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In the first approximation, charge carriers in graphene behave like massless relativistic particles with a conical energy spectrum E=vFℏk where the Fermi velocity vF plays the role of the effective speed of light and k is the wave vector. Because graphene’s spectrum is filled with electronic states up to the Fermi energy, their Coulomb interaction has to be taken into account. To do this, the standard approach of Landau’s Fermi-liquid theory, proven successful for normal metals, fails in graphene, especially at E close to the neutrality point, where the density of states vanishes. This leads to theoretical divergences that have the same origin as those in quantum electrodynamics and other interacting-field theories. In the latter case, the interactions are normally accounted for by using the renormalization group theory1, that is, by defining effective models with a reduced number of degrees of freedom and treating the effect of high-energy excitations perturbatively. This approach was also applied to graphene by using as a small parameter either the effective coupling constant α=e2/ℏvF (refs 7, 8) or the inverse of the number of fermion species in graphene Nf=4 (refs 9, 10). The resulting many-body spectrum is shown in Fig. 1.

Figure 1
figure 1

Sketch of graphene’s electronic spectrum with and without taking into account e–e interactions. The outer cone is the single-particle spectrum E=vFℏk, and the inner cone illustrates the many-body spectrum predicted by the renormalization group theory and observed in the current experiments. We need to consider this image as follows. Electron–electron (e–e) interactions reduce the density of states at low Eand lead to an increase in vF that slowly (logarithmically) diverges at zero E. As the Fermi energy changes, vF changes accordingly but remains constant under the Fermi surface (note the principal difference from the excitation spectra that probe the states underneath the surface28).

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As for experiment, graphene placed on top of an oxidized Si wafer and with typical n≈1012 cm−2 exhibits vFwith the conventional value vF*≈1.05±0.1×106 m s−1. The value was measured by using a variety of techniques including the early transport experiments, in which Shubnikov–de Haas oscillations (SdHO) were analysed to extract vF (refs 11, 12). It has been noted that vF* is larger than vF0≈0.85±0.05×106 m s−1, where vF0 is the value accepted for metallic carbon nanotubes (see, for example, ref. 6). In agreement with this notion, the energy gaps measured in semiconducting nanotubes show a nonlinear dependence on their inverse radii, which is consistent with the larger vF in flat graphene6. The differences between vF in graphene and its rolled-up version can be attributed to e–e interactions13. Another piece of evidence came from infrared measurements14 of the Pauli blocking in graphene, which showed a sharp (15%) decrease in vF on increasing n from ≈0.5 to 2×1012 cm−2. A similar increase in vF(≈25%) for similar n has recently been found by scanning tunnelling spectroscopy15. In both cases, the changes were sharper and larger than the theory predicts for the probed relatively small intervals of n.

Here, we have studied SdHO in suspended graphene devices (inset in Fig. 2a). They were fabricated by using the procedures described previously16,17,18. After current annealing, our devices exhibited record mobilities μ∼1,000,000 cm2 Vs−1, and charge homogeneity δ n was better than 109 cm−2 such that we observed the onset of SdHO in magnetic fields B≈0.01 T and the first quantum Hall plateau became clearly visible in B below 0.1 T (see Supplementary Information). To extract the information about graphene’s electronic spectrum, we employed the following routine. SdHO were measured at various B and nas a function of temperature (T). Their amplitude was then analysed by using the standard Lifshitz–Kosevich formula T/sinh(2π2T mc/ℏe B), which holds for the Dirac spectrum19 and enables us to find the effective cyclotron mass mc at a given n. This approach was previously employed for graphene on SiO2, and it was shown that, within experimental accuracy and for a range of n∼1012 cm−2, mcwas well described by dependence mc=ℏ(πn)1/2/vF*, which corresponds to the linear spectrum11,12. With respect to the earlier experiments, our suspended devices offer critical advantages. First, in the absence of a substrate, interaction-induced spectral changes are expected to be maximal because no dielectric screening is present. Second, the high quality of suspended graphene has enabled us to probe its spectrum over a very wide range of n, which is essential as the spectral changes are expected to be logarithmic in n. Third, owing to low δ n, we can approach the Dirac point within a few millielectronvolts. This low- E regime, in which a major renormalization of the Dirac spectrum is expected, has previously been inaccessible.

