Every word, explained twice

2D membrane

A sheet that lets some molecules through and holds others back. A perfect layer of graphene is so tight that not even helium gets through, so useful membranes need openings: tiny holes made in a single layer, or the narrow gaps between flakes stacked like a pack of cards. Because such a membrane is only atoms thick, water or gas can pass far faster than through today’s plastic membranes, which could make filtering, purifying water or separating gases cheaper.

Separation membranes made from 2D materials, in two designs: nanoporous single layers (graphene, hBN, MoS₂ with pores opened by ion irradiation, oxidative etching or defects), whose pore size sets the selectivity, and laminates of stacked flakes (graphene oxide, MXenes, COF and MOF nanosheets), whose interlayer channels – often swelling in water – act as sieves. Pristine graphene is impermeable to gases including helium, yet transmits protons; atomic thickness promises high permeance, and restraining the laminate spacing can exclude hydrated ions. Against mature polyamide reverse-osmosis membranes the obstacles are large defect-free areas, swelling and stability in water, fouling and cost. The word also names any freestanding thin film, as in oxide membranes lifted off a sacrificial layer.

Engineering

Air stability

Also: ambient stability

How long a material survives in ordinary air. Some 2D crystals keep for years on a shelf; others change within hours as oxygen, water and light attack them – and a measurement on a degraded sample says little about the real material.

Resistance to reaction with oxygen, water and light under ambient conditions, which often decides what can be measured at all. Few-layer black phosphorus takes up water and degrades visibly within hours; CVD-grown TMDC monolayers age over months, oxidising along grain boundaries and collecting organic contamination. Report exposure time and handling with any result, and move to glovebox transfer and encapsulation when a material’s stability is short.

Everyday

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.

Everyday

Angle-resolved photoemission spectroscopy (ARPES)

A measurement that shines light on a crystal, catches the electrons it knocks out and records the direction and energy of each one. From that it reconstructs how the electrons were moving inside – the closest thing there is to a photograph of a material’s electronic structure.

Photoemission with energy and momentum resolution, giving the occupied band structure, Fermi surface and self-energy directly. It needs a clean, flat, conducting surface in ultrahigh vacuum; focused micro- and nano-ARPES beamlines bring the spot down to the size of an exfoliated flake, which is what makes single-domain twisted and few-layer samples measurable at all.

Experiment

Anisotropy

When a material behaves differently depending on direction – conducting better along one line than across it, for example, much as wood splits easily along the grain but hardly across it. Some 2D materials, such as black phosphorus, are strongly direction-dependent within the sheet itself.

Direction dependence of a property. All layered crystals are anisotropic between in-plane and out-of-plane directions; low-symmetry lattices such as black phosphorus, ReS₂ and CrSBr are also anisotropic within the plane, with effective mass, optical absorption or magnetic order differing along crystal axes.

Everyday

Atomic force microscopy (AFM)

A tip on a tiny springy arm is moved across a surface and its bending is measured, mapping height point by point – much as a fingertip reads Braille. It works on anything, conducting or not, which is why it is the routine way to check how thick a flake is.

Force microscopy in contact, tapping or non-contact mode, used for thickness, roughness, bubbles and polymer residue, with variants for surface potential (KPFM) and local conductivity (c-AFM). Apparent height is not the crystallographic thickness: the tip interacts differently with flake and substrate, so a monolayer commonly reads between 0.4 and 1 nm. Quote the mode, and calibrate against a step of known height.

Experiment

Atomic layer deposition (ALD)

A way to coat a surface one layer of atoms at a time. Two chemicals take turns: the first sticks to every free spot on the surface but not to itself, the excess is pumped away, and the second reacts with what stuck, leaving one thin layer of, say, aluminium oxide. Repeating the pair builds the film with a precision of a fraction of a nanometre, even over deep, steep structures. It is how the gate insulators of modern computer chips are made – and on 2D materials it runs into a problem: their clean surfaces give the first chemical nothing to stick to.

Thin-film growth by alternating, self-limiting surface reactions of two precursors separated by purges – for Al₂O₃ trimethylaluminium and water, for HfO₂ a hafnium amide or chloride with water or ozone – typically at 100 to 300 °C, giving about 0.1 nm per cycle with sub-nanometre control and conformal coverage. It is the standard route to high-κ gate dielectrics in silicon CMOS. On the dangling-bond-free basal planes of graphene, TMDCs and hBN the first precursor finds no reactive sites, so nucleation starts only at steps, defects and residues, giving islands and pinholes. Seed layers (evaporated and oxidised Al, molecular layers), ozone or plasma functionalisation, low-temperature physisorption, or transferred and native-oxide dielectrics are used instead, each at a cost in thickness, interface traps or channel damage.

EngineeringExperiment

Band alignment

How the energy levels of two materials line up when they are put together: whether one material’s electrons sit above or below the other’s. Electrons roll ‘downhill’ to the lower level, so the alignment decides which way charge moves across the junction, and so decides what the stack can be used for.

The relative positions of the band edges of two materials in contact, classified as straddling (type I), staggered (type II) or broken (type III). Van der Waals stacking avoids chemical bonding and strain, so alignments start close to the isolated layers’ ionisation potentials and electron affinities, shifted by interface dipoles and by the dielectric environment. A type-II alignment puts electron and hole in different layers, which is what makes interlayer excitons and gate-tunable photovoltaic junctions possible.

Everyday

Band gap

Also: bandgap, energy gap

The energy an electron needs to jump from a full band, where it is stuck, into an empty one where it can move and carry current – like a car in a packed car park, which can only drive off once it is lifted to the empty deck above. Metals have no gap and always conduct; insulators have a large gap and hardly conduct; semiconductors sit in between, which is what makes them switchable.

The energy range between the valence-band maximum and the conduction-band minimum in which a crystal has no electronic states. It is direct when both extrema lie at the same crystal momentum and indirect otherwise; monolayer MoS₂ is direct while bulk MoS₂ is indirect. In 2D the quasiparticle and optical gaps differ strongly because excitons are tightly bound.

Everyday

Band inversion

When spin–orbit coupling pushes two bands past each other, so the one that should sit on top ends up underneath. The swap cannot be undone without closing the gap, and that is what makes a material topological rather than ordinary.

An exchange in the ordering of bands of opposite parity, usually driven by strong spin–orbit coupling, that changes the topological index while the bulk gap stays open elsewhere. Because the index then differs from that of vacuum, the boundary has to carry gapless states: inversion at Γ in HgTe quantum wells and in the Bi₂Se₃ family is the mechanism behind the quantum spin Hall and three-dimensional topological insulator phases.

Theory

Band structure

Also: electronic band structure

The map of the energies an electron may have in a crystal. The allowed energies come in ranges called bands, with forbidden gaps between them – like the decks of a multi-storey car park, where a car can stand on a deck but never between two. Whether a material conducts, gives off light or lets its electrons move easily can all be read from it, and in a layered material the map changes with the number of layers.

The electron energies E(k) of a periodic crystal as functions of crystal momentum, one branch per band, usually plotted along high-symmetry lines of the Brillouin zone. Band edges, slopes and curvatures give the gap, group velocities and effective masses; crossings and inversions give the topology. In layered crystals interlayer hybridisation makes it depend on thickness: the valence-band maximum of MoS₂ moves from Γ in the bulk to K in the monolayer, turning an indirect gap into a direct one.

Everyday

Battery electrode

A battery stores energy by moving ions – in phones and cars, lithium ions – between two electrodes, the anode and the cathode. Charging pushes the ions into the anode, where they slip between the layers of a material such as graphite; using the battery lets them flow back. Layered and 2D materials matter because ions can enter and leave the gaps between layers, and because thin conductive flakes can help electrons reach every grain of an electrode.

A lithium- or sodium-ion cell shuttles ions through an electrolyte between a negative electrode (the anode on discharge) and a positive one (the cathode), while electrons pass through the external circuit. Graphite, the standard anode, intercalates lithium between its layers up to LiC₆, about 372 mAh/g; layered oxides such as LiCoO₂ and NMC dominate cathodes. 2D materials enter as conductive additives (graphene flakes replacing part of the carbon black), as higher-capacity anodes that alloy or convert (phosphorus, antimony, SnS₂, SnO), as hosts for larger ions such as sodium, which graphite takes up poorly, and as MXene electrodes. Their weaknesses are large volume change, irreversible first-cycle loss from the high surface area, low packing density and cost, so gains at cell level are usually far smaller than those claimed for the material.

Everyday

Berezinskii–Kosterlitz–Thouless transition

Also: BKT transition, 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.

Theory

Berry curvature

A twist in the way an electron’s wave changes as the electron moves through a crystal. It acts like a hidden magnetic field, pushing moving electrons sideways even when no real magnetic field is applied.

The imaginary part of the quantum geometric tensor – an effective magnetic field in momentum space responsible for anomalous and valley Hall effects.

Theory

Bridgman growth

Also: Bridgman–Stockbarger method

A way to grow a large single crystal from a melt. The ingredients are melted in a sealed tube with a pointed bottom, and the tube is lowered slowly out of the hot part of a furnace. The melt freezes first in the tip, where one small crystal wins out, and the rest of the melt freezes onto it as the tube moves on. It gives the big crystals, a centimetre or more across, from which flakes of InSe, Bi₂Se₃ or PbI₂ are peeled.

Directional solidification of a melt in a sealed ampoule, usually conical at the bottom, moved through a temperature gradient at about a millimetre per hour (Bridgman, 1925; Stockbarger’s two-zone furnace with a baffle that sharpens the gradient, 1936). Competitive growth in the tip selects one grain, which seeds the boule. It suits compounds that melt congruently, such as Bi₂Se₃, Bi₂Te₃, SnSe and PbI₂; incongruently melting ones such as InSe are grown from off-stoichiometric melts. It is fast and gives large crystals, but freezes in the segregation, stoichiometry drift, vacancies and stacking disorder of growth from the melt, so its crystals are often less pure and less ordered than those from flux or vapour transport.

Experiment

Chalcogen vacancy

A spot in a TMDC crystal where a sulfur, selenium or tellurium atom is missing. It is the most commonly reported flaw in these materials, and it is easily confused with an oxygen atom sitting in its place.

A missing S, Se or Te atom – the most commonly reported point defect in TMDCs, often confused with substitutional oxygen.

Experiment

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.

Everyday

Charge carrier mobility

Also: carrier mobility

How easily electrons, or the gaps they leave behind (holes), move through a material when pushed by a voltage. Higher mobility means faster, more efficient electronics – but it is easily ruined by dirt, defects and a poor substrate.

Drift velocity per unit electric field, usually quoted in cm²/V·s. In 2D materials it is limited intrinsically by phonon scattering and extrinsically by charged impurities, roughness, remote phonons and disorder; field-effect values extracted from two-terminal devices can be distorted by contact resistance.

Everyday

Charge density wave (CDW)

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 superconductivity for the same electrons.

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, doping and pressure, and the ordered state frequently coexists or competes with superconductivity in the same phase diagram.

TheoryExperiment

Charge transfer

Electrons moving from one material to its neighbour simply because they are more comfortable there. It is how a layer gets doped without adding a single impurity atom to it – the charge comes from the molecule, substrate or layer next door.

Equilibration of electrochemical potential across an interface, moving electrons to whichever side has the deeper states and leaving a dipole and band bending behind. It dopes 2D layers without substitutional disorder, from adsorbed molecules and substrate surface states to the partner layer in a type-II stack, where photoexcited carriers separate in tens of femtoseconds – faster than they recombine, which is what makes such stacks useful for light harvesting and detection.

Everyday

Chemical bond

The glue that holds atoms together. In a covalent bond two atoms share electrons, as within a sheet of graphene; in an ionic bond one atom hands electrons to another and the resulting charges attract, as in table salt; in a metal the outer electrons are shared by all the atoms at once. Much weaker van der Waals forces hold molecules and the layers of a layered crystal together. A 2D material is a crystal with strong bonds within its layers and only weak ones between them – which is why it can be peeled.

The attractive interactions that hold atoms in molecules and solids, classified by how electrons are shared: covalent bonds, from overlapping orbitals holding shared electron pairs, strong and directional (sp² carbon, Mo–S in TMDCs); ionic bonds, from electron transfer and electrostatic attraction (halides, oxides); metallic bonding, with electrons delocalised in a partly filled band; and the weak, non-directional van der Waals interaction from correlated charge fluctuations, besides hydrogen bonds. Real bonds are mixtures, characterised by electronegativity differences and charge analyses. Layered crystals combine in-plane bonds of several electronvolts with interlayer binding of only about 20 meV per Ų (some 0.3 J/m²), the criterion high-throughput screens use to identify exfoliable compounds.

Everyday

Chemical vapour deposition (CVD)

Also: chemical vapor deposition

A way of growing thin films from gases. The ingredients arrive as vapour, react on a hot surface and leave a solid layer behind. It is how large sheets of graphene and other 2D materials are made for industry.

Growth of films from gas-phase precursors that react or decompose on a heated substrate. For 2D materials this includes graphene on copper foil, powder-source CVD of TMDCs from metal oxides and chalcogens, and metal-organic CVD (MOCVD) for wafer-scale TMDC films; grain size, nucleation density and residues are the usual quality limits.

Everyday

Chern number

A whole number that adds up the twist of the electron waves in a band – the twist Berry curvature describes – over every way the electrons can move. Like the number of holes in a doughnut, it cannot change a little, so the effects it controls, such as a sideways resistance locked to an exact value, are extremely robust.

The integral of Berry curvature over a band, an integer that fixes the quantised Hall conductance contributed by that band.

Theory

Chirality

The property of an object that cannot be turned into its own mirror image, like a left and a right hand or a corkscrew that turns one way. Molecules can be chiral, and so can crystals: in tellurium the atoms form screw-shaped chains that wind either to the left or to the right. Electrons, spins and vibrations can take on a handedness too, and chiral materials can treat left- and right-turning light, or spins pointing one way or the other, differently.

