Exciton
An electron paired with the ‘hole’ it left behind – the empty place, which acts like a positive charge. The two attract each other and form something like a tiny atom. In very thin materials these pairs hold together unusually strongly, so they dominate how the material absorbs and emits light – even at room temperature.
A bound state of a conduction-band electron and a valence-band hole. In 2D semiconductors reduced dielectric screening raises binding energies to hundreds of meV in freestanding TMDC monolayers, so excitons dominate optical spectra at room temperature and follow a non-hydrogenic Rydberg series.
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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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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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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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Photon
The smallest possible amount of light – one indivisible packet of it. The packets of a given colour all carry the same energy, and blue ones carry more than red: that is why ultraviolet light gives you sunburn and red light, however bright, does not. A semiconductor absorbs only photons with enough energy to cross its band gap and gives light back at about the gap’s energy – which is why LEDs of different colours are made from different semiconductors.
The quantum of the electromagnetic field, with energy E = hν and momentum h/λ but no mass or charge. Visible photons carry about 1.7–3.1 eV, comparable to the band gaps of semiconducting TMDCs, but a momentum far smaller than the crystal momenta across the Brillouin zone, so optical transitions are vertical and an indirect gap needs a phonon to make up the difference. Single-photon emission is established by antibunching in photon-correlation measurements.
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Charge transfer
Electrons moving from one material to its neighbour simply because they are more comfortable there. It is how a layer gets doped without adding a single impurity atom to it – the charge comes from the molecule, substrate or layer next door.
Equilibration of electrochemical potential across an interface, moving electrons to whichever side has the deeper states and leaving a dipole and band bending behind. It dopes 2D layers without substitutional disorder, from adsorbed molecules and substrate surface states to the partner layer in a type-II stack, where photoexcited carriers separate in tens of femtoseconds – faster than they recombine, which is what makes such stacks useful for light harvesting and detection.
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Pump–probe spectroscopy
A way to film events that last a millionth of a millionth of a second. A first, strong flash of laser light – the pump – disturbs the sample; a second, weaker flash – the probe – arrives a precisely set moment later and measures how the sample looks then. Repeating this with the delay changed step by step builds up a slow-motion film of how the electrons and atoms settle back. The delay is set by sending one flash along a longer path: a tenth of a millimetre more makes it a third of a picosecond later.
A stroboscopic measurement in which an ultrashort pump pulse excites the sample and a delayed probe pulse records the change, as a function of a delay set by an optical delay line, with a time resolution limited by the pulse durations – tens of femtoseconds routinely, a few with dedicated sources. The probe can be optical reflectivity or transmission, terahertz conductivity, photoemission (time-resolved ARPES), or X-ray or electron diffraction, giving access to carrier thermalisation and cooling, exciton formation, interlayer charge transfer, coherent phonons, and the melting, recovery or switching of ordered phases such as charge density waves and excitonic insulators. Its power lies in separating processes by timescale: electrons respond within femtoseconds, the lattice within picoseconds.
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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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Van der Waals gap
The thin empty slot between two neighbouring layers of a layered crystal, bridged only by weak van der Waals attraction. It is why the layers slide and peel apart, and it is room into which atoms, ions and even whole molecules can be slipped.
The region between adjacent layers that no covalent bond crosses, typically about 0.3 nm from the outer atoms of one layer to the next. Interlayer binding energies across very different layered compounds fall in a narrow range around 20 meV per Å2 , which sets the exfoliation energy. The gap carries the stacking and sliding degrees of freedom and hosts intercalants – from lithium ions to organic molecules large enough to push the layers apart and decouple them electronically.
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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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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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Moiré superlattice
The larger pattern that appears when two lattices are laid on top of each other slightly rotated, or with slightly different spacings – like two fine mesh curtains overlapping. Electrons feel this larger pattern, and it can reshape how they behave.
The long-wavelength interference pattern formed by two lattices with a small twist or lattice mismatch; its period sets a new, much larger unit cell for electrons.
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Magic angle
Stack two sheets of graphene and turn one very slightly. At about 1.1 degrees the pattern they make together slows the electrons almost to a standstill, so they start acting collectively – and the pair can even become a superconductor.
The twist angle (about 1.1° for bilayer graphene) at which interlayer tunnelling makes the lowest moiré bands nearly flat.
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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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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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Commensurate and incommensurate
Whether two repeating patterns fit each other. If one pattern repeats after exactly a whole number of steps of the other – every three atoms, say – the two are commensurate, and together they repeat again after a short distance. If the ratio is not a simple fraction, they are incommensurate and never line up exactly. The question comes up whenever a crystal grows a second pattern of its own – a charge density wave, a magnetic spiral – and whenever two different layers are stacked, as in a moiré.
A modulation is commensurate when its wavevector is a rational fraction of a reciprocal lattice vector, so that crystal and modulation share a finite supercell, and incommensurate when the ratio is irrational, leaving no exact common period. The charge density wave of 1T-TaS2 passes from incommensurate (below about 550 K) to nearly commensurate – commensurate domains separated by discommensurations – and locks into the commensurate √13 × √13 Star-of-David phase below about 180 K; RTe3 keeps an incommensurate wave, and NiI2 an incommensurate spin spiral. For two lattices – graphene on hBN, twisted bilayers – the same competition between elastic energy and interlayer adhesion decides whether the layers stretch into commensurate domains or keep their own lattices.
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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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Van der Waals heterostructure
A stack of different 2D materials placed on top of each other, like a sandwich built one atom-thin slice at a time. Because the layers only stick together weakly, almost any combination can be stacked, which lets researchers build materials that do not exist in nature.
A vertical assembly of dissimilar 2D layers bound by van der Waals forces, so no lattice matching is required. Interfaces can be atomically sharp, and twist angle, stacking order and dielectric environment become design parameters. Stacks are built by dry transfer or by sequential growth.
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