The map of the distinct ways an electron – or a vibration – can travel through a crystal, each set by a direction and a wavelength. Because the atoms repeat, waves whose momenta differ by certain fixed steps look the same to the crystal, so all of them fit into one cell of this map: the Brillouin zone. are drawn along paths through it. For graphene and the other layers it is a hexagon, with Γ at the centre and the K and K′ points at its corners – where graphene’s and the of the sit.
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The moiré is the beat of the two reciprocal lattices: each short reciprocal vector of the bottom layer beats with the nearest one of the top, and the two shortest beats span the moiré.
The Brillouin zone holds every distinct crystal momentum once: momenta outside it repeat ones inside, shifted by a step of the reciprocal lattice. Band structures are plotted along lines between its special points – for graphene, in a simple tight-binding model, the bands touch at K and have saddle points at M. A moiré’s long period folds the bands into a much smaller mini zone.
Momentum in a repeating crystal
A wave travelling through a crystal – an electron, or a vibration of the atoms – can be labelled by its momentum: which way it runs and how short its wavelength is. In empty space every momentum describes a different wave. In a crystal the atoms repeat, and two waves whose momenta differ by one of a special set of steps look the same at every atom and describe the same state. Those steps form a lattice of their own, in momentum rather than in space – the reciprocal lattice – and the label, defined only up to them, is called the crystal momentum.
So only one cell of momentum space is needed, and the Brillouin zone is the natural choice: every momentum closer to the centre, Γ, than to any other point of the reciprocal lattice. The shorter a crystal’s repeat in space, the larger its zone in momentum. Each band of a band structure, and each branch of the , runs once across it, and plots usually follow straight lines between a few points of high symmetry, where bands tend to have their highest and lowest points and their crossings.
Γ, K and M
The honeycomb layers – graphene, hexagonal boron nitride, the transition metal dichalcogenides – share a hexagonal Brillouin zone. Its centre is Γ, the midpoints of its edges are the M points, and its six corners alternate between K and K′. Corners of the same kind are one point seen three times, shifted by steps of the reciprocal lattice, so there are only two distinct corners. For graphene, with carbon atoms 1.42 Å apart, they lie 1.70 Å−1 from the centre (Castro Neto and colleagues, 2009).
Much of what makes these layers special happens at those points. Graphene’s two bands touch at K and K′, forming its Dirac cones. The of MoS2 and its relatives sit at the same corners, as two valleys that circularly polarised light can address one at a time. At M graphene’s bands have saddle points, and with them a . Angle-resolved maps bands across the zone directly, converting the angle at which an electron leaves the crystal into the momentum it had inside.
Folding into a mini zone
Give a crystal a second, longer period – a from two mismatched or twisted layers, a , a patterned gate – and its Brillouin zone shrinks to match. The bands do not disappear: they fold back into the small zone, branches from far apart land on top of each other, and where they meet, the new periodic potential mixes them, opening gaps or making new crossings.
Graphene on hexagonal boron nitride shows it. The two lattices differ by 1.8%, so even aligned layers form a moiré, at most about 14 nm long, and its mini zone is small enough that the folded bands produce a second set of Dirac points, seen by as dips in the at an energy set by the moiré period (Yankowitz and colleagues, 2012). In twisted bilayer graphene near 1.1°, the moiré period is about 13 nm and the mini zone nearly three thousand times smaller in area than graphene’s; at a discrete set of the lowest folded band becomes almost flat (Bistritzer and MacDonald, 2011), the starting point for its .
For specialists
The Wigner–Seitz cell of the reciprocal lattice: all wavevectors closer to Γ than to any other reciprocal-lattice vector G, so that each crystal momentum – defined only modulo G by Bloch’s theorem – appears once. For a hexagonal lattice it is a hexagon with Γ at the centre, M at the edge midpoints and K, K′ alternating at the corners; with graphene’s carbon–carbon distance of 1.42 Å, K lies 4π/(3√3 a) ≈ 1.70 Å−1 from Γ. A superlattice of period L folds the bands into a mini zone of size about 2π/L: graphene aligned on hBN, whose lattice is 1.8% longer, forms a moiré up to about 14 nm long whose folded bands make secondary Dirac points, and twisted bilayer graphene near 1.1° has a mini zone nearly three thousand times smaller in area than graphene’s, in which the lowest bands flatten at the magic angles.
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