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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Chemical bond
The glue that holds atoms together. In a covalent bond two atoms share electrons, as within a sheet of graphene. In an ionic bond the atom that pulls harder on electrons – the one with the higher electronegativity – takes them from the other, and the resulting charges attract, as in table salt. In a metal the outer electrons are shared by all the atoms at once. Much weaker van der Waals forces hold molecules and the layers of a layered crystal together. A 2D material is a crystal with strong bonds within its layers and only weak ones between them – which is why it can be peeled.
The attractive interactions that hold atoms in molecules and solids, classified by how electrons are shared: covalent bonds, from overlapping orbitals holding shared electron pairs, strong and directional (sp2 carbon, Mo–S in TMDCs); ionic bonds, from electron transfer and electrostatic attraction (halides, oxides); metallic bonding, with electrons delocalised in a partly filled band; and the weak, non-directional van der Waals interaction from correlated charge fluctuations, besides hydrogen bonds. Real bonds are mixtures, characterised by electronegativity differences and charge analyses. Layered crystals combine in-plane bonds of several electronvolts with interlayer binding of only about 20 meV per Å2 (some 0.3 J/m2 ), the criterion high-throughput screens use to identify exfoliable compounds.
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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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Orbitals and hybridisation
The shapes an electron’s cloud can take around an atom. Electrons do not circle the nucleus like planets; each sits in an orbital, a region with a definite energy and shape – a sphere (s), a dumbbell (p) or a clover-like shape (d). When atoms bond, their orbitals overlap and can blend into new ones pointing at the neighbours, which is hybridisation: in graphene each carbon mixes one s and two p orbitals into three bonds in the plane (sp2 ), leaving one p orbital standing up out of the sheet. Which orbitals hold the outermost electrons decides much of what a material does. Electrons in an orbital still carry a kind of circling motion, angular momentum, and the magnetism that comes from it rather than from their spin is called orbital magnetism.
Single-electron states of an atom labelled by angular momentum – s, p, d, f – whose overlap between neighbours forms bonds and bands. Hybridisation mixes orbitals on one atom, as in graphene’s sp2 carbon, with three in-plane σ bonds 1.42 Å long and a half-filled π band from the out-of-plane p orbital, where the Dirac cones lie; or between atoms, as in the p–d hybridisation of transition-metal compounds. In 2H TMDCs a hybridisation gap splits a d band off from the rest, and whether it is half-filled or full separates metallic NbSe2 and TaS2 from semiconducting MoS2 . “Orbital” also names the electron’s motion: orbital magnetism comes from circulating electrons rather than from spin.
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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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Density functional theory (DFT)
The workhorse calculation of materials physics. Rather than following every electron, it works with the electron density and a recipe for how electrons avoid one another – which makes a whole crystal cheap enough to compute, and makes the accuracy depend entirely on that recipe.
Ground-state electronic structure from the density, exact in principle by Hohenberg–Kohn and made practical by the Kohn–Sham equations with an approximate exchange–correlation functional. Two dimensions need care: a vacuum gap wide enough that periodic images stop interacting or a truncated Coulomb interaction, a dispersion correction for interlayer binding, and the knowledge that semi-local functionals underestimate gaps – hybrid functionals narrow that error at far higher cost, and quantitative gaps and optical spectra need GW and Bethe–Salpeter on top.
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Tight-binding model
A stripped-down model in which electrons sit on atoms and hop to their neighbours with a fixed probability. It predicts little on its own, but once its handful of numbers is fitted it reproduces bands cheaply enough to handle millions of atoms – a twisted bilayer, for instance.
An expansion of the Hamiltonian in localised orbitals with hopping integrals between them, fitted to first-principles bands or constrained by symmetry in the Slater–Koster scheme. Its value in 2D is scale: moiré supercells of 104 –105 atoms, disorder averaging and transport are out of reach for DFT and routine here. Three bands built from the metal d orbitals already capture the band edges and spin–orbit splitting of a group-VI TMDC monolayer.
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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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Electron–phonon coupling
The push and pull between a crystal’s electrons and the vibrations of its atoms. Vibrating atoms scatter moving electrons and slow them down, which is why a metal conducts worse when warm; the electrons in turn shake the atoms and lose energy to them as heat. The same coupling can also draw electrons together: it is the glue that pairs them in ordinary superconductors, and when it is strong it can make the whole crystal buckle into a charge density wave.
