Weyl and Dirac semimetal

Everyday term

In plain words

Crystals in which the highest filled band of electron energies and the lowest empty one touch only at isolated points, instead of overlapping broadly or leaving a gap between them. Electrons near such a point behave as if they had no mass – the crystal is a three-dimensional cousin of graphene – and its surface carries loose-ended arcs of electron states that no ordinary metal can have.

Going deeper

A Dirac point is a fourfold band touching protected by inversion and time-reversal symmetry together. Break either and it splits into two Weyl nodes of opposite chirality, which act as sources and sinks of Berry curvature and are joined across each surface by open Fermi arcs.

Band touchings that cannot be removed

In most crystals the conduction and either overlap broadly, giving a metal, or are separated everywhere, giving an . In a Weyl or Dirac semimetal they touch at isolated points, with energy rising linearly in every direction away from the touching, so carriers there behave like relativistic particles.

A Weyl node is a twofold touching carrying a , plus or minus. It is protected: no small perturbation can open a gap at it, because a node can only disappear by meeting a partner of opposite chirality, and the two chiralities must always balance across the Brillouin zone. A Dirac point is two Weyl nodes of opposite chirality sitting on top of each other, held together by inversion and . Break inversion, as in TaAs, or time reversal, with magnetism, and they separate.

Arcs on the surface

Each Weyl node acts as a source or sink of in momentum space – a monopole. A consequence appears at any surface: the states there form open arcs that begin at the projection of one node and end at the projection of its partner. No ordinary metal can have such an open ; a closed loop is the only option.

These Fermi arcs are what confirmed the first Weyl semimetal, TaAs, in 2015, through angle-resolved that mapped the and found arcs ending on the projected nodes. Where a crystalline symmetry such as a mirror protects a whole line of touchings rather than isolated points, the same reasoning gives nodal-line semimetals with flat drumhead surface states.

What follows, and what to watch for

The chirality of the nodes leads to the chiral anomaly: parallel electric and magnetic fields pump electrons from one node to the other, giving a negative longitudinal . Magnetic Weyl semimetals show large responses, and combined with linear bands produces unusually strong nonlinear optical effects.

The experimental caveats are real. Negative magnetoresistance can also come from current jetting in small crystals with point contacts, so careful geometry and consistency checks are needed. Most examples are three-dimensional crystals, several of them layered – the telluride family and MoTe₂ among them – and in the 2D limit the same physics reappears differently: -WTe₂ is a rather than a semimetal.

For specialists

Semimetals whose bands touch at isolated points with linear dispersion in every direction. A Dirac point is fourfold degenerate and protected by time-reversal and inversion symmetry together; breaking either splits it into pairs of twofold Weyl nodes of opposite chirality, which act as sources and sinks of Berry curvature and are joined across the surface by Fermi arcs. Where a mirror or another crystalline symmetry protects a whole line of band touchings instead of isolated points, the same physics gives a nodal-line semimetal with drumhead surface states. The chiral anomaly, large anomalous Hall response and strongly non-linear optical effects follow from that structure.

Where this comes from

Discovery of a Weyl fermion semimetal and topological Fermi arcs Xu et al. · Science 349, 613 (2015) cited by 3,278
Weyl and Dirac semimetals in three-dimensional solids Armitage et al. · Reviews of Modern Physics 90, 015001 (2018) cited by 4,563
Observation of the quantum spin Hall effect up to 100 kelvin in a monolayer crystal Wu et al. · Science 359, 76 (2018) cited by 823

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

Chirality

The property of an object that cannot be turned into its own mirror image, like a left and a right hand or a corkscrew that turns one way. Molecules can be chiral, and so can crystals: in tellurium the atoms form screw-shaped chains that wind either to the left or to the right. Electrons, spins and vibrations can take on a handedness too, and chiral materials can treat left- and right-turning light, or spins pointing one way or the other, differently.

The absence of improper symmetry operations (mirror planes, inversion, rotoreflections), so that an object and its mirror image are distinct enantiomorphs. Chiral crystals belong to the 65 Sohncke space groups, such as trigonal tellurium (P3₁21 or P3₂21), whose helical chains give natural optical activity, current-induced magnetisation and radial spin textures through spin–orbit coupling. Chirality also describes Weyl fermions, whose chirality sets the chiral anomaly and the chiral magnetic effect; edge modes that propagate one way; spin spirals and skyrmions with a fixed sense of rotation from the Dzyaloshinskii–Moriya interaction; phonons carrying angular momentum in the valleys of TMDCs; and proposed chiral orders such as loop currents in kagome metals and Mn₃Si₂Te₆. Barron’s distinction between true and false chirality decides which physical effects a given handedness allows.

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

Time-reversal symmetry

The idea that the laws governing electrons would look just as valid if you ran a film of them backwards. Reversing time flips every motion and every spin, so a material with no magnetism of its own usually keeps this symmetry, while a magnet – or a magnetic field – breaks it. Which case applies decides a lot: some protected edge currents need the symmetry, and others need it broken.

The antiunitary operation t → −t, which reverses momenta and spins; for spin-½ electrons T² = −1, giving Kramers degeneracy of every state at time-reversal-invariant momenta. It is broken by magnetic order or applied fields. Its presence protects the helical edge states of quantum spin Hall insulators and makes Berry curvature odd in k, so valley Hall but not anomalous Hall effects survive; its breaking allows Chern insulators, the anomalous Hall effect and Kerr rotation, the standard probes of spontaneous breaking in correlated phases.

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

Surface state

Electron states that exist only at the surface of a crystal, because the crystal stops there. Atoms at a surface have lost their neighbours on one side, so electrons can take energies there that the inside of the crystal does not allow. Usually such states are a nuisance that traps charge. In a topological insulator they are the point: the inside insulates while the surface is guaranteed to conduct, with each electron’s spin tied to its direction of motion.

States localised at a crystal surface and decaying into the bulk, at energies forbidden in the bulk for that in-plane momentum. Ordinary Tamm and Shockley states come from the broken periodicity, dangling bonds or reconstruction, and can be removed or shifted by passivation. Topological surface states are required by a non-trivial bulk invariant: a single spin–momentum-locked Dirac cone in Bi₂Se₃-family topological insulators, protected against backscattering by time-reversal symmetry, and Fermi arcs joining the projections of Weyl points in Weyl semimetals. In thin films the top and bottom surface states hybridise and open a gap below a critical thickness, about six quintuple layers for Bi₂Se₃. In a 2D material the boundary of a topological phase is its edge, and the counterpart of the surface state is the edge state.

Full explanation

Magnetoresistance

A change in electrical resistance when a magnetic field is applied. It is how the read head of a hard disk senses its bits, and in some layered magnets the change is enormous – a stack can go from barely conducting to conducting – which is how the magnetic state of a few atomic layers is read out electrically.

The field dependence of resistivity. Ordinary orbital magnetoresistance is positive and usually small, though enormous in compensated semimetals such as WTe₂; giant and tunnelling magnetoresistance come from spin-dependent transport across magnetic layers; colossal magnetoresistance from a field-driven change of electronic state. In van der Waals stacks a few layers of CrI₃ act as a spin filter whose resistance changes by orders of magnitude as adjacent layers align, which is how layer-by-layer magnetic order became measurable in transport.

Full explanation

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.

Full explanation