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Experimental observation of the quantum Hall effect and Berrys phase in graphene
Yuanbo Zhang1, Yan-Wen Tan1, Horst L. Stormer1,2 & Philip Kim1

When electrons are conned in two-dimensional materials, quantum-mechanically enhanced transport phenomena such as the quantum Hall effect can be observed. Graphene, consisting of an isolated single atomic layer of graphite, is an ideal realization of such a two-dimensional system. However, its behaviour is expected to differ markedly from the well-studied case of quantum wells in conventional semiconductor interfaces. This difference arises from the unique electronic properties of graphene, which exhibits electronhole degeneracy and vanishing carrier mass near the point of charge neutrality1,2. Indeed, a distinctive half-integer quantum Hall effect has been predicted35 theoretically, as has the existence of a non-zero Berrys phase (a geometric quantum phase) of the electron wavefunctiona consequence of the exceptional topology of the graphene band structure6,7. Recent advances in micromechanical extraction and fabrication techniques for graphite structures812 now permit such exotic two-dimensional electron systems to be probed experimentally. Here we report an experimental investigation of magneto-transport in a high-mobility single layer of graphene. Adjusting the chemical potential with the use of the electric eld effect, we observe an unusual halfinteger quantum Hall effect for both electron and hole carriers in graphene. The relevance of Berrys phase to these experiments is conrmed by magneto-oscillations. In addition to their purely scientic interest, these unusual quantum transport phenomena may lead to new applications in carbon-based electronic and magneto-electronic devices. The low-energy band structure of graphene can be approximated as cones located at two inequivalent Brillouin zone corners (Fig. 1a, left inset). In these cones, the two-dimensional (2D) energy dispersion relation is linear and the electron dynamics can be treated as relativistic, in which the Fermi velocity v F of the graphene substitutes for the speed of light. In particular, at the apex of the cones (termed the Dirac point), electrons and holes (particles and antiparticles) are degenerate. Landau-level (LL) formation for electrons in this system under a perpendicular magnetic eld, B, has been studied theoretically using an analogy to 2 1-dimensional quantum electrodynamics2,3, in which the Landau level energy is given by q h F 1 En sgnn 2e v2 jnjB Here e and h h=2p are electron charge and Plancks constant divided  by 2p, and the integer n represents an electron-like (n . 0) or a holelike (n , 0) LL index. Crucially, a single LL with n 0 and E 0 0 also occurs. When only low-lying LLs (jnj , 104 for B 10 T) are occupied, the separation of E n is much larger than the Zeeman spin splitting, so each LL has a degeneracy g s 4, accounting for spin degeneracy and sublattice degeneracy. Previous studies of mesoscopic graphite samples consisting of a few layers of graphene exhibited magneto-oscillations associated with the LL formation by
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electron-like and hole-like carriers tuned by the electric eld effect8,9,11. However, the quantum Hall effect (QHE) was not observed in these samples, possibly as a result of their low mobility and/or the residual three-dimensional nature of the specimens. The high-mobility graphene samples used in our experiments were extracted from Kish graphite (Toshiba Ceramics) on degenerately doped Si wafers with a 300-nm SiO2 coating layer, by using micromechanical manipulation similar to that described in ref. 8.

Figure 1 | Resistance, carrier density, and mobility of graphene measured at 1.7 K at different gate voltages. a, Changes in resistance as a function of gate voltage in a graphene device shown in the optical microscope image in the right inset. The position of the resistance peaks varies from device to device, but the peak values are always of the order of 4 kQ, suggesting a potential quantum-mechanical origin. The left inset shows a schematic diagram of the low-energy dispersion relation near the Dirac points in the graphene Brillouin zone. Only two Dirac cones are nonequivalent to each other, producing a twofold valley degeneracy in the band structure. b, Charge carrier density (open circles) and mobility (lled circles) of graphene as a function of gate voltage. The solid line corresponds to the estimated charge induced by the gate voltage, n s C gV g/e, assuming a gate capacitance C g of 115 aF mm22 obtained from geometrical considerations.

Department of Physics, 2Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027, USA.

