Unraveling the Intrinsic and Robust Nature of van Hove Singularities in Twisted Bilayer Graphene by Scanning Tunneling Microscopy and Theoretical Analysis

Extensive scanning tunneling microscopy and spectroscopy experiments complemented by first-principles and parametrized tight binding calculations provide a clear answer to the existence, origin, and robustness of van Hove singularities (vHs) in twisted graphene layers. Our results are conclusive: vHs due to interlayer coupling are ubiquitously present in a broad range (from 1° to 10°) of rotation angles in our graphene on $6H$-SiC(000-1) samples. From the variation of the energy separation of the vHs with the rotation angle we are able to recover the Fermi velocity of a graphene monolayer as well as the strength of the interlayer interaction. The robustness of the vHs is assessed both by experiments, which show that they survive in the presence of a third graphene layer, and by calculations, which test the role of the periodic modulation and absolute value of the interlayer distance. Finally, we clarify the role of the layer topographic corrugation and of electronic effects in the apparent moiré contrast measured on the STM images.

Figure 1

(a) Illustration of a MP arising from a rotation angle $\theta =9.6°$. (b) Emergence of vHs as a consequence of the rotation in reciprocal space. (c) STM images of several MP with different $\theta$. The scale bar is 5.0 nm. (d) LDOS spectra taken on the MP shown in (c). The curves are shifted vertically for clarity. The arrows point to the vHs. For $\theta =1.4°$, max (min) indicates a spectrum taken on a bright (dark) area. (e) vHs separation as a function of $\theta$. Crosses: Data from Ref. 13. Filled and open circles: This work. Colored dots refer to the spectra displayed in (d) (same color code). Open circles correspond to multiple MP discussed in Fig. 3.

Figure 2

(a) Ab initio calculations of the total DOS, interlayer distance $d=3.42\AA$. (b) DOS calculations using the TB formalism with $v_F=1.1\times {10}^6\mathrm{m}/\mathrm{s}$. The red continuous line is the LDOS in $AA$ stacked areas, the black dotted line is the total DOS. The spectra have been shifted for clarity. (c) Computed vHs splitting as a function of $\theta$. Green line: Eq. (1) with $v_F=1.0\times {10}^6\mathrm{m}/\mathrm{s}$ and $t_{\theta }=0.11\mathrm{eV}$ [13], black [gray (pink)] circles: ab initio calculation with $d=3.42\AA$ [$d=3.10\AA$], black [gray (blue)] triangles: TB calculations with $v_F=1.09\times {10}^6\mathrm{m}/\mathrm{s}$ [$v_F=0.79\times {10}^6\mathrm{m}/\mathrm{s}$]. (d) Filled gray (orange) circles: Experimental data as in Fig. 1c compared to some calculations displayed in (c) with the same color code. The dotted (blue) line is a fit of the experimental data using Eq. (1) with $v_F=1.12\times {10}^6\mathrm{m}/\mathrm{s}$ and $t_{\theta }=0.108\mathrm{eV}$.

Figure 3

(a) Left: $10.6\times 10.6{\mathrm{nm}}^2$ STM image of a MP with two periods: 1.52 and 3.7 nm. The smaller period corresponds to the MP between first and second surface layers. Right: Average spectrum taken on the green line. (b) Left: $24\times 24{\mathrm{nm}}^2$ STM image of a MP with two periods: 12.7 and 3.0 nm (blue diamond). The larger period corresponds to the MP between first and second surface layers. Right: Local spectra taken on the spots labeled on the image. Arrows point to the vHs.

Figure 4

(a) STM image at sample bias $V=+0.5\mathrm{V}$ of a MP with $P=2.32\mathrm{nm}$ ($\theta =6.08°$). The measured corrugation is 0.14 Å. (b) STM image at $V=+0.2\mathrm{V}$ of a MP with period $P=4.3\mathrm{nm}$ ($\theta =3.28°$). (c) Range of STM measured MP corrugation as a function of rotation angle $\theta$. (d) Simulated STM image for a bilayer rotated $\theta =6.01°$ at $+0.5\mathrm{eV}$. (e) Atomic (dots) and total (line) corrugation for the bilayer image shown in (d) along the long MP diagonal. The blue and red horizontal dashed lines indicate the interlayer spacing computed for $AB$ and $AA$ bilayers, respectively.