Stoichiometry

In plain words

The ratio in which the elements of a compound are combined – the numbers in its formula. MoS₂ should have two sulfur atoms for every molybdenum atom. Real crystals often deviate slightly, with a few atoms missing or a few extra squeezed in, and in a deviation of a few percent can change how a crystal conducts, glows or behaves magnetically. That is why careful work measures the composition rather than assuming the formula.

Going deeper

The formula gives the ratio of the elements in a perfect crystal – two sulfur atoms for every molybdenum in MoS₂. Real crystals lose or gain a few atoms, and the conditions during growth decide how many: a metal-rich atmosphere leaves more sulfur vacancies than a sulfur-rich one.

Recipe and reality

A chemical formula is a recipe: MoS₂ has two sulfur atoms for every molybdenum atom, CrI₃ three iodine atoms for every chromium. In a perfect crystal the ratio holds exactly. Real crystals are rarely perfect. Some sites stay empty, extra atoms slip between the layers, and atoms of one element take the place of another, so the measured composition differs slightly from the formula – the crystal is non-stoichiometric.

The deviation is set by the conditions of growth. Each kind of defect costs a formation energy that depends on how much of each element is available: growing in a sulfur-poor, metal-rich atmosphere makes cheap, a sulfur-rich one suppresses them. It also depends on the , because many defects carry charge. That is why the same compound made by different methods, or in different laboratories, can behave differently.

Why a few percent matters

In a , one missing atom in ten thousand is already a large level. Chalcogen vacancies make many n-type; extra titanium atoms sitting between the layers of TiS₂ give it so many electrons that it behaves like a metal, which long clouded whether TiS₂ is a semiconductor at all. In magnets and , the content of the alkali or rare-earth element shifts the – in AV₃Sb₅ and EuSn₂As₂ variations between samples are a main source of disagreement. Layered materials add a twist: atoms can sit in the without disturbing the layers much, so large deviations can hide in a crystal that looks perfect.

Measuring it

gives the composition near the surface, energy-dispersive X-ray spectroscopy in an electron microscope gives it at a chosen spot, and Rutherford backscattering or electron-probe microanalysis give more accurate averages. None is better than about a percent without careful standards, which is not enough to see the deviations that matter most for doping. In practice small deviations are often inferred from properties – carrier density, , transition temperature – and the growth conditions are reported so that others can reproduce them.

For specialists

The proportions of the elements in a compound, ideally the integer ratios of its formula. Real crystals are often non-stoichiometric, through , interstitials, antisites or atoms whose concentrations depend on the chemical potentials during growth and on the Fermi level, as point-defect thermodynamics describes. Deviations dope the material and can decide its ground state: -deficient TMDCs are n-type, self-intercalated titanium makes TiS₂ rather than semiconducting, the alkali content of AV₃Sb₅ and the europium content of EuSn₂As₂ shift their transitions, and melt-grown InSe varies along the boule. Composition is measured by X-ray photoelectron or energy-dispersive spectroscopy, Rutherford backscattering or electron-probe microanalysis, with accuracies of about a percent at best, so small deviations are often inferred from properties instead.

Where this comes from

Relations between the concentrations of imperfections in crystalline solids Kröger and Vink · Solid State Physics 3, 307 (1956)
Chemical potential dependence of defect formation energies in GaAs: application to Ga self-diffusion Zhang and Northrup · Physical Review Letters 67, 2339 (1991)
Impurities and electronic property variations of natural MoS₂ crystal surfaces Addou et al. · ACS Nano 9, 9124 (2015)

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.

Full explanation

Point defect

A flaw in a crystal that involves a single atomic site: an atom missing (a vacancy), an atom of the wrong kind in a site (a substitution, or an antisite when the crystal’s own two kinds of atom swap places), or an extra atom squeezed in between (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.

Full explanation

Chalcogenide, halide and pnictide

Family names for compounds, taken from the element in them that carries the negative charge. Chalcogenides contain sulfur, selenium or tellurium; halides contain fluorine, chlorine, bromine or iodine; pnictides contain phosphorus, arsenic, antimony or bismuth. Most layered crystals belong to one of these families, and the catalogue groups many of them that way.

Compounds named for their most electronegative constituent: chalcogenides of group 16 (S, Se, Te; oxides are conventionally counted apart), halides of group 17 (F, Cl, Br, I) and pnictides of group 15 (P, As, Sb, Bi; nitrides usually keep their own name). Down each group the anion grows larger and less electronegative, so bonding turns more covalent, gaps shrink and spin–orbit coupling grows. Chalcogenides give the MX₂ semiconductors and metals, halides mostly ionic, often magnetic insulators built from edge-sharing octahedra, and pnictides many metals, semimetals and the parents of the iron-based superconductors; mixed-anion compounds such as oxychalcogenides and chalcohalides combine two.

Full explanation

Metal, semimetal and insulator

The three basic ways a solid treats electricity, set by how its electrons fill the energies open to them. In a metal they fill a band of energies only partly, so the smallest push sets them moving, and it conducts at any temperature. In an insulator they fill a band completely and a wide gap separates it from the next, so nothing can move. A semimetal sits in between: two bands just touch or overlap a little, leaving a small number of mobile electrons and holes. NbSe₂ is a metal, graphene a semimetal, hBN an insulator.

A classification by where the Fermi level falls in the band structure. In a metal it crosses a partly filled band, leaving a Fermi surface, a carrier density of order one per atom (about 10²² cm⁻³) and a resistivity that falls on cooling. In a band insulator it lies in a gap of several electronvolts; a semiconductor is an insulator whose gap is small enough to be doped or gated. A semimetal has a slight overlap between conduction and valence bands (graphite, bismuth, bulk WTe₂) or bands that touch at isolated points (graphene), giving low and nearly equal densities of electrons and holes. Interactions can open gaps that band theory does not predict: Mott insulators from half-filled bands, excitonic insulators from semimetals.

Full explanation