Figure 2: Probing graphene’s electronic spectrum through SdHO.
figure 2

a, Symbols show examples of the T dependence of SdHO for n≈+1.4 and −7.0×1010 cm−2 for electrons and holes, respectively. The dependence is well described by the Lifshitz–Kosevich formula (solid curves). The dashed curves are the behaviour expected for vF = vF* (in the matching colours). The inset shows a scanning electron micrograph of one of our devices. The vertical graphene wire is ≈2 μm wide and suspended above an oxidized Si wafer attached to Au/Cr contacts. Approximately half of the 300-nm-thick SiO2 was etched away underneath the graphene structure. b, mc as a function of kF for the same device. m0 is the free-electron mass. It is the exponential dependence of the SdHO amplitude on mc that enables high accuracy of the cyclotron-mass measurements. The error bars indicate maximum and minimum values of mc that could fit data such as in a. The dashed curves are the best linear fits mc∝n1/2 at high and low n. The dotted line is for the standard value of vF=vF*. Graphene’s spectrum renormalized owing to e–e interactions is expected to result in the dependence shown by the solid curve. c, mcre-plotted in terms of varying vF. The colour scheme is to match the corresponding data in b.

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Figure 2a shows examples of the T dependence of the SdHO amplitude at low n (for details, see Supplementary Information). The curves are well described by the Lifshitz–Kosevich formula but the inferred mc are half those expected if we assume that vFretains its conventional value vF*. To emphasize this profound discrepancy with the earlier experiments, the dashed curves in Fig. 2a plot the T dependence expected under the assumption vF=vF*. The SdHO would then have to decay twice as fast with increasing T, which would result in a qualitatively different behaviour of the SdHO. From the measured mc we find vF≈1.9 and 2.2×106 m s−1 for the higher and lower |n| in Fig. 2a, respectively. We have carried out measurements of mc as in Fig. 2a for many different n, and the extracted values are presented in Fig. 2b for one of the devices. For the linear spectrum, mcis expected to increase linearly with kF=(πn)1/2. In contrast, the experiment shows a superlinear behaviour. Trying to fit the curves in Fig. 2b with the linear dependence mc(kF), we find vF≥2.5×106 m s−1 at n<1010 cm−2 and ≤1.5×106 m s−1 for n>2×1011 cm−2, as indicated by the dashed lines. The observed superlinear dependence of mc can be translated into vF varying with n. Figure 2c replots the data in Fig. 2b in terms of vF=ℏ(πn)1/2/mc, which shows a diverging-like behaviour of vF near the neutrality point. This sharp increase in vF (by nearly a factor of three with respect to vF*) contradicts to the linear model of graphene’s spectrum but is consistent with the spectrum reshaped by e–e interactions (Fig. 1).

The data for mc measured in four devices extensively studied in this work are collected in Fig. 3 and plotted on a logarithmic scale for both electrons and holes (no electron–hole asymmetry was noticed). The plot covers the experimental range of |n| from 109 to nearly 1012 cm−2. All the data fall within the range marked by the two dashed curves that correspond to constant vF = vF* and vF=3×106 m s−1. We can see a gradual increase in vF as n increases, although the logarithmic scale makes the observed threefold increase less dramatic than in the linear presentation of Fig. 2c. Note that, even for the highest n in Fig. 3, the measured mcdo not reach the values expected for vF=vF*and are better described by vF≈1.3vF*. This could be due to the fact that the highest n values we could achieve for suspended graphene were still within a sub-1012 cm−2 range, in which some enhancement in vF was reported for graphene on SiO2 (refs 14, 15). Alternatively, the difference could be due to the absence of a substrate in our case. To find out which of the effects dominates, we have studied high- μ devices made from graphene deposited on boron nitride20,21 (its dielectric constant ɛ is close to that of SiO2) and found that mc in the range of n between ≈0.1 and 1×1012 cm−2 is well described by vF≈vF* (Supplementary Information). This indicates that the observed difference in mc at high n in Fig. 3 with respect to the values expected for vF* is likely to be due to the absence of dielectric screening in suspended graphene, which maximizes the interaction effects.