The absence of improper symmetry operations (mirror planes, inversion, rotoreflections), so that an object and its mirror image are distinct enantiomorphs. Chiral crystals belong to the 65 Sohncke space groups, such as trigonal tellurium (P3₁21 or P3₂21), whose helical chains give natural optical activity, current-induced magnetisation and radial spin textures through spin–orbit coupling. Chirality also describes Weyl fermions, whose chirality sets the chiral anomaly and the chiral magnetic effect; edge modes that propagate one way; spin spirals and skyrmions with a fixed sense of rotation from the Dzyaloshinskii–Moriya interaction; phonons carrying angular momentum in the valleys of TMDCs; and proposed chiral orders such as loop currents in kagome metals and Mn₃Si₂Te₆. Barron’s distinction between true and false chirality decides which physical effects a given handedness allows.

Theory

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.

Theory

Complementary metal–oxide–semiconductor (CMOS)

The way almost every digital chip is built. Each logic gate pairs a transistor that conducts with electrons with one that conducts with holes, so one of the two is always off and the gate draws almost no current except while it switches. A 2D material becomes useful for logic only if both kinds of transistor can be made from it, or if it can be added to a silicon chip without harming it.

Logic in which n- and p-channel field-effect transistors are wired in series between supply and ground, so that in either stable state one of them is off and static power is set by leakage alone. It demands n- and p-type devices of comparable drive, threshold voltages held within tens of millivolts across a wafer, and a process compatible with the silicon line – about 400 °C at most for back-end-of-line integration. For 2D channels the p-type half is the harder one: most TMDCs conduct n-type with common contact metals, and WSe₂ is the leading p-type candidate.

Engineering

Composites and coatings

The biggest market for 2D materials so far: flakes of graphene or other layered materials mixed in small amounts into plastics, rubber, concrete, paints or battery electrodes, to make them stiffer, tougher, more conductive or less permeable. Sold by the tonne, these products use cheap flakes made in bulk rather than perfect single layers. Whether a few percent of flakes really helps depends on how well they are spread out, how they line up and how firmly they hold on to the material around them.

Bulk uses in which exfoliated or chemically derived flakes – graphene nanoplatelets, reduced graphene oxide, hBN, MXenes, clays – are dispersed in a matrix or applied as a film. In polymer composites a few weight percent can raise stiffness and toughness, cross the percolation threshold for electrical conduction – as low as about 0.1 percent by volume for thin, wide flakes – and raise thermal conductivity; aligned flakes in barrier coatings lengthen the diffusion path of water, oxygen and ions and slow corrosion; flakes used as conductive additives replace part of the carbon black in battery electrodes; MXene films give strong electromagnetic shielding. Performance depends on flake size and thickness, dispersion without restacking, alignment and bonding to the matrix, and on powders whose properties vary widely between suppliers.

Engineering

Contact resistance (Ω·µm)

The electrical resistance where a metal wire meets the material it is meant to feed current into. In devices made from 2D materials it often wastes more than the material itself.

Resistance at the metal–semiconductor junction normalised to contact width; the key figure of merit for 2D transistor contacts.

Engineering

Coordination polyhedron

Also: coordination environment

The small shape traced by the atoms that surround a metal atom in a crystal. In many layered materials each metal sits inside an octahedron – six neighbours, three above and three below, turned against each other – or a trigonal prism, where the upper three sit right above the lower three. That one turn can make the same compound a semiconductor or a metal.

The polyhedron formed by an atom’s nearest neighbours; their number is its coordination number. In MX₂ layers the metal is either trigonal prismatic (D3h, as in 2H-MoS₂) or octahedral (D3d, as in 1T-TaS₂, and distorted in 1T′-WTe₂), and the ligand-field splitting of the d levels in each geometry, together with the d-electron count, decides whether the layer is a semiconductor or a metal. Halides and many oxides build layers from MX₆ octahedra sharing edges (CrI₃, RuCl₃) or corners (layered perovskites), and that connectivity sets the lattice – honeycomb, triangular or square – that the metal ions form.

Everyday

Coulomb truncation

A correction used in computer simulations of 2D materials. Simulations usually repeat a sheet endlessly above and below, like the reflections between two facing mirrors, and those copies would wrongly pull and push on each other; truncation cuts that false interaction out.

A modification of the Coulomb interaction in periodic codes that removes spurious interaction between repeated layers, essential for GW, BSE and charged calculations in 2D.

Theory

Cpk

Also: process capability index

A single number for how comfortably a production process stays within its allowed limits – like a car parked in a garage, where what counts is the room left on the tighter side. A higher value means fewer parts fall outside specification.

A process capability index comparing the spread and centring of a process with its specification limits.

Engineering

Critical raw materials

Raw materials an economy depends on but whose supply could be interrupted – because they come mostly from one country or a few mines, or only as a by-product of mining something else. The EU, the United States and other countries keep official lists. Several 2D materials rely on such inputs: natural graphite for graphene, tungsten for WS₂ and WSe₂, borates for boron nitride, and gallium, indium or tellurium for others. Supply risk can matter as much as performance in deciding whether a material reaches the market.

Materials assessed as both economically important and at high supply risk, using indicators of production concentration, the governance of producing countries, import reliance, substitutability and end-of-life recycling (as in the EU methodology and Graedel’s criticality framework). The EU list, revised every three years, includes graphite, tungsten, boron, gallium and germanium, and the Critical Raw Materials Act of 2024 sets targets for domestic extraction, processing and recycling; the United States keeps its own list, and China introduced export licensing for graphite at the end of 2023 and for tungsten, molybdenum, tellurium, bismuth and indium products in February 2025. For 2D materials the exposure lies in feedstocks, precursors and substrates – graphite, molybdenum and tungsten compounds, chalcogens, borates, MAX-phase powders, sapphire and SiC – and is reduced by substitution, synthetic routes, recycling and diversified supply; thin-film uses consume little material, bulk uses such as composites and battery anodes a great deal.

Engineering

Curie and Néel temperature

Also: ordering temperature

The temperature at which a magnet loses its order on warming: the Curie temperature for a ferromagnet, whose spins all point one way, and the Néel temperature for an antiferromagnet, whose neighbouring spins alternate. Above it the spins point in every direction and the material is no longer magnetic.

The critical temperatures of ferromagnetic and antiferromagnetic order. In 2D magnets they depend on layer number and on magnetic anisotropy rather than on exchange alone: CrI₃ orders at 45 K as a monolayer against 61 K in the bulk, and Fe₃GeTe₂ can be pushed to room temperature by ionic gating. Quote the layer number, the substrate and the measurement with any value.

Everyday

Dangling bond

A bond left with nothing on the other end, on an atom at a surface that has lost the neighbour it was bonded to inside the crystal. Such bonds grab stray molecules and trap electrons, which is why ordinary surfaces are messy – and why layered crystals, whose sheets keep every bond inside themselves, split into faces that stay clean down to a single layer.

An unsaturated valence orbital at a surface, edge or defect where a covalent bond has been cut. In a bulk semiconductor such as silicon these states lie in the gap, trap charge and pin the Fermi level unless passivated by hydrogen or a grown oxide. The basal plane of a layered crystal has none, which is what allows van der Waals stacking without lattice matching and interfaces with few traps – and also why atomic layer deposition nucleates poorly on it. Edges, grain boundaries and vacancies do carry them, and are where 2D materials are chemically active.

Everyday

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.

Theory

Density of states

A count of how many states electrons can occupy at each energy – like a chart of how many seats each row of a stadium has. Where the count is high, many electrons can join in whatever happens at that energy; inside a band gap it is zero. Flat bands pile many states onto one energy, which is why they favour unusual behaviour, and a scanning tunnelling microscope can measure the count at a single spot.

The number of single-particle states per unit energy (and per unit area in 2D), g(E) = Σ_n ∫ δ(E − ε_n(k)) d²k/(2π)². For a parabolic 2D band it is constant, m*/(2πħ²) per spin and valley; for graphene’s Dirac cone it rises linearly from zero; van Hove singularities appear at saddle points. Its value at the Fermi level sets the electronic heat capacity, Pauli susceptibility, screening and the tendency to Stoner or superconducting instabilities; scanning tunnelling spectroscopy measures the local density of states through dI/dV.

Theory

Design of experiments (DOE)

A way of planning tests that changes several settings at once in a structured pattern, so the effect of each – and how they interact – can be found in far fewer runs than changing one thing at a time.

Structured variation of several process parameters at once to find their effects and interactions efficiently.

Engineering

Dielectric

Also: gate insulator

An electrical insulator used for what it does in an electric field rather than for carrying current – the thin layer inside a capacitor, or the insulator that separates a transistor’s gate from its channel. A good dielectric blocks current completely, withstands a strong field without breaking down and, for gates, has a high permittivity, so that a voltage on the gate pulls many charges into the channel. For 2D transistors the dielectric is as hard to get right as the 2D layer itself.

An insulator characterised by its permittivity κ, band gap and band offsets to the channel, breakdown field, leakage, and density of fixed charge and traps at the interface and in the bulk (border traps). Silicon technology moved from SiO₂ (κ ≈ 3.9) to hafnium-based high-κ oxides (κ ≈ 20–25) to keep gate control at sub-nanometre equivalent oxide thickness. For 2D channels, hBN gives clean, trap-poor interfaces but a low κ of about 3–4 and is hard to grow at wafer scale; ALD oxides nucleate poorly on basal planes; alternatives include crystalline CaF₂, native oxides such as Bi₂SeO₅ on Bi₂O₂Se, and molecular crystals. Reliability – hysteresis, bias-temperature instability and time-dependent breakdown – is the yardstick for any dielectric meant for products.

Engineering

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.

Everyday

Dirac cone

The shape of graphene’s energy landscape near its most important points: plot an electron’s energy against how it moves and you get two cones touching tip to tip. It means electrons in graphene act as if they had no mass and all move at the same speed, about a three-hundredth of the speed of light – much as light moves at one speed whatever its colour.

A linear, conical band crossing, as at the K and K′ points of graphene, where the energy grows in proportion to momentum and the Fermi velocity is about 10⁶ m/s. Carriers behave as massless Dirac fermions; the crossing is protected by symmetry and gapped by breaking sublattice symmetry or by spin–orbit coupling.

Everyday

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.

Theory

Dispersions and inks

Flakes of a 2D material floating in a liquid, and that liquid used like paint or ink. Stirring or shaking a layered crystal in the right liquid splits it into flakes that stay suspended; spinning away the thick ones leaves a dispersion of thin flakes, which can be sprayed, printed or coated onto almost anything. It is the cheapest way to make 2D materials by the kilogram, but the flakes are small and overlap in a film, so they never match a perfect continuous sheet.

Colloidal suspensions of nanosheets made by liquid-phase exfoliation of layered crystals in solvents whose surface energy matches the crystal (NMP, DMF, cyclohexanone) or in water with surfactants, by electrochemical exfoliation, or – for graphene oxide and MXenes – by chemical routes that leave hydrophilic surfaces. Centrifugation sorts flakes by size and thickness (liquid cascade centrifugation), and optical spectra give mean thickness and lateral size in the liquid. Formulated as inks with tuned viscosity and surface tension, they are inkjet-, screen- or spray-printed or coated into films limited by flake–flake junctions, porosity, residual solvent or surfactant, and oxidation in storage. Powders and dispersions sold as graphene range from monolayers to graphite, which ISO/TS 21356-1 addresses by defining how they are characterised.

EngineeringExperiment

Doping

Adding a small amount of extra electrons to a material, or taking some away to leave ‘holes’ that act as positive charges, to change how well it conducts. In silicon this is done by mixing in foreign atoms; in 2D materials it can also be done with a nearby voltage, molecules on the surface, or the material underneath.

Control of carrier type and density by substitutional impurities, surface charge transfer, electrostatic gating or the dielectric environment. Stable, spatially localised substitutional doping remains difficult in 2D semiconductors, and surface-transfer doping is often unstable in air.

Everyday

E′ and A′₁ modes

Two characteristic vibrations of a single layer of MoS₂ and its relatives, one along the sheet and one out of it. Each shows up as a peak in the Raman spectrum, and the gap between the two peaks widens as layers are added – a quick way to count them.

The in-plane and out-of-plane Raman-active vibrations of 1H TMDC monolayers, whose separation is a common layer-number indicator.

Experiment

Edge state

Also: edge channel

Electrons that can only travel along the border of a sheet while its interior stays insulating. In a topological material these border lanes are guaranteed to exist, and in the cleanest cases they carry current without losing energy to scattering.

States localised at a sample boundary and dispersing inside the bulk gap. In quantum Hall and Chern insulators they are chiral, one-way channels; in a quantum spin Hall insulator they are helical, counter-propagating with opposite spins and protected by time reversal, giving e²/h per edge in the ideal case – monolayer 1T′-WTe₂ keeps that edge conduction up to about 100 K. Trivial edges carry states too, from dangling bonds, reconstructions and graphene’s zigzag edge, so the evidence for topology is quantised conductance and its magnetic-field dependence, not conduction at the edge alone.

Everyday

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.

Everyday

Electride

A crystal in which some electrons belong to no atom at all. In table salt, each sodium atom hands an electron to a chlorine atom; in an electride the metal atoms give up electrons too, but there is no atom to take them, so they sit on their own in the empty space between the layers, playing the chlorine’s part. Electrons held that loosely leave easily, which makes such crystals unusually good at giving electrons to whatever touches them.

An ionic crystal whose anions are electrons confined in interstitial space – in the layered case a two-dimensional electron gas in the interlayer gallery. The anionic layer gives a low work function, a high carrier density and metallic conduction, and it survives exfoliation. Ca₂N was the first layered example; the same loosely bound electrons drive catalytic ammonia synthesis and electron-injection layers.

