Unit cell

Everyday term

In plain words

The smallest block of a crystal which, repeated over and over in every direction, builds the whole thing – as one motif, repeated, covers a whole roll of patterned wallpaper. Its edge lengths and angles are the numbers quoted as a crystal’s lattice parameters.

Going deeper

The unit cell is the smallest block that builds the whole crystal by repetition. In a layered material the in-plane lengths come from the bonding inside a layer, while the repeat along the stacking direction counts layers and the gaps between them.

The block that builds the crystal

A crystal is a lattice of repeated points with the same group of atoms attached to each. The unit cell is the parallelepiped that fills space by translation alone, described by three edge lengths a, b and c and the three angles α, β and γ between them, plus the positions of the atoms inside it. Those six numbers are the lattice parameters that any crystallographic description quotes.

The choice is not unique: many cells can generate the same lattice. Convention picks the smallest cell consistent with the symmetry, which is why a hexagonal crystal is described with a 120° angle rather than a rectangular box. The full set of conventions is codified in the International Tables for Crystallography.

Cells of layered materials

In a layered crystal the two in-plane parameters are set by strong bonding within a layer and hardly change between : graphene, graphite and graphene all share an in-plane spacing of about 2.46 Å. The third parameter is different in kind, because it counts how many layers pass before the pattern repeats, and each layer contributes its own thickness plus a .

That is why 2H-MoS₂ has c of about 12.3 Å, two layers of about 6.15 Å each, while 3R-MoS₂ has a c of three such layers. Polytypes share a and b and differ along c. Pressure squeezes c far more easily than a, and expands c while leaving a untouched.

Describing a single layer

A monolayer is not periodic perpendicular to itself, so it has no meaningful c. Its cell is given by a, b and γ, with the layer thickness stated separately as a measured or nominal value rather than a lattice parameter. Calculations, however, usually place the layer in a periodic box with vacuum above and below, and the vacuum height is a convergence parameter rather than a property of the material – an important distinction when comparing published numbers.

A between two twisted layers has a cell of its own, tens of nanometres across and containing thousands of atoms, which is why moiré calculations use effective models rather than treating every atom.

For specialists

The repeating parallelepiped that generates the lattice under translation, given by a, b, c and α, β, γ together with the positions of the atoms inside it. In a layered material the in-plane parameters are fixed by bonding while c depends on stacking and on the van der Waals gap, so polytypes share in-plane parameters and differ along c. A 2D cell quotes a, b and γ only, with the layer thickness stated separately.

Where this comes from

International Tables for Crystallography International Union of Crystallography · Volume A: space-group symmetry (2006) cited by 829

Chemical bond

The glue that holds atoms together. In a covalent bond two atoms share electrons, as within a sheet of graphene; in an ionic bond one atom hands electrons to another and the resulting charges attract, as in table salt; in a metal the outer electrons are shared by all the atoms at once. Much weaker van der Waals forces hold molecules and the layers of a layered crystal together. A 2D material is a crystal with strong bonds within its layers and only weak ones between them – which is why it can be peeled.

The attractive interactions that hold atoms in molecules and solids, classified by how electrons are shared: covalent bonds, from overlapping orbitals holding shared electron pairs, strong and directional (sp² carbon, Mo–S in TMDCs); ionic bonds, from electron transfer and electrostatic attraction (halides, oxides); metallic bonding, with electrons delocalised in a partly filled band; and the weak, non-directional van der Waals interaction from correlated charge fluctuations, besides hydrogen bonds. Real bonds are mixtures, characterised by electronegativity differences and charge analyses. Layered crystals combine in-plane bonds of several electronvolts with interlayer binding of only about 20 meV per Å² (some 0.3 J/m²), the criterion high-throughput screens use to identify exfoliable compounds.

Full explanation