Crystal-field splitting

Also called ligand-field splitting

In plain words

How the neighbours of an atom change the energies of its outer electrons. A lone metal atom has five of equal energy. In a crystal, the orbitals that point straight at the surrounding atoms are pushed up and those pointing between them stay lower, so the five split into groups. How electrons fill those groups decides a great deal: whether a compound is a or a metal, how magnetic an ion is, and whether the crystal distorts to lower its energy – the Jahn–Teller effect. MoS₂ is a semiconductor or a metal depending on whether its molybdenum atoms sit in a prism or an of sulfur.

As the site uses it

The splitting comes from strong spin–orbit coupling on bismuth combined with the polar crystal field.

BiTeX · Bismuth tellurohalides

A clean example of a second-order Jahn–Teller ferroelectric: d¹⁰ Cu⁺ gains energy by moving off-centre in its sulfur cage, and the layered lattice lets that instability order cooperatively.

CuInP₂S₆, CuCrP₂S₆ · Copper thiophosphate ferroelectrics

Going deeper

Neighbouring atoms push up the d orbitals that point at them, splitting the five into groups. How electrons fill the groups sets the magnetism – three parallel spins on each chromium in CrI₃ – and the conduction: the same two d electrons give semiconducting MoS₂ in a trigonal prism and metallic MoS₂ in an octahedron.

Orbitals among neighbours

An atom’s five d orbitals have the same energy when the atom is alone. In a crystal it has neighbours – in most layered compounds six of them, arranged as an octahedron or a trigonal prism – and their electrons repel the metal’s own. Orbitals whose lobes point straight at a neighbour are pushed up in energy; those pointing between neighbours are pushed up less. The five orbitals split into groups, and the gap between them is the crystal-field splitting.

In an octahedron the result is three lower orbitals, called t2g, and two upper ones, eg. A trigonal prism – the shape around molybdenum in ordinary MoS₂ – makes three levels instead: a single dz² orbital lowest, then two pairs (Kim and colleagues, 2024). Real bonds are partly , so chemists often speak of the ligand field, but the pattern is the same.

Filling the levels

Electrons then fill the split levels, and two tendencies compete. Hund’s rule favours putting electrons in separate orbitals with parallel , which maximises the ; the splitting favours filling the lower group first, even if that means pairing spins. When the splitting wins, an ion is low-spin; when Hund’s rule wins, high-spin.

Chromium in CrI₃ shows the simple case. Each Cr³⁺ ion has three d electrons and sits in an octahedron of iodine; the three go into the three t2g orbitals with parallel spins, giving a spin of 3/2, and with no orbital degeneracy left, the orbital moment is quenched (Lado and Fernández-Rossier, 2017). Molybdenum in MoS₂ has two d electrons. In the trigonal prism they fill the lone dz² level and the falls in the gap above, so -MoS₂ is a semiconductor; in the octahedral 1T form the same two electrons only partly fill the three t2g orbitals, so 1T-MoS₂ is a metal (Kim and colleagues, 2024).

When the crystal bends

If electrons fill a degenerate set of orbitals unevenly, the crystal can lower its energy by distorting so that the degeneracy is lifted – the Jahn–Teller effect. High-spin Mn³⁺, with three electrons in t2g and one in the two eg orbitals, is the textbook case: its oxygen octahedra stretch along one axis, lengthening two of the six Mn–O bonds. When many such ions distort together the runs through the whole lattice; in layered manganese oxide stacked with graphene, that cooperative distortion has been turned to advantage, steadying a zinc-ion through more than 5,000 charging cycles (Wang and colleagues, 2025).

A related, second-order effect pushes an ion off-centre even when its d shell is full, by mixing in empty orbitals; in CuInP₂S₆ it lies behind the copper displacements that make the layers .

For specialists

The lifting of d-orbital degeneracy by the electrostatic and covalent field of the surrounding ligands. In an octahedron the d levels split into a lower t2g triplet and an upper eg doublet; in the trigonal prism of 2H-MoS₂ into dz², (dx²−y², dxy) and (dxz, dyz). Mo⁴⁺ (d²) fills dz² in the prism, leaving a gap, but only partly fills t2g in the octahedron, so 1T-MoS₂ is . The competition with Hund’s exchange decides between high- and low-spin states: Cr³⁺ (d³) in the iodine octahedra of CrI₃ puts three parallel electrons in t2g, giving S = 3/2 and a quenched orbital moment. A single electron in a degenerate level drives a Jahn–Teller distortion, as for high-spin Mn³⁺ (t2g³ eg¹), whose octahedra elongate along one axis.

Where this comes from

Phase engineering of two-dimensional transition metal dichalcogenides Kim, Sung and Lee · Small Science 4, 2300093 (2024)
On the origin of magnetic anisotropy in two dimensional CrI₃ Lado and Fernández-Rossier · 2D Materials 4, 035002 (2017) cited by 741
Cooperative Jahn-Teller effect and engineered long-range strain in manganese oxide/graphene superlattice for aqueous zinc-ion batteries Wang et al. · Nature Communications 16, 5191 (2025)

In the news

The newest items in the site’s news feed that use the term, one from each source.

