Wigner crystal

Say it Wigner VIG-ner

Also called generalised Wigner crystal

In plain words

A crystal made of electrons themselves. When electrons are few and far apart, their mutual repulsion can win over their motion, and they freeze into a regular pattern that keeps them as far from each other as possible – a state Eugene Wigner predicted in 1934. It needs very low densities and temperatures, but in stacks of 2D the moiré pattern helps hold the electrons in place, and these ‘generalised’ Wigner crystals have been seen at fractional fillings and even imaged directly.

Not to be confused with

Charge density wave (CDW)

Both arrange electrons into a regular pattern. In a a metal’s mobile electrons only bunch into a ripple locked to the crystal, helped by a distortion of the atoms; in a Wigner crystal the electrons themselves freeze in place, held by their own repulsion, at densities far too low for a metal. A generalised Wigner crystal on a moiré lattice is driven by repulsion like the one but locked to a lattice like the other.

As the site uses it

WSe₂/WS₂ moiré superlattices are well described by triangular-lattice Hubbard models with tunable U/t and have become a testbed for correlated insulators and generalised Wigner crystals.

WSe₂ · Tungsten diselenide

Going deeper

Electrons crystallise when they are dilute enough that repulsion outweighs motion. A moiré lattice helps by offering sites, so that a fraction of them, here one in three, fills in a regular pattern.

When repulsion wins

In a metal the electrons move fast and their mutual repulsion is a small correction. Thin them out and the balance shifts: in two dimensions the kinetic energy falls in proportion to the density, the repulsion only with its square root. Below a certain density repulsion wins, and the electrons settle into a lattice that keeps each as far as possible from its neighbours – triangular in a plane.

The measure physicists use is r_s, the average spacing between electrons in units of their effective Bohr radius. Calculations put the crystal at r_s of roughly 30 to 40 for a clean two-dimensional gas, so dilute that in ordinary semiconductors a strong magnetic field was long needed to help, by taking away the electrons’ freedom to move.

Moiré makes it easier

A moiré superlattice gives the electrons a lattice of preferred sites, one per moiré cell. When only a fraction of the sites is filled – a third, half or two thirds – repulsion arranges the electrons into a regular pattern that fits the moiré: a generalised Wigner crystal. In 2020 such states showed up as fillings in WSe₂/WS₂ moiré stacks, and in 2021 they were imaged with a looking through a graphene layer, electron by electron.

Without a moiré

favour the crystal even without help: their electrons are heavy and the surrounding hBN screens their repulsion only weakly. In 2021 a monolayer of MoSe₂ at low density showed the optical signature of an electron lattice, a new line made possible by the lattice’s periodicity, and stacks of two such layers showed Wigner crystals in both. Melting them by temperature or density is now an experiment rather than a thought experiment.

For specialists

The ground state of a dilute electron gas in which Coulomb repulsion dominates the kinetic energy and the electrons localise, on a in two dimensions. Its stability is set by r_s, the mean spacing in units of the effective Bohr radius; quantum puts crystallisation of a clean 2D gas near r_s ≈ 30–40. Heavy and in TMDC monolayers bring this within reach, and a moiré potential lowers the bar further, giving generalised Wigner crystals with the superlattice at fractional fillings such as 1/3 and 2/3, detected optically, by microwave impedance and by STM through a sensing layer.

Where this comes from

On the interaction of electrons in metals Wigner · Physical Review 46, 1002 (1934)
Phase diagram of the low-density two-dimensional homogeneous electron gas Drummond and Needs · Physical Review Letters 102, 126402 (2009)
Mott and generalized Wigner crystal states in WSe₂/WS₂ moiré superlattices Regan et al. · Nature 579, 359 (2020) cited by 995
Signatures of Wigner crystal of electrons in a monolayer semiconductor Smoleński et al. · Nature 595, 53 (2021)
Imaging two-dimensional generalized Wigner crystals Li et al. · Nature 597, 650 (2021)

