�� 2005 Nature Publishing Group Two-dimensional gas of massless Dirac fermions in graphene K. S. Novoselov1, A. K. Geim1, S. V. Morozov2, D. Jiang1, M. I. Katsnelson3, I. V. Grigorieva1, S. V. Dubonos2 & A. A. Firsov2 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 chemistry1���3. The ideas underlying quantum electrodynamics also influence the theory of condensed matter4,5, but quantum relativistic effects are usually minute in the known experimental systems that can be described accurately by the non-relativistic Schrodinger �� equation. Here we report an experimental study of a condensed-matter system (graphene, a single atomic layer of carbon6,7) 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 * 106 m s21. 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 recently6,7, 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 procedures7. 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 recently7���10) or even of other devices con- sisting of just two layers of graphene (see below). Figure 1 shows the electric field effect7���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 �� 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 �� aV g with a 7.3 �� 1010 cm22 V21, in agreement with the theoretical estimate n/V g 7.2 �� 1010 cm22 V21 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 carrier mobilities m �� j/ne, which reached 15,000 cm2 V21 s21 for both electrons and holes, were independent of temperature T between 10 and 100 K 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 different magnetic fields B, gate voltages and temperatures. Unlike ultrathin graphite7, graphene exhibits only one set of SdHO for both electrons and holes. By using standard fan diagrams7,8 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 �� bn, with b 1.04 �� 10215 Tm2 (^2%). Theoretically, for any two- dimensional (2D) system b is defined only by its degeneracy f so that B F �� f 0n/f, where f 0 �� 4.14 �� 10215 Tm2 is the flux quantum. Comparison with the experiment yields f �� 4, in agreement with the double-spin and double-valley degeneracy expected for graphene11,12 (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 film7. 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 100 V) becomes practically invisible at 80 K, whereas the first one (V g , 10V) clearly survives at 140 K 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(2p2k 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 �� (h/2pe)S(E) and m c �� (h2/2p)���S(E)/���E, LETTERS 1Manchester Centre for Mesoscience and Nanotechnology, University of Manchester, Manchester M13 9PL, UK. 2Institute for Microelectronics Technology, 142432 Chernogolovka, Russia. 3Institute 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) ��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 dependences m c / n 1/2 and B F �� (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 �� hkc * for both electrons and holes with a common origin at E �� 0 (refs 11, 12). Furthermore, the above equations also imply m c �� E/c * 2 �� (h 2n/4pc * 2)1/2 and the best fit to our data yields c * 106 m s21, in agreement with band structure calculations11,12. The semiclassical model employed is fully justi- fied by a recent theory for graphene14, which shows that SdHO���s amplitude can indeed be described by the above expression T/sinh(2p2k BTm c/heB) with m c �� 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 �� (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 �� 1/2). The transition from the lowest hole (n �� 21/2) to the lowest electron (n �� ��1/2) Landau level (LL) in graphene requires the same number of carriers (Dn �� 4B/ f 0 1.2 �� 1012 cm22) 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 groups15,16, stimulated by our work on thin graphite films7 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 energy14���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- strings18,19. For 2D massless Dirac fermions, the theorem guarantees the existence of Landau states at E �� 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 0.2mm). 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 2 T, respectively. The induced carrier concentrations n are described in ref. 7 n/V g �� 1 01/te, where 1 0 and 1 are the permittivities of free space and SiO2, respectively, and t 300 nm is the thickness of SiO2 on top of the Si wafer used as a substrate. R H �� 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 �� 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 doping7, 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 �� 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 | Quantum oscillations in graphene. SdHO at constant gate voltage V g �� 260 Vas a function of magnetic field B (a) and at constant B �� 12 Tas 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 �� mB) were found more informative. In b, SdHOs in graphene are more sensitive to Tat high carrier concentrations: blue, T �� 20 K green, T �� 80 K red, T �� 140 K. 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 �� fB/f 0 ), which for the observed periodicity of ,15.8 Vat B �� 12 T yields a quadruple degeneracy. Arrows in a indicate integer n (for example, n �� 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 localization magnetoresistance, which are normally intrinsic for 2D materials with so high resistivity. LETTERS NATURE|Vol 438|10 November 2005 198