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Dodecagonal 30° twisted bilayer graphene is just two graphene honeycomb nets with an exact 30° rotation with respect to the other. If you treat it as a 2D pattern rather than a 3D stacked structure, it would have 12-fold rotational symmetry and apparently some quasicrystal properties, including:

  1. the diffraction pattern (and thus the Fourier transform) has clean, crisp 12-fold symmetrical spot pattern, even though there is no translational symmetry
  2. the pattern can be tiled

The field of tilings is well outside the scope of my understanding of mathematics and so I'd like to make some inroads using a familiar context, and bilayer graphene (and 2D materials in general) is somewhat familiar to me.

So I'd like to ask within the context of the paper and figures cited below:

Question:

What is 12-fold Stampfli-inflation tiling and where/how can I recognize it in this analysis of dodecagonal 30° twisted bilayer graphene?

How can I clearly see the "inflation" and the "self-similarity" at $\sqrt{2 + \sqrt{3}}$ and $2 + \sqrt{3}$? I see the drawn boundaries denoting these in the last figures (Fig S7 from the supllement), but is it possible to some how say Here, this is the self-similarity they are talking about.


Figure 1 a, b from Ahn et al 2018 "Dirac electrons in a dodecagonal graphene quasicrystal" in Science 361, (6404) pp.782-786

FIGURE 1. A LEED (low energy electron diffraction) pattern and a TEM image of graphene quasicrystal. (A) A LEED pattern of graphene quasicrystal. (B) A Fourier transformed pattern of graphene quasicrystal (see also Fig. S2 in SI).

Figure 1 c through f from Ahn et al 2018 "Dirac electrons in a dodecagonal graphene quasicrystal" in Science 361, (6404) pp.782-786

(C)-(D) An atomic structure model of twisted bilayer graphene with a rotational angle of 30°. (E) Atomic structures and TEM images of Stampfli tiles (rhombuses (red), equilateral triangles (green), squares (blue)). (F) A false colored TEM image of graphene quasicrystal mapped with 12-fold Stampfli-inflation tiling.

From the supplement:

Figure S7 c and d from the supplemental material to Ahn et al 2018 "Dirac electrons in a dodecagonal graphene quasicrystal" in Science 361, (6404) pp.782-786

FIGURE S7. TEM images of graphene quasicrystal. The TEM images in (A)-(B) and (D)-(E) are measured after transferring graphene quasicrystal on a TEM grid. A cross-sectional scanning transmission electron microscope (STEM) bright field image in (C) is measured on SiC before transferring graphene quasicrystal. (A) A large-scale TEM image, where its selected area electron diffraction (SAED) pattern is shown in the inset. (B) An enlarged atomic scale TEM image, where the Fourier-transformed image of the TEM image is in the inset. (C) A cross-sectional STEM bright field image of graphene quasicrystal on SiC, where bilayer graphene are clearly observed. (D)-(E) Two different Stampfli-inflation tiling, compared with the TEM image. (D) 12-fold inflation rule with a scaling factor of $\sqrt{2 + \sqrt{3}}$, (E) Dodecagonal inflation rule with a scaling factor of $2 + \sqrt{3}$.

All of this is from the paper Ahn et al. (2018) Dirac electrons in a dodecagonal graphene quasicrystal Science 361, (6404) pp.782-786 (also in arXiv). The text refers to figures in the paper's supplemental information or "SI" as well.

The graphene quasicrystal can be spatially mapped onto a quasicrystal lattice model constructed by dodecagonal compound tessellations (see Figs. 1D-E and Figs. S3D-E in (*supplemental information))) (23-26). Squares, rhombuses, equilateral triangles with different orientations can fill the entire space with a 12-fold rotationally symmetric pattern without translational symmetry. Since the Stampfli tiles have a fractal structure with self-similarity, the same pattern emerges at a larger scale with an irrational scaling factor. For the graphene quasicrystal, the Stampfli tiles have the scaling factor of $\sqrt{2 + \sqrt{3}}$ (Figs. 1D-E and Fig. S3 in SI)23. (note: maybe it's supposed to be S7?) As shown in the atomic model (Figs. 1C-D), the graphene quasicrystal results in Stampfli tiles such as equilateral triangles and rhombuses. In the false colored TEM image of the graphene quasicrystal transferred from a SiC wafer to a TEM grid (Fig. 1F and Fig. S3D in SI), the Stampfli tiles including squares (blue), rhombuses (red), equilateral triangles (green) with different orientations were obviously observed. The LEED pattern and TEM image clearly support that the twisted bilayer graphene with a rotational angle of 30 ̊ has a quasicrystalline order with 12-fold rotational symmetry. We note that, for TEM experiments, the graphene quasicrystal grown on a SiC wafer should be transferred on other substrates and should be robust to chemical treatments in air. The successful TEM experiments support that graphene quasicrystal can be isolated from a substrate and are chemically and structurally stable at room temperature in air. We expect that the robustness of graphene quasicrystal can lead to further studies on the physical properties and applications.

Having established quasicrystalline order in our sample, now we discuss its characteristic electronic structures...

23P. Stampfli, A dodecagonal quasi-periodic lattice in 2 dimensions. Helv. Phys. Acta 59, 1260-1263 (1986) (roughly here)

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    $\begingroup$ Maybe this paper helps? arxiv.org/abs/2102.06046. If you would like to learn about substitution/inflation tilings in general, then Baake and Grimm's book "Aperiodic Order Volume 1" is a good place to start. I presume that when they say there is a self-similar structure, it means they are saying there is some inflation rule which produces a tiling that is 'mutually locally derivable' to the point set produced by the bi-layer of graphene. $\endgroup$
    – Dan Rust
    Commented Apr 4, 2022 at 10:58
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    $\begingroup$ A couple minor points. (1) Even with the nonplanar structure the bilayer has an improper 12-fold axis, so it will produce the observed quasilattice diffraction pattern. (2) The only quasiperiodic ratios have the form $a+b\sqrt3$ with $a$ and $b$ integers; the nested square roots do not enter because they do not correspond to length ratios along parallel lines. $\endgroup$ Commented Aug 19, 2022 at 22:16

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