Journal 17 Oct 2026 Advanced Functional Materials

Steering no Dimerization Through Boron‐Enhanced Spin–Orbit Coupling in Ultrathin PdCuB Nanosheets for Durable N2‐Selective Nitrate Reduction

The selectivity bottleneck of electrochemical nitrate‐to‐N₂ conversion is kinetic rather than thermodynamic: dimerization of two adsorbed *NO intermediates is spin‐forbidden, imposing a kinetic barrier invisible to conventional thermodynamic analyses. (SOC) offers the principal channel for the…

Theory

Preprintnot yet peer reviewed 12 Oct 2026 arXiv

High-mobility spin transport mediated by indirect excitons in a van der Waals heterostructure

Efficient with high spin is of both fundamental and practical significance for spintronics. Spin can be by moving particles. However, particle scattering limits spin transfer and prevents high spin mobilities. In this work, we study the kinetics of spin transport mediated by a…

TheoryExperiment

Preprintnot yet peer reviewed 8 Oct 2026 arXiv

Emergent gold-induced interfacial magnetism in monolayer FePS₃

Owing to their atomically clean interfaces and extreme thinness, are ideal hosts for strong , whereby contact with an adjacent material reshapes their ground state. Here, we demonstrate such an effect within a hybrid , between the 2D FePS₃…

ExperimentMPS₃

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Orbitals and hybridisation

The shapes an electron’s cloud can take around an atom. Electrons do not circle the nucleus like planets; each sits in an orbital, a region with a definite energy and shape – a sphere (s), a dumbbell (p) or a clover-like shape (d). When atoms bond, their orbitals overlap and can blend into new ones pointing at the neighbours, which is hybridisation: in graphene each carbon mixes one s and two p orbitals into three bonds in the plane (sp²), leaving one p orbital standing up out of the sheet. Which orbitals hold the outermost electrons decides much of what a material does. Electrons in an orbital still carry a kind of circling motion, angular momentum, and the magnetism that comes from it rather than from their spin is called orbital magnetism.

Single-electron states of an atom labelled by angular momentum – s, p, d, f – whose overlap between neighbours forms bonds and bands. Hybridisation mixes orbitals on one atom, as in graphene’s sp² carbon, with three in-plane σ bonds 1.42 Å long and a half-filled π band from the out-of-plane p orbital, where the Dirac cones lie; or between atoms, as in the p–d hybridisation of transition-metal compounds. In 2H TMDCs a hybridisation gap splits a d band off from the rest, and whether it is half-filled or full separates metallic NbSe₂ and TaS₂ from semiconducting MoS₂. “Orbital” also names the electron’s motion: orbital magnetism comes from circulating electrons rather than from spin.

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

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 the atom that pulls harder on electrons – the one with the higher electronegativity – takes them from the other, 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

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

Magnetic moment

The strength and direction of the tiny magnet an atom or an electron carries. Every electron is a small magnet through its spin, and its motion around an atom can add more. In most atoms the electrons pair up and their magnets cancel; atoms with unpaired electrons – chromium, manganese, iron, nickel, the rare earths – keep a moment. How these moments line up decides whether a material is a ferromagnet, an antiferromagnet or no magnet at all.

The magnetic dipole of an atom, ion or electron, measured in Bohr magnetons (μB ≈ 9.27 × 10⁻²⁴ J/T). In insulating magnets it is a local moment set mainly by the spins of unpaired d or f electrons – about one Bohr magneton per unpaired electron for 3d ions, whose orbital moment the crystal field largely quenches: 3 μB for Cr³⁺, 5 μB for Mn²⁺. Ordered moments from neutron diffraction often come out below these values, 4.04 μB per manganese in MnBi₂Te₄. In metallic magnets the moment belongs partly to itinerant electrons and need not be near a whole number, as with 1.11 μB per iron in Fe₃GeTe₂.

Full explanation

Fermi level

Roughly, the energy up to which a material’s electron states are filled – like the water line in a partly filled glass. Where it sits relative to the band gap decides how many charges can move and whether they are electrons or holes, the empty places electrons leave behind. A gate voltage raises and lowers it, like pouring water in or out.

The electrochemical potential of electrons: the energy at which a state has 50 % occupation in thermal equilibrium. Its position relative to the band edges sets carrier density and type, and where electron and hole densities balance – at the Dirac point of graphene, near mid-gap in an undoped semiconductor – the material is at charge neutrality, where a gated 2D device conducts least; at metal–semiconductor contacts, interface states can pin it and fix the Schottky barrier.

Full explanation

Battery electrode

A battery stores energy by moving ions – in phones and cars, lithium ions – between two electrodes, the anode and the cathode. Charging pushes the ions into the anode, where they slip between the layers of a material such as graphite; using the battery lets them flow back. Layered and 2D materials matter because ions can enter and leave the gaps between layers, and because thin conductive flakes can help electrons reach every grain of an electrode.

