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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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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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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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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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, so 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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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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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 energy difference between magnetisation directions, arising from spin–orbit coupling and dipolar effects; it opens the spin-wave gap that allows 2D magnetic order.
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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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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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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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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. Both kinds of order have been found in sheets only one layer thick.
Magnetically ordered states with parallel (ferromagnetic) or antiparallel (antiferromagnetic) alignment of neighbouring moments. 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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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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Edge state
Electrons that can only travel along the border of a sheet while its interior stays insulating. In a topological material these border lanes are guaranteed to exist, and in the cleanest cases they carry current without losing energy to scattering.
States localised at a sample boundary and dispersing inside the bulk gap. In quantum Hall and Chern insulators they are chiral, one-way channels; in a quantum spin Hall insulator they are helical, counter-propagating with opposite spins and protected by time reversal, giving e2 /h per edge in the ideal case – monolayer 1T′-WTe2 keeps that edge conduction up to about 100 K. Trivial edges carry states too, from dangling bonds, reconstructions and graphene’s zigzag edge, so the evidence for topology is quantised conductance and its magnetic-field dependence, not conduction at the edge alone.
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Stacking fault
A mistake in the order in which a crystal’s layers are stacked – one layer shifted or turned from where the pattern says it should sit, like one misaligned sheet in a neatly squared stack of paper. Layered crystals get them easily, because the weak bonds between layers barely care. But the stacking can change a material’s magnetism, symmetry or electronic behaviour, so faults can make one crystal behave unlike the next.
A planar defect in which the stacking sequence is interrupted – for example …ABCABABC… in place of …ABCABCABC… – without a change in the layers themselves. In van der Waals crystals the energy cost is small, so faults form during growth, on cooling through structural transitions, and during cleaving and handling. They produce diffuse streaks along c* in diffraction and can alter interlayer exchange, symmetry-allowed responses and transition temperatures, as in α-RuCl3 , where faulted regions order near 14 K rather than 7 K.
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Exfoliation
Peeling thin layers off a bulk crystal. The famous version uses sticky tape: press it on a crystal, pull it away, repeat, and some flakes end up only one layer thick. Other versions do the same in a liquid to make large amounts of flakes at once.
Separation of layers from a layered bulk crystal by overcoming interlayer van der Waals bonding – mechanically with adhesive tape or stamps, which gives the highest-quality flakes but low yield, or in liquids by sonication, shear or electrochemical intercalation, which scales but yields smaller, more defective nanosheets with broad size distributions.
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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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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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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, which makes 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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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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