Everyday

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.

Everyday

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.

Theory

Encapsulation

Sealing a delicate 2D material between protective layers so that air, water and dirt cannot reach it. Wrapping it in boron nitride, itself a 2D material, is the standard way to get the cleanest results.

Enclosure of a 2D layer between barrier layers – most often hexagonal boron nitride for the highest quality, or deposited oxides and polymers for larger areas – to suppress oxidation, adsorbates, charge disorder and substrate roughness. It also seals in whatever contamination was present when it was applied.

Everyday

Endothermic versus exothermic transport

Whether a crystal-growing reaction absorbs or releases heat decides which end of the sealed tube the crystals form at: the cooler end for reactions that absorb heat, the hotter end for reactions that release it.

Endothermic transport reactions deposit material at the cooler end of the ampoule; exothermic ones deposit at the hotter end.

Experiment

Epitaxy

Growing a crystal on top of another so that its atoms line up with the ones underneath and carry on the pattern of the crystal below. It is how most semiconductor chips, lasers and LEDs are built, and it works cleanly only when the two crystals’ atomic spacings match closely – a restriction that layered 2D materials largely escape.

Growth of a crystalline film whose orientation is fixed by a single-crystal substrate: homoepitaxy on the same material, heteroepitaxy on a different one. A covalently bonded film strains to the substrate’s lattice until, beyond a critical thickness that falls steeply with mismatch, misfit dislocations relax it, so mismatches above a few percent permit only very thin coherent films. In van der Waals epitaxy the film is aligned but not bonded and mismatch is largely irrelevant; in remote epitaxy the substrate’s potential aligns the film through a graphene interlayer.

Everyday

Equivalent oxide thickness (EOT)

A way of comparing insulating layers: the thickness of plain silicon dioxide that would do the same job. A smaller number means the gate controls the transistor more strongly.

The SiO₂ thickness that would give the same gate capacitance as the actual dielectric stack.

Engineering

Etching

Removing material on purpose, with a chemical or a plasma. In chip-making it carves patterns: a stencil of light-sensitive polymer protects some areas while a plasma eats away the rest. For 2D materials etching has three jobs: cutting flakes into shaped devices, thinning a crystal one layer at a time, and – for MXenes – making the material in the first place, by dissolving one kind of atom out of a layered ceramic so that the remaining sheets come apart.

Material removal by chemical reaction in a liquid (wet etching) or by reactive and physical processes in a plasma (dry etching, including reactive ion etching with oxygen-, fluorine- or chlorine-based chemistries, and atomic layer etching). For 2D materials it patterns channels and Hall bars through resist masks – oxygen plasma for graphene, fluorine-based plasmas for hBN and TMDCs – exposes edges in encapsulated stacks for edge contacts, thins crystals layer by layer, and selectively removes the A element, usually aluminium, from MAX phases with hydrofluoric acid or fluoride-containing acids to make MXenes. Its side effects are edge roughness and damage, residues, doping and plasma-induced defects in neighbouring layers, so the etched edge and its electronic states have to be considered in narrow devices.

ExperimentEngineering

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.

Theory

Exciton

An electron paired with the ‘hole’ it left behind – the empty place, which acts like a positive charge. The two attract each other and form something like a tiny atom. In very thin materials these pairs hold together unusually strongly, so they dominate how the material absorbs and emits light – even at room temperature.

A bound state of a conduction-band electron and a valence-band hole. In 2D semiconductors reduced dielectric screening raises binding energies to hundreds of meV in freestanding TMDC monolayers, so excitons dominate optical spectra at room temperature and follow a non-hydrogenic Rydberg series.

Everyday

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.

Theory

Exfoliation

Also: Scotch-tape method

Peeling thin layers off a bulk crystal. The famous version uses sticky tape: press it on a crystal, pull it away, repeat, and some flakes end up only one layer thick. Other versions do the same in a liquid to make large amounts of flakes at once.

Separation of layers from a layered bulk crystal by overcoming interlayer van der Waals bonding – mechanically with adhesive tape or stamps, which gives the highest-quality flakes but low yield, or in liquids by sonication, shear or electrochemical intercalation, which scales but yields smaller, more defective nanosheets with broad size distributions.

Everyday

FEOL and BEOL

Also: front end of line, back end of line

The two halves of making a chip: the front end builds the transistors in the silicon itself, and the back end adds the layers of metal wiring on top – at lower temperatures, so nothing underneath is damaged.

Front end of line forms the transistors in the silicon; back end of line builds the metal interconnect layers above them at lower temperatures.

Engineering

Fermi level

Also: Fermi energy

Roughly, the energy up to which a material’s electron states are filled – like the water line in a partly filled glass. Where it sits relative to the band gap decides how many charges can move and whether they are electrons or holes, the empty places electrons leave behind. A gate voltage raises and lowers it, like pouring water in or out.

The electrochemical potential of electrons: the energy at which a state has 50 % occupation in thermal equilibrium. Its position relative to the band edges sets carrier density and type; at metal–semiconductor contacts, interface states can pin it and fix the Schottky barrier.

Everyday

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.

Theory

Fermi-level pinning

When the junction between a metal and a semiconductor behaves the same whichever metal is used. The choice of metal ought to set the height of the energy step electrons must climb to get into the semiconductor, but stray electron states at the interface take up charge and hold the step at much the same height every time. It makes low-resistance contacts hard to engineer.

The fixing of the Fermi level at a metal–semiconductor interface by interface states, so the Schottky barrier barely depends on the metal work function.

Experiment

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.

Everyday

Ferromagnet and antiferromagnet

In a ferromagnet the tiny magnets of the atoms all point the same way, as in a fridge magnet. In an antiferromagnet neighbours point in opposite directions and cancel out. Both kinds of order have been found in sheets only one layer thick.

Magnetically ordered states with parallel (ferromagnetic) or antiparallel (antiferromagnetic) alignment of neighbouring moments. Long-range order in 2D requires magnetic anisotropy to evade the Mermin–Wagner theorem; CrI₃ monolayers are Ising-like ferromagnets that couple antiferromagnetically between layers, and the MPS₃ compounds are antiferromagnets.

Everyday

Field-effect transistor (FET)

The electronic switch at the heart of every chip. It works like a tap: a voltage on one terminal, the gate, opens or closes a channel through which current flows. Researchers test new 2D materials by making them into the channel of such a switch.

A three-terminal device in which a gate voltage modulates the carrier density of a semiconductor channel between source and drain through a thin dielectric. For 2D channels the key figures of merit are on/off ratio, subthreshold swing, mobility, contact resistance, hysteresis and threshold-voltage variability.

Everyday

Flat band

A range of electron energies so narrow that the electrons barely move on their own. With their motion frozen out, the way they repel each other takes over – as in a packed train carriage, where nobody can walk anywhere and everything depends on how people get on with their neighbours. That is where unusual states such as superconductivity and magnetism can appear.

A band whose kinetic-energy width is small compared with the interaction energy, so electron–electron interactions dominate the physics.

Theory

Fractional Chern insulator

A state in which electrons act together so strongly that a disturbance in them behaves like a particle carrying a fraction – a third, say – of an electron’s charge, although no electron has been split. Such states were long thought to need enormous magnetic fields; in some twisted and stacked 2D materials they appear without any.

A lattice analogue of a fractional quantum Hall state that forms in a partially filled Chern band, potentially without any external magnetic field.

Theory

G and 2D bands

The two main peaks in the Raman spectrum of graphene. Their shape, position and relative height reveal how many layers there are and whether the graphene is stretched or electrically doped.

Graphene’s first-order in-plane Raman mode and its second-order double-resonant overtone, which together report layer number, strain and doping.

Experiment

Gate-all-around and CFET

Also: GAA

Newer transistor designs in which the controlling gate wraps completely around thin stacked channels – like gripping a hose with the whole hand rather than pinching it from one side. In the CFET version the two kinds of transistor a circuit needs are also stacked on top of each other to save space.

Transistor architectures in which the gate surrounds stacked channel sheets; complementary FETs stack n- and p-type devices vertically.

Engineering

Gating

Also: electrostatic 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.

Experiment

Grain boundary

The seam where two crystal grains grown in different orientations meet. A film made of many grains is only as good as its seams: they scatter electrons, weaken the sheet and are where damage usually begins.

A line defect joining two domains of different orientation, built in 2D from rows of non-hexagonal rings – pentagon–heptagon pairs in graphene, dislocation cores of four- to eight-membered rings in TMDCs, with the mirror twin boundary as a special case. Its effect depends on misorientation and core structure more than on grain size: some boundaries barely scatter carriers while others block transport, which is why wafer-scale growth aims for aligned single-orientation domains rather than merely larger ones.

Everyday

GW and the Bethe–Salpeter equation

Two calculations that take over where density functional theory, the standard one, stops. The first corrects the energy cost of adding or removing an electron; the second adds the attraction between an electron and the hole it leaves behind, which is what decides the colour of light a thin crystal absorbs.

GW replaces Kohn–Sham eigenvalues with quasiparticle energies from a screened-exchange self-energy, opening the gaps that semi-local DFT underestimates; solving the Bethe–Salpeter equation on top adds the electron–hole interaction and produces optical spectra with bound excitons. In 2D both converge painfully slowly with vacuum size and k-point sampling, because screening is non-analytic near q = 0 – a Coulomb cutoff and dense sampling around the band edge are not optional.

Theory

Hall effect

Send a current along a strip in a magnetic field and the moving charges are pushed towards one edge, so a small voltage appears across the strip. Its size tells how many charge carriers there are, and its sign whether they are electrons or holes, which makes it the standard way to count them. In a magnetic material a sideways voltage appears even without an outside field – the anomalous Hall effect, a handy way to see whether a tiny flake is magnetic.

The transverse voltage that the Lorentz force produces when a current flows in a perpendicular magnetic field. For one type of carrier in a 2D sheet the Hall resistance is B/ne, independent of thickness, so its slope gives the sheet density n and its sign the carrier type; combined with the sheet resistance it gives the Hall mobility. Two carrier types make the Hall resistance non-linear in B and call for a two-band fit. In magnetic conductors an extra term that follows the magnetisation – the anomalous Hall effect – comes from the Berry curvature of the bands (intrinsic) or from skew and side-jump scattering (extrinsic); quantised, it becomes the quantum anomalous Hall effect.

Experiment

Honeycomb lattice

A flat net of hexagons, like a beehive, with an atom at every corner. Graphene is the famous example, but the same pattern gives boron nitride its band gap, gives graphene its massless electrons, and gives some magnets interactions that cannot all be satisfied at once.

A triangular Bravais lattice with a two-site basis, so every site has three nearest neighbours on the other sublattice. Equivalent sublattices with nearest-neighbour hopping give Dirac cones at K and K′ (graphene); inequivalent ones open a gap with valley-contrasting Berry curvature (hBN, TMDC monolayers seen from above); buckling mixes in spin–orbit coupling (the Xenes); and bond-dependent exchange on the same geometry gives the exactly solvable Kitaev model behind the α-RuCl₃ spin-liquid programme.

Everyday

Hot pick-up

Assembling stacks of 2D materials while they are warm. The warmth lets trapped dirt and bubbles be pushed out from between the layers as they come together – much as bubbles are smoothed out from under a phone’s screen protector – giving cleaner interfaces.

Assembly of van der Waals stacks at elevated temperature so that contamination is expelled from the interface as layers make contact.

Experiment

Huang–Rhys factor

A number describing how strongly the light emitted or absorbed by a single defect is tied to vibrations of the surrounding crystal. A small value means sharp, clean emission – what you want from a source of single photons.

The mean number of phonons emitted in an optical transition of a localised defect, which sets the balance between zero-phonon line and phonon sideband.

Theory

Hydrothermal and solvothermal synthesis

Making crystals in a sealed steel pot, an autoclave, in which water – or another solvent – is heated well above its normal boiling point. The pressure keeps it liquid, and hot, pressurised water dissolves and moves substances that barely dissolve at all under normal conditions, so crystals can grow from it at a few hundred degrees instead of the far higher temperatures a furnace would need. Nature grows quartz and many minerals this way; in the laboratory it gives nanosheets of BiOCl, LDHs and tellurene, and the framework materials known as COFs and MOFs.

Crystallisation from solution in a sealed vessel above the solvent’s boiling point at autogenous pressure – water for hydrothermal, organic solvents such as ethylene glycol, DMF or mesitylene–dioxane for solvothermal synthesis – usually at 100 to 250 °C in Teflon-lined steel autoclaves, and at higher temperatures and pressures for minerals and quartz. Raised solubility and transport allow crystalline products at low temperature, and pH, solvent, surfactants and capping agents steer the shape towards thin plates. It yields BiOX and LDH nanosheets, SnS₂ and tellurene flakes and crystalline COFs and MOFs, whose reversible bond formation in solution corrects errors as they grow. The products are typically small, polydisperse crystals or powders whose surfaces carry ligands and solvent.

Experiment

Hysteresis

When a device gives different readings depending on whether a voltage is swept up or down, because charges are trapped and released along the way. It is a sign of an unclean or unstable device.

Different transfer curves for forward and reverse gate sweeps, caused by charge trapping in adsorbates, dielectrics or defects.

Experiment

Intercalation

Slipping atoms or molecules into the gaps between the layers of a layered material, like sliding cards between the pages of a book. It can change a material’s properties completely, or push the layers apart so they separate more easily. A lithium-ion battery charges this way, with lithium slipping in between the carbon layers of its graphite electrode.

Insertion of guest species – ions such as Li⁺, molecules or solvents – into the van der Waals gaps of a layered host. It changes carrier density, stacking and phase, for example driving 2H-to-1T′ transitions in MoS₂; it enables electrochemical exfoliation and underlies ion storage in batteries and MXene supercapacitors.

