# Twisted bilayers and moiré superlattices

> Twisted bilayer graphene and graphene on hBN in six panels: tear-and-stack assembly, the moiré pattern and its period, the two layers in 3D, the magic angle of 1.1°, the moiré of graphene on hBN from the 1.8 % lattice mismatch, and the moiré seen by STM.

- Page: https://www.labcanvas.com/templates/graphene-research/twisted-bilayers-moire-superlattices
- Open in the LabCanvas editor (LabCanvas account): https://www.labcanvas.com/editor.html?template=twisted-bilayer-graphene-moire
- Updated: 2026-10-14
- Category: Graphene research
- Method: Moiré
- Starting materials: Exfoliated graphene flakes and hBN on a polymer stamp
- Product: Twisted bilayers at a set angle, and aligned graphene on hBN
- Key result: Lattices, stacking maps and STM images computed by LabCanvas from the lattice constants and the twist; the tear-and-stack sizes and angle are drawn, not to scale
- Figure: 1920 × 1080 px, 45 editable layers (6 generated objects with settings)

## Steps

1. **Tear and stack**: Successive transfers with accurate rotational alignment built Bernal bilayers and aligned double-bilayer devices (Kim et al. 2016) and bilayer moiré crystals with twist angles controlled below 1° (Kim et al. 2017); in hBN-assisted tear-and-stack the angle is fixed when the stack is assembled (Varma Sangani et al. 2020; Zhang et al. 2026)
2. **The moiré pattern**: Two honeycomb lattices, one turned by θ, give AA, AB and BA stacking regions repeating with the period L = a / (2 sin(θ/2))
3. **Two layers**: The layers drawn apart in 3D at graphite’s 3.35 Å spacing, one turned by θ
4. **Magic angle**: At the magic angles the moiré bands of a twisted bilayer become flat (Bistritzer & MacDonald 2011); bilayers twisted to about 1.1° became superconducting (Cao et al. 2018)
5. **Graphene on hBN**: hBN’s lattice constant of 2.504 Å against graphene’s 2.460 Å (Liu et al. 2022) gives a moiré even when the layers are aligned
6. **Seen by STM**: STM images the moiré of graphene on hBN, whose wavelength follows the twist and the 1.8 % mismatch (Yankowitz et al. 2012)

## Strengths

- The twist is a new control knob for the bands
- Tear-and-stack fixes the angle at assembly (Zhang et al. 2026)
- hBN gives flat, clean substrates

## Limitations

- Small angles relax and vary across a device
- Exact angles are hard to reach and to keep

## Uses

- Superconducting and flat-band states
- Moiré superlattice devices
- Talks on twistronics

## References

1. Bistritzer & MacDonald, “Moiré bands in twisted double-layer graphene”, PNAS 2011. https://doi.org/10.1073/pnas.1108174108 (primary)
2. Cao et al., “Unconventional superconductivity in magic-angle graphene superlattices”, Nature 2018. https://doi.org/10.1038/nature26160 (primary)
3. Kim et al., “van der Waals heterostructures with high accuracy rotational alignment”, Nano Lett. 2016. https://doi.org/10.1021/acs.nanolett.5b05263 (primary)
4. Kim et al., “Tunable moiré bands and strong correlations in small-twist-angle bilayer graphene”, Proc. Natl Acad. Sci. USA 2017. https://doi.org/10.1073/pnas.1620140114 (primary)
5. Yankowitz et al., “Emergence of superlattice Dirac points in graphene on hexagonal boron nitride”, Nat. Phys. 2012. https://doi.org/10.1038/nphys2272 (primary)
6. Varma Sangani et al., “Facile deterministic cutting of 2D materials for twistronics using a tapered fibre scalpel”, Nanotechnology 2020. https://doi.org/10.1088/1361-6528/ab8b93 (further reading)
7. Zhang et al., “Dynamic twisting and imaging of moiré crystals”, Sci. Adv. 2026. https://doi.org/10.1126/sciadv.aeg5362 (further reading)
8. Liu et al., “Influence of hexagonal boron nitride on electronic structure of graphene”, Molecules 2022. https://doi.org/10.3390/molecules27123740 (further reading)
9. Hunt et al., “Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure”, Science 2013. https://doi.org/10.1126/science.1237240 (further reading)