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Interference-induced colour shifts, cross-correlated with an atomic force microscopy prole, allow us to identify the number of deposited graphene layers from optical images of the samples (Supplementary Information). After a suitable graphene sample has been selected, electron beam lithography followed by thermally evaporated Au/Cr (30 nm and 5 nm, respectively) denes multiple electrodes for transport measurement (Fig. 1a, right inset). With the use of a Hall-bar-type electrode conguration, the magnetoresistance R xx and Hall resistance R xy are measured. Applying a gate voltage, V g, to the Si substrate controls the charge density in the graphene samples. Figure 1a shows the gate modulation of R xx at zero magnetic eld in a typical graphene device whose lateral size is ,3 mm. Whereas R xx remains in the ,100-Q range at high carrier density, a sharp peak at ,4 kQ is observed at V g < 0. Although different samples show slightly different peak values and peak positions, similar behaviours were observed in three other graphene samples that we measured. The existence of this sharp peak is consistent with the reduced carrier density as E F approaches the Dirac point of grapheme, at which the density of states vanishes. Thus, the gate voltage corresponding to the charge-neutral Dirac point, V Dirac, can be determined from this peak position. A separate Hall measurement provides a measure for the sheet carrier density, n s, and for the mobility, m, of the sample, as shown in Fig. 1b, assuming a simple Drude model. The sign of n s changes at V g V Dirac, indicating that E F does indeed cross the charge-neutral point. Mobilities are higher than 104 cm2 V21 s21 for the entire gate voltage range, considerably exceeding the quality of graphene samples studied previously8,9. The exceptionally high-mobility graphene samples allow us to

investigate transport phenomena in the extreme magnetic quantum limit, such as the QHE. Figure 2a shows R xy and R xx for the sample of Fig. 1 as a function of magnetic eld B at a xed gate voltage V g . V Dirac. The overall positive R xy indicates that the contribution is mainly from electrons. At high magnetic eld, R xy(B) exhibits plateaux and R xx is vanishing, which are the hallmark of the QHE. At least two well-dened plateaux with values (2e 2/h)21 and (6e 2/h)21, followed by a developing (10e 2/h)21 plateau, are observed before the QHE features transform into Shubnikov de Haas (SdH) oscillations at lower magnetic eld. The quantization of R xy for these rst two plateaux is better than 1 part in 104, precise within the instrumental uncertainty. We observed the equivalent QHE features for holes with negative R xy values (Fig. 2a, inset). Alternatively, we can probe the QHE in both electrons and holes by xing the magnetic eld and changing V g across the Dirac point. In this case, as V g increases, rst holes (V g , V Dirac) and later electrons (V g . V Dirac) ll successive Landau levels and exhibit the QHE. This yields an antisymmetric (symmetric) pattern of R xy (R xx) in Fig. 2b, with R xy quantization in accordance with R21 ^g s n 1=2e2 =h xy 2 where n is a non-negative integer and ^ stands for electrons and holes, respectively. This quantization condition can be translated to the quantized lling factor v ^g s(n 1/2) in the usual QHE language. In addition, there is an oscillatory structure developed near the Dirac point. Although this structure is reproducible for any given sample, its shape varies from device to device, suggesting potentially mesoscopic effects depending on the details of the sample geometry13. Although the QHE has been observed in many 2D

Figure 2 | Quantized magnetoresistance and Hall resistance of a graphene device. a, Hall resistance (black) and magnetoresistance (red) measured in the device in Fig. 1 at T 30 mK and V g 15 V. The vertical arrows and the numbers on them indicate the values of B and the corresponding lling factor n of the quantum Hall states. The horizontal lines correspond to h/e 2n values. The QHE in the electron gas is shown by at least two quantized plateaux in R xy, with vanishing R xx in the corresponding magnetic eld regime. The inset shows the QHE for a hole gas at V g 24 V, measured at 1.6 K. The quantized plateau for lling factor n 2 is well dened, and the second and third plateaux with n 6 and n 10 are also resolved. b, Hall
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resistance (black) and magnetoresistance (orange) as a function of gate voltage at xed magnetic eld B 9 T, measured at 1.6 K. The same convention as in a is used here. The upper inset shows a detailed view of high-lling-factor plateaux measured at 30 mK. c, A schematic diagram of the Landau level density of states (DOS) and corresponding quantum Hall conductance (j xy) as a function of energy. Note that, in the quantum Hall states, j xy 2R 21. The LL index n is shown next to the DOS peak. In our xy experiment the Fermi energy E F can be adjusted by the gate voltage, and R 21 xy changes by an amount g se 2/h as E F crosses a LL.

