Band 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.
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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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Berezinskii–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.
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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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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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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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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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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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Skyrmion and the Dzyaloshinskii–Moriya interaction
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, Fe3 GeTe2 hosts bubbles and, in WTe2 /Fe3 GeTe2 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.
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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.
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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.
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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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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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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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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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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.
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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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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.
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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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Majorana mode
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.
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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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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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