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In two dimensions, a saddle point in the electronic band structure leads to a divergence in the density of states, also known as a Van Hove singularity1 (VHS). When the Fermi energy (EF) is close to the VHS, interactions, however weak, are magnified by the enhanced density of states (DOS), resulting in instabilities, which can give rise to new phases of matter2,3,4,5 with desirable properties. This implies the possibility of engineering material properties by bringing EF and the VHS together. However, in most materials one cannot change the position of the VHS in the band structure. Instead, it may be possible to tune EF through the VHS by chemical doping8 or by gating7. In this regard, graphene, the recently discovered two-dimensional form of carbon, is quite special5,9. It has linearly dispersing bands at the K (K′) point in the Brillouin zone, the so-called Dirac points, and a DOS that is linear and vanishes at Dirac point. The fact that this material is truly two-dimensional and has a low DOS means that it cannot screen applied electric fields, allowing for strong gating and ambipolar behaviour7. However, although the band structure of graphene5 contains a VHS, its large distance from the Dirac point makes it prohibitively difficult to reach by either gating or chemical doping. We show that by introducing a rotation between stacked graphene layers, it is possible to induce VHSs that are within the range of EF achievable by gate tuning. As the samples studied here are not intentionally doped, EF is within a few millielectronvolts of the Dirac point.

Rotation between graphene layers is often observed as a Moiré pattern on graphite surfaces10, as illustrated in Fig. 1. Graphite consists of stacked layers of graphene, the lattice structure of which contains two interpenetrating triangular sublattices, denoted A and B. In the most common (Bernal) stacking, adjacent layers are shifted by one atomic spacing, so that B atoms of layer 2 (B2) sit directly on top of A atoms of layer 1 (A1) and B1 and A2 atoms are in the centre of the hexagons of the opposing layer. Assuming a rotation through an angle θ about an A1 (B2) site in bilayer graphene, a set of conditions for commensurate periodic structures leading to Moiré patterns is easily derived11 cos(θi)=(3i2+3i+1/2)/(3i2+3i+1), with i being an integer (i=0, θ=60∘ corresponds to AA stacking and , θ=0∘ to AB stacking) and lattice constant of the superlattice , where a0∼2.46 Å is the atomic lattice constant. For the extended Moiré pattern in Fig. 1, θi=1.79∘ is obtained from the superlattice period, L=7.7±0.3 nm, corresponding to i=18.

Figure 1: STM of a graphene flake on a freshly cleaved surface of highly oriented pyrolitic graphite revealing a Moiré pattern.
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a, Large-area scan of a graphene flake. Regions M2 and G flank the boundary of the Moire pattern. b, Zoom-in of the frame in a, showing a Moiré pattern with period 7.7±0.3 nm. Inset: Fourier transforms of the superstructure. c, Zoom-in to the centre of the pattern, region M1. d, Atomic-resolution image of a bright spot. Inset: Fourier transforms of the atomic-resolution image. e,f, Atomic-resolution image on bright and dark regions of the pattern, showing a well-ordered triangular lattice within the bright spots (e) and a less-ordered honeycomb-like structure in between (f). The former indicates Bernal-stacked layers16,23, whereas the latter suggests slipped stacking, resulting from a small-angle rotation between layers. Tunnelling current 20 pA, sample bias voltage 300 mV.

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Thus far, most work on Moiré patterns focused on structural aspects revealed by scanning tunnelling microscopy (STM), but their effect on the electronic properties has not been addressed. By using scanning tunnelling spectroscopy (STS), we find that rotation markedly alters the DOS. Figure 2 shows the tunnelling differential conductance, dI/dV, a quantity proportional to the local DOS (ref. 12). In regions inside the Moiré pattern (M1 and M2), the spectra develop two sharp peaks flanking the Dirac points with energy separation ΔEvhs∼82 meV. Below, we show that these peaks correspond to rotation-induced VHSs. The effect of the VHS on the DOS is evident in the energy dependence of the dI/dV maps. Close to the VHS, the maps develop a strong density modulation (Figs 2f and 3c), characteristic of charge-density waves (CDWs), suggesting an impending Fermi-surface instability. In contrast, for energies away from the VHS, the charge density becomes nearly homogeneous.