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systems, the QHE observed in graphene is distinctively different those conventional QHEs because the quantization condition (equation (2)) is shifted by a half-integer. These unusual quantization conditions are a result of the topologically exceptional electronic structure of grapheme, which we discuss below. The sequence of half-integer multiples of quantum Hall plateaux has been predicted by several theories that combine relativistic Landau levels with the particlehole symmetry of graphene35. This can be easily understood from the calculated LL spectrum (equation (1)) as shown in Fig. 2c. Here we plot the density of states of the g s-fold degenerate (spin and sublattice) of LLs and the corresponding Hall conductance (j xy 2R 21, for R xx ! 0) in the quantum Hall xy regime as a function of energy. j xy exhibits QHE plateaux when E F (tuned by V g) falls between LLs, and jumps by an amount of g se 2/h when E F crosses a LL. Time-reversal invariance guarantees particle hole symmetry; j xy is therefore an odd function in energy across the Dirac point2. However, in graphene, the n 0 LL is robustthat is, E 0 0 regardless of the magnetic eldprovided that the sublattice symmetry is preserved2. Thus, the rst plateau of R 21 for electron xy and hole is situated exactly at ^g se 2/2h. As E F crosses the next electron (hole) LL, R 21 increases (decreases) by an amount g se 2/h, xy which yields the quantization condition in equation (2). As noted by several workers, a consequence of the combination of time-reversal symmetry with the novel Dirac point structure can be viewed in terms of Berrys phase arising from the band degeneracy point7,14. A direct implication of Berrys phase in graphene is discussed in the context of the quantum phase of a spin-1/2 pseudo-spinor that describes the sublattice symmetry6,15. This phase is already implicit in the half-integer-shifted quantization rules of the QHE. It can further be probed in the magnetic eld regime, in which a semi-classical magneto-oscillation description holds16,17: DRxx RB; Tcos2pBF =B 1=2 b 3 Here R(B,T) is the SdH oscillation amplitude, B F is the frequency of the SdH oscillation in 1/B, and b is the associated Berrys phase, in the range 0 , b , 1. Berrys phase b 0 (or, equivalently, b 1) corresponds to the trivial case. A deviation from this value is indicative of new physics with b 1/2, implying the existence of Dirac particles7. Experimentally, this phase shift in the semi-classical regime can be obtained from an analysis of the SdH fan diagram, in

which the sequence of values of 1/B n of the nth minimum in R xx are plotted against their index n (Fig. 3b). The intercept of linear t to the data with the n-index axis yields Berrys phase, modulo an integer. The resulting b is very close to 0.5 (Fig. 3b, upper inset), providing further manifestation of the existence of a non-zero Berrys phase in graphene and the presence of Dirac particles. Such a non-zero Berrys phase was not observed in the previous few-layer graphite specimens8,11,18, although there have been claims of hints of a phase shift in earlier measurements on bulk graphite17. Our data for graphene provide indisputable evidence for such an effect in a solid-state system. The non-zero Berrys phase observed in the SdH fan diagram is related to the vanishing mass at the Dirac point. We can extract this effective carrier mass m c from the temperature dependence of the well-developed SdH oscillations at low B eld (Fig. 3a, left inset) by using the standard SdH formalism19. Indeed, the analysis at different gate voltages yields a strong suppression of m c near the Dirac point. Whereas the high-density (n s , 5 1012 cm22) carrier gas shows m c , 0.04m e, the mass drops to m c , 0.007m e near the Dirac point (n s , 2 1011 cm22), where m e is the mass of the free electron. Overall, the observed gate voltage-dependent effectivep can be mass  tted to a ctitious relativistic mass: mc EF =v2 ph2 ns =v2 F F by using v F as the only tting parameter (Fig. 3a, right inset). In accordance with the Berrys phase argument, this procedure extrapolates to a vanishing mass at the Dirac point. Thus, we have experimentally discovered an unusual QHE in highquality graphene samples. In contrast with conventional 2D systems, in graphene the observed quantization condition is described by halfinteger rather than integer values. The measured phase shift in magneto-oscillation can be attributed to the peculiar topology of the graphene band structure with a linear dispersion relation and vanishing mass near the Dirac point, which can be described in terms of ctitious relativistic carriers. The unique behaviour of electrons in this newly discovered 2 1-dimensional quantum electrodynamics system not only opens up many interesting questions in mesoscopic transport in electronic systems with non-zero Berrys phase but may also provide the basis for new applications in carbonbased electric and magnetic eld-effect devices, such as ballistic metallic/semiconducting graphene ribbon devices9 and electric eld effective spin transport devices using a spin-polarized edge state20.

