Twisted bilayer graphene: the magic angle and moiré superlattices

Lay one graphene sheet on another and turn it by a small angle θ. Where the two honeycombs line up atom on atom the stacking is AA; a little further on, half the atoms of the top layer sit over hexagon centres of the bottom one, which is the AB (Bernal) stacking of graphite, and then BA. These regions repeat in a triangular moiré pattern with period

L = a / (2 sin(θ/2)), where a = 2.46 Å is graphene’s lattice constant.

The smaller the angle, the larger the moiré: about 2.8 nm at 5° and about 12.8 nm at 1.1°, where one moiré cell holds roughly 11,000 carbon atoms (calculated). The layers stay about 0.335 nm apart, as in graphite (Sun et al. 2017). A similar moiré forms between graphene and hexagonal boron nitride even when they are aligned, because hBN’s lattice constant (2.504 Å) is 1.8 % larger than graphene’s (2.460 Å), giving a period of about 14 nm (Liu et al. 2022; Yankowitz et al. 2012).

Each graphene layer has Dirac cones. In a twisted bilayer, electrons hop between the layers, and the moiré periodicity folds the bands into moiré Bloch bands. Bistritzer and MacDonald (2011) showed in a continuum model that as the twist angle is reduced the layers become more strongly coupled and the Dirac velocity crosses zero several times: at a discrete set of magic angles the velocity vanishes, the lowest moiré band flattens and the density of states at the Dirac point is strongly enhanced.

Cao et al. (2018) confirmed experimentally that near the first magic angle, about 1.1°, the bands near zero Fermi energy become flat and show insulating states at half-filling that are not expected without correlations between electrons, consistent with Mott-like insulators.

1. Exfoliate. Graphene and hBN flakes are peeled from crystals onto substrates.
2. Tear and stack. A polymer stamp carrying hBN picks up part of a graphene flake, tearing it; the substrate is turned by the chosen angle and the rest of the same flake is picked up, so both layers have the same crystal orientation before the twist. Successive transfers with accurate rotational alignment (Kim et al. 2016) gave bilayers 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).
3. Gate. Contacts and a gate are added, so electrostatic doping tunes the carrier density and therefore the filling of the flat bands (Cao et al. 2018).

• Flat bands near zero Fermi energy at twist angles of about 1.1° (Cao et al. 2018).
• Correlated insulators at half-filling of the flat bands, consistent with Mott-like insulator states of electrons localised in the moiré superlattice (Cao et al. 2018, correlated insulator).
• Superconductivity. Doping away from the insulating states gives zero-resistance states with critical temperatures up to 1.7 K; the temperature–density phase diagram has dome-shaped superconducting regions similar to those of the cuprates (Cao et al. 2018).
• Strong pairing. Given its small Fermi surface, its critical temperature puts TBG among the superconductors with the strongest pairing between electrons (Cao et al. 2018).
• Tunable. The filling is set by a gate and the bandwidth by the twist angle (Cao et al. 2018, correlated insulator).

Twisted bilayer graphene is a purely carbon-based, precisely tunable two-dimensional superconductor (Cao et al. 2018). Correlated insulators and superconductivity appear in the same device as a gate changes the filling, and the twist angle is a control knob that conventional heterostructures do not have (Cao et al. 2018, correlated insulator). That makes it a model system for strongly correlated electrons and for questions about high-temperature superconductors.

It also turned the twist angle into a design parameter for all van der Waals heterostructures (Geim & Grigorieva 2013): aligned graphene on hBN, for example, gives a moiré superlattice in which a Hofstadter butterfly was observed (Hunt et al. 2013).

• Correlated electron physics. Mott-like insulators and unconventional superconductivity in one tunable device (Cao et al. 2018).
• Moiré superlattices. Graphene on hBN, with superlattice Dirac points (Yankowitz et al. 2012) and massive Dirac fermions (Hunt et al. 2013).
• Device assembly. Methods to set and hold the twist angle (Kim et al. 2016; Varma Sangani et al. 2020).
• Imaging. Scanning tunnelling microscopy shows the moiré directly (Yankowitz et al. 2012).

For a paper, slide or poster, show the two honeycombs with one turned, the resulting moiré with its AA, AB and BA regions and the period L, and, for the physics, the flat bands at the magic angle. The moiré figure above shows tear-and-stack assembly, the pattern, the two layers in 3D, the magic angle and graphene on hBN, and opens as an editable LabCanvas figure. The structure drawing is a high-resolution SVG of the figure editor’s magic-angle stacking map, in which you can change the twist or switch to the atoms of both layers. The 3D model opens in the LabCanvas 3D Studio, where you can change the twist angle and export a high-resolution image. A twisted bilayer has no ordinary crystal file, so there is no crystal structure drawing on this page.