Everyday

Interface trap density (Dit)

How many electrical ‘traps’ sit at the boundary between a transistor’s channel and its insulating layer. Traps catch and release charge, which makes switching slower, noisier and less predictable.

The density of electronic states at the channel–dielectric interface that capture charge, degrading switching and stability.

Engineering

Interlayer coupling

How strongly neighbouring sheets in a stack affect one another. The pull holding them together is weak, but electrons can still hop from one sheet to the next and whole sheets can vibrate against each other – which is why one, two and many layers of the same crystal behave differently, and why twisting or sliding one sheet changes the whole stack.

The electronic hybridisation, mechanical force constants and electrostatic interaction between adjacent layers of a van der Waals crystal. Electronically it is hopping between orbitals that reach out of the plane, a few tenths of an electronvolt in graphite, which splits and shifts bands and makes the band structure depend on thickness, stacking and twist. Mechanically it sets the interlayer shear and breathing modes, whose Raman frequencies of a few tens of cm⁻¹ measure it directly. Far weaker than in-plane bonding, it still decides the direct–indirect gap crossover of TMDCs, the flat bands of twisted bilayers and interlayer magnetic order.

Everyday

Interlayer exciton

An electron in one layer bound to a hole in the layer next door. Because the two sit in different sheets, the pair lives far longer than an ordinary exciton and can be pushed around with an electric field.

An exciton whose electron and hole sit in adjacent layers of a type-II heterostructure. The spatial separation gives lifetimes orders of magnitude longer than intralayer excitons and a permanent out-of-plane dipole that makes the energy gate-tunable; in a twisted bilayer the moiré potential traps them on a regular grid, turning a heterobilayer into an array of nearly identical emitters.

Theory

Inversion symmetry

A crystal has inversion symmetry if turning it inside out through a point – replacing every atom by the one directly opposite – leaves it looking the same. Many of the most useful effects in 2D materials appear only when that symmetry is missing.

Invariance under r → −r about a centre. Its absence is the precondition for piezoelectricity, a switchable polarisation, second-harmonic generation, Rashba splitting and valley-contrasting optical selection rules, and it decides whether Weyl or Dirac nodes are allowed. In layered crystals it follows layer number and stacking rather than chemistry alone: a 2H TMDC monolayer lacks it, the bilayer has it, and sliding one layer over another removes it again.

Everyday

Ising superconductivity

A kind of superconductivity in some single-layer materials in which electron spins are locked pointing out of the sheet. The locking lets it survive magnetic fields along the sheet far stronger than would normally destroy superconductivity.

Superconductivity in which spin–orbit coupling pins electron spins out of plane with opposite orientation in opposite valleys, protecting pairing against in-plane magnetic fields.

Theory

Janus 2D material

A single layer whose two faces are made of different atoms – selenium on top and sulfur underneath, for example – named after the two-faced Roman god. Because top and bottom differ, the sheet carries a built-in electric field from one face to the other, which a symmetric layer cannot have. That lopsidedness allows effects a symmetric sheet forbids, such as producing a voltage when it is squeezed from above.

A monolayer in which the two outer atomic planes are different elements, as in MoSSe, where one chalcogen sheet is S and the other Se. Mirror symmetry through the metal plane is broken (D₃h → C₃v for the 1H structure), giving an intrinsic out-of-plane dipole, out-of-plane piezoelectricity, Rashba-type spin splitting and an out-of-plane second-harmonic response. Most are made by replacing one face of a parent monolayer; RhSeCl is a rare bulk crystal whose layers are Janus by nature.

Everyday

Josephson effect

A current that flows between two superconductors through a thin barrier with no voltage at all, carried by pairs of electrons tunnelling across – slipping through a barrier that everyday physics says they cannot pass. How large that current is, and how it reacts to a magnetic field, is one of the sharpest ways to find out what kind of superconductor you are holding.

Phase-coherent tunnelling of Cooper pairs across a weak link, giving a supercurrent I = I_c·sin Δφ and, under a voltage, an oscillation at 2eV/h. In van der Waals stacks the weak link can be a graphene channel, a semiconducting monolayer or a twisted interface, and the current–phase relation, the Fraunhofer pattern and the temperature dependence of the critical current report on ballistic transport, edge modes and pairing symmetry.

Everyday

Kagome lattice

A pattern of corner-sharing triangles, named after a Japanese basket weave. Electrons travelling across it interfere in a way that leaves some of them almost unable to move, which is why kagome metals so often show charge order, magnetism and superconductivity in the same crystal.

A net of corner-sharing triangles whose tight-binding spectrum contains a flat band, Dirac points at the zone corners and van Hove singularities at the zone boundary. Filling near those features favours charge order, unconventional pairing and, once spin–orbit coupling is included, topological gaps – the combination seen in the layered AV₃Sb₅ family.

Everyday

Knock-on damage

Damage done by the beam of an electron microscope when fast electrons knock atoms out of the sample. Imaging with lower-energy electrons avoids much of it.

Atom displacement caused by momentum transfer from high-energy electrons – for sulfur in TMDCs it becomes severe at high accelerating voltages.

Experiment

Layer transfer

Moving an atomically thin layer from where it was made to where it is needed – from the copper foil it grew on to a silicon wafer, or from a flake on tape onto another flake. The layer is too thin to pick up on its own, so it is carried on a polymer film or a soft stamp, put down, and the carrier removed. Each step risks tears, wrinkles, trapped bubbles and a film of leftover polymer, which is why transfer is often the dirtiest step in making a 2D device.

The step that moves a 2D layer from its growth substrate or exfoliation source to a target substrate or stack. Wet transfer coats the layer with a supporting polymer, usually PMMA, detaches it by etching the growth substrate or by electrochemical bubbling, and floats it onto the target before the polymer is dissolved. Dry transfer picks layers up with a viscoelastic PDMS stamp or a polymer film such as polycarbonate, often using the van der Waals adhesion of a top flake, and releases them by heating. The costs are cracks and wrinkles, trapped water, hydrocarbons and polymer residue, strain and doping, and – at wafer scale – yield that compounds with every transfer, which is why growth directly on the target and transfer-free routes are pursued.

ExperimentEngineering

Linear and circular dichroism

When a material absorbs light differently depending on how the light is polarised. In linear dichroism the difference is between light vibrating along one direction of the crystal and across it, as in a polarising filter; in circular dichroism, between light whose vibration turns one way and the other. Shining polarised light on a flake this way reveals hidden directions in it – its crystal axes, its magnetic order, which valley its electrons sit in – without touching it.

Polarisation-dependent absorption: linear dichroism between orthogonal linear polarisations, from in-plane anisotropy of the lattice or of an order that breaks rotational symmetry; circular dichroism between left and right circular polarisations, from broken mirror or time-reversal symmetry. In black phosphorus, ReS₂ and the transition metal trichalcogenides linear dichroism follows the crystal axes; in FePS₃ it appears with zigzag antiferromagnetic order and tracks it with temperature. In monolayer TMDCs valley-selective circular dichroism at K and K′ underlies optical valley polarisation, and magnetic circular dichroism – reflective in the visible, XMCD at X-ray absorption edges – measures magnetisation, with sum rules separating spin and orbital moments.

Experiment

Magic angle

Stack two sheets of graphene and turn one very slightly. At about 1.1 degrees the pattern they make together slows the electrons almost to a standstill, so they start acting collectively – and the pair can even become a superconductor.

The twist angle (about 1.1° for bilayer graphene) at which interlayer tunnelling makes the lowest moiré bands nearly flat.

Theory

Magnetic anisotropy energy

The extra energy needed to point a material’s magnetisation in a direction it does not prefer. In 2D magnets that preference is what keeps the magnetic order from falling apart.

The energy difference between magnetisation directions, arising from spin–orbit coupling and dipolar effects; it opens the spin-wave gap that allows 2D magnetic order.

Theory

Magnetometry

Measuring how magnetic a sample is. For a crystal you can hold, this is routine: it is moved through a coil or past a superconducting sensor called a SQUID, which picks up its tiny magnetic field. A flake one atom thick is a different matter – its magnetism is far too weak for such instruments and is swamped by the substrate it sits on – so 2D magnets are usually studied with light, with currents, or with single-spin sensors in diamond that hover nanometres above the flake.

Measurement of magnetic moment or stray field. SQUID and vibrating-sample magnetometry of bulk crystals, with sensitivities around 10⁻⁸ emu (10⁻¹¹ A·m²), give magnetisation against field and temperature, from which ordering temperatures, saturation moments and anisotropy follow; torque and Hall magnetometry and magnetic force microscopy extend this to small samples. Monolayer flakes lie far below SQUID sensitivity and are swamped by substrate diamagnetism and magnetic contamination, so their magnetism is read optically (MOKE, RMCD), electrically (anomalous Hall effect, tunnelling magnetoresistance) or with scanning nitrogen-vacancy magnetometry, which images stray fields quantitatively at the nanoscale. Claims of room-temperature ferromagnetism in thin films based on bulk magnetometry, as for monolayer VSe₂, need element-specific (XMCD) or local confirmation.

Experiment

Magneto-optical effect (MOKE)

Also: Kerr rotation

Light reflected from a magnet comes back with its polarisation – the direction its waves vibrate in, which polarised sunglasses filter – slightly turned, by an amount that follows the magnetisation. Because it needs nothing but a focused laser beam, it can read the magnetic state of a flake far narrower than a hair without touching it.

Rotation and ellipticity of reflected polarised light in a magnetised medium (the Kerr effect), and the differential absorption of circular polarisations (RMCD). Sensitivity down to a single layer is what established magnetic order in CrI₃ and Cr₂Ge₂Te₆, and both give layer-resolved hysteresis loops, domain images and the sign of interlayer coupling without any contacts.

Experiment

Magnetoresistance

A change in electrical resistance when a magnetic field is applied. It is how the read head of a hard disk senses its bits, and in some layered magnets the change is enormous – a stack can go from barely conducting to conducting – which is how the magnetic state of a few atomic layers is read out electrically.

The field dependence of resistivity. Ordinary orbital magnetoresistance is positive and usually small, though enormous in compensated semimetals such as WTe₂; giant and tunnelling magnetoresistance come from spin-dependent transport across magnetic layers; colossal magnetoresistance from a field-driven change of electronic state. In van der Waals stacks a few layers of CrI₃ act as a spin filter whose resistance changes by orders of magnitude as adjacent layers align, which is how layer-by-layer magnetic order became measurable in transport.

Everyday

Magnon

Also: spin wave

A ripple of tilted spins – the atoms’ tiny magnets – travelling through a magnet, like a stadium wave in which each spectator only stands up and sits down yet the wave runs all the way round. Because the ripple carries information without moving any electrons, it is one route to lower-power devices.

A quantised spin-wave excitation of a magnetically ordered state. Its dispersion is set by exchange, anisotropy and dipolar terms, and in a 2D magnet the anisotropy gap at zero wavevector is what holds order against thermal fluctuations – so the magnon spectrum measures the same anisotropy that lets the material evade the Mermin–Wagner theorem.

TheoryExperiment

Majorana mode

Also: Majorana zero mode, Majorana bound state

A state that can appear at the ends of certain superconducting wires, or in the whirlpools of a superconductor, and that is its own antimatter twin. Two of them together make up one ordinary electron state split in half and kept in two separate places, so no disturbance at either place alone can read or destroy what it holds – which is why they are pursued for quantum computing.

A zero-energy state equal to its own conjugate, appearing at defects of a topological superconductor – wire ends, vortex cores, domain walls. A pair defines one non-local fermionic state, so exchanging them realises non-Abelian statistics and topologically protected operations. Recipes combine strong spin–orbit coupling, magnetism and superconductivity, which is what makes van der Waals stacks attractive; proximity-induced pairing on a topological surface is the canonical route, and a zero-bias conductance peak on its own is not proof.

Theory

Memristor and neuromorphic computing

A memristor is a device whose resistance remembers the voltage it has seen, so the same spot both stores a number and multiplies by it. Chips built from arrays of them work more like a brain than like a processor with memory bolted on the side – which is what neuromorphic means.

Non-volatile resistive switching used as memory and as an analogue synaptic weight, so that a crossbar performs multiply–accumulate where the data already sits. Two-dimensional materials contribute switching layers under a nanometre thick – vacancy or metal-atom migration in a monolayer TMDC gives non-volatile switching – along with gate-tunable synaptic transistors and back-end-compatible integration above logic.

Everyday

Mermin–Wagner theorem

A result showing that in a strictly two-dimensional material, heat at any temperature above absolute zero destroys the long-range order of atomic magnets that are free to point in any direction. Real 2D magnets exist because their atomic magnets prefer a particular direction, which breaks the theorem’s assumptions.

No spontaneous breaking of a continuous symmetry at finite temperature in 2D with short-range interactions; magnetic anisotropy evades it.

Theory

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.

Everyday

Metal-induced gap states

Also: MIGS

Electron states that leak from a metal contact into the semiconductor it touches. They make the contact behave the same whichever metal is used, which is a big reason why contacts to 2D semiconductors are hard to improve.

Evanescent metal wavefunctions penetrating a semiconductor’s gap at a contact; a major cause of Fermi-level pinning.

Theory

Mirror symmetry

A crystal has a mirror symmetry if reflecting it in a plane leaves it unchanged. For a 2D sheet the question that matters most is whether it looks the same from above and from below – which decides how its electrons’ spins behave, which band crossings are protected, and even how a calculation of it has to be set up.

Invariance under reflection through a plane. For a layer the horizontal mirror in the plane of the sheet matters most: it forbids an out-of-plane dipole, so a slab without it needs a dipole correction in a periodic calculation, and in a 2H monolayer that also lacks inversion it keeps spin–orbit-split spins pointing out of plane, the origin of Ising pairing. Vertical mirrors can protect band crossings – mirror-protected nodal lines in PbTaSe₂, mirror Chern numbers in the SnTe class of topological crystalline insulators – and losing a mirror at a structural transition is how some chain compounds become ferroelectric.

