Disorder and localisation

In plain words

Every real crystal has some disorder: missing or misplaced atoms, impurities, stray charges in the , a random mixture of elements in an . 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.

Going deeper

In a perfectly regular crystal an electron wave runs through unhindered; random bumps and dips scatter it and, if strong enough, trap it by interference so that it no longer carries current. In graphene, charges in the substrate split the sheet into puddles of electrons and holes, which a clean hBN substrate smooths out.

Scattering and resistance

An electron wave travels through a perfectly periodic crystal without being scattered; it is the departures from perfection that resist it. , impurities, atoms of an alloy placed at random, rough interfaces and charged defects all scatter carriers and set the residual resistance that remains at the lowest temperatures. In a the surroundings count as much as the layer: charged impurities and roughness in a silicon dioxide substrate limit the of graphene, and in graphene they break the sheet near charge neutrality into puddles that are rich in . between flat, clean hBN layers removes much of this outside disorder.

When electrons get trapped

Philip Anderson showed in 1958 that strong enough disorder does more than scatter: waves bouncing off random obstacles interfere so that they cancel everywhere except in a small region, and the electron is localised. A material that should be a metal then insulates, conducting only by hopping between localised states with the help of heat. Scaling theory added in 1979 that in two dimensions, without , any disorder at all localises non-interacting electrons in the end.

The first sign is weak localisation: waves travelling the same loop in opposite directions interfere constructively and return to their starting point slightly more often, so the resistance rises gently on cooling and falls again in a small magnetic field, which spoils the interference. Strong spin–orbit coupling reverses the sign – weak antilocalisation – a common signature in .

Disorder as a confounder

Disorder often hides behind other explanations. It broadens lines, smears over a range of temperatures, and can make a magnet glassy. In candidate , atoms swapped between sites can produce the same broad continua of excitations as the exotic state being looked for, and in oxides and perovskites cation disorder changes gaps and lifetimes. The word is also used for atomic positions: in an order–disorder transition, such as that of the copper ions in CuInP₂S₆, atoms that hop between several sites at high temperature settle into one on cooling.

For specialists

Deviations from perfect periodicity – point defects, substitutional or alloy disorder, interface roughness, charged impurities and 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.

Where this comes from

Absence of diffusion in certain random lattices Anderson · Physical Review 109, 1492 (1958)
Scaling theory of localization: absence of quantum diffusion in two dimensions Abrahams et al. · Physical Review Letters 42, 673 (1979)
Observation of electron–hole puddles in graphene using a scanning single-electron transistor Martin et al. · Nature Physics 4, 144 (2008)

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

Alloy and solid solution

A crystal in which two similar kinds of atom share the same sites at random – some molybdenum sites taken by tungsten, say, or some sulfur sites by selenium – so that one crystal structure holds a mixture. By choosing the mix, its properties can be set almost anywhere between those of the two pure compounds, much as mixing two paints gives any shade between them. Chemists call it a solid solution; alloy is the everyday word.

A single-phase crystal in which two or more species share a sublattice at random, such as Mo_1−xW_xS₂ or MoS_2(1−x)Se_2x, with lattice constants close to Vegard’s law and band gaps that vary continuously with x, often with a small bowing. Complete miscibility needs similar radii, the same structure type and favourable mixing energetics; otherwise the mixture separates into phases or orders. Random disorder adds alloy scattering and inhomogeneous broadening.

Full explanation

Point defect

A flaw in a crystal that involves a single atomic site: an atom missing (a vacancy), an atom of the wrong kind in a site (a substitution, or an antisite when the crystal’s own two kinds of atom swap places), or an extra atom squeezed in between (an interstitial). Every real crystal has some. In a sheet a few atoms thick each one sits at the surface, so a handful of them can change how the material conducts, glows or reacts far more than in a thick crystal.

A defect confined to one or a few lattice sites – vacancy, interstitial, substitutional impurity, antisite or adatom – or a small complex of them. What it does is set by the levels it introduces: shallow levels dope, deep levels trap carriers and act as non-radiative recombination centres, and an isolated deep level in a wide gap can emit single photons. Formation energies and charge states depend on the chemical conditions of growth and on the Fermi level. In 2D materials every defect is a surface defect, less screened and more exposed to the environment, and densities of 10¹² to 10¹³ per cm² are typical of TMDC monolayers.

Full explanation

Electron hole

Usually just called a hole: the empty place an electron leaves behind in an otherwise full band. It behaves like a particle of its own with a positive charge. When a neighbouring electron steps into the gap, the gap moves the other way – like the empty space in a queue of cars, which travels backwards as each car edges forward. Many materials conduct mainly with holes, and chips need both kinds of carrier.

An unoccupied state in an otherwise filled band, usually near the valence-band maximum, treated as a quasiparticle with positive charge and a positive effective mass set by the band curvature. Holes carry the current in p-type semiconductors and give a positive Hall coefficient. In 2D semiconductors good p-type transport is harder to obtain than n-type, largely because metal contacts pin the Fermi level nearer the conduction band.

Full explanation

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

Quantum spin liquid

A magnet whose atomic magnets never settle into a pattern, even at absolute zero. Quantum effects keep them fluctuating and entangled – linked so that none has a direction of its own – so instead of freezing into order the material stays restless, like a liquid that never turns solid. Disturbances in it can behave like fractions of an electron, carrying its magnetism but not its charge.

A ground state of a frustrated magnet with no symmetry breaking down to zero temperature, marked by long-range entanglement, an emergent gauge structure and fractionalised excitations such as spinons or Majorana fermions. Honeycomb materials with bond-dependent Kitaev exchange – α-RuCl₃ above all – are the leading layered candidates, but the evidence is indirect: a scattering continuum and thermal Hall signals, against residual magnetic order in zero field.

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