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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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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Magnetic moment
The strength and direction of the tiny magnet an atom or an electron carries. Every electron is a small magnet through its spin, and its motion around an atom can add more. In most atoms the electrons pair up and their magnets cancel; atoms with unpaired electrons – chromium, manganese, iron, nickel, the rare earths – keep a moment. How these moments line up decides whether a material is a ferromagnet, an antiferromagnet or no magnet at all.
The magnetic dipole of an atom, ion or electron, measured in Bohr magnetons (μB ≈ 9.27 × 10−24 J/T). In insulating magnets it is a local moment set mainly by the spins of unpaired d or f electrons – about one Bohr magneton per unpaired electron for 3d ions, whose orbital moment the crystal field largely quenches: 3 μB for Cr3+ , 5 μB for Mn2+ . Ordered moments from neutron diffraction often come out below these values, 4.04 μB per manganese in MnBi2 Te4 . In metallic magnets the moment belongs partly to itinerant electrons and need not be near a whole number, as with 1.11 μB per iron in Fe3 GeTe2 .
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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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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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Photoluminescence (PL)
Light given off by a material after it has absorbed light – what a highlighter pen does under a black light. In 2D semiconductors the colour and brightness of that glow report how many layers there are, how clean the sample is and whether it is carrying extra charge.
Radiative recombination following optical excitation. In monolayer TMDCs the spectrum is dominated by excitons and trions rather than free carriers, so peak energy, linewidth and quantum yield track layer number, strain, dielectric environment, doping and defect density. The indirect-to-direct crossover makes the monolayer far brighter than the bilayer, and chemical treatment of defects can bring the yield close to unity.
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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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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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Chirality
The property of an object that cannot be turned into its own mirror image, like a left and a right hand or a corkscrew that turns one way. Molecules can be chiral, and so can crystals: in tellurium the atoms form screw-shaped chains that wind either to the left or to the right. Electrons, spins and vibrations can take on a handedness too, and chiral materials can treat left- and right-turning light, or spins pointing one way or the other, differently.
The absence of improper symmetry operations (mirror planes, inversion, rotoreflections), so that an object and its mirror image are distinct enantiomorphs. Chiral crystals belong to the 65 Sohncke space groups, such as trigonal tellurium (P31 21 or P32 21), whose helical chains give natural optical activity, current-induced magnetisation and radial spin textures through spin–orbit coupling. Chirality also describes Weyl fermions, whose chirality sets the chiral anomaly and the chiral magnetic effect; edge modes that propagate one way; spin spirals and skyrmions with a fixed sense of rotation from the Dzyaloshinskii–Moriya interaction; phonons carrying angular momentum in the valleys of TMDCs; and proposed chiral orders such as loop currents in kagome metals and Mn3 Si2 Te6 . Barron’s distinction between true and false chirality decides which physical effects a given handedness allows.
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Interlayer exciton
An electron in one layer bound to a hole in the layer next door. Because the two sit in different sheets, the pair lives far longer than an ordinary exciton and can be pushed around with an electric field.
An exciton whose electron and hole sit in adjacent layers of a type-II heterostructure. The spatial separation gives lifetimes orders of magnitude longer than intralayer excitons and a permanent out-of-plane dipole that makes the energy gate-tunable; in a twisted bilayer the moiré potential traps them on a regular grid, turning a heterobilayer into an array of nearly identical emitters.
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Exchange interaction
The force that lines up the tiny magnets of neighbouring atoms in a magnetic material – all the same way in a ferromagnet, alternating in an antiferromagnet. Despite what it does, it is not magnetic in origin. It comes from the electric repulsion between electrons, combined with a quantum rule that keeps two electrons from sharing the same state. Together these make their energy depend on whether their spins point the same way. It is far stronger than the magnetic pull between atomic magnets, which is why iron stays magnetic up to 770 °C.
The spin-dependent energy that arises from the Coulomb repulsion together with the antisymmetry of the many-electron wavefunction, modelled as a coupling J between neighbouring spins in Heisenberg, Ising or XY form. In insulators it is mostly mediated by ligands – superexchange, whose sign follows the Goodenough–Kanamori rules: antiferromagnetic for metal–ligand–metal angles near 180°, often ferromagnetic near 90°, as in the edge-sharing octahedra of CrI3 – in metals by conduction electrons (RKKY, double exchange), and with strong spin–orbit coupling it becomes anisotropic or bond-dependent (Kitaev). In van der Waals magnets the intralayer exchange of a few meV sets the ordering scale, while a much weaker interlayer exchange decides ferro- or antiferromagnetic stacking and depends on the stacking itself; ordering a monolayer additionally requires anisotropy, by the Mermin–Wagner theorem.
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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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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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Spin–orbit coupling
A link between an electron’s spin – a tiny built-in magnet – and the way it moves. It is strongest in heavy atoms, and in some 2D materials it is strong enough to split energy levels and make spin useful for devices.
