Two-dimensional (2D) material
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.
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Spin–orbit coupling
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.
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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.
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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.
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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.
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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.
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Monolayer
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.
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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.
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Edge state
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.
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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 Bi2 Se3 -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 Bi2 Se3 . In a 2D material the boundary of a topological phase is its edge, and the counterpart of the surface state is the edge state.
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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.
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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.
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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.
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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.
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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 Mo1−x Wx S2 or MoS2(1−x) Se2x , 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.
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Fermi level
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.
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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.
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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.
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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.
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