Abstract
Quantum electrodynamics (resulting from the merger of quantum mechanics and relativity theory) has provided a clear understanding of phenomena ranging from particle physics to cosmology and from astrophysics to quantum chemistry^{1,2,3}. 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 nonrelativistic Schrödinger equation. Here we report an experimental study of a condensedmatter 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} m s^{1}. Our study reveals a variety of unusual phenomena that are characteristic of twodimensional 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 halfinteger filling factors; and third, the cyclotron mass m_{c} of massless carriers in graphene is described by E = m_{c}c_{*}^{2}. This twodimensional system is not only interesting in itself but also allows access to the subtle and rich physics of quantum electrodynamics in a benchtop experiment.
Main
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 singlewall carbon nanotubes or as a giant flat fullerene molecule. This material has not been studied experimentally before and, until recently^{6,7}, was presumed 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 atomicforce microscopy. The selected graphene films were further processed into multiterminal 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’ (singlelayer) 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^{7,8,9,10}) or even of other devices consisting of just two layers of graphene (see below).
Figure 1 shows the electric field effect^{7,8,9} in graphene. Its conductivity σ 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 = αV_{g} with α ≈ 7.3 × 10^{10} cm^{2} V^{1}, in agreement with the theoretical estimate n/V_{g} ≈ 7.2 × 10^{10} cm^{2} V^{1} 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 σ(V_{g}) we found carrier mobilities µ = σ/ne, which reached 15,000 cm^{2} V^{1} s^{1} 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 Shubnikovde Haas oscillations (SdHOs). Figure 2 shows examples of these oscillations for different magnetic 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} = βn, with β ≈ 1.04 × 10^{15} T m^{2} (± 2%). Theoretically, for any twodimensional (2D) system β is defined only by its degeneracy f so that B_{F} = φ_{0}n/f, where φ_{0} = 4.14 × 10^{15} T m^{2} is the flux quantum. Comparison with the experiment yields f = 4, in agreement with the doublespin and doublevalley 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 ρ_{xx}(B) exhibits maxima rather than minima at integer values of the Landau filling factor ν (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} ≈ 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 Tdependence of SdHO's amplitude at various gate voltages and magnetic fields. The results could be fitted accurately (Fig. 3c) by the standard expression , 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 squareroot dependence m_{c}∝n^{1/2} (Fig. 3d).
To explain the observed behaviour of m_{c}, we refer to the semiclassical expressions B_{F} = (ℏ/2πe)S(E) and m_{c} = (ℏ^{2}/2π)∂S(E)/∂E, where S(E) = πk^{2} is the area in kspace 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 = ℏ kc_{*} 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^{2}n/4πc_{*}^{2})^{1/2} and the best fit to our data yields c_{*} ≈ 10^{6} m s^{1}, in agreement with band structure calculations^{11,12}. The semiclassical model employed is fully justified by a recent theory for graphene^{14}, which shows that SdHO's amplitude can indeed be described by the above expression T/sinh(2π^{2}k_{B}Tm_{c}/ℏeB) 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 highfield limit, at which SdHOs evolve into the quantum Hall effect (QHE). Figure 4 shows the Hall conductivity σ_{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 σ_{xy} = (4e^{2}/h)N, where N is integer. On the contrary, the plateaux correspond to halfinteger ν 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 (ν = 1/2) to the lowest electron (ν = +1/2) Landau level (LL) in graphene requires the same number of carriers (Δn = 4B/φ_{0} ≈ 1.2 × 10^{12} cm^{2}) as the transition between other nearest levels (compare the distances between minima in ρ_{xx}). This results in a ladder of equidistant steps in σ_{xy} that are not interrupted when passing through zero. To emphasize this highly unusual behaviour, Fig. 4 also shows σ_{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 twolayer 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 halfinteger 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 singleparticle and is intimately related to subtle properties of massless Dirac fermions, in particular to the existence of both electronlike and holelike Landau states at exactly zero energy^{14,15,16,17}. 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 superstrings^{18,19}. 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).
