Phase transition

Everyday term

In plain words

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 when cooled. Each happens at a definite transition temperature. In quantum materials, transitions mark where new kinds of order appear: , magnetism, superconductivity, . A map of which state appears where, against temperature and a second knob such as or pressure, is a phase diagram.

Going deeper

Below a transition temperature an order parameter – a magnetisation, a polarisation, a gap – grows from zero. First-order transitions jump and show hysteresis; continuous ones set in smoothly with growing fluctuations. A phase diagram maps which state appears where, against temperature and a second knob that in 2D materials is easy to turn.

Order appears

Above its a magnet’s moments point every which way; below it they line up. The magnetisation that appears is the order parameter: zero in the state, growing as the temperature falls below the transition. Every ordered state has one – the electric polarisation of a ferroelectric, the amplitude of a charge density wave, the of a superconductor – and in each case the ordered state has less symmetry than the disordered one: a direction, a period or a phase has been chosen.

Layered materials show transitions of every kind: CrI₃ becomes magnetic below about 45 K, bulk NbSe₂ superconducts below about 7 K, FeSe turns near 90 K and -TaS₂ passes through a sequence of charge-density-wave phases between 550 K and 180 K.

Sudden or gradual

Transitions come in two kinds. In a first-order transition the state jumps: ice and water coexist at the melting point, latent heat is absorbed, and on heating and cooling the change happens at different temperatures, giving – as in the lock-in of the charge density wave in 1T-TaS₂. In a continuous transition the order grows smoothly from zero, and close to the transition fluctuations become large and long-ranged. There, very different materials behave identically: the way quantities change near the transition depends only on dimension and symmetry, not on chemistry. Kenneth Wilson explained this universality with the renormalisation group, for which he received the 1982 Nobel Prize.

Two dimensions and phase diagrams

Dimension matters a great deal. In two dimensions thermal fluctuations are so strong that a magnet whose can point in any direction cannot order at all at finite temperature – the – so need , and some transitions take the unusual form instead.

In return, are easy to tune. A gate changes the electron density continuously, and pressure squeeze the lattice, and a twist changes the , so a single device can trace out a whole phase diagram: superconducting domes beside in twisted graphene, a gate-induced superconducting dome in MoS₂. Where a transition is pushed to zero temperature by such a knob, quantum fluctuations take over – a quantum phase transition, near which unusual states are often found.

For specialists

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, order or Berezinskii–Kosterlitz–Thouless quasi-order. Transition temperatures change with layer number, , strain and twist, which makes 2D materials tunable platforms for phase diagrams, including quantum phase transitions driven at zero temperature by non-thermal parameters.

Where this comes from

Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models Mermin and Wagner · Physical Review Letters 17, 1133 (1966) cited by 8,354
Ordering, metastability and phase transitions in two-dimensional systems Kosterlitz and Thouless · Journal of Physics C 6, 1181 (1973) cited by 9,550
The renormalization group: critical phenomena and the Kondo problem Wilson · Reviews of Modern Physics 47, 773 (1975)

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 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

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

Nematic order

When the electrons in a crystal pick out one direction over another that the crystal itself treats as equal. The name comes from liquid crystals, whose rod-shaped molecules line up along one direction while staying free to flow, as in a display screen. In some superconductors and related materials the electrons do something similar: a crystal with a square or hexagonal pattern suddenly conducts better along one axis than across it, although its atoms have hardly moved.

An electronic state that breaks the rotational symmetry of the lattice while keeping its translational symmetry, named by analogy with nematic liquid crystals. Its order parameter is a direction rather than a density wave: in a tetragonal crystal it lowers four-fold to two-fold symmetry, in a hexagonal one six- or three-fold to two-fold. It appears in the iron-based superconductors – FeSe becomes nematic at about 90 K without magnetic order, with a small orthorhombic distortion that the electrons drive – in kagome metals such as CsV₃Sb₅ inside the charge-ordered state, and as two-fold anisotropy of the superconducting state itself in magic-angle graphene and few-layer NbSe₂. Elastoresistance, polarised Raman scattering, scanning tunnelling microscopy and angle-dependent transport detect it; telling spontaneous nematicity from the effect of strain is the central difficulty.

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

Berezinskii–Kosterlitz–Thouless transition

How a flat, two-dimensional system that cannot have perfect order can still switch between an almost-ordered state and a disordered one. At low temperature, tiny whirlpools in the pattern of its spins – or of a superconductor – exist only in tightly bound pairs; above a critical temperature they break free and scramble the order.

A phase transition without a local order parameter in two-dimensional systems with a continuous planar symmetry – XY magnets, superfluid and superconducting films – driven by the unbinding of vortex–antivortex pairs. Below T_BKT correlations decay as a power of distance (quasi-long-range order), above it exponentially. The superfluid stiffness jumps to zero from the universal value 2k_BT_BKT/π, and in superconducting films the current–voltage curve follows V ∝ I³ at T_BKT. It evades the Mermin–Wagner theorem, which forbids true long-range order but not this.

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

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

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