A lithium- or sodium-ion cell shuttles ions through an electrolyte between a negative electrode (the anode on discharge) and a positive one (the cathode), while electrons pass through the external circuit. Graphite, the standard anode, intercalates lithium between its layers up to LiC₆, about 372 mAh/g; layered oxides such as LiCoO₂ and NMC dominate cathodes. 2D materials enter as conductive additives (graphene flakes replacing part of the carbon black), as higher-capacity anodes that alloy or convert (phosphorus, antimony, SnS₂, SnO), as hosts for larger ions such as sodium, which graphite takes up poorly, and as MXene electrodes. Their weaknesses are large volume change, irreversible first-cycle loss from the high surface area, low packing density and cost, so gains at cell level are usually far smaller than those claimed for the material.

Full explanation

Ferroelectricity

A material that holds an electric polarisation – one side slightly positive, the other slightly negative – with no voltage applied, and that can be flipped by a large enough field – the coercive field – and stay flipped. That memory of the last field applied is what makes it useful for storing data.

A spontaneous, switchable electric polarisation produced by a structural distortion that breaks inversion symmetry. In 2D the usual obstacle – depolarising fields destroying the polarisation as a film thins – is sidestepped in three ways: in-plane polarisation, layered compounds whose dipoles survive to the monolayer such as CuInP₂S₆ and In₂Se₃, and sliding ferroelectricity, where the polarisation belongs to the interface between layers rather than to the layers themselves.

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

Layer transfer

Moving an atomically thin layer from where it was made to where it is needed – from the copper foil it grew on to a silicon wafer, or from a flake on tape onto another flake. The layer is too thin to pick up on its own, so it is carried on a polymer film or a soft stamp, put down, and the carrier removed. Each step risks tears, wrinkles, trapped bubbles and a film of leftover polymer, which is why transfer is often the dirtiest step in making a 2D device.

The step that moves a 2D layer from its growth substrate or exfoliation source to a target substrate or stack. Wet transfer coats the layer with a supporting polymer, usually PMMA, detaches it by etching the growth substrate or by electrochemical bubbling, and floats it onto the target before the polymer is dissolved. Dry transfer picks layers up with a viscoelastic PDMS stamp or a polymer film such as polycarbonate, often using the van der Waals adhesion of a top flake, and releases them by heating. The costs are cracks and wrinkles, trapped water, hydrocarbons and polymer residue, strain and doping, and – at wafer scale – yield that compounds with every transfer, which is why growth directly on the target and transfer-free routes are pursued.

Full explanation

Proximity effect

A property borrowed from a neighbour. Where two materials touch, the electrons near the boundary feel both, so one can take on some of the other’s character. Graphene on a magnet becomes slightly magnetic, on a semiconductor made of heavy atoms it gains spin–orbit coupling, and between two superconductors it carries a supercurrent. In a thick crystal only a thin skin is affected, but a 2D layer is all skin, so a single neighbour can change it throughout – one of the main reasons for stacking different layers together.

The induction of an order or interaction in one material by contact with another, through tunnelling and orbital hybridisation across the interface. Superconducting correlations leak into a normal conductor by Andreev reflection over a coherence length, which lets graphene, semiconductor and topological-insulator weak links carry Josephson currents and, on a topological surface, was proposed as a route to Majorana states. Exchange and spin–orbit coupling are induced by wavefunction overlap and decay within a few ångström, so an atomically thin layer lies entirely within their reach: valley splittings of a few meV in WSe₂ on CrI₃, and spin–orbit coupling of the order of a millielectronvolt in graphene on WS₂ or WSe₂. The size depends on twist angle, stacking and band alignment, and is usually inferred indirectly – from valley splitting in optics, weak antilocalisation and spin-relaxation anisotropy in transport, or induced gaps in tunnelling spectroscopy.

Full explanation

Ferromagnet and antiferromagnet

In a ferromagnet the tiny magnets of the atoms all point the same way, as in a fridge magnet. In an antiferromagnet neighbours point in opposite directions and cancel out; if the opposing magnets differ in size they cancel only in part, and the material is a ferrimagnet. Ferromagnets and antiferromagnets have both been found in sheets only one layer thick.

Magnetically ordered states with parallel (ferromagnetic) or antiparallel (antiferromagnetic) alignment of neighbouring moments; antiparallel moments of unequal size make a ferrimagnet. Long-range order in 2D requires magnetic anisotropy to evade the Mermin–Wagner theorem; CrI₃ monolayers are Ising-like ferromagnets that couple antiferromagnetically between layers, and the MPS₃ compounds are antiferromagnets.

Full explanation

In plain words

As the site uses it

Going deeper

Orbitals among neighbours

Filling the levels

When the crystal bends

For specialists

Where this comes from

In the news

Steering no Dimerization Through Boron‐Enhanced Spin–Orbit Coupling in Ultrathin PdCuB Nanosheets for Durable N2‐Selective Nitrate Reduction

High-mobility spin transport mediated by indirect excitons in a van der Waals heterostructure

Emergent gold-induced interfacial magnetism in monolayer FePS3

Emergent gold-induced interfacial magnetism in monolayer FePS₃