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Observation of Van Hove singularities in twisted graphene layers

Guohong Li, A. Luican, J. M. B. Lopes dos Santos, A. H. Castro Neto, A. Reina, J. Kong, E. Y. Andrei

arXiv:0912.2102v1cond-mat.mes-hall

TL;DR

Van Hove singularities can be difficult to bring near the Fermi energy, limiting control of conditions associated with electronic instabilities. Here, scanning tunneling spectroscopy reveals Van Hove singularities in twisted graphene, whose energy position can be tuned by changing the angle between layers.

  • Problem

    Controlling the position of Van Hove singularities near the Fermi energy is difficult, despite their association with possible new phases of matter.

    Source: p. 2

  • Method

    The authors use scanning tunneling spectroscopy to measure the density of states in regions of rotated graphene layers.

    Source: p. 3

  • Results

    Rotated regions show two sharp density-of-states peaks flanking the Dirac points, and the Van Hove singularity position varies with the relative layer angle.

    Source: pp. 3, 6

  • Takeaways & Limitations

    Changing the relative layer angle offers a way to bring graphene’s Van Hove singularities closer to the Fermi energy.

    Source: p. 6

    Van Hove singularities form only with finite interlayer coupling; they do not form in decoupled graphene flakes even when a Moiré pattern is visible.

    Source: p. 5

Abstract

from arXiv · show

Electronic instabilities at the crossing of the Fermi energy with a Van Hove singularity in the density of states often lead to new phases of matter such as superconductivity, magnetism or density waves. However, in most materials this condition is difficult to control. In the case of single-layer graphene, the singularity is too far from the Fermi energy and hence difficult to reach with standard doping and gating techniques. Here we report the observation of low-energy Van Hove singularities in twisted graphene layers seen as two pronounced peaks in the density of states measured by scanning tunneling spectroscopy. We demonstrate that a rotation between stacked graphene layers can generate Van Hove singularities, which can be brought arbitrarily close to the Fermi energy by varying the angle of rotation. This opens intriguing prospects for Van Hove singularity engineering of electronic phases.

Figure Captions

The captions document moiré-associated stacking textures and spectroscopic features, including peaks near the Dirac point, energy-dependent spatial patterns, and angle-dependent Van Hove singularity separation.

Figure Captions

  • A 7.7±0.3 nm moiré pattern has bright regions with a triangular lattice indicating Bernal stacking and darker regions with a honeycomb-like structure suggesting slipped stacking from a small-angle rotation.The atomic-resolution images distinguish the two stacking textures.Source: Figure Captions · pp. 9–10, Figure 1
  • ~82 meV separates two sharp peaks flanking the Dirac point inside the moiré pattern; the peaks are absent outside it, where spectra resemble graphite.Spectra evolve monotonically between the rotated and unrotated regions.Source: Figure 2 · p. 10, Figure Captions · p. 10
  • At -58 meV, near a Van Hove singularity, dI/dV maps show strong density modulation, unlike maps at 77 meV farther from the singularity.The caption describes the modulation as suggesting an imminent Fermi-surface instability.Source: Figure 2 · p. 10, Figure Captions · pp. 10–11
  • At VHS energies of 1 mV and -11 mV, dI/dV maps show localized states, whereas the map at 90 mV is smooth and indicates extended states.The maps compare energies at and far from the VHS.Source: Figure Captions · p. 11
  • Bilayer and trilayer calculations both fit the measured angle-dependent ΔEvhs, with slightly different t⊥ values; the comparison includes experimental data.The bilayer and trilayer cases use different stated interlayer hopping values and are plotted against rotation angle.Source: Figure 4 · p. 11, Figure Captions · p. 11
  • A CVD graphene moiré pattern with a 12±0.5 nm superlattice constant corresponds to a rotation angle of about 1.16°.Source: Figure 3 · p. 14

1. Experimental details

Experiments used a custom low-temperature, high-field scanning tunnelling microscope to study cleaved HOPG and CVD graphene prepared on Ni and transferred to an Au grid. Main-text data were acquired at 4.4 K, with dI/dV measured using lock-in detection.