To explain the halfinteger QHE qualitatively, we invoke the formal expression^{2,14,15,16,17} for the energy of massless relativistic fermions in quantized fields, . In quantum electrodynamics, the sign ± describes two spins, whereas in graphene it refers to ‘pseudospins’. The latter have nothing to do with the real spin but are ‘built in’ to the Diraclike spectrum of graphene; their origin can be traced to the presence of two carbon sublattices. The above formula shows that the lowest LL (N = 0) appears at E = 0 (in agreement with the index theorem) and accommodates fermions with only one (minus) projection of the pseudospin. All other levels N ≥ 1 are occupied by fermions with both (± ) pseudospins. This implies that for N = 0 the degeneracy is half of that for any other N. Alternatively, one can say that all LLs have the same ‘compound’ degeneracy but the zeroenergy LL is shared equally by electrons and holes. As a result the first Hall plateau occurs at half the normal filling and, oddly, both ν = 1/2 and +1/2 correspond to the same LL (N = 0). All other levels have normal degeneracy 4B/φ_{0} and therefore remain shifted by the same 1/2 from the standard sequence. This explains the QHE at ν = N + 1/2 and, at the same time, the ‘odd’ phase of SdHO (minima in ρ_{xx} correspond to plateaux in ρ_{xy} and therefore occur at halfinteger ν; see Figs 2 and 4), in agreement with theory^{14,15,16,17}. Note, however, that from another perspective the phase shift can be viewed as the direct manifestation of Berry's phase acquired by Dirac fermions moving in magnetic field^{20,21}.
Finally, we return to zerofield behaviour and discuss another feature related to graphene's relativisticlike spectrum. The spectrum implies vanishing concentrations of both carriers near the Dirac point E = 0 (Fig. 3e), which suggests that lowT resistivity of the zerogap semiconductor should diverge at V_{g} ≈ 0. However, neither of our devices showed such behaviour. On the contrary, in the transition region between holes and electrons graphene's conductivity never falls below a welldefined value, practically independent of T between 4 K and 100 K. Figure 1c plots values of the maximum resistivity ρ_{max} found in 15 different devices at zero B, which within an experimental error of ∼15% all exhibit ρ_{max} ≈ 6.5 kΩ independently of their mobility, which varies by a factor of 10. Given the quadruple degeneracy f, it is obvious to associate ρ_{max} with h/fe^{2} = 6.45 kΩ, where h/e^{2} is the resistance quantum. We emphasize that it is the resistivity (or conductivity) rather than the resistance (or conductance) that is quantized in graphene (that is, resistance R measured experimentally scaled in the usual manner as R = ρL/w with changing length L and width w of our devices). Thus, the effect is completely different from the conductance quantization observed previously in quantum transport experiments.
However surprising it may be, the minimum conductivity is an intrinsic property of electronic systems described by the Dirac equation^{22,23,24,25}. It is due to the fact that, in the presence of disorder, localization effects in such systems are strongly suppressed and emerge only at exponentially large length scales. Assuming the absence of localization, the observed minimum conductivity can be explained qualitatively by invoking Mott's argument^{26} that the mean free path l of charge carriers in a metal can never be shorter than their wavelength λ_{F}. Then, σ = neµ can be rewritten as σ = (e^{2}/h)k_{F}l, so σ cannot be smaller than ∼e^{2}/h for each type of carrier. This argument is known to have failed for 2D systems with a parabolic spectrum in which disorder leads to localization and eventually to insulating behaviour^{22,23}. For 2D Dirac fermions, no localization is expected^{22,23,24,25} and, accordingly, Mott's argument can be used. Although there is a broad theoretical consensus^{15,16,23,24,25,26,27,28} that a 2D gas of Dirac fermions should exhibit a minimum conductivity of about e^{2}/h, this quantization was not expected to be accurate and most theories suggest a value of ∼e^{2}/πh, in disagreement with the experiment.
Thus, graphene exhibits electronic properties that are distinctive for a 2D gas of particles described by the Dirac equation rather than the Schrödinger equation. The work shows a possibility of studying phenomena of the quantum field theory in a condensedmatter experiment.
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Acknowledgements
We thank L. Glazman, V. Falko, S. Sharapov and A. Castro Neto for discussions. K.S.N. was supported by Leverhulme Trust. S.V.M., S.V.D. and A.A.F. acknowledge support from the Russian Academy of Science and INTAS. This research was funded by the EPSRC (UK).
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Correspondence to K. S. Novoselov or A. K. Geim.
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