In plain words
The small shape traced by the atoms that surround a metal atom in a crystal. In many layered materials each metal sits inside an octahedron – six neighbours, three above and three below, turned against each other – or a trigonal prism, where the upper three sit right above the lower three. That one turn can make the same compound a or a metal.
Going deeper
Counting neighbours
The atoms nearest to a given atom mark out a small polyhedron around it: four neighbours make a tetrahedron, as in silicon, six an octahedron, eight a cube. Chemists call this the atom’s coordination and the count its coordination number. In a layered MX2 crystal the question usually concerns the metal, sandwiched between two sheets of anions with three neighbours above and three below. If the upper three sit directly over the lower three they form a trigonal prism; if the upper triangle is turned by 60° they form an octahedron. MoS2 in its usual form is trigonal prismatic, while SnS2, ZrS2 and HfS2 are octahedral.
Octahedra also build most layered and many oxides. In CrI3 and RuCl3 each metal sits in an octahedron of halide ions, and neighbouring octahedra share edges, so the metals form a ; in the layered perovskites the octahedra share corners and form a square net. Edge-sharing and corner-sharing, on a material’s page, describe exactly this – and the lattice the metals end up on decides much of their magnetism.
Why the shape decides the electronics
The neighbours’ negative charge pushes on the metal’s five d orbitals unequally, depending on where each orbital points, and splits their energies – the ligand-field splitting. The pattern depends on the shape. A trigonal prism leaves one orbital alone at the bottom, below the other four; an octahedron makes a lower group of three below an upper group of two. What the layer does then follows from counting electrons. Molybdenum in MoS2 has two d electrons, which exactly fill the prism’s single low orbital, leaving a gap: 2H-MoS2 is a semiconductor. Put the same two electrons into an octahedron’s group of three and the group is only partly filled, so 1T-MoS2 is a metal.
The rule of thumb works across the table. Zirconium and hafnium have no d electrons left to place, so their octahedral disulfides are semiconductors; niobium and tantalum have one, which only partly fills a band in either shape, so their dichalcogenides are metals prone to and ; tungsten ditelluride sits in a distorted octahedron and is a .
Distortions, mixtures and how to tell
Octahedra often distort. The metals shift towards one another and pair up into chains or clusters: MoTe2 and WTe2 form zigzag chains in their 1T′ and Td phases, ReS2 forms diamond-shaped clusters of four atoms, and Nb3Cl8 forms triangles of three. Each distortion lowers the symmetry of the layer, which is where much of the in-plane and several of the unusual electronic states in the catalogue come from.
Which coordination a has is a measurement, not an assumption. Chemically exfoliated MoS2, for example, is a patchwork of trigonal prismatic and octahedral regions that share one continuous lattice. Atomic-resolution shows the arrangement directly, picks up the extra modes of the distorted phases, and separates the two by a shift of the molybdenum core levels, which gives the fraction of each.
For specialists
The polyhedron formed by an atom’s nearest neighbours; their number is its coordination number. In MX2 layers the metal is either trigonal prismatic (D3h, as in 2H-MoS2) or octahedral (D3d, as in 1T-TaS2, and distorted in 1T′-WTe2), and the ligand-field splitting of the d levels in each geometry, together with the d-electron count, decides whether the layer is a semiconductor or a metal. Halides and many oxides build layers from MX6 octahedra sharing edges (CrI3, RuCl3) or corners (layered perovskites), and that connectivity sets the lattice – honeycomb, triangular or square – that the metal ions form.