Hall effect

In plain words

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 , 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 is magnetic.

Going deeper

A magnetic field pushes moving charges sideways, so a voltage builds up across a current-carrying strip. Its slope against field counts the carriers – the fewer there are, the steeper – and its sign tells electrons from holes. In a magnet an extra part follows the magnetisation and stays at zero field, tracing a loop.

Counting carriers with a magnet

Edwin Hall found the effect in 1879, in a thin gold leaf. A carrier moving through a magnetic field feels a sideways force; charge piles up on one edge until the electric field of that charge balances the force, and the voltage across the sample measures it. The fewer the carriers, the faster each must move to carry the same current and the larger the voltage, so the signal is biggest in exactly the dilute systems that 2D are.

For a sheet the numbers are simple: 10¹² carriers per cm² in a field of 1 tesla give a Hall resistance of about 620 Ω, whatever the thickness. Electrons and holes moving with the same current are pushed towards the same edge but carry opposite charge, so the sign of the voltage tells which of them carries the current.

Hall bars and their pitfalls

In practice a flake is patterned into a Hall bar: a strip with current contacts at the ends and pairs of voltage contacts along the sides, so that the resistance along the strip and across it are measured together, free of the . Irregular flakes use the arrangement of four contacts around the edge instead.

Three things commonly go wrong. Voltage contacts that are not exactly opposite pick up part of the ordinary resistance; reversing the field and keeping only the part that changes sign removes it. With electrons and holes both present, as in , a single density read off the slope means little, and the curve has to be fitted with two carrier types. And the Hall mobility of a channel need not agree with its , which also carries the effects of contacts and traps, so the two should not be compared as if they were one number.

The anomalous Hall effect

In a magnetic conductor the Hall voltage has a second part that follows the magnetisation rather than the applied field. It traces a loop as the field sweeps and survives at zero field, so it reads out the magnetic state of a flake far too small for a . Atomically thin Fe₃GeTe₂, for example, was followed this way, including an ionic gate raising its to room temperature.

Its origin took decades to settle. Part comes from the itself – the of the occupied states acts like a magnetic field in momentum space – and part from -dependent scattering off impurities. In a magnetic the band-structure part becomes exactly quantised, giving the effect seen in MnBi₂Te₄ and in magnetically doped (Bi,Sb)₂Te₃.

For specialists

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.

Where this comes from

On a new action of the magnet on electric currents Hall · American Journal of Mathematics 2, 287 (1879)
Anomalous Hall effect Nagaosa et al. · Reviews of Modern Physics 82, 1539 (2010)
Gate-tunable room-temperature ferromagnetism in two-dimensional Fe₃GeTe₂ Deng et al. · Nature 563, 94 (2018) cited by 2,639

Electron hole

Usually just called a hole: the empty place an electron leaves behind in an otherwise full band. It behaves like a particle of its own with a positive charge. When a neighbouring electron steps into the gap, the gap moves the other way – like the empty space in a queue of cars, which travels backwards as each car edges forward. Many materials conduct mainly with holes, and chips need both kinds of carrier.

An unoccupied state in an otherwise filled band, usually near the valence-band maximum, treated as a quasiparticle with positive charge and a positive effective mass set by the band curvature. Holes carry the current in p-type semiconductors and give a positive Hall coefficient. In 2D semiconductors good p-type transport is harder to obtain than n-type, largely because metal contacts pin the Fermi level nearer the conduction band.

Full explanation

Resistivity and sheet resistance

Two ways to put a number on how strongly a material resists a current. Resistivity belongs to the material itself, whatever the size of the piece: copper’s is tiny, glass’s enormous. For a sheet so thin that its thickness is hard to pin down, the sheet resistance is used instead: the resistance of a square of it, measured from one edge to the opposite one, which comes out the same for a square of any size. It is quoted in ohms per square.

Resistivity ρ relates electric field to current density, in Ω·m or Ω·cm, so that a bar of length L and cross-section A has resistance ρL/A. For a film of thickness t the sheet resistance is ρ/t, quoted in Ω per square, and a strip L long and W wide has the sheet resistance times L/W. For an atomically thin layer it is the natural quantity, since the thickness is a convention, and it equals 1/neμ for sheet density n and mobility μ. Both are measured with four probes – a Hall bar, a collinear four-point probe or the van der Pauw method for arbitrary shapes – so that the voltage is sensed away from the current contacts and the contact resistance drops out.

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

Magnetometry

Measuring how magnetic a sample is. For a crystal you can hold, this is routine: it is moved through a coil or past a superconducting sensor called a SQUID, which picks up its tiny magnetic field. A flake one atom thick is a different matter – its magnetism is far too weak for such instruments and is swamped by the substrate it sits on – so 2D magnets are usually studied with light, with currents, or with single-spin sensors in diamond that hover nanometres above the flake.

Measurement of magnetic moment or stray field. SQUID and vibrating-sample magnetometry of bulk crystals, with sensitivities around 10⁻⁸ emu (10⁻¹¹ A·m²), give magnetisation against field and temperature, from which ordering temperatures, saturation moments and anisotropy follow; torque and Hall magnetometry and magnetic force microscopy extend this to small samples. Monolayer flakes lie far below SQUID sensitivity and are swamped by substrate diamagnetism and magnetic contamination, so their magnetism is read optically (MOKE, RMCD), electrically (anomalous Hall effect, tunnelling magnetoresistance) or with scanning nitrogen-vacancy magnetometry, which images stray fields quantitatively at the nanoscale. Claims of room-temperature ferromagnetism in thin films based on bulk magnetometry, as for monolayer VSe₂, need element-specific (XMCD) or local confirmation.

Full explanation

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

Spin

A built-in property of every electron that makes it a tiny magnet. Measure it along any direction and you only ever find one of two answers, ‘up’ or ‘down’. Despite the name, nothing is actually spinning – the word stuck from an early picture – but the magnetism is real: countless electron spins lined up are what make a fridge magnet stick. Spintronics tries to carry information in spin rather than in charge.

The intrinsic angular momentum of the electron, ħ/2, with a magnetic moment of almost exactly one Bohr magneton; a measurement along any axis yields one of two values. In solids spin couples to orbital motion through spin–orbit coupling and to other spins through exchange; in 2D materials it sets magnetic order, spin–valley locking and how long a spin survives as a carrier of information.

Full explanation