, we studied Shubnikov-de Haas
oscillations (SdHOs). Figure 2 shows examples of these oscillations
for differentmagnetic fields B, gate voltages and temperatures. Unlike
ultrathin graphite
7
, graphene exhibits only one set of SdHO for both
electrons and holes. By using standard fan diagrams
7,8
we have
determined the fundamental SdHO frequency B
F
for various V
g
.
The resulting dependence of B
F
on n is plotted in Fig. 3a. Both
carriers exhibit the same linear dependence B
F
¼ bn,with
b < 1.04 £ 10
215
Tm
2
(^2%). Theoretically, for any two-
dimensional (2D) system b is defined only by its degeneracy f so
that B
F
¼ f
0
n/f, where f
0
¼ 4.14 £ 10
215
Tm
2
is the flux quantum.
Comparison with the experiment yields f ¼ 4, in agreement with the
double-spin and double-valley degeneracy expected for graphene
11,12
(see caption to Fig. 2). Note, however, an anomalous feature of SdHO
in graphene, which is their phase. In contrast to conventional metals,
graphene’s longitudinal resistance r
xx
(B) exhibits maxima rather
than minima at integer values of the Landau filling factor n (Fig. 2a).
Figure 3b emphasizes this fact by comparing the phase of SdHO in
graphene with that in a thin graphite film
7
. The origin of the ‘odd’
phase is explained below.
Another unusual feature of 2D transport in graphene clearly
reveals itself in the dependence of SdHO on T (Fig. 2b). Indeed,
with increasing T the oscillations at high V
g
(high n) decay more
rapidly. One can see that the last oscillation (V
g
< 100V) becomes
practically invisible at 80 K, whereas the first one (V
g
, 10V) clearly
survives at 140K and remains notable even at room temperature. To
quantify this behaviour we measured the T-dependence of SdHO’s
amplitude at various gate voltages and magnetic fields. The results
could be fitted accurately (Fig. 3c) by the standard expression
T/sinh(2p
2
k
B
Tm
c
/
heB), which yielded m
c
varying between ,0.02
and 0.07m
0
(m
0
is the free electron mass). Changes in m
c
are well
described by a square-root dependence m
c
/ n
1/2
(Fig. 3d).
To explain the observed behaviour of m
c
, we refer to the semi-
classical expressions B
F
¼ (
h/2pe)S(E) and m
c
¼ (
h
2
/2p)›S(E)/›E,
LETTERS
1
Manchester Centre for Mesoscience and Nanotechnology, University of Manchester, Manchester M13 9PL, UK.
2
Institute for Microelectronics Technology, 142432
Chernogolovka, Russia.
3
Institute for Molecules and Materials, Radboud University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands.
Vol 438|10 November 2005|doi:10.1038/nature04233
197

' 2005 Nature Publishing Group

where S(E) ¼ pk
2
is the area in k-space of the orbits at the Fermi
energy E(k) (ref. 13). If these expressions are combined with the
experimentally found dependencesm
c
/ n
1/2
and B
F
¼ (h/4e)n it is
straightforward to show that S must be proportional to E
2
, which
yields E / k. The data in Fig. 3 therefore unambiguously prove the
linear dispersion E ¼
hkc
*
for both electrons and holes with a
common origin at E ¼ 0 (refs 11, 12). Furthermore, the above
equations also imply m
c
¼ E/c
*
2
¼ (h
2
n/4pc
*
2
)
1/2
and the best fit
to our data yields c
*
< 10
6
ms
21
, in agreement with band structure
calculations
11,12
. The semiclassical model employed is fully justi-
fied by a recent theory for graphene
14
, which shows that SdHO’s
amplitude can indeed be described by the above expression
T/sinh(2p
2
k
B
Tm
c
/
heB)withm
c
¼ E/c
*
2
. Therefore, even though
the linear spectrum of fermions in graphene (Fig. 3e) implies zero
rest mass, their cyclotron mass is not zero.
