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.
Full explanation →
Photon
The smallest possible amount of light – one indivisible packet of it. The packets of a given colour all carry the same energy, and blue ones carry more than red: that is why ultraviolet light gives you sunburn and red light, however bright, does not. A semiconductor absorbs only photons with enough energy to cross its band gap and gives light back at about the gap’s energy – which is why LEDs of different colours are made from different semiconductors.
The quantum of the electromagnetic field, with energy E = hν and momentum h/λ but no mass or charge. Visible photons carry about 1.7–3.1 eV, comparable to the band gaps of semiconducting TMDCs, but a momentum far smaller than the crystal momenta across the Brillouin zone, so optical transitions are vertical and an indirect gap needs a phonon to make up the difference. Single-photon emission is established by antibunching in photon-correlation measurements.
Full explanation →
Fermi-level pinning
When the junction between a metal and a semiconductor behaves the same whichever metal is used. The choice of metal ought to set the height of the energy step electrons must climb to get into the semiconductor, but stray electron states at the interface take up charge and hold the step at much the same height every time. It makes low-resistance contacts hard to engineer.
The fixing of the Fermi level at a metal–semiconductor interface by interface states, so the Schottky barrier barely depends on the metal work function.
Full explanation →
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.
Full explanation →
Effective mass
How heavy an electron seems as it moves through a crystal. Pushed by an electric field, it speeds up as if its mass were different from a free electron’s – lighter in some materials, heavier in others, and in graphene as if it had no mass at all. Light carriers generally make for faster devices.
The mass m* with which a carrier near a band edge responds to forces as if it were free, set by the band curvature: 1/m* = (1/ħ2 ) d2 E/dk2 . It enters the mobility (μ = eτ/m*), the density of states, confinement energies and tunnelling rates, and in anisotropic crystals it is a tensor – in black phosphorus the armchair and zigzag masses differ several-fold. Graphene’s linear bands have no curvature, and its carriers are described as massless Dirac fermions; their cyclotron mass, the Fermi energy divided by the square of the Fermi velocity, grows with the square root of the carrier density.
Full explanation →
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 – 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.
Full explanation →
Fermi level
Roughly, the energy up to which a material’s electron states are filled – like the water line in a partly filled glass. Where it sits relative to the band gap decides how many charges can move and whether they are electrons or holes, the empty places electrons leave behind. A gate voltage raises and lowers it, like pouring water in or out.
The electrochemical potential of electrons: the energy at which a state has 50 % occupation in thermal equilibrium. Its position relative to the band edges sets carrier density and type; at metal–semiconductor contacts, interface states can pin it and fix the Schottky barrier.
Full explanation →
Fermi surface and nesting
The Fermi surface is the boundary between the states a metal’s electrons fill and the ones they leave empty, drawn on a map of every way an electron can move through the crystal; its shape decides how the metal conducts. Nesting is when large flat stretches of that boundary can be slid onto each other by one single shift – a metal like that is unstable and tends to settle into a regular ripple of charge or spin.
The constant-energy surface separating occupied from empty states, whose geometry governs transport, screening and instabilities. Nesting – parallel sheets connected by a single wavevector q – peaks the bare susceptibility at q and is the textbook route to charge and spin density waves. In real layered metals the ordering wavevector often follows momentum-dependent electron–phonon coupling instead, so a nesting figure alone does not settle the mechanism.
Full explanation →
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.
Full explanation →
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.
Full explanation →
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.
Full explanation →
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 – 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.
Full explanation →
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.
Full explanation →
Pump–probe spectroscopy
A way to film events that last a millionth of a millionth of a second. A first, strong flash of laser light – the pump – disturbs the sample; a second, weaker flash – the probe – arrives a precisely set moment later and measures how the sample looks then. Repeating this with the delay changed step by step builds up a slow-motion film of how the electrons and atoms settle back. The delay is set by sending one flash along a longer path: a tenth of a millimetre more makes it a third of a picosecond later.
A stroboscopic measurement in which an ultrashort pump pulse excites the sample and a delayed probe pulse records the change, as a function of a delay set by an optical delay line, with a time resolution limited by the pulse durations – tens of femtoseconds routinely, a few with dedicated sources. The probe can be optical reflectivity or transmission, terahertz conductivity, photoemission (time-resolved ARPES), or X-ray or electron diffraction, giving access to carrier thermalisation and cooling, exciton formation, interlayer charge transfer, coherent phonons, and the melting, recovery or switching of ordered phases such as charge density waves and excitonic insulators. Its power lies in separating processes by timescale: electrons respond within femtoseconds, the lattice within picoseconds.
Full explanation →
Quasiparticle gap versus optical gap
Two answers to ‘how much energy does it take to excite this material?’. Adding a free electron and a free hole costs more; creating a bound electron–hole pair with light costs less, and in 2D materials the difference is large.
The quasiparticle gap is the energy to add an electron and a hole independently; the optical gap is smaller by the exciton binding energy.
Full explanation →
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.
Full explanation →
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.
Full explanation →
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.
Full explanation →