Dirac cone

Everyday term

In plain words

The shape of graphene’s energy landscape near its most important points: plot an electron’s energy against how it moves and you get two cones touching tip to tip. It means electrons in graphene act as if they had no mass and all move at the same speed, about a three-hundredth of the speed of light – much as light moves at one speed whatever its colour.

Going deeper

Left: in an ordinary semiconductor the bands curve, and a gap separates them. Right: in graphene the filled and empty bands are two cones that touch at a single point. Their sides are straight, so every electron near the tip moves at the same speed, about 10⁶ m/s, whatever its energy.

Straight lines instead of a parabola

Near the edge of an ordinary band, energy grows with the square of momentum – the way a ball’s kinetic energy grows with the square of its speed – so the band is a parabola and the electron behaves as if it had an . In graphene the valence and meet at isolated points, and near those points energy rises in proportion to momentum in every direction. Plotted against the two in-plane momenta, the bands form two cones touching at their tips.

The slope of a cone is a velocity, and it is the same for every electron near the tip: about 10⁶ m/s, roughly three hundred times slower than light. Particles whose energy is proportional to momentum and which have no rest mass are described by the Dirac equation of relativistic quantum mechanics rather than by the ordinary picture of a massive electron – hence the name.

Why graphene has them

Graphene’s is two interleaved triangular sublattices built from identical atoms. That symmetry forces the two bands to touch at the corners of the Brillouin zone, the K and K′ points that form graphene’s two . Anything that makes the two sublattices different opens a gap. In hexagonal boron nitride one sublattice is boron and the other nitrogen, and the gap is about 6 eV; aligning graphene with an hBN opens a small one; the of carbon opens one too, but so small that it is invisible in practice.

How the cones show up in measurements

Angle-resolved maps the cone directly. A moves the through the Dirac point: the conductivity is lowest there, and the carriers change from holes to electrons, which gives graphene its V-shaped response to a gate. In a strong magnetic field the Dirac electrons form with one level pinned at zero energy, producing a half-integer quantum Hall effect – the measurement that confirmed the massless Dirac picture in 2005.

For specialists

A linear, conical band crossing, as at the K and K′ points of graphene, where the energy grows in proportion to momentum and the Fermi velocity is about 10⁶ m/s. Carriers behave as massless Dirac fermions; the crossing is protected by symmetry and gapped by breaking sublattice symmetry or by spin–orbit coupling.

Where this comes from

Two-dimensional gas of massless Dirac fermions in graphene Novoselov et al. · Nature 438, 197 (2005) cited by 21,603
Hexagonal boron nitride is an indirect bandgap semiconductor Cassabois, Valvin and Gil · Nature Photonics 10, 262 (2016) cited by 1,413

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) d²E/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

Honeycomb lattice

A flat net of hexagons, like a beehive, with an atom at every corner. Graphene is the famous example, but the same pattern gives boron nitride its band gap, gives graphene its massless electrons, and gives some magnets interactions that cannot all be satisfied at once.

A triangular Bravais lattice with a two-site basis, so every site has three nearest neighbours on the other sublattice. Equivalent sublattices with nearest-neighbour hopping give Dirac cones at K and K′ (graphene); inequivalent ones open a gap with valley-contrasting Berry curvature (hBN, TMDC monolayers seen from above); buckling mixes in spin–orbit coupling (the Xenes); and bond-dependent exchange on the same geometry gives the exactly solvable Kitaev model behind the α-RuCl₃ spin-liquid programme.

Full explanation

Substrate

The base a thin film or flake sits on – often a polished slice of silicon, glass or sapphire. For a crystal one atom thick, the substrate is not just a table: its bumps, stray charges and vibrations reach right into the sheet and change how it conducts and glows. That is why the same material can behave differently on two substrates, and why the flattest, cleanest one – boron nitride – gives the best results.

The material beneath a 2D layer. It acts on the layer through surface roughness, charged impurities, surface optical phonons, dielectric screening, strain and charge transfer, and during growth it sets orientation through epitaxy. Replacing SiO₂ with hexagonal boron nitride, which is atomically flat and nearly free of dangling bonds and charge traps, improved the mobility and charge homogeneity of graphene by close to an order of magnitude.

Full explanation

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 Fe₃GeTe₂, 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.

Full explanation

Quantum Hall effect

In a strong magnetic field at low temperature, electrons flowing along a flat sheet are pushed to one side, so a voltage builds up across it. That sideways voltage, divided by the current, stops changing smoothly and locks onto exact steps set only by fundamental constants – the same in every sample, and so precise that they are used to define the ohm.

Quantisation of the Hall conductance in units of e²/h when a 2D electron system sits in a strong perpendicular field and the Fermi level lies between Landau levels, with the longitudinal resistance vanishing. Graphene shows a half-integer sequence with a level pinned at zero energy, a direct consequence of its Dirac dispersion and π Berry phase, and the effect survives to room temperature there. The zero-field analogues are the quantum anomalous and quantum spin Hall effects.

Full explanation