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 ; Bi2 Se3 -family crystals are three-dimensional topological insulators with surface Dirac cones.
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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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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.
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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 (P31 21 or P32 21), 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 Mn3 Si2 Te6 . Barron’s distinction between true and false chirality decides which physical effects a given handedness allows.
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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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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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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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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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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 AV3 Sb5 and the europium content of EuSn2 As2 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.
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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.
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Nanosheet
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.
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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.
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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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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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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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