Crystals in layers

Look down on a layer and its atoms form a flat net that repeats – a lattice. Three nets turn up again and again in 2D materials, and each leaves its mark on what the material does.

In a triangular lattice every atom has six neighbours. It is the densest way to lay equal balls flat, the pattern marbles make when you push them together on a table, and the metal atoms of MoS₂ sit on one. Leave out one site in every three and the triangles open into hexagons: the honeycomb, in which each atom has three neighbours. Graphene and boron nitride are honeycombs, and the honeycomb is what gives graphene its electrons that behave as if they had no mass. Leave out one site in four instead and you get the kagome lattice, named after a Japanese basket weave: triangles that share corners around hexagonal holes. Electrons on a kagome net can get stuck in place, which makes kagome metals a hunting ground for magnetism and superconductivity.

Material pages describe the net by its unit cell, in ångströms. One ångström, Å, is a tenth of a nanometre – about the size of an atom. ‘Hexagonal, a = 3.16 Å’ for MoS₂ means a cell shaped like a rhombus, with 120° between its sides, each 3.16 Å long; copied side by side, it rebuilds the whole layer. Black phosphorus is the odd one out: its cell is a rectangle, so its two in-plane directions differ, and so does how it conducts heat, light and current along each.

Some layers are a single sheet of atoms: graphene, boron nitride. Most are thicker. A transition metal dichalcogenide such as MoS₂ is a sandwich three atoms thick – a plane of metal atoms between two planes of sulfur, selenium or tellurium – a little over 0.3 nanometres from the top atoms to the bottom ones. The chromium halides are similar sandwiches with iodine, bromine or chlorine outside, and some layers are five atoms thick or more.

Inside the sandwich, each metal atom is bonded to six atoms around it, three above and three below, and those six can be arranged in two ways. If the upper three sit directly above the lower three, the six form a trigonal prism – the shape of one piece of a Toblerone bar. That is the 1H form, and MoS₂ built this way is a semiconductor. If the upper three are turned by 60°, the six form an octahedron. That is the 1T form, and the same MoS₂ becomes a metal.

The shape traced by an atom’s neighbours is its coordination polyhedron, and material pages name it in their structure notes: ‘trigonal-prismatic’ or ‘octahedral’. Which one a layer takes depends on its elements and on how many electrons the metal brings, and it can sometimes be switched: slipping lithium between the layers of MoS₂ turns its layers from the 1H form to the metallic 1T. The same atoms, arranged a little differently, make a semiconductor or a metal.

Between one layer and the next lies the van der Waals gap, bridged only by the weak attraction met in level 1. Graphite’s sheets sit 0.335 nanometres apart, centre to centre. In MoS₂ the layers sit about 0.62 nanometres apart, while the atoms of one layer span only about 0.32 of that; the rest is the gap.

That distinction matters when you read a thickness. A material page’s thickness per layer – about 0.65 nm for MoS₂ – is the step from one layer to the next in a stack, as an atomic force microscope measures it, not the size of the atoms themselves. A single layer lying on a surface often measures a little thicker again, because of molecules trapped underneath it.

The gap is also a place where things can go. Atoms, ions and even small molecules can slip in between the layers – intercalation – pushing them apart and often changing the material completely. This is how a lithium-ion battery charges: lithium ions slide in between the carbon layers of its graphite electrode, and slide out again as it discharges. Intercalation can also loosen a crystal’s layers so that they come apart in a liquid, a common way of making flakes in bulk.

Weak as it is, the attraction across the gap is not nothing. It holds the stack together, lets the layers slide over one another – which is why graphite and MoS₂ are used as lubricants – and lets electrons hop from layer to layer, so a stack of two or three layers never behaves quite like one.

Layers can sit on one another in more than one way, and the way they sit is part of what a material is. Graphite is the simplest case. Its honeycomb sheets do not lie exactly on top of each other: each is shifted so that half its atoms sit over the middle of a hexagon below. Call the possible positions A, B and C, and ordinary graphite stacks ABAB…, while a rarer form stacks ABCABC….

Different stackings of the same layer are called polytypes, and material pages name them with a number and a letter. The number counts the layers in one repeat; the letter gives the symmetry of the whole stack – H for hexagonal, R for rhombohedral, T for trigonal. In 2H-MoS₂, the usual form of the mineral, each layer is turned by 180° against the one below, and the pattern repeats every two layers. In 3R-MoS₂ every layer is shifted by the same step in the same direction, and the pattern repeats every three. The turnable model on MoS₂’s page shows both.

The stacking changes real properties. In 2H-MoS₂, neighbouring layers point opposite ways, so effects that need a lopsided crystal cancel in pairs; in 3R they all point the same way and add up, so a thick 3R crystal makes far more light at double the frequency, and sliding its layers can store an electric polarisation. A mistake in the order – one layer out of place – is a stacking fault. Stacking faults are common in soft layered crystals and can change, for example, the temperature at which a magnet orders.

Symmetry sounds abstract, but material pages mention it constantly, because it decides which effects a crystal is allowed to show at all. A symmetry is a move that leaves the crystal looking exactly the same: a turn, a reflection, a shift by one cell.

Two matter most for layers. Mirror symmetry asks whether a sheet looks the same reflected in its own middle plane – the same from above as from below. Graphene and MoS₂ do; a layer with different atoms on its two faces does not. Inversion symmetry asks whether the crystal looks the same when every atom is swapped for the one directly opposite a central point. Graphene has such an inversion centre. A single layer of MoS₂ does not, and that lack is what lets it turn light into light of double the frequency, produce a voltage when squeezed, and keep its two valleys of electrons apart. Stack two layers of 2H-MoS₂ and inversion symmetry returns; stack three and it is lost again.

The full list of a crystal’s symmetries is summed up in a short code, its space group. ‘P6₃/mmc’, the space group of bulk 2H-MoS₂ and of graphite, reads: P for a plain lattice; 6₃ for a six-fold screw – turn by 60° and shift half a cell along the axis; then mirror and glide planes. You do not need to decode every symbol to use them. The same code on two pages means the same set of symmetries, and a material page often gives one code for the bulk crystal and another, with fewer symmetries, for a single layer.

Two last ways of building a layer show how far the rules can be bent. In an alloy, or solid solution, two similar kinds of atom share the same sites at random: some of the sulfur sites in MoS₂ hold selenium instead, or some of the molybdenum sites hold tungsten. The crystal keeps its structure and its symmetry, but its properties move smoothly with the mix – the band gap of an alloy of MoS₂ and MoSe₂ falls steadily from one parent’s value towards the other’s as selenium is added.

A Janus layer uses the same two elements in quite a different way: all the selenium on one face and all the sulfur on the other. Named after the two-faced Roman god, it no longer looks the same from above and below, so its mirror symmetry is gone. Its two faces hold on to electrons differently, so the layer carries a built-in electric field across it and shows effects a symmetric layer cannot, such as a voltage when it is pressed from above. Most Janus layers are made by swapping one face of a finished single layer, but a few crystals, such as RhSeCl, grow with Janus layers by nature.

Telling the two apart is a real problem: an alloy and a Janus layer can hold exactly the same atoms in exactly the same amounts. Only measurements that respond to symmetry, or images that show single atoms, can say which you have. That is the habit this level has been building: in a 2D material, where the atoms sit matters as much as which atoms are there.