A flat net of hexagons, like a beehive, with an atom at every corner. Graphene is the famous example, but the same pattern gives boron nitride its , gives graphene its electrons, and gives some magnets interactions that cannot all be satisfied at once.
Going deeper
The honeycomb is a triangular lattice with two sites per cell, so every A site has three B neighbours. When the two sites are identical, as in graphene, the bands touch at the zone corners K and K′; when they differ, as in hBN, a gap opens there.
Two sites in every cell
A honeycomb is not a Bravais lattice on its own: no single translation maps every corner onto every other. It is a with a basis of two sites, conventionally A and B. Each A site has three nearest neighbours, all B sites, at 120° to each other – the geometry of sp2 bonding in graphene.
The reciprocal lattice is also triangular, and the Brillouin zone is a hexagon. Its six corners fall into two inequivalent sets, K and K′, related by time reversal. Much of the physics of honeycomb materials happens near those corners, which is why the index – whether an electron sits near K or near K′ – becomes a useful label.
What the geometry does to electrons
In 1947 Wallace applied a nearest-neighbour to a single graphite layer. With identical A and B sites, the conduction and valence bands touch at K and K′ and disperse linearly away from them: the Dirac cones behind graphene’s massless carriers. The touching is protected as long as the two sublattices remain equivalent.
Make them different and a gap opens. In hBN the two sites hold boron and nitrogen, and the gap is about 6 eV; a seen from above is a honeycomb of metal and sites with a gap near 2 eV. The gapped valleys carry opposite , the basis of valley-selective optics. such as silicene, germanene and stanene add , which in principle opens a gap.
Honeycomb magnets and beyond
The same geometry shapes magnetism. In CrI3 and α-RuCl3 the magnetic ions sit on a honeycomb. Kitaev showed in 2006 that on a honeycomb whose coupling depends on the direction of each bond form an exactly solvable model with a ground state and fractionalised excitations. α-RuCl3 is the best-studied candidate, although it orders magnetically at about 7 K, and how close it comes to Kitaev’s model under a magnetic field is still debated.
Honeycombs also appear in photonic crystals, arrays of cold atoms and molecules arranged on metal surfaces, where Dirac cones can be engineered and tuned more freely than in any natural crystal.
For specialists
A triangular Bravais lattice with a two-site basis, so every site has three nearest neighbours on the other sublattice. Equivalent sublattices with nearest-neighbour hopping give Dirac cones at K and K′ (graphene); inequivalent ones open a gap with valley-contrasting Berry curvature (hBN, TMDC monolayers seen from above); buckling mixes in spin–orbit coupling (the Xenes); and on the same geometry gives the exactly solvable Kitaev model behind the α-RuCl3 spin-liquid programme.