Everyday

Mirror twin boundary

A line where two regions of a crystal meet as mirror images of each other. In some 2D films these lines conduct electricity even where the rest of the film does not.

A line defect in TMDCs where two domains meet as mirror images, common in MBE-grown films and often metallic.

Experiment

Moiré superlattice

Also: moiré pattern

The larger pattern that appears when two lattices are laid on top of each other slightly rotated, or with slightly different spacings – like two fine mesh curtains overlapping. Electrons feel this larger pattern, and it can reshape how they behave.

The long-wavelength interference pattern formed by two lattices with a small twist or lattice mismatch; its period sets a new, much larger unit cell for electrons.

Theory

Molecular beam epitaxy (MBE)

Growth in ultrahigh vacuum by aiming beams of atoms at a heated surface, one element at a time and slowly enough that they settle into place layer by layer. It is the slowest and cleanest way to build a film, and the surface can be watched while it grows.

Ultrahigh-vacuum growth from effusion or cracker sources with in-situ RHEED monitoring, usually at chalcogen-rich flux ratios for TMDCs. It gives coverage over a whole wafer on weakly interacting substrates, atomically abrupt interfaces and access to metastable phases and doping profiles that CVD cannot reach; domains stay small and mirror twin boundaries are common, so grain structure belongs in any report of an MBE film.

Experiment

Molecular dynamics and machine-learned potentials

Molecular dynamics is a computer simulation that follows every atom in a material through time, like the frames of a film: work out the force on each atom, nudge it a tiny step, repeat millions of times. It shows how a crystal melts, cracks or carries heat. The hard part is the forces. Exact quantum calculations are slow, so machine-learned potentials are trained on a few thousand of them and then reproduce the forces far faster.

Molecular dynamics integrates Newton’s equations for all atoms with a time step of about a femtosecond, giving trajectories from which structure, diffusion, thermal conductivity and phase behaviour follow. Forces come from empirical force fields, from DFT at every step (ab initio MD, limited to hundreds of atoms and picoseconds), or from machine-learned interatomic potentials – neural-network, kernel or equivariant graph models fitted to DFT energies and forces – that approach DFT accuracy at a small fraction of the cost within the configurations they were trained on.

Theory

Monolayer

Also: single layer

One single layer of a layered material. Two stacked layers are a bilayer; a handful are called few-layer. Many properties change between one, two and several layers, so the exact count matters.

A single structural layer of a layered crystal – one atom thick for graphene and hBN, three atomic planes for a TMDC such as MoS₂. Band structure, screening and symmetry depend on layer number (monolayer 2H-MoS₂ lacks the inversion symmetry of the bilayer), so monolayer, bilayer and few-layer samples are distinct systems.

Everyday

Monolithic 3D integration

Building a second layer of electronic devices directly on top of a finished chip, instead of making two chips separately and joining them.

Building additional active device layers directly on top of a finished circuit rather than bonding separate chips.

Engineering

Monte Carlo simulation

A way to predict how a material behaves at a given temperature by proposing random changes over and over and accepting them with a probability set by their energy – a game of chance, hence the casino’s name. Run long enough, the averages match what an experiment at that temperature would measure.

Stochastic sampling of a thermodynamic ensemble, most often Metropolis sampling of a spin Hamiltonian. In 2D magnetism it is the standard route from exchange and anisotropy parameters – usually computed with DFT – to a Curie or Néel temperature, and it is where Mermin–Wagner physics becomes concrete: without anisotropy the simulated order melts away as the simulated system grows.

Theory

Multiferroic

A material that is magnetic and electrically polarised at the same time, so a magnetic field can shift its charge and an electric field can turn its magnetism. The combination is rare, because the two kinds of order usually want incompatible atoms.

A material carrying more than one ferroic order – usually ferroelectric together with (anti)ferromagnetic – plus magnetoelectric coupling between them. Type-I multiferroics have independent origins and weak coupling; in type-II the polarisation is produced by the magnetic order itself, typically a spiral, and the coupling is strong. NiI₂ has been reported to keep a type-II order, with spiral magnetism and polarisation appearing together, down to the single layer – although the optical evidence is disputed.

Everyday

Multi-project wafer (MPW) run

A shared production run in which designs from several companies or labs are made on the same wafers, so each pays only a fraction of the cost.

A shared fabrication run in which several customers’ designs are processed on the same wafers to split cost.

Engineering

MXene

A large family of 2D carbides and nitrides – thin sheets of metal atoms bonded to carbon or nitrogen. They are made by chemically etching one kind of atom out of a bulk crystal, and they conduct electricity well, which makes them useful for batteries, coatings and electromagnetic shielding.

Two-dimensional transition-metal carbides, nitrides and carbonitrides M_n+1X_nT_x, obtained by selectively etching the A-element layers from MAX phases, commonly in fluoride-containing acids. Surface terminations T_x (–O, –OH, –F, –Cl) set their properties; Ti₃C₂T_x is the most studied, with metallic conductivity and hydrophilic surfaces that allow water-based processing.

Everyday

Nanosheet

Also: flake

A general word for a very thin sheet of material, often just a few atoms thick and up to a few micrometres – thousandths of a millimetre – across. It is used especially for flakes made in large quantities in liquids.

A particle whose lateral dimensions far exceed its nanometre-scale thickness; the term is used especially for liquid-exfoliated and chemically derived dispersions, whose lateral-size and thickness distributions must be specified rather than assumed. In transistor engineering ‘nanosheet’ also denotes the stacked silicon channel sheets of gate-all-around devices.

Everyday

Nematic order

Also: electronic nematicity

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.

Theory

Neutron scattering

Shooting neutrons at a crystal and watching how they bounce off. Neutrons carry no charge, so they pass deep into a sample, but they carry a tiny magnetic moment, so they feel the magnetic moments of atoms. That makes them the standard tool for finding how the spins in a magnet are arranged and – from the energy the neutrons lose – how the spins wave and wobble. The catch is that neutrons come only from research reactors and large accelerators, and the samples have to be large crystals.

Elastic and inelastic scattering of thermal and cold neutrons from nuclei and, through the neutron’s magnetic moment, from unpaired electrons. Diffraction gives the nuclear structure, with sensitivity to light elements and isotopes, and the magnetic structure from magnetic Bragg peaks below the ordering temperature (Shull, 1949); polarised neutrons separate magnetic from nuclear scattering. Inelastic scattering on triple-axis and time-of-flight spectrometers maps phonon and magnon dispersions, exchange constants and continua – the magnon gaps of CrI₃, the scattering continuum of α-RuCl₃. Weak fluxes require gram-scale samples or many co-aligned crystals, so 2D materials are measured in bulk form; monolayers are out of reach, and bulk results are transferred to thin layers with care.

Experiment

On/off ratio

How much more current a transistor lets through when it is switched on than when it is switched off. A switch for digital logic needs at least ten thousand, and far more in chips that spend most of their time idle; graphene, which has no band gap, manages only about ten, which is why it cannot replace silicon in logic.

The ratio of the drain current with the transistor switched on to the current with it switched off, at a stated supply voltage. Below threshold the current falls at best tenfold per 60 mV of gate voltage at room temperature, so a ratio of 10⁶ needs at least 0.36 V of swing, and the off current is floored by leakage over the gap, band-to-band and source-to-drain tunnelling, and gate leakage. Monolayer MoS₂ transistors reached about 10⁸; gapless graphene stays near 10 at room temperature. Benchmarks compare the on current at a fixed off current and supply voltage rather than the ratio alone.

Engineering

Optical contrast

How much a thin flake stands out from the surface it sits on under a microscope. On the right base – usually silicon coated with a thin layer of glassy oxide – even a single layer of atoms becomes visible through an ordinary optical microscope, thanks to the same effect that paints colours on a soap bubble.

The difference in reflected intensity between a flake and its substrate, enhanced by thin-film interference in the oxide layer.

Experiment

Passivation

Treating a surface so that it stops reacting with its surroundings and stops trapping electrons – by covering it with a thin protective layer, or by attaching atoms that tie up its loose bonds. Aluminium does it by itself: a skin of oxide a few nanometres thick is why aluminium window frames do not corrode away. A 2D material is all surface, so passivating it can decide whether it glows brightly, conducts well or survives in air at all.

The chemical or physical deactivation of a surface or interface: saturating dangling bonds and trap states (hydrogen on silicon, sulfur on III–V surfaces), compensating defects (superacid or thiol treatment of TMDC vacancies), or sealing with a dielectric such as Al₂O₃ or hBN. In 2D semiconductors it raises photoluminescence yield, lowers interface trap density and hysteresis, and slows oxidation; in perovskite solar cells a thin 2D perovskite layer passivates the surface of the 3D absorber.

Engineering

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.

Everyday

Phase-change material (PCM)

A material that can be switched between a disordered and an ordered arrangement of its atoms by a pulse of heat or light, and then stays that way. Because the two states reflect light and carry current very differently, the switch stores a bit – which is how rewritable DVDs and Blu-ray discs hold their data.

A compound cycled between amorphous and crystalline states by melt-quench and crystallisation pulses, giving a large non-volatile contrast in resistivity and in refractive index. Ge–Sb–Te alloys are the established family for memory; layered antimony chalcogenides extend the same trick to programmable photonics, where the figure of merit is index change against optical loss rather than a resistance ratio.

Everyday

Phonon

A packet of vibration travelling through a crystal – the way its atoms jiggle together. Sound travelling through a solid is made of phonons. They also carry heat, slow electrons down by jostling them, and are what Raman spectroscopy measures.

A quantised collective lattice vibration. 2D materials host flexural (out-of-plane) acoustic modes with quadratic dispersion; electron–phonon scattering sets intrinsic mobility limits, and phonon frequencies measured by Raman spectroscopy report layer number, strain and doping.

Everyday

Photocatalysis and electrocatalysis

Using light, or an applied voltage, to drive a chemical reaction at a surface – most often splitting water to make hydrogen. Thin materials are natural candidates because nearly every atom is a surface atom, and because their edges are usually far more active than their flat faces.

Surface reactions driven by photons (photocatalysis) or by electrode potential (electrocatalysis), judged by overpotential, Tafel slope and turnover per site rather than by raw current. In MoS₂ the basal plane is close to inert while the edge sites carry the hydrogen-evolution activity – the result that made layered chalcogenides catalyst candidates – so activity scales with edge length, defect density and phase, and any comparison needs an active-site count rather than a geometric area.

Everyday

Photodetector

A device that turns light into an electrical signal. A single layer absorbs only a few percent of the light that hits it, but it responds quickly, can be tuned by a gate voltage and can be placed on surfaces an ordinary detector cannot reach, such as a flexible sheet or an optical chip.

A device converting incident photons into current or voltage. A 2D channel trades absorption per pass for gate tunability, short transit times and transfer onto arbitrary substrates or waveguides; the response may be photoconductive with gain, photovoltaic at a junction, or bolometric. Responsivity alone says little – it is only comparable alongside the mechanism, the bandwidth and the illuminated area.

Everyday

Photoluminescence (PL)

Light given off by a material after it has absorbed light – what a highlighter pen does under a black light. In 2D semiconductors the colour and brightness of that glow report how many layers there are, how clean the sample is and whether it is carrying extra charge.

Radiative recombination following optical excitation. In monolayer TMDCs the spectrum is dominated by excitons and trions rather than free carriers, so peak energy, linewidth and quantum yield track layer number, strain, dielectric environment, doping and defect density. The indirect-to-direct crossover makes the monolayer far brighter than the bilayer, and chemical treatment of defects can bring the yield close to unity.

Experiment

Photon

Also: light quantum

The smallest possible amount of light – one indivisible packet of it. The packets of a given colour all carry the same energy, and blue ones carry more than red: that is why ultraviolet light gives you sunburn and red light, however bright, does not. A semiconductor absorbs only photons with enough energy to cross its band gap and gives light back at about the gap’s energy – which is why LEDs of different colours are made from different semiconductors.

The quantum of the electromagnetic field, with energy E = hν and momentum h/λ but no mass or charge. Visible photons carry about 1.7–3.1 eV, comparable to the band gaps of semiconducting TMDCs, but a momentum far smaller than the crystal momenta across the Brillouin zone, so optical transitions are vertical and an indirect gap needs a phonon to make up the difference. Single-photon emission is established by antibunching in photon-correlation measurements.

Everyday

Photovoltaic effect

Light absorbed in a material frees electrons, and a built-in asymmetry – usually a junction between two different layers – pushes them one way rather than the other, producing a voltage. That is how a solar cell works.

Separation of photogenerated carriers by a built-in field at a p–n junction, a Schottky contact or a type-II van der Waals interface. Monolayers absorb roughly an order of magnitude more sunlight per unit thickness than GaAs or Si, so a stack a few nanometres thick reaches useful power per unit mass even though its absolute absorption stays small; in a gate-tunable heterojunction the alignment, and with it the sign and size of the photoresponse, can be switched electrically.

Everyday

Piezoelectricity

Squeeze the crystal and it produces a voltage; apply a voltage and it changes shape. It makes the spark in a push-button lighter and keeps time in a quartz watch. It only works in crystals with no centre of symmetry, which is why a single layer of MoS₂ is piezoelectric and two stacked layers are not.

Linear coupling between strain and polarisation, allowed only in non-centrosymmetric crystals. Odd-layer 2H TMDCs lack inversion symmetry and are piezoelectric while even-layer stacks are not, so the response alternates with layer number and disappears in the bulk. Measurements on monolayer MoS₂ agree with the first-principles coefficients, and the same coupling gives piezotronic control of a contact by strain.

Everyday

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.

