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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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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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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Polaron
An electron that drags a dent in the crystal along with it, like a ball rolling across a mattress. It pushes the nearby atoms slightly aside, and electron and distortion then travel together as one heavier, slower object.
A quasiparticle made of a carrier dressed in the lattice distortion it induces. Coupling strength sets its character: a large Fröhlich polaron stays delocalised with a renormalised mass, a small polaron self-traps on one site and moves by hopping. In 2D the weaker dielectric screening and the missing third dimension change the binding, and first-principles calculations now resolve polaron structures in monolayers that a bulk formula misses.
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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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Phonon
A packet of vibration travelling through a crystal – the way its atoms jiggle together. Sound travelling through a solid is made of phonons. They also carry heat, slow electrons down by jostling them, and are what Raman spectroscopy measures.
A quantised collective lattice vibration. 2D materials host flexural (out-of-plane) acoustic modes with quadratic dispersion; electron–phonon scattering sets intrinsic mobility limits, and phonon frequencies measured by Raman spectroscopy report layer number, strain and doping.
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Magnon
A ripple of tilted spins – the atoms’ tiny magnets – travelling through a magnet, like a stadium wave in which each spectator only stands up and sits down yet the wave runs all the way round. Because the ripple carries information without moving any electrons, it is one route to lower-power devices.
A quantised spin-wave excitation of a magnetically ordered state. Its dispersion is set by exchange, anisotropy and dipolar terms, and in a 2D magnet the anisotropy gap at zero wavevector is what holds order against thermal fluctuations – so the magnon spectrum measures the same anisotropy that lets the material evade the Mermin–Wagner theorem.
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Polariton
A wave that is part light and part motion in a material – electrons or atoms swinging back and forth – travelling as one. In thin crystals such waves can be squeezed to wavelengths many times shorter than the light that made them, so they can guide and focus infrared light into spaces far smaller than light normally allows.
A hybrid mode of photons with a polarisation excitation: free-carrier plasma oscillations (plasmon polaritons), optical phonons (phonon polaritons) or excitons (exciton polaritons). In van der Waals crystals they confine light to wavelengths tens to hundreds of times below that in free space. hBN supports hyperbolic phonon polaritons in its two Reststrahlen bands, α-MoO3 in-plane anisotropic ones and graphene gate-tunable plasmons; TMDC monolayers in optical cavities form exciton polaritons. They are imaged in real space by scattering-type near-field microscopy (s-SNOM), which launches them from a sharp tip and maps their interference fringes.
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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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Raman spectroscopy
Shining a laser on a material and measuring the tiny fraction of light that comes back at slightly different colours. The shifts reveal how the atoms vibrate, which tells researchers how many layers a flake has and whether it is strained or damaged.
Inelastic light scattering that measures phonon energies. In 2D materials peak positions, widths and intensity ratios – graphene’s G and 2D bands, or the E′ and A′1 modes of TMDCs – report layer number, strain, doping and defect density; laser heating and substrate interference must be controlled.
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Neutron scattering
Shooting neutrons at a crystal and watching how they bounce off. Neutrons carry no charge, so they pass deep into a sample, but they carry a tiny magnetic moment, so they feel the magnetic moments of atoms. That makes them the standard tool for finding how the spins in a magnet are arranged and – from the energy the neutrons lose – how the spins wave and wobble. The catch is that neutrons come only from research reactors and large accelerators, and the samples have to be large crystals.
Elastic and inelastic scattering of thermal and cold neutrons from nuclei and, through the neutron’s magnetic moment, from unpaired electrons. Diffraction gives the nuclear structure, with sensitivity to light elements and isotopes, and the magnetic structure from magnetic Bragg peaks below the ordering temperature (Shull, 1949); polarised neutrons separate magnetic from nuclear scattering. Inelastic scattering on triple-axis and time-of-flight spectrometers maps phonon and magnon dispersions, exchange constants and continua – the magnon gaps of CrI3 , the scattering continuum of α-RuCl3 . Weak fluxes require gram-scale samples or many co-aligned crystals, so 2D materials are measured in bulk form; monolayers are out of reach, and bulk results are transferred to thin layers with care.