1. Experimental details

  • A home-built STM operated at 4 K/2 K and 13 T/15 T; samples included air-cleaved HOPG and CVD graphene grown on Ni, transferred to an Au grid, and cleaned in H2 at 250°C.The STM used a commercial RHK SPM 1000 controller and mechanically cut Pt-Ir tips.Source: §1 · p. 17
  • Main-text measurements were taken at 4.4 K, and dI/dV was measured by lock-in detection with a 340 Hz bias-voltage modulation.A superconducting magnet applied the field perpendicular to the sample surface in persistent mode.Source: §1 · p. 17
  • Large-area coarse STM nanopositioners were used to search for the moiré pattern.Source: §1 · p. 17

2. Tight binding model for twisted graphene bilayers

Rotation shifts the layers’ Dirac points, so interlayer coupling lifts degeneracies where the cones intersect and produces extrema associated with Van Hove singularities, while linear dispersion near zero energy is retained.

2. Tight binding model for twisted graphene bilayers

  • The twisted-bilayer model retains hopping from each atom to its closest atoms in the other layer, with geometry-based positions and graphite Slater–Koster hopping amplitudes periodic in the Moiré pattern.The supplied passages contrast this construction with an unrotated bilayer, where interlayer hopping makes the touching bands quadratic.Source: §2 · p. 18
  • At small twist angles, three Fourier amplitudes—G=0, G=−G1, and G=−G1−G2—dominate and have equal modulus.G1 and G2 are reciprocal vectors of the Moiré superlattice.Source: §2 · pp. 18–19
  • A quasi-free-electron calculation captures the lowest bands by coupling three momentum states in each layer, for twelve states overall including positive and negative energies.The layer-2 low-energy states couple to three layer-1 momenta at equal distance ΔK from its Dirac point, with corresponding energies ±vFΔK.Source: §2 · p. 19
  • Twisted-layer coupling lifts cone-intersection degeneracies, producing an energy maximum along the line joining K and Kθ and a minimum along the cones’ intersection, while preserving linear dispersion near zero energy.Unlike the unrotated bilayer, the shifted Dirac cones couple zero-energy states to finite-energy states.Source: §2 · p. 19

3. Scanning tunneling spectroscopy of Moiré pattern in magnetic fields

A small twist between graphene layers strongly changes the density of states and Landau-level spectrum, producing sharp zero-field Van Hove singularities and altering the finite-field sequence.

3. Scanning tunneling spectroscopy of Moiré pattern in magnetic fields

  • At small rotation angles, twisted-graphene Landau levels form a small modulation atop Van Hove singularity peaks, unlike the single-layer square-root sequence in field and level index [16].For a decoupled layer, the Landau levels instead form a single sequence with square-root dependence.Source: §3 · p. 20
  • In zero field, the density of states develops sharp Van Hove singularities, showing that a small twist drastically changes the electronic structure rather than revealing an isolated graphene layer.The passages contrast this result with the expectation that a small twist would decouple the layers.Source: §3 · p. 20
  • At large angles, square-root Landau levels reappear between Van Hove singularities with reduced Fermi velocity; this contrasts with the cited report’s decoupling interpretation, whose spectra lacked Van Hove singularities and had period-independent levels [17].The passage states that Landau levels should depend on twist angle for the reported Moiré patterns, with weaker dependence only at very large rotations.Source: §3 · p. 20

4. Large angle moiré pattern in CVD graphene

In CVD graphene twisted by ~3.48°, the moiré pattern is accompanied by two widely separated Van Hove singularities, while spectra show no detectable spatial variation across the pattern.

  • ~430mV separates the two VHS in the CVD sample at a rotation angle of ~3.48° (i=9).Source: §4 · p. 21
  • Spectra at moiré peaks and valleys are identical within experimental noise, indicating no detectable charge-density-wave localization at this large twisting angle.Source: §4 · pp. 21–22
  • The moiré pattern has a lattice constant of 3.9±0.2 nm.Source: §4 · p. 21
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