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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Electron hole
Usually just called a hole: the empty place an electron leaves behind in an otherwise full band. It behaves like a particle of its own with a positive charge. When a neighbouring electron steps into the gap, the gap moves the other way – like the empty space in a queue of cars, which travels backwards as each car edges forward. Many materials conduct mainly with holes, and chips need both kinds of carrier.
An unoccupied state in an otherwise filled band, usually near the valence-band maximum, treated as a quasiparticle with positive charge and a positive effective mass set by the band curvature. Holes carry the current in p-type semiconductors and give a positive Hall coefficient. In 2D semiconductors good p-type transport is harder to obtain than n-type, largely because metal contacts pin the Fermi level nearer the conduction band.
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Unit cell
The smallest block of a crystal which, repeated over and over in every direction, builds the whole thing – as one motif, repeated, covers a whole roll of patterned wallpaper. Its edge lengths and angles are the numbers quoted as a crystal’s lattice parameters.
The repeating parallelepiped that generates the lattice under translation, given by a, b, c and α, β, γ together with the positions of the atoms inside it. In a layered material the in-plane parameters are fixed by covalent bonding while c depends on stacking and on the van der Waals gap, so polytypes share in-plane parameters and differ along c. A 2D cell quotes a, b and γ only, with the layer thickness stated separately.
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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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Residual resistivity ratio (RRR)
A quick score for how clean a metal crystal is: its electrical resistance at room temperature divided by its resistance when cooled close to absolute zero. Cooling stops the atoms jiggling, so the resistance that remains comes from impurities and defects. A pure crystal loses almost all of its resistance and scores in the hundreds or thousands; a dirty one keeps most of it and scores near one.
The ratio R(300 K)/R(T → 0) of a metallic sample, where the low-temperature value is the residual resistivity from static disorder once phonon scattering has frozen out (Matthiessen’s rule). It is a standard, geometry-independent proxy for crystal quality and mean free path, from order one in disordered films to 103 –105 in the purest metals. Its low-temperature reference is ambiguous in superconductors, where the normal-state value just above T_c is used, and in materials whose resistivity turns up at low temperature.
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Semiconductor
A material whose ability to carry electricity can be switched on and off, for example by an applied voltage. That switching is what every transistor in a computer chip relies on. Silicon is the classic example; several 2D materials, such as MoS2 , are semiconductors too.
A material with a band gap of a few electronvolts or less, whose conductivity can be tuned over many orders of magnitude by doping, gating, temperature or light. 2D semiconductors attract interest because their sub-nanometre thickness preserves electrostatic gate control in very short channels.
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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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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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Magnetoresistance
A change in electrical resistance when a magnetic field is applied. It is how the read head of a hard disk senses its bits, and in some layered magnets the change is enormous – a stack can go from barely conducting to conducting – which is how the magnetic state of a few atomic layers is read out electrically.
The field dependence of resistivity. Ordinary orbital magnetoresistance is positive and usually small, though enormous in compensated semimetals such as WTe2 ; giant and tunnelling magnetoresistance come from spin-dependent transport across magnetic layers; colossal magnetoresistance from a field-driven change of electronic state. In van der Waals stacks a few layers of CrI3 act as a spin filter whose resistance changes by orders of magnitude as adjacent layers align, which is how layer-by-layer magnetic order became measurable in transport.
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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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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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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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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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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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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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Fermi surface and nesting
The Fermi surface is the boundary between the states a metal’s electrons fill and the ones they leave empty, drawn on a map of every way an electron can move through the crystal; its shape decides how the metal conducts. Nesting is when large flat stretches of that boundary can be slid onto each other by one single shift – a metal like that is unstable and tends to settle into a regular ripple of charge or spin.
The constant-energy surface separating occupied from empty states, whose geometry governs transport, screening and instabilities. Nesting – parallel sheets connected by a single wavevector q – peaks the bare susceptibility at q and is the textbook route to charge and spin density waves. In real layered metals the ordering wavevector often follows momentum-dependent electron–phonon coupling instead, so a nesting figure alone does not settle the mechanism.
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Resistivity and sheet resistance
Two ways to put a number on how strongly a material resists a current. Resistivity belongs to the material itself, whatever the size of the piece: copper’s is tiny, glass’s enormous. For a sheet so thin that its thickness is hard to pin down, the sheet resistance is used instead: the resistance of a square of it, measured from one edge to the opposite one, which comes out the same for a square of any size. It is quoted in ohms per square.
Resistivity ρ relates electric field to current density, in Ω·m or Ω·cm, so that a bar of length L and cross-section A has resistance ρL/A. For a film of thickness t the sheet resistance is ρ/t, quoted in Ω per square, and a strip L long and W wide has the sheet resistance times L/W. For an atomically thin layer it is the natural quantity, since the thickness is a convention, and it equals 1/neμ for sheet density n and mobility μ. Both are measured with four probes – a Hall bar, a collinear four-point probe or the van der Pauw method for arbitrary shapes – so that the voltage is sensed away from the current contacts and the contact resistance drops out.
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Strong electron correlation
What happens when electrons in a material repel each other so strongly that they can no longer be treated as moving independently. Each electron’s motion then depends on where all the others are, and the result can be an insulator where simple theory predicts a metal, or magnetism, superconductivity and other collective states.
The regime in which the Coulomb repulsion U between electrons is comparable to or larger than their kinetic energy, set by the bandwidth W, so single-particle band theory fails. At half filling with U well above W the Hubbard model gives a Mott insulator with local moments and antiferromagnetic exchange of order t2 /U; doping or tuning U/W gives correlated metals, heavy-fermion behaviour and unconventional superconductivity. In 2D, weak screening keeps U large and moiré superlattices shrink W, so twisted bilayers make U/W tunable by gate and twist. Semi-local DFT misses the physics; DFT+U, DMFT, exact diagonalisation and quantum Monte Carlo are the usual tools.
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Excitonic insulator
A state in which electrons and the holes they leave behind bind into pairs so strongly that the material opens a gap and stops conducting – not because of its chemistry, but because the pairs have all settled together into one shared, collective state. Telling this apart from an ordinary distortion of the crystal is notoriously difficult.
A ground state predicted for a semimetal with small band overlap, or a semiconductor whose gap is smaller than the exciton binding energy, in which spontaneous exciton condensation reconstructs the bands and opens a gap. The experimental difficulty is that the transition is usually accompanied by a lattice distortion of the same symmetry, so photoemission, ultrafast and pressure experiments have to separate an electronic condensate from a conventional Peierls-like instability – the argument that surrounds Ta2 NiSe5 and 1T-TiSe2 .
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