Dirac cones reshaped by interaction effects in suspended graphene

In the first approximation, charge carriers in graphene behave like massless relativistic particles with a conical energy spectrum E=v_Fℏk where the Fermi velocity v_F 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/ℏv_F (refs 7, 8) or the inverse of the number of fermion species in graphene N_f=4 (refs 9, 10). The resulting many-body spectrum is shown in Fig. 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=v_Fℏk, 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 v_F that slowly (logarithmically) diverges at zero E. As the Fermi energy changes, v_F 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 v_Fwith the conventional value v_F*≈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 v_F (refs 11, 12). It has been noted that v_F* is larger than v_F⁰≈0.85±0.05×10⁶ m s⁻¹, where v_F⁰ 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 v_F in flat graphene⁶. The differences between v_F 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 v_F on increasing n from ≈0.5 to 2×10¹² cm⁻². A similar increase in v_F(≈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 previously^16,17,18. 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 m_c/ℏe B), which holds for the Dirac spectrum¹⁹ and enables us to find the effective cyclotron mass m_c 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, m_cwas well described by dependence m_c=ℏ(πn)^1/2/v_F*, which corresponds to the linear spectrum^11,12. 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.

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 v_F = v_F* (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, m_c as a function of k_F for the same device. m₀ is the free-electron mass. It is the exponential dependence of the SdHO amplitude on m_c that enables high accuracy of the cyclotron-mass measurements. The error bars indicate maximum and minimum values of m_c that could fit data such as in a. The dashed curves are the best linear fits m_c∝n^1/2 at high and low n. The dotted line is for the standard value of v_F=v_F*. Graphene’s spectrum renormalized owing to e–e interactions is expected to result in the dependence shown by the solid curve. c, m_cre-plotted in terms of varying v_F. 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 m_c are half those expected if we assume that v_Fretains its conventional value v_F*. To emphasize this profound discrepancy with the earlier experiments, the dashed curves in Fig. 2a plot the T dependence expected under the assumption v_F=v_F*. 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 m_c we find v_F≈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 m_c 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, m_cis expected to increase linearly with k_F=(πn)^1/2. In contrast, the experiment shows a superlinear behaviour. Trying to fit the curves in Fig. 2b with the linear dependence m_c(k_F), we find v_F≥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 m_c can be translated into v_F varying with n. Figure 2c replots the data in Fig. 2b in terms of v_F=ℏ(πn)^1/2/m_c, which shows a diverging-like behaviour of v_F near the neutrality point. This sharp increase in v_F (by nearly a factor of three with respect to v_F*) 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 m_c 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 v_F = v_F* and v_F=3×10⁶ m s⁻¹. We can see a gradual increase in v_F 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 m_cdo not reach the values expected for v_F=v_F*and are better described by v_F≈1.3v_F*. 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 v_F 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 nitride^20,21 (its dielectric constant ɛ is close to that of SiO₂) and found that m_c in the range of n between ≈0.1 and 1×10¹² cm⁻² is well described by v_F≈v_F* (Supplementary Information). This indicates that the observed difference in m_c at high n in Fig. 3 with respect to the values expected for v_F* is likely to be due to the absence of dielectric screening in suspended graphene, which maximizes the interaction effects.

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 v_F equal to v_F* 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 ɛ_G(n), graphene’s intrinsic dielectric constant. The best-fit curves yield ɛ_G≈2.2and 4.9 at low and high ends of the n range.

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To explain the observed changes in v_F, 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 v_F. However, the fact that the increase in v_Fis 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 equations^8,9,10,

where ɛ=(1+ɛ_s)/2 describes the effect of a substrate with a dielectric constant ɛ_s. Equation (1) can be considered as the leading term in the renormalization group theory expansion in powers of α=e²/ɛℏv_F, whereas (2) corresponds to a similar expansion in powers of 1/N_f (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 ɛ_G(n)for the graphene layer and add it to ɛ (for suspended graphene ɛ=ɛ_G). Then, integrating equation (1), we obtain the logarithmic dependence⁸

where n₀ is the concentration that corresponds to the ultraviolet cutoff energy Λ, and v_F(n₀) is the Fermi velocity near the cutoff. We assume v_F(n₀)≡v_F⁰, its accepted value in graphene structures with weak e–e interaction.

Both approximations result in a similar behaviour of v_F(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 v_F⁰ 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 m_c(n) and v_F(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 ɛ_G∼3.5over the whole range of our n. More detailed analysis (dotted curves in Fig. 3) yields ɛ_G≈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 ɛ_G=1+πN_fe²/8ℏv_F. 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 ɛ_G has recently become a subject of considerable debate^{23,24,25,26,27}. Our data clearly show no anomalous screening, contrary to the recent report²⁷ that suggested ɛ_G≈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 phases^28,29,30,31, especially in the least-screened case of suspended graphene with ɛ=1. Our experiments shows the diverging behaviour of v_Fbut 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 v_F increases near the neutrality point, which leads to smaller and smaller α=e²/ℏv_F at low E and, consequently, prevents the emergence of the predicted many-body gapped states.