The relativistic interaction between an electron’s spin and its orbital motion, growing steeply with atomic number. In TMDC monolayers with broken inversion symmetry it splits the valence band by roughly 150–450 meV, producing spin–valley locking, and it underlies Ising superconductivity, topological gaps and magnetic anisotropy.
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Spin–valley locking
In certain single-layer materials an electron’s spin – its built-in magnetic direction – is tied to the valley it sits in: flip one and you flip the other. Each valley responds only to light whose waves corkscrew one way – clockwise or anticlockwise – so both can be addressed with light.
In TMDC monolayers, spin–orbit coupling and broken inversion symmetry tie the spin of band-edge states to their valley.
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Ising superconductivity
A kind of superconductivity in some single-layer materials in which electron spins are locked pointing out of the sheet. The locking lets it survive magnetic fields along the sheet far stronger than would normally destroy superconductivity.
Superconductivity in which spin–orbit coupling pins electron spins out of plane with opposite orientation in opposite valleys, protecting pairing against in-plane magnetic fields.
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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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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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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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Point defect
A flaw in a crystal that involves a single atomic site. An atom can be missing (a vacancy), of the wrong kind (a substitution, or an antisite when the crystal’s own two kinds of atom swap places), or squeezed in between the others (an interstitial). Every real crystal has some. In a sheet a few atoms thick each one sits at the surface, so a handful of them can change how the material conducts, glows or reacts far more than in a thick crystal.
A defect confined to one or a few lattice sites – vacancy, interstitial, substitutional impurity, antisite or adatom – or a small complex of them. What it does is set by the levels it introduces: shallow levels dope, deep levels trap carriers and act as non-radiative recombination centres, and an isolated deep level in a wide gap can emit single photons. Formation energies and charge states depend on the chemical conditions of growth and on the Fermi level. In 2D materials every defect is a surface defect, less screened and more exposed to the environment, and densities of 1012 to 1013 per cm2 are typical of TMDC monolayers.
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Sensor
A device that turns something in its surroundings – a gas, a magnetic field, a pressure, a molecule in blood – into an electrical signal. 2D materials make unusually sensitive sensors because every atom is on the surface: a few molecules landing on a single layer change how well it conducts. Graphene also makes very good magnetic-field sensors. The hard part is not sensitivity but selectivity – responding to one thing and ignoring everything else – and staying stable for years.
Transducers in which a stimulus changes an electrical property of a 2D layer: chemiresistive and field-effect gas and humidity sensors, in which adsorbates transfer charge and shift the conductance (graphene registered single NO2 molecules in 2007); Hall sensors, in which graphene’s low carrier density and high mobility give a large, temperature-stable Hall voltage; biosensors with receptors on a graphene or MoS2 channel that bind target molecules; and strain, pressure and light sensors. Sensitivity comes from the all-surface geometry and the low density of states; the limits are selectivity, drift and hysteresis from humidity and contamination, recovery time, device-to-device variation and calibration, addressed by functionalisation, encapsulation and arrays read by pattern recognition.
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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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Skyrmion and the Dzyaloshinskii–Moriya interaction
A skyrmion is a tiny whirl in the magnetisation of a material: the magnetic direction turns smoothly from pointing down at its centre to pointing up at its rim. It behaves like a particle – a small current can move it, and it is hard to destroy, because undoing the whirl would mean flipping a whole region at once – which makes skyrmions candidates for dense, low-power magnetic memory. Most are held together by the Dzyaloshinskii–Moriya interaction, a twisting force between neighbouring magnetic atoms that appears only where a mirror symmetry is missing.
A particle-like spin texture whose magnetisation wraps the unit sphere once, giving it an integer topological charge. Bloch-type skyrmions form in bulk chiral magnets (MnSi, 2009) and Néel-type ones at interfaces and in polar crystals. They are stabilised by the antisymmetric Dzyaloshinskii–Moriya interaction – proportional to the cross product of neighbouring spins, arising from spin–orbit coupling where inversion symmetry is broken – competing with exchange, anisotropy and the applied field; dipolar-stabilised skyrmion bubbles form without it. Among 2D magnets, Fe3 GeTe2 hosts bubbles and, in WTe2 /Fe3 GeTe2 heterostructures, interface-induced Néel skyrmions. Their emergent magnetic field can add a topological contribution to the Hall effect, and their current-driven motion by spin–orbit torques underlies racetrack-memory proposals.
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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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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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Rashba effect
Some crystals are not the same seen from above and from below, because they sit on a substrate, have a field across them, or are simply built that way. In such a crystal a moving electron feels a magnetic field that depends on which way it is going. Its spin then winds around its direction of motion, which is what lets an electric field steer spins.
A momentum-dependent spin splitting from spin–orbit coupling in a structure without inversion symmetry, H = α(σ × k)·ẑ, locking spin perpendicular to in-plane momentum and winding it around the Fermi contour. The coefficient α follows the potential gradient and the atomic spin–orbit strength, so it is tunable by gating, by the substrate and by the choice of heavy elements; BiTeI splits its bands far enough for the two spin-split Fermi surfaces to be resolved separately.
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