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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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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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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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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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.
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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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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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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.
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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.
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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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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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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.
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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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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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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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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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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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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.
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