Exchange interaction

In plain words

The force that lines up the tiny magnets of neighbouring atoms in a magnetic material – all the same way in a , alternating in an antiferromagnet. Despite what it does, it is not magnetic in origin: it comes from the electric repulsion between electrons combined with a quantum rule that keeps two electrons from sharing the same state, which makes their energy depend on whether their point the same way. It is far stronger than the magnetic pull between atomic magnets, which is why iron stays magnetic up to 770 °C.

Going deeper

Exchange makes neighbouring spins line up parallel, as in a ferromagnet, or alternate, as in an antiferromagnet. In insulators it acts through the atom between two magnetic ones, and the bond angle sets its sign. In layered magnets it is strong within each layer and weak between layers, so the stacking of the layers can switch how they couple.

Not a magnetic force

Two neighbouring atomic magnets pull on each other magnetically, but far too weakly to explain magnetism at room temperature: that interaction is worth about a kelvin or less. The real cause, found by and Dirac in 1926, is electric. Electrons repel each other, and quantum mechanics forbids two of them with the same spin from occupying the same place. Electrons with parallel spins therefore keep further apart and repel each other less – or, depending on the orbitals involved, opposite spins win because they let the electrons spread over both atoms. The resulting energy difference between parallel and antiparallel spins is the exchange interaction, worth tens to hundreds of kelvin.

Its sign decides the kind of magnet: one sign lines the spins up into a ferromagnet, the other alternates them into an antiferromagnet, and competing couplings to more distant neighbours produce spirals and frustrated states.

Through the atom between

In magnetic the magnetic atoms rarely touch; their electrons interact through the atoms in between. In this superexchange the sign depends on the geometry: when the bond from one metal through a ligand to the next is straight, the coupling is usually antiferromagnetic; when it bends at about 90°, as between the octahedra of CrI₃ and Cr₂Ge₂Te₆, it is often ferromagnetic. These Goodenough–Kanamori rules explain why so many layered and chalcogenides are ferromagnetic within their layers.

In metals the conduction electrons carry the coupling instead, by the RKKY or double-exchange mechanisms, and in heavy elements strong makes it depend on the direction of the spins relative to the bond. That bond-dependent form, proposed by Kitaev, is what makes α-RuCl₃ a candidate .

Exchange in layered magnets

A magnet has two very different couplings. Within a layer, exchange is strong and sets the temperature scale of the order. Between layers it is far weaker and can have either sign – which is why CrI₃ is ferromagnetic within each layer but its layers alternate in thin , and why a change in stacking under pressure can switch them to parallel. Because the coupling between layers is so weak, a gate, or a neighbouring layer can change it too, which makes these magnets attractive for devices.

Exchange alone cannot order a , though: by the an isotropic magnet in two dimensions has no long-range order at any finite temperature, and it is magnetic anisotropy that lets CrI₃ order as a monolayer. Computed exchange constants also need care, because values depend on the choice of U; spectra from neutron or scattering are the check.

For specialists

The spin-dependent energy that arises from the Coulomb repulsion together with the antisymmetry of the many-electron wavefunction, modelled as a coupling J between neighbouring spins in Heisenberg, Ising or XY form. In insulators it is mostly mediated by ligands – superexchange, whose sign follows the Goodenough–Kanamori rules: antiferromagnetic for metal–ligand–metal angles near 180°, often ferromagnetic near 90°, as in the edge-sharing octahedra of CrI₃ – in metals by conduction electrons (RKKY, double exchange), and with strong spin–orbit coupling it becomes or bond-dependent (Kitaev). In van der Waals magnets the intralayer exchange of a few meV sets the ordering scale, while a much weaker interlayer exchange decides ferro- or antiferromagnetic stacking and depends on the stacking itself; ordering a monolayer additionally requires anisotropy, by the Mermin–Wagner theorem.

Where this comes from

Mehrkörperproblem und Resonanz in der Quantenmechanik Heisenberg · Zeitschrift für Physik 38, 411 (1926)
Antiferromagnetism. Theory of superexchange interaction Anderson · Physical Review 79, 350 (1950)
Superexchange interaction and symmetry properties of electron orbitals Kanamori · Journal of Physics and Chemistry of Solids 10, 87 (1959)
Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit Huang et al. · Nature 546, 270 (2017) cited by 5,987
Switching 2D magnetic states via pressure tuning of layer stacking Song et al. · Nature Materials 18, 1298 (2019)

Spin

A built-in property of every electron that makes it a tiny magnet. Measure it along any direction and you only ever find one of two answers, ‘up’ or ‘down’. Despite the name, nothing is actually spinning – the word stuck from an early picture – but the magnetism is real: countless electron spins lined up are what make a fridge magnet stick. Spintronics tries to carry information in spin rather than in charge.

