Coordination polyhedron
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.
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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.
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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.
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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) = Σn ∫ δ(E − εn (k)) d2 k/(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.
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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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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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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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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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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.
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Quantum spin liquid
A magnet whose atomic magnets never settle into a pattern, even at absolute zero. Quantum effects keep them fluctuating and entangled – linked so that none has a direction of its own – so instead of freezing into order the material stays restless, like a liquid that never turns solid. Disturbances in it can behave like fractions of an electron, carrying its magnetism but not its charge.
A ground state of a frustrated magnet with no symmetry breaking down to zero temperature, marked by long-range entanglement, an emergent gauge structure and fractionalised excitations such as spinons or Majorana fermions. Honeycomb materials with bond-dependent Kitaev exchange – α-RuCl3 above all – are the leading layered candidates, but the evidence is indirect: a scattering continuum and thermal Hall signals, against residual magnetic order in zero field.
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Van der Waals force
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.
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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.
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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.
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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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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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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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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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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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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 Fe3 GeTe2 , 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.
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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.
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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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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.
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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.
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