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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Two-dimensional (2D) material
A crystal so thin that it is only one or a few atoms thick – a sheet rather than a lump. Graphene, a single layer of carbon atoms, is the best-known example: about 300,000 of its layers stacked up would be as thick as one sheet of paper. At that thickness a material can conduct, glow or respond to magnetism quite differently from the same substance in bulk.
A crystalline material whose thickness is one or a few unit cells, so that electrons, phonons and other excitations are confined in one direction. Most are obtained from layered bulk crystals in which strong in-plane bonds coexist with weak van der Waals bonding between layers.
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Ferromagnet and antiferromagnet
In a ferromagnet the tiny magnets of the atoms all point the same way, as in a fridge magnet. In an antiferromagnet neighbours point in opposite directions and cancel out; if the opposing magnets differ in size they cancel only in part, and the material is a ferrimagnet. Ferromagnets and antiferromagnets have both been found in sheets only one layer thick.
Magnetically ordered states with parallel (ferromagnetic) or antiparallel (antiferromagnetic) alignment of neighbouring moments; antiparallel moments of unequal size make a ferrimagnet. Long-range order in 2D requires magnetic anisotropy to evade the Mermin–Wagner theorem; CrI3 monolayers are Ising-like ferromagnets that couple antiferromagnetically between layers, and the MPS3 compounds are antiferromagnets.
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Anisotropy
When a material behaves differently depending on direction – conducting better along one line than across it, for example, much as wood splits easily along the grain but hardly across it. Some 2D materials, such as black phosphorus, are strongly direction-dependent within the sheet itself.
Direction dependence of a property. All layered crystals are anisotropic between in-plane and out-of-plane directions; low-symmetry lattices such as black phosphorus, ReS2 and CrSBr are also anisotropic within the plane, with effective mass, optical absorption or magnetic order differing along crystal axes.
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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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Magnetic anisotropy energy
The extra energy needed to point a material’s magnetisation in a direction it does not prefer. In 2D magnets that preference is what keeps the magnetic order from falling apart. The direction a magnet prefers is its easy axis and the one it resists most its hard axis; when any direction within the layer will do, it has an easy plane.
The energy difference between magnetisation directions – the easy axis or easy plane it prefers and the hard directions it resists – arising from spin–orbit coupling and dipolar effects; it opens the spin-wave gap that allows 2D magnetic order.
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Hysteresis
When a device gives different readings depending on whether a voltage is swept up or down, because charges are trapped and released along the way. It is a sign of an unclean or unstable device.
Different transfer curves for forward and reverse gate sweeps, caused by charge trapping in adsorbates, dielectrics or defects.
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Magnetometry
Measuring how magnetic a sample is, or how readily a magnetic field magnetises it – its magnetic susceptibility. For a crystal you can hold, this is routine: it is moved through a coil or past a superconducting sensor called a SQUID, which picks up its tiny magnetic field. A flake one atom thick is a different matter: its magnetism is far too weak for such instruments and is swamped by the substrate it sits on. So 2D magnets are usually studied with light, with currents, or with single-spin sensors in diamond that hover nanometres above the flake.
Measurement of magnetic moment or stray field. SQUID and vibrating-sample magnetometry of bulk crystals, with sensitivities around 10−8 emu (10−11 A·m2 ), give magnetisation and magnetic susceptibility (magnetisation per unit field) against field and temperature, from which ordering temperatures, saturation moments and anisotropy follow; torque and Hall magnetometry and magnetic force microscopy extend this to small samples. Monolayer flakes lie far below SQUID sensitivity and are swamped by substrate diamagnetism and magnetic contamination, so their magnetism is read optically (MOKE, RMCD), electrically (anomalous Hall effect, tunnelling magnetoresistance) or with scanning nitrogen-vacancy magnetometry, which images stray fields quantitatively at the nanoscale. Claims of room-temperature ferromagnetism in thin films based on bulk magnetometry, as for monolayer VSe2 , need element-specific (XMCD) or local confirmation.
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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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Nanosheet
A general word for a very thin sheet of material, often just a few atoms thick and up to a few micrometres – thousandths of a millimetre – across. It is used especially for flakes made in large quantities in liquids.
A particle whose lateral dimensions far exceed its nanometre-scale thickness; the term is used especially for liquid-exfoliated and chemically derived dispersions, whose lateral-size and thickness distributions must be specified rather than assumed. In transistor engineering ‘nanosheet’ also denotes the stacked silicon channel sheets of gate-all-around devices.
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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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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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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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Topological superconductivity
Superconductivity whose paired electrons carry a twist that cannot be undone smoothly – the superconducting cousin of a topological insulator. Inside, the material behaves like an ordinary superconductor, but its edges, surfaces and the cores of its vortices must hold special states, among them Majorana modes, which could store quantum information in a way that local disturbances cannot easily spoil. Few materials are clearly topological superconductors on their own, so researchers also build them by bringing an ordinary superconductor into contact with a topological or magnetic material.
