Density functional theory (DFT)

In plain words

The workhorse calculation of materials physics. Rather than following every electron, it works with the electron density and a recipe for how electrons avoid one another – which makes a whole crystal cheap enough to compute, and makes the accuracy depend entirely on that recipe.

Going deeper

Left: a DFT calculation is a loop. From a guessed electron density it builds a potential, solves one-electron equations, and forms a new density, repeating until nothing changes. Everything approximate is hidden in the exchange–correlation part. Right: codes that assume periodic crystals see a monolayer as an infinite stack of copies, so enough vacuum must separate them that they stop interacting.

Trading electrons for a density

Solving the quantum mechanics of every electron in a crystal exactly is hopeless: the effort grows exponentially with their number. Density functional theory rests on two results. Hohenberg and Kohn showed that the ground-state electron density alone – a function of just three coordinates – determines every ground-state property. Kohn and Sham then showed how to find that density by solving equations for independent electrons moving in an effective potential, which is tractable for hundreds or thousands of atoms.

The catch is a single unknown term, the exchange–correlation energy, which contains all the subtle many-electron physics. It has to be approximated. Local and semi-local functionals (LDA, GGA such as PBE) are fast and give good structures and vibrations; hybrid functionals, which mix in some exact exchange, are more accurate for many properties and far more expensive.

What it is good for, and what it is not

DFT is the standard tool for , formation energies, relative phase stability, , magnetic ground states and , and for screening many hypothetical materials quickly. Its best-known failure is the : the Kohn–Sham eigenvalues are not true excitation energies, and semi-local functionals typically underestimate gaps by tens of percent. are absent altogether. When quantitative gaps or optical spectra matter, DFT is the starting point for and Bethe–Salpeter calculations rather than the answer.

Traps specific to 2D materials

Most codes treat the simulation cell as the repeating unit of an infinite crystal, so a is really a stack of copies separated by vacuum. That vacuum has to be thick enough – often 15–20 Å, and far more for charged or excited-state calculations – or a must be applied. A layer without needs a dipole correction. Interlayer binding in bilayers and bulk crystals comes almost entirely from , which standard functionals miss, so a dispersion correction is required. must be switched on for heavy elements, and magnetic usually need an on-site U whose value changes and noticeably.

For specialists

Ground-state electronic structure from the density, exact in principle by Hohenberg–Kohn and made practical by the Kohn–Sham equations with an approximate exchange–correlation functional. Two dimensions need care: a vacuum gap wide enough that periodic images stop interacting or a truncated Coulomb interaction, a dispersion correction for interlayer binding, and the knowledge that semi-local functionals underestimate gaps – hybrid functionals narrow that error at far higher cost, and quantitative gaps and optical spectra need GW and Bethe–Salpeter on top.

Where this comes from

Inhomogeneous electron gas Hohenberg and Kohn · Physical Review 136, B864 (1964) cited by 53,624
Self-consistent equations including exchange and correlation effects Kohn and Sham · Physical Review 140, A1133 (1965) cited by 64,435

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

GW and the Bethe–Salpeter equation

Two calculations that take over where density functional theory, the standard one, stops. The first corrects the energy cost of adding or removing an electron; the second adds the attraction between an electron and the hole it leaves behind, which is what decides the colour of light a thin crystal absorbs.

GW replaces Kohn–Sham eigenvalues with quasiparticle energies from a screened-exchange self-energy, opening the gaps that semi-local DFT underestimates; solving the Bethe–Salpeter equation on top adds the electron–hole interaction and produces optical spectra with bound excitons. In 2D both converge painfully slowly with vacuum size and k-point sampling, because screening is non-analytic near q = 0 – a Coulomb cutoff and dense sampling around the band edge are not optional.

Full explanation

Mirror symmetry

A crystal has a mirror symmetry if reflecting it in a plane leaves it unchanged. For a 2D sheet the question that matters most is whether it looks the same from above and from below – which decides how its electrons’ spins behave, which band crossings are protected, and even how a calculation of it has to be set up.

Invariance under reflection through a plane. For a layer the horizontal mirror in the plane of the sheet matters most: it forbids an out-of-plane dipole, so a slab without it needs a dipole correction in a periodic calculation, and in a 2H monolayer that also lacks inversion it keeps spin–orbit-split spins pointing out of plane, the origin of Ising pairing. Vertical mirrors can protect band crossings – mirror-protected nodal lines in PbTaSe₂, mirror Chern numbers in the SnTe class of topological crystalline insulators – and losing a mirror at a structural transition is how some chain compounds become ferroelectric.

Full explanation

Strong electron correlation

What happens when electrons in a material repel each other so strongly that they can no longer be treated as moving independently. Each electron’s motion then depends on where all the others are, and the result can be an insulator where simple theory predicts a metal, or magnetism, superconductivity and other collective states.

The regime in which the Coulomb repulsion U between electrons is comparable to or larger than their kinetic energy, set by the bandwidth W, so single-particle band theory fails. At half filling with U well above W the Hubbard model gives a Mott insulator with local moments and antiferromagnetic exchange of order t²/U; doping or tuning U/W gives correlated metals, heavy-fermion behaviour and unconventional superconductivity. In 2D, weak screening keeps U large and moiré superlattices shrink W, so twisted bilayers make U/W tunable by gate and twist. Semi-local DFT misses the physics; DFT+U, DMFT, exact diagonalisation and quantum Monte Carlo are the usual tools.

Full explanation

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.

Full explanation