Credit: Computed by LabCanvas after Bistritzer & MacDonald, PNAS 2011 (doi:10.1073/pnas.1108174108), Cao et al., Nature 2018, Kim et al., Nano Lett. 2016, Yankowitz et al., Nat. Phys. 2012, Liu et al., Molecules 2022, and the papers in the details.

## Questions

**What does the “Twisted bilayers and moiré superlattices” template show?** Twisted bilayer graphene and graphene on hBN in six panels: tear-and-stack assembly, the moiré pattern and its period, the two layers in 3D, the magic angle of 1.1°, the moiré of graphene on hBN from the 1.8 % lattice mismatch, and the moiré seen by STM.

**What are the steps shown in “Twisted bilayers and moiré superlattices”?** 1. Tear and stack: Successive transfers with accurate rotational alignment built Bernal bilayers and aligned double-bilayer devices (Kim et al. 2016) and bilayer moiré crystals with twist angles controlled below 1° (Kim et al. 2017); in hBN-assisted tear-and-stack the angle is fixed when the stack is assembled (Varma Sangani et al. 2020; Zhang et al. 2026). 2. The moiré pattern: Two honeycomb lattices, one turned by θ, give AA, AB and BA stacking regions repeating with the period L = a / (2 sin(θ/2)). 3. Two layers: The layers drawn apart in 3D at graphite’s 3.35 Å spacing, one turned by θ. 4. Magic angle: At the magic angles the moiré bands of a twisted bilayer become flat (Bistritzer & MacDonald 2011); bilayers twisted to about 1.1° became superconducting (Cao et al. 2018). 5. Graphene on hBN: hBN’s lattice constant of 2.504 Å against graphene’s 2.460 Å (Liu et al. 2022) gives a moiré even when the layers are aligned. 6. Seen by STM: STM images the moiré of graphene on hBN, whose wavelength follows the twist and the 1.8 % mismatch (Yankowitz et al. 2012).

**What are the starting materials and the product?** Starting materials: Exfoliated graphene flakes and hBN on a polymer stamp. Product: Twisted bilayers at a set angle, and aligned graphene on hBN.

**Which papers is “Twisted bilayers and moiré superlattices” based on?** Bistritzer & MacDonald, “Moiré bands in twisted double-layer graphene”, PNAS 2011 (doi:10.1073/pnas.1108174108); Cao et al., “Unconventional superconductivity in magic-angle graphene superlattices”, Nature 2018 (doi:10.1038/nature26160); Kim et al., “van der Waals heterostructures with high accuracy rotational alignment”, Nano Lett. 2016 (doi:10.1021/acs.nanolett.5b05263); Kim et al., “Tunable moiré bands and strong correlations in small-twist-angle bilayer graphene”, Proc. Natl Acad. Sci. USA 2017 (doi:10.1073/pnas.1620140114); Yankowitz et al., “Emergence of superlattice Dirac points in graphene on hexagonal boron nitride”, Nat. Phys. 2012 (doi:10.1038/nphys2272); Varma Sangani et al., “Facile deterministic cutting of 2D materials for twistronics using a tapered fibre scalpel”, Nanotechnology 2020 (doi:10.1088/1361-6528/ab8b93); Zhang et al., “Dynamic twisting and imaging of moiré crystals”, Sci. Adv. 2026 (doi:10.1126/sciadv.aeg5362); Liu et al., “Influence of hexagonal boron nitride on electronic structure of graphene”, Molecules 2022 (doi:10.3390/molecules27123740); Hunt et al., “Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure”, Science 2013 (doi:10.1126/science.1237240).

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