Charge density wave (CDW)

In plain words

A state in which the electrons in a crystal bunch up into a regular ripple instead of spreading out evenly, pulling the atoms slightly out of place as they go. It sets in below a certain temperature and often competes with for the same electrons.

Going deeper

In a charge-density wave the conduction electrons settle into a periodic ripple and pull the atoms with them, locking in a new, longer period. The ordered state usually competes with superconductivity: as pressure or doping suppresses it, superconductivity often strengthens.

A ripple in the electron sea

Below a , the density of conduction electrons in some metals stops being uniform and takes on a periodic modulation, with a wavelength that need not match the lattice spacing. The lattice follows: atoms shift slightly towards the maxima, so a new, longer periodicity appears in the crystal structure, visible as extra spots in diffraction.

The classical explanation is : if large parts of the Fermi surface can be connected by one wavevector, electronic energy is gained by opening a gap there. In layered , where charge-density waves have been studied since the 1970s, the modern picture gives at least as much weight to strongly momentum-dependent , in which a particular softens towards zero frequency at the ordering wavevector.

Why thin layers change it

Thinning a crystal changes the Fermi surface, the screening and the coupling to the , so move – and not always downwards. NbSe₂ orders at a higher temperature than the bulk, while its superconductivity is weakened, an illustration that the two orders draw on the same electrons. In -TaS₂ the layer count changes which of several ordered states appears, and in TiSe₂ pressure and both suppress the charge-density wave and reveal superconductivity beneath it.

Because a monolayer can be gated, the electron density can be tuned continuously through such a phase diagram in one sample, instead of comparing separately grown crystals.

How it is detected

Diffraction – electron, X-ray or neutron – shows the superlattice reflections directly, and images the modulation in real space along with the domain walls and between patches. Transport shows an anomaly at the transition, because part of the Fermi surface is gapped; scattering picks up new modes folded in by the longer period, and the softening of the driving phonon appears in inelastic scattering.

Care is needed with claims about the mechanism. Nesting, phonon softening and lattice distortion often appear together, and telling which drives which requires momentum-resolved data rather than a transition temperature alone.

For specialists

A periodic modulation of conduction-electron density locked to a periodic lattice distortion, driven by Fermi-surface nesting, momentum-dependent electron–phonon coupling or both. Transition temperature and ordering wavevector depend on layer number, and pressure, and the ordered state frequently coexists or competes with superconductivity in the same phase diagram.

Where this comes from

Charge-density waves and superlattices in the metallic layered transition metal dichalcogenides Wilson et al. · Advances in Physics 24, 117 (1975) cited by 2,237
Strongly enhanced charge-density-wave order in monolayer NbSe₂ Xi et al. · Nature Nanotechnology 10, 765 (2015) cited by 876

Phase transition

A sudden change in the state of a material when temperature, pressure or another control is varied – ice melting into water, a magnet losing its magnetism when heated, a metal becoming a superconductor when cooled. Each happens at a definite transition temperature. In quantum materials, transitions mark where new kinds of order appear: charge density waves, magnetism, superconductivity, ferroelectricity. A map of which state appears where, against temperature and a second knob such as doping or pressure, is a phase diagram.

A non-analytic change of thermodynamic state at a transition point, described by an order parameter that vanishes in the disordered phase and is finite in the ordered one – magnetisation, polarisation, charge-density-wave amplitude, superconducting gap. First-order transitions show latent heat, phase coexistence and hysteresis; continuous ones show a diverging correlation length and susceptibility and universal critical exponents set by dimension and symmetry (Landau theory, the renormalisation group). In two dimensions fluctuations are stronger: the Mermin–Wagner theorem forbids breaking a continuous symmetry at finite temperature, leaving anisotropic, Ising-like order or Berezinskii–Kosterlitz–Thouless quasi-order. Transition temperatures change with layer number, gating, strain and twist, which makes 2D materials tunable platforms for phase diagrams, including quantum phase transitions driven at zero temperature by non-thermal parameters.

Full explanation

Fermi surface and nesting

The Fermi surface is the boundary between the states a metal’s electrons fill and the ones they leave empty, drawn on a map of every way an electron can move through the crystal; its shape decides how the metal conducts. Nesting is when large flat stretches of that boundary can be slid onto each other by one single shift – a metal like that is unstable and tends to settle into a regular ripple of charge or spin.

The constant-energy surface separating occupied from empty states, whose geometry governs transport, screening and instabilities. Nesting – parallel sheets connected by a single wavevector q – peaks the bare susceptibility at q and is the textbook route to charge and spin density waves. In real layered metals the ordering wavevector often follows momentum-dependent electron–phonon coupling instead, so a nesting figure alone does not settle the mechanism.

Full explanation

Electron–phonon coupling

The push and pull between a crystal’s electrons and the vibrations of its atoms. Vibrating atoms scatter moving electrons and slow them down, which is why a metal conducts worse when warm; the electrons in turn shake the atoms and lose energy to them as heat. The same coupling can also draw electrons together: it is the glue that pairs them in ordinary superconductors, and when it is strong it can make the whole crystal buckle into a charge density wave.

The change in electronic energies when atoms are displaced, described by matrix elements between electron states linked by a phonon and summarised by the dimensionless constant λ obtained from the Eliashberg function α²F(ω). It limits room-temperature mobility through phonon scattering, gives metals their linear-in-temperature resistivity at high temperature, renormalises quasiparticle bands (the kinks seen in ARPES), mediates the attraction behind conventional superconductivity and, where it is strong at particular wavevectors, softens phonons into a charge density wave. In 2D materials it is computed from first principles with density functional perturbation theory and Wannier interpolation, and it can be tuned by doping, strain and the dielectric environment.

Full explanation

Substrate

The base a thin film or flake sits on – often a polished slice of silicon, glass or sapphire. For a crystal one atom thick, the substrate is not just a table: its bumps, stray charges and vibrations reach right into the sheet and change how it conducts and glows. That is why the same material can behave differently on two substrates, and why the flattest, cleanest one – boron nitride – gives the best results.

The material beneath a 2D layer. It acts on the layer through surface roughness, charged impurities, surface optical phonons, dielectric screening, strain and charge transfer, and during growth it sets orientation through epitaxy. Replacing SiO₂ with hexagonal boron nitride, which is atomically flat and nearly free of dangling bonds and charge traps, improved the mobility and charge homogeneity of graphene by close to an order of magnitude.

Full explanation

Gating

Changing a material by putting a voltage on a nearby electrode – a gate – separated from it by an insulator. The voltage pulls electrons into the material or pushes them out, as in every transistor. Because a 2D material is so thin, the gate reaches all of it, so the number of electrons can be dialled up and down continuously: a single device can be turned from insulator to metal, its magnetism strengthened, or superconductivity switched on. Two gates, above and below, can also apply an electric field across the layer.

Electrostatic control of carrier density and electric field through a gate coupled capacitively across a dielectric: the induced density equals the gate capacitance times the voltage beyond threshold, divided by the electron charge, reaching a few 10¹³ cm⁻² with oxide or hBN gates before breakdown. Dual gating sets density and perpendicular displacement field independently, opening the gap of bilayer graphene and tuning moiré flat bands. Ionic-liquid and solid-electrolyte gating form an electric double layer about a nanometre thick and reach 10¹⁴–10¹⁵ cm⁻², enough to induce superconductivity in MoS₂ or raise the Curie temperature of Fe₃GeTe₂, but can intercalate or react electrochemically and work only while the ions are mobile. Hysteresis from traps, contact effects and quantum capacitance are the usual caveats.

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

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