TheoryExperiment

Polariton

A wave that is part light and part motion in a material – electrons or atoms swinging back and forth – travelling as one. In thin crystals such waves can be squeezed to wavelengths many times shorter than the light that made them, so they can guide and focus infrared light into spaces far smaller than light normally allows.

A hybrid mode of photons with a polarisation excitation: free-carrier plasma oscillations (plasmon polaritons), optical phonons (phonon polaritons) or excitons (exciton polaritons). In van der Waals crystals they confine light to wavelengths tens to hundreds of times below that in free space. hBN supports hyperbolic phonon polaritons in its two Reststrahlen bands, α-MoO₃ in-plane anisotropic ones and graphene gate-tunable plasmons; TMDC monolayers in optical cavities form exciton polaritons. They are imaged in real space by scattering-type near-field microscopy (s-SNOM), which launches them from a sharp tip and maps their interference fringes.

Experiment

Polaron

An electron that drags a dent in the crystal along with it, like a ball rolling across a mattress. It pushes the nearby atoms slightly aside, and electron and distortion then travel together as one heavier, slower object.

A quasiparticle made of a carrier dressed in the lattice distortion it induces. Coupling strength sets its character: a large Fröhlich polaron stays delocalised with a renormalised mass, a small polaron self-traps on one site and moves by hopping. In 2D the weaker dielectric screening and the missing third dimension change the binding, and first-principles calculations now resolve polaron structures in monolayers that a bulk formula misses.

Theory

Polytype

One of several ways the same layered compound can arrange or stack its atoms. The chemistry is identical, yet one arrangement may be a semiconductor and another a metal.

One of several stacking or coordination variants of the same layered compound, e.g. 1H/2H, 3R, 1T and 1T′ TMDCs.

Experiment

Pulsed laser deposition (PLD)

Growth by firing short, intense laser pulses at a target so that a plume of material blows off it and settles on a nearby substrate. Its main attraction is that a complicated composition transfers to the film almost unchanged.

Ablation of a target by nanosecond pulses, with the plume condensing on a heated substrate. It transfers multi-element stoichiometry congruently, tolerates reactive background gases and reaches metastable phases, at the cost of droplets, a narrow useful plume and an energetic flux that can damage a van der Waals surface; for chalcogenides the chalcogen still has to be replenished during growth.

Experiment

Pump–probe spectroscopy

Also: ultrafast spectroscopy, time-resolved spectroscopy

A way to film events that last a millionth of a millionth of a second. A first, strong flash of laser light – the pump – disturbs the sample; a second, weaker flash – the probe – arrives a precisely set moment later and measures how the sample looks then. Repeating this with the delay changed step by step builds up a slow-motion film of how the electrons and atoms settle back. The delay is set by sending one flash along a longer path: a tenth of a millimetre more makes it a third of a picosecond later.

A stroboscopic measurement in which an ultrashort pump pulse excites the sample and a delayed probe pulse records the change, as a function of a delay set by an optical delay line, with a time resolution limited by the pulse durations – tens of femtoseconds routinely, a few with dedicated sources. The probe can be optical reflectivity or transmission, terahertz conductivity, photoemission (time-resolved ARPES), or X-ray or electron diffraction, giving access to carrier thermalisation and cooling, exciton formation, interlayer charge transfer, coherent phonons, and the melting, recovery or switching of ordered phases such as charge density waves and excitonic insulators. Its power lies in separating processes by timescale: electrons respond within femtoseconds, the lattice within picoseconds.

Experiment

Quantum emitter

Also: single-photon emitter

A single point in a material that gives out light one photon – one indivisible packet of light – at a time, and never two at once: a building block for quantum communication and sensing. A 2D host is attractive because the emitter sits right at the surface, so almost all of its light gets out.

A localised optical transition emitting antibunched light, identified by g⁽²⁾(0) well below 0.5. In monolayer TMDCs the emitters are excitons trapped at strain wells or defects; hBN hosts bright emitters that work at room temperature and are attributed to defect centres. Being one layer deep helps extraction and makes the emitter responsive to strain, field and cavity coupling, at the cost of spectral diffusion.

Everyday

Quantum Hall effect

In a strong magnetic field at low temperature, electrons flowing along a flat sheet are pushed to one side, so a voltage builds up across it. That sideways voltage, divided by the current, stops changing smoothly and locks onto exact steps set only by fundamental constants – the same in every sample, and so precise that they are used to define the ohm.

Quantisation of the Hall conductance in units of e²/h when a 2D electron system sits in a strong perpendicular field and the Fermi level lies between Landau levels, with the longitudinal resistance vanishing. Graphene shows a half-integer sequence with a level pinned at zero energy, a direct consequence of its Dirac dispersion and π Berry phase, and the effect survives to room temperature there. The zero-field analogues are the quantum anomalous and quantum spin Hall effects.

Everyday

Quantum metric

Every electron in a crystal is a wave whose shape depends on how the electron is moving. The quantum metric measures how much that shape changes when the motion changes a little. It sounds abstract, but it decides, for example, whether electrons in a flat band – which barely move on their own – can still flow together as a superconductor.

The real part of the quantum geometric tensor, measuring how Bloch states change across the Brillouin zone; it bounds superfluid stiffness and affects fractional-state stability.

Theory

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.

Theory

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.

Everyday

Quasiparticle

An electron moving through a crystal is never alone: it pushes and pulls on all the other electrons and atoms around it, and drags a little cloud of disturbance along. Physicists treat the electron together with its cloud as a new particle – a quasiparticle – with its own mass and lifetime. The idea goes further: holes, excitons, vibrations of the lattice and waves of spins also behave like particles, and describing a solid as a gas of such quasiparticles is how most of its properties are understood.

An elementary excitation of an interacting many-body system that behaves like a particle with a defined energy–momentum relation, charge or spin, and a finite lifetime, as in Landau’s Fermi-liquid theory. Dressed electrons and holes (renormalised by screening, phonons or correlations, with effective masses that can reach hundreds of free-electron masses in heavy-fermion systems), excitons, trions and polarons, and bosonic collective modes such as phonons, magnons, plasmons and polaritons are all quasiparticles. Their energies are poles of the Green’s function; the GW approximation computes quasiparticle band structures, which in 2D semiconductors are strongly renormalised by the reduced screening, so that the quasiparticle gap exceeds the optical gap by the large exciton binding energy.

Theory

Quasiparticle gap versus optical gap

Two answers to ‘how much energy does it take to excite this material?’. Adding a free electron and a free hole costs more; creating a bound electron–hole pair with light costs less, and in 2D materials the difference is large.

The quasiparticle gap is the energy to add an electron and a hole independently; the optical gap is smaller by the exciton binding energy.

Theory

Raman spectroscopy

Shining a laser on a material and measuring the tiny fraction of light that comes back at slightly different colours. The shifts reveal how the atoms vibrate, which tells researchers how many layers a flake has and whether it is strained or damaged.

Inelastic light scattering that measures phonon energies. In 2D materials peak positions, widths and intensity ratios – graphene’s G and 2D bands, or the E′ and A′₁ modes of TMDCs – report layer number, strain, doping and defect density; laser heating and substrate interference must be controlled.

Everyday

Rashba effect

When a crystal is not the same seen from above and from below – because it sits on a substrate, has a field across it, or is simply built that way – a moving electron feels a magnetic field that depends on which way it is going. Its spin then winds around its direction of motion, which is what lets an electric field steer spins.

A momentum-dependent spin splitting from spin–orbit coupling in a structure without inversion symmetry, H = α(σ × k)·ẑ, locking spin perpendicular to in-plane momentum and winding it around the Fermi contour. The coefficient α follows the potential gradient and the atomic spin–orbit strength, so it is tunable by gating, by the substrate and by the choice of heavy elements; BiTeI splits its bands far enough for the two spin-split Fermi surfaces to be resolved separately.

Theory

REACH nanoforms

European rules requiring companies to register chemicals – including nano-sized forms such as many 2D-material powders – with information on their form and safety before making or importing them in quantity in the EU.

The EU chemicals regulation’s requirement to register and characterise nanoform variants of substances, including many 2D-material powders.

Engineering

Residual resistivity ratio (RRR)

A quick score for how clean a metal crystal is: its electrical resistance at room temperature divided by its resistance when cooled close to absolute zero. Cooling stops the atoms jiggling, so the resistance that remains comes from impurities and defects. A pure crystal loses almost all of its resistance and scores in the hundreds or thousands; a dirty one keeps most of it and scores near one.

The ratio R(300 K)/R(T → 0) of a metallic sample, where the low-temperature value is the residual resistivity from static disorder once phonon scattering has frozen out (Matthiessen’s rule). It is a standard, geometry-independent proxy for crystal quality and mean free path, from order one in disordered films to 10³–10⁵ in the purest metals. Its low-temperature reference is ambiguous in superconductors, where the normal-state value just above T_c is used, and in materials whose resistivity turns up at low temperature.

Experiment

Resistivity and sheet resistance

Two ways to put a number on how strongly a material resists a current. Resistivity belongs to the material itself, whatever the size of the piece: copper’s is tiny, glass’s enormous. For a sheet so thin that its thickness is hard to pin down, the sheet resistance is used instead: the resistance of a square of it, measured from one edge to the opposite one, which comes out the same for a square of any size. It is quoted in ohms per square.

Resistivity ρ relates electric field to current density, in Ω·m or Ω·cm, so that a bar of length L and cross-section A has resistance ρL/A. For a film of thickness t the sheet resistance is ρ/t, quoted in Ω per square, and a strip L long and W wide has the sheet resistance times L/W. For an atomically thin layer it is the natural quantity, since the thickness is a convention, and it equals 1/neμ for sheet density n and mobility μ. Both are measured with four probes – a Hall bar, a collinear four-point probe or the van der Pauw method for arbitrary shapes – so that the voltage is sensed away from the current contacts and the contact resistance drops out.

Experiment

Rytova–Keldysh potential

The formula for how strongly an electron and a hole attract each other inside a very thin sheet. Much of the electric field between them passes through the air or material outside the sheet rather than through the sheet itself, so the pull changes with distance in an unusual way – and the pair’s energy levels do not form the neat ladder of a hydrogen atom.

The effective electron–hole interaction in a thin dielectric sheet: logarithmic at short range, Coulomb-like at long range, which makes 2D exciton series non-hydrogenic.

Theory

Salt-assisted CVD

Adding ordinary salts to the gas-phase growth of 2D films. The salt helps the metal ingredients evaporate at lower temperatures, which gives larger, more even crystals.

CVD in which alkali halides form volatile oxyhalides with metal oxides, lowering growth temperatures and enlarging domains.

Experiment

Scanning tunnelling microscopy (STM)

Also: scanning tunnelling spectroscopy

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.

Experiment

Second-harmonic generation (SHG)

Shine intense light on a crystal and a little of it comes back at exactly twice the frequency – half the wavelength, which is how a green laser pointer turns invisible infrared into green light. It happens only if the crystal has no centre of symmetry, which makes it a quick, contactless test of symmetry, layer number and crystal orientation.

A second-order nonlinear response, forbidden in centrosymmetric media, so in 2H TMDCs it appears for odd layer numbers and vanishes for even ones. The polarisation dependence gives crystal orientation, and so the twist angle in a stack; the intensity follows layer number and strain; and it detects broken inversion symmetry in ferroelectric and magnetic layers where no other quick probe exists.

Experiment

Self-flux growth

Growing crystals from a molten bath made mostly of one of the crystal’s own ingredients – extra selenium, for example – much as sugar crystals grow from a cooling syrup. Because the bath is part of the crystal already, no foreign solvent atoms get trapped inside.

Crystal growth from a melt rich in one of the crystal’s own constituents, such as excess selenium for WSe₂.

Experiment

Semiconductor

A material whose ability to carry electricity can be switched on and off, for example by an applied voltage. That switching is what every transistor in a computer chip relies on. Silicon is the classic example; several 2D materials, such as MoS₂, are semiconductors too.

A material with a band gap of a few electronvolts or less, whose conductivity can be tuned over many orders of magnitude by doping, gating, temperature or light. 2D semiconductors attract interest because their sub-nanometre thickness preserves electrostatic gate control in very short channels.

Everyday

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.

Everyday

Shielding effectiveness (dB)

How much a protective layer weakens electromagnetic signals passing through it, on a logarithmic scale: 20 dB blocks 99 % of the power, and 60 dB lets through only a millionth.

The attenuation of electromagnetic radiation by a shield; 20 dB blocks 99% of power, 60 dB blocks 99.9999%.

Engineering

Silicon photonics

Also: integrated photonics

Guiding light instead of electricity through tiny channels on a silicon chip, made with the same factories and tools as ordinary computer chips. Light carries data faster and with less loss over distance, so such chips already connect the servers in data centres. Silicon itself is poor at a few jobs – it hardly emits light, cannot detect the infrared light used in fibre networks, and switches light only weakly – and 2D materials laid on top of the light channels are one way to add the missing functions.

Photonic integrated circuits in silicon-on-insulator, in which submicron silicon waveguides, with their high index contrast to the buried oxide, route light at telecom wavelengths through splitters, rings and gratings made with CMOS tools; germanium detectors and silicon carrier-depletion modulators are standard, and lasers are bonded or grown from III–V semiconductors. 2D materials couple to the evanescent field of the waveguide: graphene gives broadband, fast modulators and photodetectors that can be integrated at wafer scale, thin MoTe₂ and black phosphorus emit and detect in the near infrared, and low-loss phase-change Sb₂S₃ and Sb₂Se₃ give non-volatile switches and tuning. The obstacles are wafer-scale transfer or growth, contacts, optical loss and reproducibility to foundry standards.

Engineering

Single crystal

A piece of material in which the atoms are lined up in one unbroken pattern from one side to the other, with every row pointing the same way. Most solids are not like that: they are polycrystalline, made of many small crystals stuck together at random angles, like a pavement of tiles laid every which way. The boundaries between those grains scatter electrons, weaken the material and blur measurements, so physics experiments and the best devices want single crystals – a centimetre-sized block to peel flakes from, or a sheet grown across a whole wafer in one orientation.