The change in electronic energies when atoms are displaced, described by matrix elements between electron states linked by a phonon and summarised by the dimensionless constant λ obtained from the Eliashberg function α2 F(ω). It limits room-temperature mobility through phonon scattering, gives metals their linear-in-temperature resistivity at high temperature, renormalises quasiparticle bands (the kinks seen in ARPES), mediates the attraction behind conventional superconductivity and, where it is strong at particular wavevectors, softens phonons into a charge density wave. In 2D materials it is computed from first principles with density functional perturbation theory and Wannier interpolation, and it can be tuned by doping, strain and the dielectric environment.
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Charge carrier mobility
How easily electrons, or the gaps they leave behind (holes), move through a material when pushed by a voltage. Higher mobility means faster, more efficient electronics – but it is easily ruined by dirt, defects and a poor substrate.
Drift velocity per unit electric field, usually quoted in cm2 /V·s. In 2D materials it is limited intrinsically by phonon scattering and extrinsically by charged impurities, roughness, remote phonons and disorder; field-effect values extracted from two-terminal devices can be distorted by contact resistance.
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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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Ferroelectricity
A material that holds an electric polarisation – one side slightly positive, the other slightly negative – with no voltage applied, and that can be flipped by a large enough field – the coercive field – and stay flipped. That memory of the last field applied is what makes it useful for storing data.
A spontaneous, switchable electric polarisation produced by a structural distortion that breaks inversion symmetry. In 2D the usual obstacle – depolarising fields destroying the polarisation as a film thins – is sidestepped in three ways: in-plane polarisation, layered compounds whose dipoles survive to the monolayer such as CuInP2 S6 and In2 Se3 , and sliding ferroelectricity, where the polarisation belongs to the interface between layers rather than to the layers themselves.
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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. It is 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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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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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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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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Flat band
A range of electron energies so narrow that the electrons barely move on their own. With their motion frozen out, the way they repel each other takes over – as in a packed train carriage, where nobody can walk anywhere and everything depends on how people get on with their neighbours. That is where unusual states such as superconductivity and magnetism can appear.
A band whose kinetic-energy width is small compared with the interaction energy, so electron–electron interactions dominate the physics.
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Triangular lattice and frustration
A triangular lattice is a flat net of triangles, each atom with six neighbours. Put a tiny magnet that wants to point opposite to its neighbours on every corner, and a triangle cannot oblige: once two corners point up and down, the third has no good choice. That standoff is called frustration. It can stop a magnet from ordering at all, leaving room for unusual states such as quantum spin liquids.
A Bravais lattice with six nearest neighbours per site. With antiferromagnetic nearest-neighbour exchange the three spins of a triangle cannot all be antiparallel: Ising spins have a macroscopically degenerate ground state, while Heisenberg spins settle into the 120° state. This geometric frustration suppresses ordering and enhances quantum fluctuations; frustration also arises from competing exchange paths and from bond-dependent Kitaev interactions on the honeycomb lattice. Its strength is gauged by the ratio |θ_CW|/T_N.
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Honeycomb lattice
A flat net of hexagons, like a beehive, with an atom at every corner. Graphene is the famous example, but the same pattern gives boron nitride its band gap, gives graphene its massless electrons, and gives some magnets interactions that cannot all be satisfied at once.
A triangular Bravais lattice with a two-site basis, so every site has three nearest neighbours on the other sublattice. Equivalent sublattices with nearest-neighbour hopping give Dirac cones at K and K′ (graphene); inequivalent ones open a gap with valley-contrasting Berry curvature (hBN, TMDC monolayers seen from above); buckling mixes in spin–orbit coupling (the Xenes); and bond-dependent exchange on the same geometry gives the exactly solvable Kitaev model behind the α-RuCl3 spin-liquid programme.
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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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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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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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Photovoltaic effect
Light absorbed in a material frees electrons, and a built-in asymmetry – usually a junction between two different layers – pushes them one way rather than the other, producing a voltage. That is how a solar cell works. The current that light drives is the photocurrent. A crystal without a centre of symmetry needs no junction at all: in it, light alone can push a current through the bulk.
Separation of photogenerated carriers by a built-in field at a p–n junction, a Schottky contact or a type-II van der Waals interface. Monolayers absorb roughly an order of magnitude more sunlight per unit thickness than GaAs or Si, so a stack a few nanometres thick reaches useful power per unit mass even though its absolute absorption stays small; in a gate-tunable heterojunction the alignment, and with it the sign and size of the photoresponse, can be switched electrically. In crystals without inversion symmetry a bulk photovoltaic effect needs no junction at all: the shift current flows with no bias, as in ferroelectric SbSI below its 295 K transition.
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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. It is 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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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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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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