The intrinsic angular momentum of the electron, ħ/2, with a magnetic moment of almost exactly one Bohr magneton; a measurement along any axis yields one of two values. In solids spin couples to orbital motion through spin–orbit coupling and to other spins through exchange; in 2D materials it sets magnetic order, spin–valley locking and how long a spin survives as a carrier of information.

Full explanation

Metal, semimetal and insulator

The three basic ways a solid treats electricity, set by how its electrons fill the energies open to them. In a metal they fill a band of energies only partly, so the smallest push sets them moving, and it conducts at any temperature. In an insulator they fill a band completely and a wide gap separates it from the next, so nothing can move. A semimetal sits in between: two bands just touch or overlap a little, leaving a small number of mobile electrons and holes. NbSe₂ is a metal, graphene a semimetal, hBN an insulator.

A classification by where the Fermi level falls in the band structure. In a metal it crosses a partly filled band, leaving a Fermi surface, a carrier density of order one per atom (about 10²² cm⁻³) and a resistivity that falls on cooling. In a band insulator it lies in a gap of several electronvolts; a semiconductor is an insulator whose gap is small enough to be doped or gated. A semimetal has a slight overlap between conduction and valence bands (graphite, bismuth, bulk WTe₂) or bands that touch at isolated points (graphene), giving low and nearly equal densities of electrons and holes. Interactions can open gaps that band theory does not predict: Mott insulators from half-filled bands, excitonic insulators from semimetals.

Full explanation

Coordination polyhedron

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 semiconductor or a metal.

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.

Full explanation

Chalcogenide, halide and pnictide

Family names for compounds, taken from the element in them that carries the negative charge. Chalcogenides contain sulfur, selenium or tellurium; halides contain fluorine, chlorine, bromine or iodine; pnictides contain phosphorus, arsenic, antimony or bismuth. Most layered crystals belong to one of these families, and the catalogue groups many of them that way.

Compounds named for their most electronegative constituent: chalcogenides of group 16 (S, Se, Te; oxides are conventionally counted apart), halides of group 17 (F, Cl, Br, I) and pnictides of group 15 (P, As, Sb, Bi; nitrides usually keep their own name). Down each group the anion grows larger and less electronegative, so bonding turns more covalent, gaps shrink and spin–orbit coupling grows. Chalcogenides give the MX₂ semiconductors and metals, halides mostly ionic, often magnetic insulators built from edge-sharing octahedra, and pnictides many metals, semimetals and the parents of the iron-based superconductors; mixed-anion compounds such as oxychalcogenides and chalcohalides combine two.

Full explanation

Quantum spin liquid

A magnet whose atomic magnets never settle into a pattern, even at absolute zero. Quantum effects keep them fluctuating and entangled – linked so that none has a direction of its own – so instead of freezing into order the material stays restless, like a liquid that never turns solid. Disturbances in it can behave like fractions of an electron, carrying its magnetism but not its charge.

A ground state of a frustrated magnet with no symmetry breaking down to zero temperature, marked by long-range entanglement, an emergent gauge structure and fractionalised excitations such as spinons or Majorana fermions. Honeycomb materials with bond-dependent Kitaev exchange – α-RuCl₃ above all – are the leading layered candidates, but the evidence is indirect: a scattering continuum and thermal Hall signals, against residual magnetic order in zero field.

Full explanation

Density functional theory (DFT)

The workhorse calculation of materials physics. Rather than following every electron, it works with the electron density and a recipe for how electrons avoid one another – which makes a whole crystal cheap enough to compute, and makes the accuracy depend entirely on that recipe.

Ground-state electronic structure from the density, exact in principle by Hohenberg–Kohn and made practical by the Kohn–Sham equations with an approximate exchange–correlation functional. Two dimensions need care: a vacuum gap wide enough that periodic images stop interacting or a truncated Coulomb interaction, a dispersion correction for interlayer binding, and the knowledge that semi-local functionals underestimate gaps – hybrid functionals narrow that error at far higher cost, and quantitative gaps and optical spectra need GW and Bethe–Salpeter on top.

Full explanation