In the news

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

Preprintnot yet peer reviewed 8 Oct 2026 arXiv

Ambipolar metal-insulator transition in bilayer MoSe2

When Coulomb interactions dominate kinetic energy, dilute two-dimensional carriers form a Wigner solid. Both its melting and -driven localization can drive density-tuned metal-insulator transitions, but their relative roles and interplay have been debated for decades. Atomically thin semiconductors offer a new…

ExperimentMoSe₂

Preprintnot yet peer reviewed 30 Sep 2026 arXiv

Quantum Well Resonant Tunneling Diode Probe of Correlated States in Twisted Bilayer MoS₂

Moiré superlattices formed in transition metal dichalcogenides (TMDs) offer a versatile platform for exploring emergent quantum phases arising from strong . In this work, we develop a new experimental platform, the resonant tunneling diode (QWRTD), to probe the electronic landscape…

TheoryExperimentMoS₂

Preprintnot yet peer reviewed 29 Sep 2026 arXiv

Interlayer Fermi Polarons in Bilayer MoTe₂

Atomic bilayers of transition metal dichalcogenides (TMDs) host quantum phases governed by the layer degree of freedom, including bilayer Wigner crystals, , and . These phases are probed primarily through exciton spectroscopy, yet it remains poorly understood how excitons…

TheoryMoTe₂

Search the news for this term

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

Scanning tunnelling microscopy (STM)

A needle sharpened to a single atom is brought so close to a surface that electrons tunnel across the gap – a quantum effect by which they pass through a barrier they could never climb over. The current maps the surface atom by atom, and how it changes with voltage tells you which energies electrons are allowed to have at that exact spot.

Tunnelling between a sharp tip and a conducting surface, giving atomic-resolution topography and, in spectroscopy mode, the local density of states against energy. On 2D materials it resolves defect levels, moiré potentials, edge modes and quasiparticle gaps site by site. The sample has to be clean and conducting, so it is run on epitaxial films or on flakes placed on graphite, in vacuum and often at cryogenic temperature.

Full explanation

Sensor

A device that turns something in its surroundings – a gas, a magnetic field, a pressure, a molecule in blood – into an electrical signal. 2D materials make unusually sensitive sensors because every atom is on the surface: a few molecules landing on a single layer change how well it conducts. Graphene also makes very good magnetic-field sensors. The hard part is not sensitivity but selectivity – responding to one thing and ignoring everything else – and staying stable for years.

Transducers in which a stimulus changes an electrical property of a 2D layer: chemiresistive and field-effect gas and humidity sensors, in which adsorbates transfer charge and shift the conductance (graphene registered single NO₂ molecules in 2007); Hall sensors, in which graphene’s low carrier density and high mobility give a large, temperature-stable Hall voltage; biosensors with receptors on a graphene or MoS₂ channel that bind target molecules; and strain, pressure and light sensors. Sensitivity comes from the all-surface geometry and the low density of states; the limits are selectivity, drift and hysteresis from humidity and contamination, recovery time, device-to-device variation and calibration, addressed by functionalisation, encapsulation and arrays read by pattern recognition.

Full explanation

Triangular lattice and frustration

A triangular lattice is a flat net of triangles, each atom with six neighbours. Put a tiny magnet that wants to point opposite to its neighbours on every corner, and a triangle cannot oblige: once two corners point up and down, the third has no good choice. That standoff is called frustration. It can stop a magnet from ordering at all, leaving room for unusual states such as quantum spin liquids.

A Bravais lattice with six nearest neighbours per site. With antiferromagnetic nearest-neighbour exchange the three spins of a triangle cannot all be antiparallel: Ising spins have a macroscopically degenerate ground state, while Heisenberg spins settle into the 120° state. This geometric frustration suppresses ordering and enhances quantum fluctuations; frustration also arises from competing exchange paths and from bond-dependent Kitaev interactions on the honeycomb lattice. Its strength is gauged by the ratio |θ_CW|/T_N.

Full explanation

Effective mass

How heavy an electron seems as it moves through a crystal. Pushed by an electric field, it speeds up as if its mass were different from a free electron’s – lighter in some materials, heavier in others, and in graphene as if it had no mass at all. Light carriers generally make for faster devices.

