Quantum Hall effect

Everyday term

In plain words

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.

Going deeper

In a strong perpendicular field, electrons in the interior of a 2D sample orbit on the spot, and only one-way channels at the edge carry current. The Hall resistance then locks onto plateaus at h/νe², with the longitudinal resistance vanishing – in graphene at the half-integer sequence ν = 2, 6, 10.

Landau levels and edge channels

A magnetic field perpendicular to a 2D electron system bends carriers into circular orbits, and quantum mechanics allows only certain orbit energies: the Landau levels. In the interior, an electron in a completed orbit goes nowhere, so the bulk is whenever the lies between levels. At the boundary the orbits cannot close; they skip along the edge, always in the same direction set by the field.

These carry the current. Because there is no channel going the other way on the same edge, an electron has nothing to scatter into, and transport along the edge is dissipationless. The number of channels equals the number of filled Landau levels, which is why the Hall conductance comes in multiples of e²/h and the longitudinal resistance drops to zero.

Why it is so exact

The plateaus are flat because localises the states between Landau levels: adding carriers fills localised states that carry no current, so the measured conductance does not change until the next level is reached. The quantised value depends only on e and h, not on the material, its geometry or its purity, and has been reproduced to parts in a billion.

That precision made it metrological. From 1990 the ohm was maintained through the quantum Hall effect, and since the 2019 redefinition of the SI, with e and h fixed by definition, it realises the ohm directly. Its discovery in 1980 brought the 1985 Nobel Prize.

What graphene changed

Graphene shows a different sequence. Its Dirac dispersion gives a Landau level pinned at zero energy, shared between , so plateaus appear at ν = ±2, ±6, ±10 – a half-integer sequence in terms of the four-fold and degeneracy – and the π Berry phase of its carriers shows up in the phase of the oscillations.

Because the level spacing in graphene is large, the effect survives to room temperature in strong fields, unlike the millikelvin conditions usual in . In cleaner samples the degeneracies lift and the fractional quantum Hall effect appears, where interactions between electrons produce plateaus at fractional filling. The zero-field relatives – the and effects – reproduce the edge-channel picture without a magnet.

For specialists

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.

Where this comes from

New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance von Klitzing et al. · Physical Review Letters 45, 494 (1980) cited by 7,111
Experimental observation of the quantum Hall effect and Berry’s phase in graphene Zhang et al. · Nature 438, 201 (2005) cited by 13,491

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

Edge state

Electrons that can only travel along the border of a sheet while its interior stays insulating. In a topological material these border lanes are guaranteed to exist, and in the cleanest cases they carry current without losing energy to scattering.

States localised at a sample boundary and dispersing inside the bulk gap. In quantum Hall and Chern insulators they are chiral, one-way channels; in a quantum spin Hall insulator they are helical, counter-propagating with opposite spins and protected by time reversal, giving e²/h per edge in the ideal case – monolayer 1T′-WTe₂ keeps that edge conduction up to about 100 K. Trivial edges carry states too, from dangling bonds, reconstructions and graphene’s zigzag edge, so the evidence for topology is quantised conductance and its magnetic-field dependence, not conduction at the edge alone.

Full explanation

Disorder and localisation

Every real crystal has some disorder: missing or misplaced atoms, impurities, stray charges in the substrate, a random mixture of elements in an alloy. Electrons moving through it scatter off these imperfections, which raises the resistance. If the disorder is strong enough, electron waves become trapped in small regions – localised – and the material stops conducting, even if it would be a metal when perfect. In a sheet one atom thick, disorder in its surroundings matters especially.

Deviations from perfect periodicity – point defects, substitutional or alloy disorder, interface roughness, charged impurities and strain variations in the substrate – that scatter carriers and, beyond a threshold, localise their wavefunctions through interference (Anderson localisation). Scaling theory predicts that non-interacting electrons in two dimensions without spin–orbit coupling are localised by any disorder; its precursor, weak localisation, appears as a logarithmic rise of resistance at low temperature that a magnetic field suppresses, while strong spin–orbit coupling gives weak antilocalisation. In graphene on SiO₂, charged impurities break the sheet near charge neutrality into electron–hole puddles, which hBN encapsulation reduces. Strong disorder leads to hopping conduction, broadens optical lines and smears phase transitions; in frustrated magnets and correlated oxides, structural or cation disorder can mimic exotic states.

Full explanation

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

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