Angle-dependent van Hove singularities and their breakdown in twisted graphene bilayers

The creation of van der Waals heterostructures based on a graphene monolayer and other two-dimensional crystals has attracted great interest because the atomic registry of the two-dimensional crystals can modify the electronic spectra and properties of graphene. A twisted graphene bilayer can be viewed as a special van der Waals structure composed of two mutually misoriented graphene layers, where the sublayer graphene not only plays the role of a substrate, but also acts in an equivalent role as the top graphene layer in the structure. Here we report the electronic spectra of slightly twisted graphene bilayers studied by scanning tunneling microscopy and spectroscopy. Our experiment demonstrates that twist-induced van Hove singularities are ubiquitously present for rotation angles θ of less than about 3.5°, corresponding to moiré-pattern periods $D$ longer than 4 nm. However, they totally vanish for θ>5.5° $(D<2.5\text{nm})$. Such a behavior indicates that the continuum models, which capture moiré-pattern periodicity more accurately at small rotation angles, are no longer applicable at large rotation angles.

Figure 1

(a) Schematic structural model of two misoriented honeycomb lattices with a twist angle θ (left) and the schematic Dirac cones, $K$ and $K_{\theta }$, of the two graphene layers in the reciprocal space (right). (b) Electronic band structure of twisted bilayer graphene along a line joining the two Dirac points (left) and the corresponding low-energy density of states calculated numerically according to the well-known formula $\frac{S}{4{\pi }^2}\oint 1/{\nabla }_{k}E\left(\vec{k}\right)dk$ (right). In the calculation, θ = 3.9° and $t_{\theta }=0.04\mathrm{eV}$ are used. (c), (e), (g) The 11 × 11 ${\mathrm{nm}}^2$ STM topographies of graphene bilayers on Rh foils showing three different periods of moiré patterns. (c) θ = 5.2°, $D=2.7$ nm ($V_{\mathrm{sample}}$ = −0.5 V, $I=0.22$ nA). (e) θ = 3.9°, $D=3.6$ nm ($V_{\mathrm{sample}}=0.7$ V, $I=0.22$ nA). (g) θ = 3.4°, $D=4.1$ nm ($V_{\mathrm{sample}}$ = −0.5 V, $I=0.19$ nA). (d), (f), and (h) show the tunneling spectra, i.e., dI/dV-$V$ curves, acquired on different positions of the moiré patterns in (c), (e), and (g), respectively. The spectra have been vertically offset for clarity. The two peaks flanking the Dirac point of the spectra are attributed to the twist-induced VHSs of the graphene bilayer. The positions of the two VHSs are not always symmetric around the Fermi level, which may arise from a different charge transfer between the graphene and the substrate on different samples.

Figure 2

(a) A STM image of a graphene bilayer shows a moiré pattern with the twisted angle ∼2.7° ($V_{\mathrm{sample}}$ = −0.50 eV, $I$ = 0.29 nA). The period of the moiré pattern is about 5.22 nm. (b) Tunneling spectra, i.e., dI/dV-$V$ curves, acquired on the top and troughs of the moiré pattern at positions indicated in (a). The spectra have been vertically offset for clarity. The two peaks flanking the Dirac point are attributed to the two VHSs. (c) Tunneling spectra recorded on the top of the moiré pattern, as indicated in (a).

Figure 3

Angle-dependent VHSs in twisted graphene bilayers. Solid blue circles are the average experimental data measured in different twisted graphene bilayers. Error bars in energy represent the standard deviation of the $\Delta E_{\mathrm{VHS}}$ observed in our experiment. The gray dotted line is a linear fit of the experimental data using the empirical equation $\Delta E_{\mathrm{VHS}}=\hslash {\nu }_{\mathrm{F}}\Delta K-2t_{\theta }$, with ${\nu }_{\mathrm{F}}=0.70\pm 0.02$ m/s and $t_{\theta }=0.04\pm 0.01$ eV.

Figure 4

(a)–(c) The 11.6 × 11.6 ${\mathrm{nm}}^2$ STM topographies of three graphene bilayers with different twisted angles. (a) θ = 4.8°, $D$ = 3.0 nm ($V_{\mathrm{sample}}$ = −0.5 V, $I$ = 0.20 nA). (b) θ = 4.0°, $D$ = 3.5 nm ($V_{\mathrm{sample}}$ = 0.37 V, $I$ = 0.24 nA). (c) θ = 3.7°, $D$ = 3.8 nm ($V_{\mathrm{sample}}$ = 0.41 V, $I$ = 0.21 nA). (d)–(f) Typical dI/dV-$V$ curves acquired on the moiré pattern in (a)–(c), respectively. These spectra show no discernible structures and are almost identical to those of the graphene monolayer.

Figure 5

(a), (d) 13.4 × 13.4 ${\mathrm{nm}}^2$ STM images of twisted graphene bilayers grown on a Rh foil. (a) θ = 3.7°, $D$ = 3.84 nm ($V_{\mathrm{sample}}$ = 0.7 eV, $I$ = 0.22 nA). (d) θ = 3.6°, $D$ = 3.89 nm ($V_{\mathrm{sample}}$ = −0.62 eV, $I$ = 0.20 nA). (b) and (e) show zoom-in topographies of the red frames in (a) and (d) with atomic-resolution images of the honeycomb lattice and triangular lattice, respectively. The hexagonal structure of graphene is overlaid onto the STM images. (c), (f) Tunneling spectra, i.e., dI/dV-$V$ curves, acquired on the moiré pattern in (b) and (e) respectively.

Figure 6

(a), (d) 9.2 × 9.2 ${\mathrm{nm}}^2$ STM images of twisted graphene bilayers grown on a Rh foil. (a) θ = 4.0°, $D$ = 3.53 nm, $V_{\mathrm{sample}}$ = 0.37 eV, $I$ = 0.24 nA. (d) θ = 6.6°, $D$ = 2.13 nm, $V_{\mathrm{sample}}$ = −0.50 eV, $I$ = 0.29 nA. (b), (e) Zoom-in topographies of the red frames in (a) and (d) showing atomic-resolution images of the honeycomb lattice and triangular lattice. The hexagonal structure of graphene is overlaid onto the STM images. (c), (f) Tunneling spectra, i.e., dI/dV-$V$ curves, acquired on the moiré pattern in (e) and (e), respectively.

Figure 7

Four 3 × 3 nm topographies of a twisted graphene bilayer in the same area. The voltage bias is shown in the upper right corner of the STM images and the constant current is 0.15 nA.

Figure 8

The number of coupled graphene bilayers (open bars with blue lines) and decoupled graphene bilayers (red bars) vs the rotation angles observed in our experiments.