The mass m* with which a carrier near a band edge responds to forces as if it were free, set by the band curvature: 1/m* = (1/ħ²) d²E/dk². It enters the mobility (μ = eτ/m*), the density of states, confinement energies and tunnelling rates, and in anisotropic crystals it is a tensor – in black phosphorus the armchair and zigzag masses differ several-fold. Graphene’s linear bands have no curvature, and its carriers are described as massless Dirac fermions; their cyclotron mass, the Fermi energy divided by the square of the Fermi velocity, grows with the square root of the carrier density.

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

Commensurate and incommensurate

Whether two repeating patterns fit each other. If one pattern repeats after exactly a whole number of steps of the other – every three atoms, say – the two are commensurate, and together they repeat again after a short distance. If the ratio is not a simple fraction, they are incommensurate and never line up exactly. The question comes up whenever a crystal grows a second pattern of its own – a charge density wave, a magnetic spiral – and whenever two different layers are stacked, as in a moiré.

A modulation is commensurate when its wavevector is a rational fraction of a reciprocal lattice vector, so that crystal and modulation share a finite supercell, and incommensurate when the ratio is irrational, leaving no exact common period. The charge density wave of 1T-TaS₂ passes from incommensurate (below about 550 K) to nearly commensurate – commensurate domains separated by discommensurations – and locks into the commensurate √13 × √13 Star-of-David phase below about 180 K; RTe₃ keeps an incommensurate wave, and NiI₂ an incommensurate spin spiral. For two lattices – graphene on hBN, twisted bilayers – the same competition between elastic energy and interlayer adhesion decides whether the layers stretch into commensurate domains or keep their own lattices.

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

Strong electron correlation

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 insulator where simple theory predicts a metal, or magnetism, superconductivity and other collective states.

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 antiferromagnetic exchange of order t²/U; doping 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.

Full explanation

Quantum well and quantum dot

What you get when electrons are squeezed into a space only a few nanometres across: in one direction, a thin layer, for a quantum well; in all three, a tiny speck, for a quantum dot. Squeezing raises their energy in fixed steps, so the size decides the colour of light given off – which is how quantum-dot TV screens get their pure colours. A single layer of a 2D material is a natural quantum well.

Structures that confine carriers on the scale of their de Broglie wavelength in one (well) or all three (dot) dimensions, quantising their motion into subbands or discrete levels whose spacing scales roughly as 1/L² for a deep well. Wells are made by sandwiching a narrow-gap layer between wider-gap barriers, as in HgTe/CdTe, or occur naturally in layered 2D perovskites and in monolayers; dots are colloidal nanocrystals, lithographically or electrostatically defined regions, or localised strain and defect sites in TMDC monolayers that act as single-photon sources.

Full explanation

Excitonic insulator

A state in which electrons and the holes they leave behind bind into pairs so strongly that the material opens a gap and stops conducting – not because of its chemistry, but because the pairs have all settled together into one shared, collective state. Telling this apart from an ordinary distortion of the crystal is notoriously difficult.

A ground state predicted for a semimetal with small band overlap, or a semiconductor whose gap is smaller than the exciton binding energy, in which spontaneous exciton condensation reconstructs the bands and opens a gap. The experimental difficulty is that the transition is usually accompanied by a lattice distortion of the same symmetry, so photoemission, ultrafast and pressure experiments have to separate an electronic condensate from a conventional Peierls-like instability – the argument that surrounds Ta₂NiSe₅ and 1T-TiSe₂.

Full explanation

In plain words

Not to be confused with

Charge density wave (CDW)

As the site uses it

Going deeper

When repulsion wins

Moiré makes it easier

Without a moiré

For specialists

Where this comes from

In the news

Ambipolar metal-insulator transition in bilayer MoSe2

Quantum Well Resonant Tunneling Diode Probe of Correlated States in Twisted Bilayer MoS2

Interlayer Fermi Polarons in Bilayer MoTe2

Quantum Well Resonant Tunneling Diode Probe of Correlated States in Twisted Bilayer MoS₂

Interlayer Fermi Polarons in Bilayer MoTe₂