Strong electron correlation

Also called strongly correlated electrons

In plain words

What happens when electrons in a material repel each other so strongly that they can no longer be treated as moving independently. Each electron’s motion then depends on where all the others are, and the result can be an where simple theory predicts a metal, or magnetism, and other collective states.

Going deeper

Left: with one electron per atom and weak repulsion, electrons hop freely and a half-filled band conducts. Right: when putting two electrons on one atom costs more energy, U, than hopping gains, each electron stays on its own atom, the band splits in two and the crystal insulates – a Mott insulator, whose trapped spins usually order antiferromagnetically.

When band theory stops working

Band theory treats each electron as moving in the average field of all the others, and for most metals and that works well. It fails when repulsion dominates. If hopping onto a neighbouring atom that already holds an electron costs more energy – the repulsion U – than the hopping gains, roughly the bandwidth W, electrons stay put, one per atom, and the material insulates although its band would be exactly half full. Nevill Mott and Rudolf suggested in 1937 that this is why nickel oxide insulates, and such materials are now called Mott insulators. John Hubbard’s model of 1963 captures the competition with just two numbers: how easily electrons hop, and what it costs to put two on one atom.

Stuck electrons still have , and brief virtual hops between neighbours favour antiparallel spins, so Mott insulators are usually . Adding or removing a few electrons produces some of the strangest metals known – heavy-fermion compounds whose carriers behave as if hundreds of times heavier than free electrons, and in the copper oxides high-temperature superconductivity.

Why two dimensions make it easier

Two knobs push the ratio of U to W up, and have both. leaves the repulsion strong, because a thin sheet has little around it to weaken the field between two electrons. And narrow bands make W small: the d bands of many layered are narrow to begin with, and fold graphene or bands into minibands only a few millielectronvolts wide, so even modest repulsion wins. In twisted bilayer graphene correlated insulators appear at integer numbers of electrons per moiré cell, with superconductivity close by; in TMDC moiré Mott insulators and electron crystals appear at integer and fractional fillings.

Because a gate sets the filling and the twist sets the bandwidth, such bilayers are used as tunable simulators of the Hubbard model. The catalogue has natural examples too: -TaS₂, whose charge density wave leaves one electron per star-shaped cluster; Nb₃Cl₈, a cluster Mott insulator; and α-RuCl₃, a Mott insulator in which strong shapes the magnetism.

Calculating it, and what counts as evidence

Standard density-functional approximations spread electrons out and so tend to make correlated materials too metallic. adds an on-site penalty whose value has to be chosen and stated; dynamical mean-field theory treats the fluctuations on each atom exactly and yields spectra that can be compared with ; small systems are solved exactly by diagonalisation, and larger ones by quantum , which for most fermion problems away from half filling is limited by the sign problem.

Experimentally, the signature is an insulator where bands predict a metal, whose gap is not a : local moments in the magnetic susceptibility, a gap that closes or shifts with filling, split upper and lower bands in tunnelling or photoemission spectra, and unusually heavy carriers. A charge density wave, or a stacking arrangement can also make an insulator, so a Mott claim has to rule those out – the nature of the insulating state of 1T-TaS₂ is still argued over for exactly this reason.

For specialists

The regime in which the Coulomb repulsion U between electrons is comparable to or larger than their kinetic energy, set by the bandwidth W, so single-particle band theory fails. At half filling with U well above W the Hubbard model gives a Mott insulator with local moments and of order t²/U; or tuning U/W gives correlated metals, heavy-fermion behaviour and unconventional superconductivity. In 2D, weak screening keeps U large and moiré superlattices shrink W, so twisted bilayers make U/W tunable by gate and twist. Semi-local DFT misses the physics; DFT+U, DMFT, exact diagonalisation and quantum Monte Carlo are the usual tools.

Where this comes from

Electron correlations in narrow energy bands Hubbard · Proceedings of the Royal Society of London A 276, 238 (1963) cited by 6,705
Correlated insulator behaviour at half-filling in magic-angle graphene superlattices Cao et al. · Nature 556, 80 (2018) cited by 4,899

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

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

Dielectric screening

The way a material weakens the pull between two charges inside it: its own electrons shift slightly and partly cancel the field between them. A sheet one atom thick has almost nothing around it to do this, so charges in it attract each other far more strongly than in a thick crystal – which is why light makes tightly bound pairs in 2D materials, and why whatever a sheet rests on changes its properties.

The reduction of the Coulomb interaction by the polarisation of bound electrons and ions and, in metals and doped layers, by free carriers, described by a dielectric function ε(q, ω). In a monolayer the field lines between distant charges run through the surroundings, so ε(q) approaches 1 as q → 0 and the screening is non-local and set by the environment. Exciton binding energies reach hundreds of meV, and a change of substrate or encapsulation shifts the quasiparticle gap by 100 meV or more while the optical gap barely moves.

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

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

Disorder and localisation

Every real crystal has some disorder: missing or misplaced atoms, impurities, stray charges in the substrate, a random mixture of elements in an alloy. Electrons moving through it scatter off these imperfections, which raises the resistance. If the disorder is strong enough, electron waves become trapped in small regions – localised – and the material stops conducting, even if it would be a metal when perfect. In a sheet one atom thick, disorder in its surroundings matters especially.

Deviations from perfect periodicity – point defects, substitutional or alloy disorder, interface roughness, charged impurities and strain variations in the substrate – that scatter carriers and, beyond a threshold, localise their wavefunctions through interference (Anderson localisation). Scaling theory predicts that non-interacting electrons in two dimensions without spin–orbit coupling are localised by any disorder; its precursor, weak localisation, appears as a logarithmic rise of resistance at low temperature that a magnetic field suppresses, while strong spin–orbit coupling gives weak antilocalisation. In graphene on SiO₂, charged impurities break the sheet near charge neutrality into electron–hole puddles, which hBN encapsulation reduces. Strong disorder leads to hopping conduction, broadens optical lines and smears phase transitions; in frustrated magnets and correlated oxides, structural or cation disorder can mimic exotic states.

Full explanation

Exchange interaction

The force that lines up the tiny magnets of neighbouring atoms in a magnetic material – all the same way in a ferromagnet, 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 spins 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.

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 anisotropic 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.

Full explanation