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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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GW and the Bethe–Salpeter equation
Two calculations that take over where density functional theory, the standard one, stops. The first corrects the energy cost of adding or removing an electron; the second adds the attraction between an electron and the hole it leaves behind, which is what decides the colour of light a thin crystal absorbs.
GW replaces Kohn–Sham eigenvalues with quasiparticle energies from a screened-exchange self-energy, opening the gaps that semi-local DFT underestimates; solving the Bethe–Salpeter equation on top adds the electron–hole interaction and produces optical spectra with bound excitons. In 2D both converge painfully slowly with vacuum size and k-point sampling, because screening is non-analytic near q = 0 – a Coulomb cutoff and dense sampling around the band edge are not optional.
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Quasiparticle gap versus optical gap
Two answers to ‘how much energy does it take to excite this material?’. Adding a free electron and a free hole costs more; creating a bound electron–hole pair with light costs less, and in 2D materials the difference is large.
The quasiparticle gap is the energy to add an electron and a hole independently; the optical gap is smaller by the exciton binding energy.
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Scanning tunnelling microscopy (STM)
A needle sharpened to a single atom is brought so close to a surface that electrons tunnel across the gap – a quantum effect by which they pass through a barrier they could never climb over. The current maps the surface atom by atom, and how it changes with voltage tells you which energies electrons are allowed to have at that exact spot.
Tunnelling between a sharp tip and a conducting surface, giving atomic-resolution topography and, in spectroscopy mode, the local density of states against energy. On 2D materials it resolves defect levels, moiré potentials, edge modes and quasiparticle gaps site by site. The sample has to be clean and conducting, so it is run on epitaxial films or on flakes placed on graphite, in vacuum and often at cryogenic temperature.
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Effective mass
How heavy an electron seems as it moves through a crystal. Pushed by an electric field, it speeds up as if its mass were different from a free electron’s – lighter in some materials, heavier in others, and in graphene as if it had no mass at all. Light carriers generally make for faster devices.
The mass m* with which a carrier near a band edge responds to forces as if it were free, set by the band curvature: 1/m* = (1/ħ2 ) d2 E/dk2 . It enters the mobility (μ = eτ/m*), the density of states, confinement energies and tunnelling rates, and in anisotropic crystals it is a tensor – in black phosphorus the armchair and zigzag masses differ several-fold. Graphene’s linear bands have no curvature, and its carriers are described as massless Dirac fermions; their cyclotron mass, the Fermi energy divided by the square of the Fermi velocity, grows with the square root of the carrier density.
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Trion
An exciton that has picked up an extra electron or hole, which makes it electrically charged. It shows up as a separate, slightly lower-energy glow when a thin semiconductor carries extra charge.
A charged exciton (an exciton bound to an extra electron or hole), visible as a lower-energy PL peak in doped monolayers.
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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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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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Dielectric screening
The way a material weakens the pull between two charges inside it: its own electrons shift slightly and partly cancel the field between them. A sheet one atom thick has almost nothing around it to do this, so charges in it attract each other far more strongly than in a thick crystal – which is why light makes tightly bound pairs in 2D materials, and why whatever a sheet rests on changes its properties.
The reduction of the Coulomb interaction by the polarisation of bound electrons and ions and, in metals and doped layers, by free carriers, described by a dielectric function ε(q, ω). In a monolayer the field lines between distant charges run through the surroundings, so ε(q) approaches 1 as q → 0 and the screening is non-local and set by the environment. Exciton binding energies reach hundreds of meV, and a change of substrate or encapsulation shifts the quasiparticle gap by 100 meV or more while the optical gap barely moves.
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