Figure 2: STS inside and outside a Moiré pattern with θ=1.79∘.
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a, Topography near the boundary of the pattern. b, Tunnelling spectra in the centre of the pattern (M1 in Fig. 1c) and at its edge (M2), showing two sharp peaks flanking the Dirac point with ΔEvhs∼82 meV. The peaks are absent outside the pattern area (G), where the spectrum is typical of graphite24. c, Spatial dependence of tunnelling spectra along a line connecting point M2 inside the pattern to point G outside it at positions marked by the white dots in a, showing monotonic evolution between the rotated and unrotated regions. Tunnelling current 22 pA, sample bias voltage 300 mV, a.c. bias modulation 5 mVrms at 340 Hz. e, Comparison of tunnelling spectra on bright and dark regions within the pattern at positions indicated in d. Although all spectra show two peaks, the peak heights and degree of asymmetry depend on their position in the pattern. f, dI/dV maps of the region shown in d. The top map corresponding to the VHS just below the Fermi energy (−58 meV) shows a strong density modulation, suggesting an imminent Fermi-surface instability. In contrast, the bottom map taken farther from the VHS (77 meV) does not show the modulation.

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Figure 3: Moiré pattern with θ=1.16∘ on CVD graphene.
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a, Topography taken at 200 mV and 20 pA, showing a pattern with a superlattice constant of 12±0.5 nm, corresponding to θ∼1.16∘. Scale bar: 2 nm. b, Scanning tunnelling spectra taken in bright and dark regions of the pattern. a.c. modulation 1 mVrms at 340 Hz. c–e, dI/dV maps of the region shown in a at three bias voltages: far from the VHS (90 mV) (c), the map is smooth, indicating extended states; and maps at the energies of the VHS, 1 mV (d) and −11 mV (e), show localized states. Scale bar: 2 nm

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To explore the angle dependence of ΔEvhs, we studied graphene layers prepared by chemical vapour deposition13 (CVD). CVD graphene layers have a strong twisting tendency revealed by Moiré patterns with a range of rotation angles. The example in Fig. 3a shows a pattern, corresponding to θ∼1.16∘. The STS spectra in this region reveal strong VHSs (Fig. 3b) with a much smaller ΔEvhs∼12 meV. The pronounced spatial modulation of the dI/dV maps at energies close to these VHSs (Fig. 3c), indicating the formation of a CDW, is significantly stronger than for the pattern in Fig. 2f, where the more widely separated VHSs are farther away from the Fermi energy. For a pattern with an even larger angle, θ∼3.4∘, and ΔEvhs∼430 meV, the localization is weaker still (see Supplementary Fig. S3), consistent with theoretical predictions14.

We next show that the VHSs are induced by the rotation and use the model developed in ref. 11 to derive the angle dependence of ΔEvhs. A rotation between two graphene layers causes a shift between the corresponding Dirac points in momentum space, so that the Dirac wavevector of the rotated layer is Kθ=K+ΔK, where ΔK=K×2sin(θ/2). If we use the same origin of momentum for the two layers, so that a uniform hopping couples states of the same momentum in both layers, the zero-energy states do not occur at k=0, but rather at k=−ΔK/2 in layer 1 and k=ΔK/2 in layer 2. Unlike in the AB stacked bilayer, there is no direct coupling of the zero-energy states of one layer to the zero-energy states of the other. As shown in ref. 11, the states near the Dirac cone of each layer couple with amplitudes of order t⊥θ≈0.4 t⊥ to states of energy ±ℏvFΔK of the opposing layer and the linear dispersion is preserved near zero energy. Here, t⊥ is the interlayer hopping for unrotated layers and vF∼106 m s−1 is the Fermi velocity. The two Dirac cones intersect near the centre of the superlattice Brillouin zone and hybridize, resulting in a saddle point in the energy dispersion and in two symmetric VHSs (Fig. 4a).

Figure 4: Energy separation between VHSs.
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a, Dispersion of lowest energy states for θ=1.79∘, t⊥≈0.27 eV, in a bilayer. A saddle point (marked ‘sp’) is visible between the two Dirac points in the negative energy band; another one exists in the positive energy band. b, Top: DOS of a bilayer with VHSs corresponding to the energy of the saddle points for θ=1.79∘,t⊥≈0.24 eV and no interlayer bias; middle: same with interlayer bias Vbias=0.15 V; bottom: same for a trilayer and no bias. Insets: The band energies along a line joining the two Dirac points for the corresponding layer configurations. c, Angle dependence of ΔEvhs. Open circles: bilayer with t⊥=0.27 eV and Vbias=0 V; filled diamonds: trilayer t⊥=0.24 eV,Vbias=0 V; crosses: experimental data. The error in peak separation was estimated from the peak width at 95% of peak height. The error bars are smaller than the symbols at low angles. Both the bilayer and trilayer fit the results, albeit with slightly different values of t⊥. For angles 2∘<θ<5∘, the peak separation is roughly described by ΔEvhs=ℏvF ΔK−2tθ, but curves upward at smaller angles.

