Moiré superlattice

Also called moiré pattern

In plain words

The larger pattern that appears when two lattices are laid on top of each other slightly rotated, or with slightly different spacings – like two fine mesh curtains overlapping. Electrons feel this larger pattern, and it can reshape how they behave.

Going deeper

A moiré pattern computed from two identical lattices turned 6° against each other. Where the dots of both layers coincide (circled) the stacking is AA; in between they interleave. The large pattern repeats with period λ = a / (2 sin θ/2), about 9.6 lattice constants at this angle – and about 52 at graphene’s magic angle of 1.1°.

Interference of two lattices

Lay two identical lattices on top of each other with a small rotation and a much larger pattern appears: regions where the atoms of both layers sit directly on top of one another (AA stacking) alternate with regions where they interleave (AB and BA). The period of that pattern is λ = a / (2 sin(θ/2)), where a is the and θ the , so the smaller the angle, the larger the pattern. For graphene (a = 0.246 nm) at 1.1°, λ is about 12.8 nm, and each moiré cell holds roughly ten thousand atoms.

A mismatch in lattice constant does the same without any twist. Graphene on hBN, whose lattice is about 1.8% larger, forms a moiré pattern of about 14 nm when the two are aligned.

Why a bigger unit cell changes the electrons

The moiré pattern acts on electrons as a periodic potential with a period of many nanometres. They respond to it as they would to an ordinary crystal lattice, but because the period is so large, the corresponding Brillouin zone is tiny and the original bands fold into many narrow minibands. At the magic angle the lowest minibands of twisted bilayer graphene become nearly flat, and interactions between electrons take over.

The large cell also brings magnetic effects within reach. The magnetic flux through one moiré cell reaches a flux quantum at fields of a few tens of tesla, rather than the tens of thousands of tesla an atomic cell would need, which is how the fractal Hofstadter butterfly spectrum became measurable.

Making and seeing them

Twisted stacks are usually assembled by , which sets the angle to within a fraction of a degree, although the angle often wanders across a device. At very small angles the lattices do not stay rigid: they reconstruct into large AB and BA domains separated by sharp domain walls. The superlattice can be imaged by , and several scanning-probe techniques, and it shows up in transport as extra resistance peaks whenever a moiré miniband is filled.

For specialists

The long-wavelength interference pattern formed by two lattices with a small twist or ; its period sets a new, much larger unit cell for electrons.

Where this comes from

Hofstadter’s butterfly and the fractal quantum Hall effect in moiré superlattices Dean et al. · Nature 497, 598 (2013) cited by 1,752

Scanning tunnelling microscopy (STM)

A needle sharpened to a single atom is brought so close to a surface that electrons tunnel across the gap – a quantum effect by which they pass through a barrier they could never climb over. The current maps the surface atom by atom, and how it changes with voltage tells you which energies electrons are allowed to have at that exact spot.

Tunnelling between a sharp tip and a conducting surface, giving atomic-resolution topography and, in spectroscopy mode, the local density of states against energy. On 2D materials it resolves defect levels, moiré potentials, edge modes and quasiparticle gaps site by site. The sample has to be clean and conducting, so it is run on epitaxial films or on flakes placed on graphite, in vacuum and often at cryogenic temperature.

Full explanation

Epitaxy

Growing a crystal on top of another so that its atoms line up with the ones underneath and carry on the pattern of the crystal below. It is how most semiconductor chips, lasers and LEDs are built, and it works cleanly only when the two crystals’ atomic spacings match closely – a restriction that layered 2D materials largely escape.

Growth of a crystalline film whose orientation is fixed by a single-crystal substrate: homoepitaxy on the same material, heteroepitaxy on a different one. A covalently bonded film strains to the substrate’s lattice until, beyond a critical thickness that falls steeply with mismatch, misfit dislocations relax it, so mismatches above a few percent permit only very thin coherent films. In van der Waals epitaxy the film is aligned but not bonded and mismatch is largely irrelevant; in remote epitaxy the substrate’s potential aligns the film through a graphene interlayer.

Full explanation