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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Curie and Néel temperature
The temperature at which a magnet loses its order on warming: the Curie temperature for a ferromagnet, whose spins all point one way, and the Néel temperature for an antiferromagnet, whose neighbouring spins alternate. Above it the spins point in every direction and the material is no longer magnetic.
The critical temperatures of ferromagnetic and antiferromagnetic order. In 2D magnets they depend on layer number and on magnetic anisotropy rather than on exchange alone: CrI3 orders at 45 K as a monolayer against 61 K in the bulk, and Fe3 GeTe2 can be pushed to room temperature by ionic gating. Quote the layer number, the substrate and the measurement with any value.
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Mermin–Wagner theorem
A result showing that in a strictly two-dimensional material, heat at any temperature above absolute zero destroys the long-range order of atomic magnets that are free to point in any direction. Real 2D magnets exist because their atomic magnets prefer a particular direction, which breaks the theorem’s assumptions.
No spontaneous breaking of a continuous symmetry at finite temperature in 2D with short-range interactions; magnetic anisotropy evades it.
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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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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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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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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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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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Magnetometry
Measuring how magnetic a sample is. 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 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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Magneto-optical effect (MOKE)
Light reflected from a magnet comes back with its polarisation – the direction its waves vibrate in, which polarised sunglasses filter – slightly turned, by an amount that follows the magnetisation. Because it needs nothing but a focused laser beam, it can read the magnetic state of a flake far narrower than a hair without touching it.
Rotation and ellipticity of reflected polarised light in a magnetised medium (the Kerr effect), and the differential absorption of circular polarisations (RMCD). Sensitivity down to a single layer is what established magnetic order in CrI3 and Cr2 Ge2 Te6 , and both give layer-resolved hysteresis loops, domain images and the sign of interlayer coupling without any contacts.
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Linear and circular dichroism
When a material absorbs light differently depending on how the light is polarised. In linear dichroism the difference is between light vibrating along one direction of the crystal and across it, as in a polarising filter; in circular dichroism, between light whose vibration turns one way and the other. Shining polarised light on a flake this way reveals hidden directions in it – its crystal axes, its magnetic order, which valley its electrons sit in – without touching it.
Polarisation-dependent absorption: linear dichroism between orthogonal linear polarisations, from in-plane anisotropy of the lattice or of an order that breaks rotational symmetry; circular dichroism between left and right circular polarisations, from broken mirror or time-reversal symmetry. In black phosphorus, ReS2 and the transition metal trichalcogenides linear dichroism follows the crystal axes; in FePS3 it appears with zigzag antiferromagnetic order and tracks it with temperature. In monolayer TMDCs valley-selective circular dichroism at K and K′ underlies optical valley polarisation, and magnetic circular dichroism – reflective in the visible, XMCD at X-ray absorption edges – measures magnetisation, with sum rules separating spin and orbital moments.
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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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Second-harmonic generation (SHG)
Shine intense light on a crystal and a little of it comes back at exactly twice the frequency – half the wavelength, which is how a green laser pointer turns invisible infrared into green light. It happens only if the crystal has no centre of symmetry, which makes it a quick, contactless test of symmetry, layer number and crystal orientation.
A second-order nonlinear response, forbidden in centrosymmetric media, so in 2H TMDCs it appears for odd layer numbers and vanishes for even ones. The polarisation dependence gives crystal orientation, and so the twist angle in a stack; the intensity follows layer number and strain; and it detects broken inversion symmetry in ferroelectric and magnetic layers where no other quick probe exists.
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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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Air stability
How long a material survives in ordinary air. Some 2D crystals keep for years on a shelf; others change within hours as oxygen, water and light attack them – and a measurement on a degraded sample says little about the real material.
Resistance to reaction with oxygen, water and light under ambient conditions, which often decides what can be measured at all. Few-layer black phosphorus takes up water and degrades visibly within hours; CVD-grown TMDC monolayers age over months, oxidising along grain boundaries and collecting organic contamination. Report exposure time and handling with any result, and move to glovebox transfer and encapsulation when a material’s stability is short.
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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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