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This mechanism of rotation-induced low-energy VHSs is not restricted to the bilayer. We extended the calculation to a trilayer consisting of an AB stacked bilayer and a third rotated layer on top and again found two VHSs in the DOS (Fig. 4b). In both cases, ΔEvhs is controlled by the energy scale ℏvFΔK∝ θ. Figure 4c shows that this model accounts well for the experimental angle dependence of ΔEvhs.

This analysis shows that the mechanism behind the formation of VHSs is quite robust, and we expect it to apply more generally whenever one layer is rotated with respect to the others, even for a graphene flake over graphite. The strong twisting-angle dependence of ΔEvhs is its unmistakable signature.

In the bilayer, the two Van Hove peaks are exactly symmetrical, whereas in the trilayer, a slight asymmetry occurs because of the third layer, but not as large as that observed experimentally. The discrepancy can be attributed to two factors. First, the experiments probe the local DOS, which varies across the unit cell of the Moiré pattern, whereas the calculation refers to the total DOS. Averaging the data over several unit cells reduces the asymmetry. Second, a bias between the layers can also enhance the asymmetry. In the AB stacked bilayer, a perpendicular electric field opens a gap in the spectrum15; not so in the twisted bilayer. As states close to zero energy result from the hybridization of zero-energy states in one layer with states of energy ±ℏvFΔK in the other, the corresponding wavefunctions have different weights in the two layers; when an electric field is applied, the two Dirac points move in opposite directions in energy11. The positive and negative energy states at the saddle points now have different weights in the two layers. As the STM probes predominantly the top layer, we expect this asymmetry in the STS. This effect of the bias in the DOS of the bilayer is shown in Fig. 4b (mid-panel).

It is important to note that the VHSs can form only in the presence of finite interlayer coupling. For vanishingly small interlayer coupling, t⊥∼0, as is the case when a detached graphene flake is found on a graphite substrate16 (see also Supplementary Information), the VHSs will not form even though a Moiré pattern may be visible in the STM images17 (Supplementary Information). It is well known that on the surface of graphite one often finds graphene flakes loosely bound to the surface where interlayer coupling is completely suppressed16. Such flakes, regardless of the presence or absence of rotation with respect to the underlying layer, show an uninterrupted honeycomb structure and a Landau-level sequence characteristic of massless Dirac fermions16. However, they do not show VHSs in their zero-field DOS.

The situation here is reminiscent of transition-metal dichalcogenides14, which form triangular lattices where both CDWs and superconductivity are observed18. In fact, these materials have large saddle regions in their band structure19, conjectured to be the origin of the CDW instability5. However, because the saddle regions and the position of EF are determined by the chemical composition, it is difficult to control or predict where instabilities occur. In contrast, in twisted graphene layers, both the position of EF and that of the VHSs can be controlled by gating and rotation respectively, providing a powerful toolkit for manipulating electronic phases. To achieve control of EF, the graphene sample can be deposited on a non-conducting substrate, or better yet suspended above a gate20,21. Although our present experimental set-up did not allow tuning of EF, it was possible to find rotated layers in which the VHSs and EF were sufficiently close to induce a strong CDW. A method to vary θ in situ (and thus to control the position of the VHSs) using an AFM was demonstrated in the discovery of superlubricity in graphite22. Superlubricity, which occurs in twisted graphene layers at small rotation angles, was attributed to decoupling of the layers. This led to the assumption that a small twist between layers would decouple them and reveal the electronic structure of an isolated graphene layer17. However, counterintuitively, we found that a rotation, however small, markedly affects the band structure and leads to the appearance of VHSs.

The experiments described here demonstrate that in graphene, unlike in any other known material, one can tune the position of the VHSs by controlling the relative angle between layers. As the VHS approaches the Fermi energy, we find a strongly localized CDW corresponding to the wavevector separation between the VHSs. This opens exciting opportunities for inducing and exploring correlated electronic phases in graphene.