A solid whose lattice is continuous and uniformly oriented throughout, without grain boundaries – unlike polycrystalline material, made of misoriented grains, or amorphous solids without long-range order. Bulk single crystals of layered compounds are grown by chemical vapour transport, flux or Bridgman methods and are the usual source of exfoliated flakes; their quality is judged by X-ray rocking-curve widths, the residual resistivity ratio and the absence of twins and stacking faults. Large single-crystal films grow either from one nucleus or from aligned nuclei that merge without boundaries, as for graphene on Cu(111) or Ge(110) and MoS₂ or hBN on vicinal sapphire or copper, where substrate steps fix one orientation and remove the twin boundaries that two antiparallel orientations would form.

Everyday

Skyrmion and the Dzyaloshinskii–Moriya interaction

Also: magnetic skyrmion

A skyrmion is a tiny whirl in the magnetisation of a material: the magnetic direction turns smoothly from pointing down at its centre to pointing up at its rim. It behaves like a particle – a small current can move it, and it is hard to destroy, because undoing the whirl would mean flipping a whole region at once – which makes skyrmions candidates for dense, low-power magnetic memory. Most are held together by the Dzyaloshinskii–Moriya interaction, a twisting force between neighbouring magnetic atoms that appears only where a mirror symmetry is missing.

A particle-like spin texture whose magnetisation wraps the unit sphere once, giving it an integer topological charge. Bloch-type skyrmions form in bulk chiral magnets (MnSi, 2009) and Néel-type ones at interfaces and in polar crystals. They are stabilised by the antisymmetric Dzyaloshinskii–Moriya interaction – proportional to the cross product of neighbouring spins, arising from spin–orbit coupling where inversion symmetry is broken – competing with exchange, anisotropy and the applied field; dipolar-stabilised skyrmion bubbles form without it. Among 2D magnets, Fe₃GeTe₂ hosts bubbles and, in WTe₂/Fe₃GeTe₂ heterostructures, interface-induced Néel skyrmions. Their emergent magnetic field can add a topological contribution to the Hall effect, and their current-driven motion by spin–orbit torques underlies racetrack-memory proposals.

Theory

Sliding ferroelectricity

A switchable electric polarisation created by sliding one atomic layer slightly across another. Sliding the layers back reverses it, so the stacking itself can store a state that can be written and erased.

Switchable polarisation produced by the relative lateral registry of van der Waals layers and reversed by sliding one layer over another.

Theory

Solid lubricant

Also: dry lubricant

A powder or coating that makes surfaces slide over each other easily without oil. Layered crystals such as graphite, MoS₂, WS₂ and hBN are natural lubricants: their layers are strongly bonded inside but only weakly stuck to each other, so they shear apart and coat the rubbing surfaces with slippery sheets. They work where oil cannot – in a vacuum, in space or at high temperatures – and are among the oldest uses of layered materials.

Lamellar solids whose weak interlayer bonding allows easy shear: graphite (which needs adsorbed water or vapour to lubricate well), MoS₂ and WS₂ (best in vacuum and dry gas, oxidising in humid air), and hBN (stable to high temperature in air). Used as burnished or sputtered films, bonded coatings and additives in oils and greases, they reach friction coefficients of a few hundredths, and in space mechanisms MoS₂ is a standard. At the nanoscale, incommensurate contact between layers can make friction nearly vanish – structural superlubricity, seen when graphite flakes are rotated out of register and in MoS₂ – while friction on atomically thin sheets rises as they get thinner because the sheet puckers ahead of the tip.

Everyday

Space group and crystal system

A short code, such as P6₃/mmc, that lists every way a crystal can be turned, reflected or shifted and still look exactly the same. The crystal system – hexagonal, tetragonal, orthorhombic, monoclinic and a few more – is the broader family the code belongs to, and it fixes the shape of the crystal’s repeating box.

The group of all symmetry operations – rotations, reflections, inversion, screw axes, glide planes and lattice translations – that map a crystal onto itself. There are 230 in three dimensions, written in Hermann–Mauguin notation: P6₃/mmc for 2H-MoS₂, Pnma for SnSe. The point group left when translations are ignored fixes the crystal system (triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal or cubic) and decides which tensor properties may be non-zero, such as second-harmonic generation, piezoelectricity or spontaneous polarisation. A single layer, periodic in two directions only, belongs to one of 80 layer groups.

Everyday

Spin

Also: electron 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.

Everyday

Spin–orbit coupling

Also: SOC

A link between an electron’s spin – a tiny built-in magnet – and the way it moves. It is strongest in heavy atoms, and in some 2D materials it is strong enough to split energy levels and make spin useful for devices.

The relativistic interaction between an electron’s spin and its orbital motion, growing steeply with atomic number. In TMDC monolayers with broken inversion symmetry it splits the valence band by roughly 150–450 meV, producing spin–valley locking, and it underlies Ising superconductivity, topological gaps and magnetic anisotropy.

Everyday

Spintronics

Electronics that uses the magnetic direction of an electron – its spin – as well as its charge. Writing information as a spin direction rather than as a pocket of charge can store it without power and switch it with less energy.

Device physics built on injecting, transporting, manipulating and detecting electron spin. Two-dimensional materials contribute long spin lifetimes and gate-tunable transport in graphene, spin–orbit coupling and exchange induced by proximity to an adjacent layer, spin–valley locking in TMDC monolayers, and van der Waals magnets that serve as tunnel barriers and free layers in an all-2D stack.

Everyday

Spin–valley locking

In certain single-layer materials an electron’s spin – its built-in magnetic direction – is tied to the valley it sits in: flip one and you flip the other. Each valley responds only to light whose waves corkscrew one way – clockwise or anticlockwise – so both can be addressed with light.

In TMDC monolayers, spin–orbit coupling and broken inversion symmetry tie the spin of band-edge states to their valley.

Theory

Stacking fault

A mistake in the order in which a crystal’s layers are stacked – one layer shifted or turned from where the pattern says it should sit, like one misaligned sheet in a neatly squared stack of paper. Layered crystals get them easily, because the weak bonds between layers barely care. But the stacking can change a material’s magnetism, symmetry or electronic behaviour, so faults can make one crystal behave unlike the next.

A planar defect in which the stacking sequence is interrupted – for example …ABCABABC… in place of …ABCABCABC… – without a change in the layers themselves. In van der Waals crystals the energy cost is small, so faults form during growth, on cooling through structural transitions, and during cleaving and handling. They produce diffuse streaks along c* in diffraction and can alter interlayer exchange, symmetry-allowed responses and transition temperatures, as in α-RuCl₃, where faulted regions order near 14 K rather than 7 K.

Experiment

Statistical process control (SPC)

Watching production measurements with statistics, so that a slow drift is spotted and corrected before it starts producing faulty parts.

Monitoring process parameters statistically to detect drift before it causes defects.

Engineering

Stoichiometry

The ratio in which the elements of a compound are combined – the numbers in its formula. MoS₂ should have two sulfur atoms for every molybdenum atom. Real crystals often deviate slightly, with a few atoms missing or a few extra squeezed in, and in 2D materials a deviation of a few percent can change how a crystal conducts, glows or behaves magnetically. That is why careful work measures the composition rather than assuming the formula.

The proportions of the elements in a compound, ideally the integer ratios of its formula. Real crystals are often non-stoichiometric, through vacancies, interstitials, antisites or intercalated atoms whose concentrations depend on the chemical potentials during growth and on the Fermi level, as point-defect thermodynamics describes. Deviations dope the material and can decide its ground state: chalcogen-deficient TMDCs are n-type, self-intercalated titanium makes TiS₂ metallic rather than semiconducting, the alkali content of AV₃Sb₅ and the europium content of EuSn₂As₂ shift their transitions, and melt-grown InSe varies along the boule. Composition is measured by X-ray photoelectron or energy-dispersive spectroscopy, Rutherford backscattering or electron-probe microanalysis, with accuracies of about a percent at best, so small deviations are often inferred from properties instead.

Experiment

Strain

Stretching or squeezing a material. Because 2D materials can be stretched much further than ordinary crystals before breaking, strain can be used as a knob to change their colour, conductivity or band gap.

Relative deformation of a lattice. 2D crystals sustain elastic strains of several percent, with graphene exceeding 10 % in nanoindentation, and strain shifts band edges, Raman modes and exciton energies. Unintended strain from substrates, bubbles and transfer is a common confounder in measurements.

Everyday

Strong electron correlation

Also: strongly correlated electrons

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.

Theory

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.

Everyday

Subthreshold swing

How sharply a transistor switches from off to on – the extra gate voltage needed to raise the current tenfold. Ordinary transistors at room temperature cannot do better than about 60 millivolts.

The gate voltage needed to change drain current tenfold below threshold; about 60 mV per decade is the room-temperature limit for conventional transistors.

Engineering

Supercapacitor

Also: electrochemical capacitor

A device that stores charge on the surface of its electrodes instead of in a chemical reaction deep inside them. It holds less energy than a battery but charges in seconds and survives far more cycles, which suits materials whose surface is nearly all of them.

Electrochemical storage by double-layer charging plus fast, surface-confined redox (pseudocapacitance). 2D electrodes suit it because stored charge follows accessible surface rather than bulk diffusion: MXene films combine metallic conductivity with redox-active surface groups and reach volumetric capacitances of order 900 F cm⁻³. The trade-offs are modest energy density and the need to stop the sheets restacking.

Everyday

Superconductivity

A state in which a material carries electric current with zero resistance, usually only when very cold. It is what lets the magnets of hospital MRI scanners carry huge currents without heating up. Some 2D materials become superconductors, and in twisted graphene the effect can be switched on and off with a voltage.

A macroscopic quantum state of paired electrons with zero DC resistance and magnetic-flux expulsion below a critical temperature. 2D examples include gate-tunable superconductivity in magic-angle graphene and Ising superconductivity in monolayer NbSe₂ and gated MoS₂; in the 2D limit the transition is of Berezinskii–Kosterlitz–Thouless type.

Everyday

Surface state

Electron states that exist only at the surface of a crystal, because the crystal stops there. Atoms at a surface have lost their neighbours on one side, so electrons can take energies there that the inside of the crystal does not allow. Usually such states are a nuisance that traps charge. In a topological insulator they are the point: the inside insulates while the surface is guaranteed to conduct, with each electron’s spin tied to its direction of motion.

States localised at a crystal surface and decaying into the bulk, at energies forbidden in the bulk for that in-plane momentum. Ordinary Tamm and Shockley states come from the broken periodicity, dangling bonds or reconstruction, and can be removed or shifted by passivation. Topological surface states are required by a non-trivial bulk invariant: a single spin–momentum-locked Dirac cone in Bi₂Se₃-family topological insulators, protected against backscattering by time-reversal symmetry, and Fermi arcs joining the projections of Weyl points in Weyl semimetals. In thin films the top and bottom surface states hybridise and open a gap below a critical thickness, about six quintuple layers for Bi₂Se₃. In a 2D material the boundary of a topological phase is its edge, and the counterpart of the surface state is the edge state.

Theory

Tear-and-stack

A way of making a twisted bilayer: tear a single flake in two, rotate one half by a chosen angle and lay it back on the other. Because both halves come from the same crystal, the twist equals the rotation that was applied.

Assembly of a twisted bilayer by picking up part of one flake, rotating it, and placing it on the remaining part.

Experiment

Technology readiness level (TRL)

A score from 1 to 9 for how close a technology is to real use: 1 means the basic idea has been observed, 9 means it is proven in products working in the field.

A 1–9 scale describing maturity from basic principles (1) to proven operational use (9).

Engineering

Temporary wafer bonding

Sticking a thin wafer or film to a sturdy carrier with an adhesive that can be removed later, so it can be handled and processed without breaking.

Attaching a wafer or film to a carrier with a removable adhesive so it can be processed and released.

Engineering

Thermal budget

How much heat, and for how long, a chip can take at a given production step before the parts already built are damaged. New materials often have to be grown or added within that limit.

The maximum temperature–time exposure a process step may impose without damaging structures already on the wafer.

Engineering

Thermal conductivity

How well a material carries heat from a hot place to a cold one. Graphene and boron nitride carry heat along their sheets better than copper, which is why they are used to spread heat away from hot spots – but across the sheets, and across every interface, heat moves far less easily.

The coefficient κ relating heat flux to temperature gradient, q = −κ∇T, carried in insulators and semiconductors almost entirely by phonons. Layered crystals are extremely anisotropic: graphite conducts about 2000 W/m·K along its layers and a few W/m·K across them; suspended graphene reaches several thousand, graphene on silicon dioxide about 600. Low κ – from heavy atoms, anharmonic bonding and interfaces, as in SnSe – is the goal for thermoelectrics, while the thermal boundary resistance at each interface often limits how fast heat leaves a 2D device.

Everyday

Thermoelectric effect

Also: Seebeck effect

Turning a temperature difference into a voltage, or a voltage into cooling. It powers space probes such as Voyager from the heat of decaying plutonium, and cools small camping fridges. Layered crystals are among the best materials for it, because heat travels through them badly while electricity still travels well.

Conversion between heat and electricity, measured by the figure of merit ZT = S²σT/κ. Low-dimensional and layered materials help by partly decoupling those terms: confinement was predicted to raise the power factor, while heavy elements, anharmonic bonding and interfaces suppress lattice thermal conductivity. SnSe reaches ZT ≈ 2.6 near 923 K along its soft in-plane direction.

Everyday

Threshold-voltage variability

How much the switch-on voltage differs from one transistor to the next. Chips with billions of transistors need them to be nearly identical, so large variation limits what can be built.

The spread of turn-on voltage across devices; it limits circuit design margins and yield.