The unusual response of massless fermions to a magnetic field is
highlighted further by their behaviour in the high-field limit, at
which SdHOs evolve into the quantum Hall effect (QHE). Figure 4
shows the Hall conductivity j
xy
of graphene plotted as a function of
electron and hole concentrations in a constant B. Pronounced QHE
plateaux are visible, but they do not occur in the expected sequence
j
xy
¼ (4e
2
/h)N, where N is integer. On the contrary, the plateaux
correspond to half-integer n so that the first plateau occurs at 2e
2
/h
and the sequence is (4e
2
/h)(N þ 1/2). The transition from the lowest
hole (n ¼ 21/2) to the lowest electron (n¼þ1/2) Landau level (LL)
in graphene requires the same number of carriers (Dn ¼ 4B/
f
0
< 1.2 £ 10
12
cm
22
) as the transition between other nearest levels
(compare the distances between minima in r
xx
). This results in a
ladder of equidistant steps in j
xy
that are not interrupted when
passing through zero. To emphasize this highly unusual behaviour,
Fig. 4 also shows j
xy
for a graphite film consisting of only two
graphene layers, in which the sequence of plateaux returns to normal
and the first plateau is at 4e
2
/h, as in the conventional QHE. We
attribute this qualitative transition between graphene and its two-
layer counterpart to the fact that fermions in the latter exhibit a finite
mass near n < 0 and can no longer be described as massless Dirac
particles.
The half-integer QHE in graphene has recently been suggested by
two theory groups
15,16
, stimulated by our work on thin graphite films
7
but unaware of the present experiment. The effect is single-particle
and is intimately related to subtle properties of massless Dirac
fermions, in particular to the existence of both electron-like and
hole-like Landau states at exactly zero energy
14–17
. The latter can be
viewed as a direct consequence of the Atiyah–Singer index theorem
that is important in quantum field theory and the theory of super-
strings
18,19
. For 2D massless Dirac fermions, the theorem guarantees
the existence of Landau states at E ¼ 0 by relating the difference in
the number of such states with opposite chiralities to the total flux
through the system (magnetic field can be inhomogeneous).
Figure 1 | Electric field effect in graphene. a, Scanning electron microscope
image of one of our experimental devices (the width of the central wire is
0.2 mm). False colours are chosen to match real colours as seen in an optical
microscope for large areas of the same material. b, c, Changes in graphene’s
conductivity j (b) and Hall coefficient R
H
(c) as a function of gate voltage
V
g
. j and R
H
were measured in magnetic fields B of 0 and 2T, respectively.
The induced carrier concentrations n are described in ref. 7; n/V
g
¼ 1
0
1/te,
where 1
0
and 1 are the permittivities of free space and SiO
2
, respectively, and
t < 300 nm is the thickness of SiO
2
on top of the Si wafer used as a substrate.
R
H
¼ 1/ne is inverted to emphasize the linear dependence n / V
g
.1/R
H
diverges at small n because the Hall effect changes its sign at about V
g
¼ 0,
indicating a transition between electrons and holes. Note that the transition
region (R
H
< 0) was often shifted from zero V
g
as a result of chemical
doping
7
, but annealing of our devices in vacuum normally allowed us to
eliminate the shift. The extrapolation of the linear slopes j(V
g
) for electrons
and holes results in their intersection at a value of j indistinguishable from
zero. d, Maximum values of resistivity r ¼ 1/j (circles) exhibited by devices
with different mobilities m (left y axis). The histogram (orange background)
shows the number P of devices exhibiting r
max
within 10% intervals around
the average value of ,h/4e
2
. Several of the devices shown were made from
two or three layers of graphene, indicating that the quantized minimum
conductivity is a robust effect and does not require ‘ideal’ graphene.
Figure 2 |Quantumoscillations in graphene. SdHO at constant gate voltage
V
g
¼ 260Vas a function of magnetic field B (a) and at constant B ¼ 12Tas
a function of V
g
(b). Because m does not change greatly with V
g
, the
measurements at constant B (at a constant q
c
t ¼ mB) were found more
informative. In b, SdHOs in graphene are more sensitive to Tat high carrier
concentrations: blue, T ¼ 20K; green, T ¼ 80K; red, T ¼ 140K. The Dj
xx
curves were obtained by subtracting a smooth (nearly linear) increase in j
with increasing V
g
and are shifted for clarity. SdHO periodicity DV
g
at
constant B is determined by the density of states at each Landau level
(aDV
g
¼ fB/f
0
), which for the observed periodicity of,15.8 Vat B ¼ 12 T
yields a quadruple degeneracy. Arrows in a indicate integer n (for example,
n ¼ 4 corresponds to 10.9 T) as found from SdHO frequency B
F
< 43.5 T.
Note the absence of any significant contribution of universal conductance
fluctuations (see also Fig. 1) and weak localizationmagnetoresistance, which
are normally intrinsic for 2D materials with so high resistivity.
LETTERS NATURE|Vol 438|10 November 2005
198