Superconductivity whose Bogoliubov quasiparticle bands carry a nontrivial topological invariant, so that a full or nodal bulk gap coexists with protected Andreev boundary states, including Majorana zero modes bound to vortices and wire ends. Intrinsic candidates include odd-parity pairing and strongly spin–orbit-coupled systems such as the surface of FeTe0 .55 Se0 .45 , PtBi2 and UTe2 ; engineered versions combine s-wave pairing with spin–orbit coupling and broken time-reversal symmetry, following the Fu–Kane proposal for topological-insulator surfaces, semiconductor nanowires and magnetic atom chains. A zero-bias conductance peak alone is not proof: disorder and ordinary Andreev bound states mimic it.
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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. 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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Charge density wave (CDW)
A state in which the electrons in a crystal bunch up into a regular ripple instead of spreading out evenly, pulling the atoms slightly out of place as they go. It sets in below a certain temperature and often competes with superconductivity for the same electrons.
A periodic modulation of conduction-electron density locked to a periodic lattice distortion, driven by Fermi-surface nesting, momentum-dependent electron–phonon coupling or both. Transition temperature and ordering wavevector depend on layer number, doping and pressure, and the ordered state frequently coexists or competes with superconductivity in the same phase diagram.
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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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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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Spintronics
Electronics that uses the magnetic direction of an electron – its spin – as well as its charge. Writing information as a spin direction rather than as a pocket of charge can store it without power and switch it with less energy.
Device physics built on injecting, transporting, manipulating and detecting electron spin. Two-dimensional materials contribute long spin lifetimes and gate-tunable transport in graphene, spin–orbit coupling and exchange induced by proximity to an adjacent layer, spin–valley locking in TMDC monolayers, and van der Waals magnets that serve as tunnel barriers and free layers in an all-2D stack.
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Van der Waals force
A weak attraction between neighbouring atoms and molecules – strong enough, added up over millions of tiny hairs, to let a gecko walk up a pane of glass. In layered crystals it is what holds the layers together, and it is weak enough that a single layer can be peeled off, which is how many 2D materials are made from ordinary crystals.
The weak, non-directional attraction arising from correlated charge fluctuations (dispersion) and related dipolar terms. In layered crystals it binds adjacent layers with energies of tens of meV per atom, roughly two orders of magnitude below in-plane covalent bonds, which enables exfoliation and the free stacking of dissimilar layers.
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Complementary metal–oxide–semiconductor (CMOS)
The way almost every digital chip is built. Each logic gate pairs a transistor that conducts with electrons with one that conducts with holes, so one of the two is always off and the gate draws almost no current except while it switches. A 2D material becomes useful for logic only if both kinds of transistor can be made from it, or if it can be added to a silicon chip without harming it.
Logic in which n- and p-channel field-effect transistors are wired in series between supply and ground, so that in either stable state one of them is off and static power is set by leakage alone. It demands n- and p-type devices of comparable drive, threshold voltages held within tens of millivolts across a wafer, and a process compatible with the silicon line – about 400 °C at most for back-end-of-line integration. For 2D channels the p-type half is the harder one: most TMDCs conduct n-type with common contact metals, and WSe2 is the leading p-type candidate.
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Space group and crystal system
A short code, such as P63 /mmc, that lists every way a crystal can be turned, reflected or shifted and still look exactly the same. The crystal system – hexagonal, tetragonal, orthorhombic, monoclinic and a few more – is the broader family the code belongs to, and it fixes the shape of the crystal’s repeating box.
The group of all symmetry operations – rotations, reflections, inversion, screw axes, glide planes and lattice translations – that map a crystal onto itself. There are 230 in three dimensions, written in Hermann–Mauguin notation: P63 /mmc for 2H-MoS2 , Pnma for SnSe. The point group left when translations are ignored fixes the crystal system (triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal or cubic) and decides which tensor properties may be non-zero, such as second-harmonic generation, piezoelectricity or spontaneous polarisation. A single layer, periodic in two directions only, belongs to one of 80 layer groups.
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Proximity effect
A property borrowed from a neighbour. Where two materials touch, the electrons near the boundary feel both, so one can take on some of the other’s character. Graphene on a magnet becomes slightly magnetic, on a semiconductor made of heavy atoms it gains spin–orbit coupling, and between two superconductors it carries a supercurrent. In a thick crystal only a thin skin is affected, but a 2D layer is all skin, so a single neighbour can change it throughout – one of the main reasons for stacking different layers together.
The induction of an order or interaction in one material by contact with another, through tunnelling and orbital hybridisation across the interface. Superconducting correlations leak into a normal conductor by Andreev reflection over a coherence length, which lets graphene, semiconductor and topological-insulator weak links carry Josephson currents and, on a topological surface, was proposed as a route to Majorana states. Exchange and spin–orbit coupling are induced by wavefunction overlap and decay within a few ångström, so an atomically thin layer lies entirely within their reach: valley splittings of a few meV in WSe2 on CrI3 , and spin–orbit coupling of the order of a millielectronvolt in graphene on WS2 or WSe2 . The size depends on twist angle, stacking and band alignment, and is usually inferred indirectly – from valley splitting in optics, weak antilocalisation and spin-relaxation anisotropy in transport, or induced gaps in tunnelling spectroscopy.
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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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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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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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