Figure 3: Interaction-induced changes in the cyclotron mass.
figure 3

Different symbols are the measurements for different devices. The random scatter characterizes the statistical uncertainty for different samples and experiments. Blue and green dashed lines are the behaviour expected for the linear spectrum with constant vF equal to vF* and 3×106 m s−1, respectively. The solid red curve is for the spectrum renormalized by e–e interactions and described by equation (2) that takes into account the intrinsic screening self-consistently. The two dotted curves show that the interaction effects can also be described by a simpler theory (equation (1)) with an extra fitting parameter ɛG(n), graphene’s intrinsic dielectric constant. The best-fit curves yield ɛG≈2.2and 4.9 at low and high ends of the n range.

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To explain the observed changes in vF, let us first note that, in principle, not only e–e interactions but also other mechanisms such as electron–phonon coupling and disorder can lead to changes in vF. However, the fact that the increase in vFis observed over such a wide range of E rules out electron–phonon mechanisms, whereas the virtual absence of disorder in our suspended graphene makes the influence of impurities also unlikely. Therefore, we focus on e–e interactions, in which case graphene’s spectrum is modified as shown in Fig. 1 and, in the first approximation, can be described by two related equations8,9,10,

where ɛ=(1+ɛs)/2 describes the effect of a substrate with a dielectric constant ɛs. Equation (1) can be considered as the leading term in the renormalization group theory expansion in powers of α=e2/ɛℏvF, whereas (2) corresponds to a similar expansion in powers of 1/Nf (refs 8, 9, 10). The diagrams that depict these approximations are given in Supplementary Information. Importantly, equation (2) includes self-consistently the screening by graphene’s charge carriers. An approximate scheme to incorporate this intrinsic screening while keeping the simplicity of equation (1) is to define an effective screening constant ɛG(n)for the graphene layer and add it to ɛ (for suspended graphene ɛ=ɛG). Then, integrating equation (1), we obtain the logarithmic dependence8

where n0 is the concentration that corresponds to the ultraviolet cutoff energy Λ, and vF(n0) is the Fermi velocity near the cutoff. We assume vF(n0)≡vF0, its accepted value in graphene structures with weak e–e interaction.

Both approximations result in a similar behaviour of vF(n) and provide good agreement with the experiment. However, equation (2) is more general and essentially requires no fitting parameters because Λ is expected to be of the order of graphene’s bandwidth and affects the fit only weakly, as log (Λ). Alternatively, Λ can be estimated from the known value of vF0 at high n≈5×1012 cm−2 as Λ=2.5±1.5 eV (ref. 22). The solid curves in Figs 2b,c and 3 show mc(n) and vF(n) calculated by integrating equation (2) and using Λ≈3 eV. The dependence captures all the main features of the experimental data. As for equations (1) and (3), they enable a reasonable fit by using ɛG∼3.5over the whole range of our n. More detailed analysis (dotted curves in Fig. 3) yields ɛG≈2.2 and 5 for n∼109 and 1012 cm−2, respectively. These values are close to those calculated in the random phase approximation, which predicts ɛG=1+πNfe2/8ℏvF. Using this expression in combination with equation (3) leads to a fit that is practically indistinguishable from the solid curve given by equation (2). This could be expected because equation (2) includes the screening self-consistently, also within the random phase approximation. The value of ɛG has recently become a subject of considerable debate23,24,25,26,27. Our data clearly show no anomalous screening, contrary to the recent report27 that suggested ɛG≈15, but in good agreement with measurements reported in ref. 28.

Finally, a large number of theories have been predicting that the diverging contribution of e–e interactions at low E may result in new electronic phases28,29,30,31, especially in the least-screened case of suspended graphene with ɛ=1. Our experiments shows the diverging behaviour of vFbut no new phases emerge, at least for n>109 cm−2 (E>4 meV). Moreover, we can also conclude that there are no insulating phases even at E as low as 0.1 meV. To this end, we refer to Supplementary Information, in which we present the data for graphene’s resistivity ρ(n) in zero B. The peak at the neutrality point continues to grow monotonically down to 2 K, and ρ(T) exhibits no sign of diverging (the regime of smearing by spatial inhomogeneity is not reached even at this T). This shows that, in neutral graphene in zero B, there is no gap larger than ≈0.1 meV. This observation is consistent with the fact that vF increases near the neutrality point, which leads to smaller and smaller α=e2/ℏvF at low E and, consequently, prevents the emergence of the predicted many-body gapped states.