Engineering

Tight-binding model

A stripped-down model in which electrons sit on atoms and hop to their neighbours with a fixed probability. It predicts little on its own, but once its handful of numbers is fitted it reproduces bands cheaply enough to handle millions of atoms – a twisted bilayer, for instance.

An expansion of the Hamiltonian in localised orbitals with hopping integrals between them, fitted to first-principles bands or constrained by symmetry in the Slater–Koster scheme. Its value in 2D is scale: moiré supercells of 10⁴–10⁵ atoms, disorder averaging and transport are out of reach for DFT and routine here. Three bands built from the metal d orbitals already capture the band edges and spin–orbit splitting of a group-VI TMDC monolayer.

Theory

Time-reversal symmetry

The idea that the laws governing electrons would look just as valid if you ran a film of them backwards. Reversing time flips every motion and every spin, so a material with no magnetism of its own usually keeps this symmetry, while a magnet – or a magnetic field – breaks it. Which case applies decides a lot: some protected edge currents need the symmetry, and others need it broken.

The antiunitary operation t → −t, which reverses momenta and spins; for spin-½ electrons T² = −1, giving Kramers degeneracy of every state at time-reversal-invariant momenta. It is broken by magnetic order or applied fields. Its presence protects the helical edge states of quantum spin Hall insulators and makes Berry curvature odd in k, so valley Hall but not anomalous Hall effects survive; its breaking allows Chern insulators, the anomalous Hall effect and Kerr rotation, the standard probes of spontaneous breaking in correlated phases.

Theory

Topological insulator

A material that is an insulator inside but conducts along its edges or surfaces, in a way that small imperfections cannot easily destroy. In a 2D material the conducting paths run along the edges of the sheet.

An insulator whose bulk bands carry a non-trivial topological invariant, which guarantees gapless boundary states. In 2D the time-reversal-invariant case is the quantum spin Hall insulator (Z₂ invariant) with helical edge channels, reported in monolayer 1T′-WTe₂; Bi₂Se₃-family crystals are three-dimensional topological insulators with surface Dirac cones.

Everyday

Topological phase

A state of matter told apart not by how its atoms are arranged but by a whole number that describes how its electrons’ waves twist across the crystal – the way a doughnut differs from a ball by its one hole. A whole number cannot change a little, so what it guarantees, such as current running along an edge without loss, survives defects and dirt until the band gap itself closes.

A gapped phase characterised by a topological invariant of its occupied bands – a Chern number, a Z₂ index, a winding number – that cannot change under deformations that keep the gap open and any protecting symmetry intact. Where regions with different invariants meet, the gap must close, which forces boundary states: chiral edge channels in Chern insulators, helical ones in quantum spin Hall insulators, Fermi arcs in Weyl semimetals, Majorana modes in topological superconductors. Topological order in the strict sense – the long-range entanglement of fractional quantum Hall states and spin liquids – is a distinct, stronger notion.

Everyday

Toxicity and safety

Whether 2D materials can harm people or the environment, and how to work with them safely. There is no single answer for “graphene”: the size, thickness, surface chemistry and dose of the flakes decide what they do in the body, and a material bound inside a plastic is a very different matter from a fine powder in the air. Some chemicals used to make 2D materials – hydrofluoric acid, hydrogen selenide – are far more dangerous than the materials themselves.

The hazard assessment of 2D materials and their production. Biological effects of graphene-family and other nanosheets depend on lateral size, thickness, stiffness, surface chemistry, dispersibility and dose rather than on the name: large, stiff platelets can frustrate macrophages in a partly asbestos-like way, while small, oxidised, well-dispersed flakes are cleared more readily, and a first controlled human inhalation study of small, thin graphene oxide in 2024 found no acute cardiorespiratory effects. Some compounds carry intrinsic toxicity (lead in halide perovskites; selenium, tellurium, arsenic, antimony), and processes use hazardous precursors – H₂S, H₂Se, HF, organometallics. Nanoforms are registered separately under REACH; workplace exposure is managed as for fine powders, and exposure from flakes bound in composites is generally considered low.

Everyday

Transfer length method (TLM)

A measurement that separates the resistance of a device’s contacts from the resistance of the material itself, by comparing devices with channels of different lengths and working out what would be left at zero length – much as weighing a jar with different amounts of jam in it reveals the weight of the empty jar.

Measuring resistance across channels of different lengths to separate contact resistance from channel resistance.

Experiment

Transition metal dichalcogenide (TMDC)

Also: TMD

A family of layered materials in which a sheet of metal atoms, such as molybdenum or tungsten, sits between two sheets of sulfur, selenium or tellurium. Molybdenum disulfide (MoS₂) is the best-known member – as a dark grey mineral it has long been used as a dry lubricant, because its layers slide so easily. The family ranges from semiconductors to metals and superconductors.

Layered compounds MX₂ in which a transition-metal plane (M = Mo, W, Nb, Ta, Pt and others) is sandwiched between two chalcogen planes (X = S, Se, Te) in trigonal-prismatic (H) or octahedral (T, T′) coordination. Properties span direct-gap semiconductors (monolayer MoS₂, WSe₂), semimetals (WTe₂) and superconductors (NbSe₂).

Everyday

Transmission electron microscopy (TEM)

Electrons are fired through a sample thin enough to be transparent to them, forming an image that can show individual atoms and which element each one is. A 2D crystal is the ideal specimen, because it is already thin enough.

Imaging with transmitted electrons, either parallel-beam (TEM) or scanned-probe (STEM), where annular dark-field contrast scales roughly with Z² and identifies single atoms chemically. It resolves point defects, grain boundaries, stacking and moiré patterns directly, at the cost of knock-on damage – which is why 2D samples are imaged at 60–80 kV – and of a clean, suspended specimen.

Experiment

Transport agent

A helper chemical, such as iodine, that turns a solid into gas at one end of a sealed glass tube and lets it crystallise again at the other end, growing crystals along the way.

A volatile reactant (I₂, Br₂, TeCl₄ and others) that carries a solid through the gas phase in chemical vapour transport.

Experiment

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.

Theory

Trion

Also: charged exciton

An exciton that has picked up an extra electron or hole, which makes it electrically charged. It shows up as a separate, slightly lower-energy glow when a thin semiconductor carries extra charge.

A charged exciton (an exciton bound to an extra electron or hole), visible as a lower-energy PL peak in doped monolayers.

Experiment

Two-dimensional (2D) material

Also: two-dimensional crystal

A crystal so thin that it is only one or a few atoms thick – a sheet rather than a lump. Graphene, a single layer of carbon atoms, is the best-known example: about 300,000 of its layers stacked up would be as thick as one sheet of paper. At that thickness a material can conduct, glow or respond to magnetism quite differently from the same substance in bulk.

A crystalline material whose thickness is one or a few unit cells, so that electrons, phonons and other excitations are confined in one direction. Most are obtained from layered bulk crystals in which strong in-plane bonds coexist with weak van der Waals bonding between layers.

Everyday

Unit cell

The smallest block of a crystal which, repeated over and over in every direction, builds the whole thing – as one motif, repeated, covers a whole roll of patterned wallpaper. Its edge lengths and angles are the numbers quoted as a crystal’s lattice parameters.

The repeating parallelepiped that generates the lattice under translation, given by a, b, c and α, β, γ together with the positions of the atoms inside it. In a layered material the in-plane parameters are fixed by covalent bonding while c depends on stacking and on the van der Waals gap, so polytypes share in-plane parameters and differ along c. A 2D cell quotes a, b and γ only, with the layer thickness stated separately.

Everyday

Valley

One of several low points in a material’s energy landscape where electrons settle, each belonging to a different direction of motion. In some 2D materials electrons in different valleys can be told apart, so the valley could carry information, much as spin or charge does.

A local band extremum at a distinct momentum (K and K′ in graphene and TMDCs); the valley index can act as a binary degree of freedom.

Theory

Van der Waals epitaxy

Growing a layered crystal on a surface it barely sticks to. Because the bond is weak, the new layer does not have to match the spacing of the atoms underneath, so many more material combinations can be grown.

Growth of a layered crystal on a substrate to which it bonds only weakly, so large lattice mismatches are tolerated.

Experiment

Van der Waals force

Also: van der Waals bonding, vdW

A weak attraction between neighbouring atoms and molecules – strong enough, added up over millions of tiny hairs, to let a gecko walk up a pane of glass. In layered crystals it is what holds the layers together, and it is weak enough that a single layer can be peeled off, which is how many 2D materials are made from ordinary crystals.

The weak, non-directional attraction arising from correlated charge fluctuations (dispersion) and related dipolar terms. In layered crystals it binds adjacent layers with energies of tens of meV per atom, roughly two orders of magnitude below in-plane covalent bonds, which enables exfoliation and the free stacking of dissimilar layers.

Everyday

Van der Waals gap

The thin empty slot between two neighbouring layers of a layered crystal, bridged only by weak van der Waals attraction. It is why the layers slide and peel apart, and it is room into which atoms, ions and even whole molecules can be slipped.

The region between adjacent layers that no covalent bond crosses, typically about 0.3 nm from the outer atoms of one layer to the next. Interlayer binding energies across very different layered compounds fall in a narrow range around 20 meV per Ų, which sets the exfoliation energy. The gap carries the stacking and sliding degrees of freedom and hosts intercalants – from lithium ions to organic molecules large enough to push the layers apart and decouple them electronically.

Everyday

Van der Waals heterostructure

Also: van der Waals stack, vdW heterostructure

A stack of different 2D materials placed on top of each other, like a sandwich built one atom-thin slice at a time. Because the layers only stick together weakly, almost any combination can be stacked, which lets researchers build materials that do not exist in nature.

A vertical assembly of dissimilar 2D layers bound by van der Waals forces, so no lattice matching is required. Interfaces can be atomically sharp, and twist angle, stacking order and dielectric environment become design parameters. Stacks are built by dry transfer or by sequential growth.

Everyday

Wafer and wafer-scale growth

A wafer is the thin, round slice of ultra-pure silicon – today 300 millimetres across – on which computer chips are made, hundreds at a time. Wafer-scale growth means making a 2D material evenly over a whole wafer rather than as scattered flakes, which is what any factory needs. For the best 2D materials that is still hard: a sheet one atom thick has to have the same thickness, quality and orientation from one edge of a dinner-plate-sized disc to the other.

The substrate format of the semiconductor industry – 200 and 300 mm silicon wafers, with sapphire, SiC and glass for particular processes – and the scale at which a material must be grown, transferred and processed to enter manufacturing. Wafer-scale growth of 2D materials uses CVD and MOCVD of TMDCs, CVD of graphene and hBN, and MBE, with targets of uniform layer number, low defect and grain-boundary density and, increasingly, a single orientation on vicinal sapphire or Cu(111). The obstacles are the growth temperature against the thermal budget of a finished wafer (growth on the back end of line needs to stay below about 400 to 450 °C), the transfer from growth substrate to device wafer, and metrology that confirms thickness and quality across the wafer at production speed.

Engineering

Weyl and Dirac semimetal

Crystals in which the highest filled band of electron energies and the lowest empty one touch only at isolated points, instead of overlapping broadly or leaving a gap between them. Electrons near such a point behave as if they had no mass – the crystal is a three-dimensional cousin of graphene – and its surface carries loose-ended arcs of electron states that no ordinary metal can have.

Semimetals whose bands touch at isolated points with linear dispersion in every direction. A Dirac point is fourfold degenerate and protected by time-reversal and inversion symmetry together; breaking either splits it into pairs of twofold Weyl nodes of opposite chirality, which act as sources and sinks of Berry curvature and are joined across the surface by Fermi arcs. Where a mirror or another crystalline symmetry protects a whole line of band touchings instead of isolated points, the same physics gives a nodal-line semimetal with drumhead surface states. The chiral anomaly, large anomalous Hall response and strongly non-linear optical effects follow from that structure.

Everyday

Xene

Also: elemental 2D material

The name for single-element 2D materials modelled on graphene: silicene from silicon, germanene from germanium, phosphorene from phosphorus and so on. Most of them are far less stable in air than graphene.

Monoelemental 2D crystals named by analogy with graphene – borophene, silicene, germanene, stanene, phosphorene, antimonene, bismuthene and tellurene. Unlike graphene most are buckled or puckered, many exist only on supporting substrates, and air stability is a major constraint.

Everyday

X-ray diffraction (XRD)

X-rays scattered from the regular planes of atoms in a crystal interfere, giving sharp peaks at angles that reveal the spacings between those planes – the same effect that throws a rainbow off the fine tracks of a CD. For a layered material the first thing it tells you is how far apart the layers sit.

Elastic scattering from lattice planes, giving phase identification, lattice parameters and – from peak width through the Scherrer relation – a lower bound on crystallite size. In layered materials the 00l series measures the interlayer spacing directly, which is how intercalation and polytype changes are confirmed. Thin films need grazing incidence to keep the beam inside the film rather than the substrate.

Experiment

X-ray photoelectron spectroscopy (XPS)

Also: ESCA

X-rays knock electrons out of the atoms at a surface, and the energies those electrons carry say which elements are present and what they are bonded to. It is the standard check that a film really is the compound you wanted and has not oxidised.

Core-level photoemission giving elemental composition and chemical state from the top few nanometres. On 2D films it fixes stoichiometry, detects oxidation and substitutional impurities, and follows doping through rigid core-level shifts. Binding-energy referencing is the usual source of error – adventitious carbon at 284.8 eV is not a reliable standard on a semiconductor – and charging has to be controlled on insulating substrates.

Experiment

Z₂ invariant

A yes-or-no label, 0 or 1, that says whether an insulator has protected conducting edges. A value of 1 marks a particular kind of topological insulator.

The topological index of time-reversal-symmetric insulators that distinguishes a quantum spin Hall insulator from a trivial one.

Theory