Figure 3 | Temperature dependence and gate-voltage dependence of the SdH oscillations in graphene. a, Temperature dependence of the SdH oscillations at V g 22.5 V. Each curve represents R xx(B) normalized to R xx(0) at a xed temperature. The curves are in order of decreasing temperature, starting from the top, as indicated by the vertical arrow. The corresponding temperatures are shown in the left inset, which represents the SdH oscillation amplitude A divided by temperature measured at a xed magnetic eld. The standard SdH t yields the effective mass. The right inset is a plot of the effective mass obtained at different gate voltages. The broken

line is a t to the single-parameter model described in the text, which yields v F 1.1 106 m s21, in reasonable agreement with published values. b, A fan diagram for SdH oscillations at different gate voltages. The location of 1/B for the nth minimum (maximum) of R xx, counting from B B F, plotted against n (n 1/2). The lines correspond to a linear t, in which the slope (lower inset) indicates B F and the n-axis intercept (upper inset) provides a direct probe of Berrys phase in the magneto-oscillation in graphene. The error bars indicate the standard deviation of tting errors.
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Received 18 July; accepted 12 September 2005.

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13. Roukes, M. L., Scherer, A. & Van der Gaag, B. P. Are transport anomalies in electron waveguides classical? Phys. Rev. Lett. 64, 1154 -1157 (1990). 14. Fang, Z. et al. The anomalous Hall effect and magnetic monopoles in momentum space. Science 302, 92 (2003). -95 15. McEuen, P. L., Bockrath, M., Cobden, D. H., Yoon, Y. & Louie, S. G. Disorder, pseudospins, and backscattering in carbon nanotubes. Phys. Rev. Lett. 83, 5098 -5101 (1999). 16. Sharapov, S. G., Gusynin, V. P. & Beck, H. Magnetic oscillations in planar systems with the Dirac-like spectrum of quasiparticle excitations. Phys. Rev. B 69, 075104 (2004). 17. Lukyanchuk, I. A. & Kopelevich, Y. Phase analysis of quantum oscillations in graphite. Phys. Rev. Lett. 93, 166402 (2004). 18. Morozov, S. V. et al. Two dimensional electron and hole gases at the surface of graphite. Preprint at khttp://xxx.lanl.gov/abs/cond-mat/0505319l (2005). 19. Shoenberg, D. Magnetic Oscillation in Metals (Cambridge Univ. Press, Cambridge, 1984). 20. Kane, C. L & Mele, E. J. Quantum spin Hall effect in graphene. Preprint at khttp://xxx.lanl.gov/abs/cond-mat/0411737l (2005).

Supplementary Information is linked to the online version of the paper at www.nature.com/nature. Acknowledgements We thank I. Aleiner, A. Millis, T. F. Heinz, A. Mitra, J. Small and A. Geim for discussions. This research was supported by the NSF Nanoscale Science and Engineering Center at Columbia University, New York State Ofce of Science (NYSTAR) and the Department of Energy (DOE). Author Information Reprints and permissions information is available at npg.nature.com/reprintsandpermissions. The authors declare no competing nancial interests. Correspondence and requests for materials should be addressed to P.K. (pkim@phys.columbia.edu).

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