# The Quantum Hall Effect

Laplace’s Mini World · Plate F36 · Energy, Flight & Sensing · Quantum

> Near absolute zero, in a strong field, a sheet of electrons shows Hall steps set by h and e alone.

- Interactive model: https://world.lapla.si/en/quantumhall/
- Embed: https://world.lapla.si/embed/en/quantumhall/
- 中文: https://world.lapla.si/zh/quantumhall/
- Updated: 2026-10-22

## 01 No Difference Found

In 2011 the UK’s National Physical Laboratory compared the quantum Hall resistance of a graphene sample directly with a gallium arsenide one and found no difference, within a relative standard uncertainty of 8.6 × 10⁻¹¹. In 2015 France’s national metrology laboratory, LNE, compared them again through a 100 Ω resistor, to 8.2 × 10⁻¹¹. Electrons in graphene behave like massless particles, quite unlike those in a semiconductor such as gallium arsenide.

In 1991 a comparison between gallium arsenide and silicon devices found no difference within 3.5 × 10⁻¹⁰. Metrology institutes now realise resistance this way routinely to 10⁻⁹ or better. In 2025 a graphene standard from Germany’s PTB took part in a BIPM key comparison for the first time; compared directly with the BIPM’s gallium arsenide standard, the two differed by −0.5 × 10⁻⁹ with an uncertainty of 0.8 × 10⁻⁹.

*Graphene − gallium arsenide: (+4.7 ± 8.6) × 10⁻¹¹ (2011) · (−0.9 ± 8.2) × 10⁻¹¹ (2015)*

## 02 A Sideways Voltage

Run a current along a conductor and add a magnetic field at right angles, and the moving electrons are pushed to one side. One side gains electrons, the other loses them, and a voltage builds across the conductor until its push balances the field’s. This is the Hall effect, found by Edwin Hall in 1879.

That voltage divided by the current is the Hall resistance. In an ordinary conductor it grows in proportion to the field and falls with the number of electrons per area: R = B/(ne). That is why the Hall effect is used to measure magnetic fields, or to count a material’s charge carriers.

*R\_xy = B/(ne) · this page’s gallium arsenide sample: n = 5.1 × 10¹¹ per cm²*

## 03 One Layer of Electrons

The quantum Hall effect needs electrons that can only move in a plane. Grow two crystals, gallium arsenide and aluminium gallium arsenide, in layers on top of each other, and electrons gather at the boundary between them, squeezed into a layer so thin they cannot move up or down, only across. This is a two-dimensional electron gas.

The first experiment, in 1980, used the electron layer in a silicon field-effect transistor. Standards then mostly moved to gallium arsenide: the samples in the British and French comparisons, for instance, held about 5 × 10¹¹ electrons per cm², with mobilities of about 3 × 10⁵ to 5 × 10⁵ cm² per volt-second. Now there is also single-layer graphene grown on silicon carbide.

*AlGaAs / electron layer / GaAs · the layers drawn far thicker than they are*

## 04 Landau Levels

A magnetic field turns a moving electron into a circle, at a rate set only by the field and the electron’s mass: ω = eB/m\*. In gallium arsenide an electron behaves as if it had 0.067 times the free electron’s mass; at 10 tesla it goes round about 4.2 trillion times a second.

Quantum mechanics allows only certain orbits, with energies spaced ħω apart: about 17 meV at 10 tesla, the thermal energy at 200 K. Each level holds a fixed number of electrons for each spin direction: eB/h per square metre, whatever the electron’s effective mass. These are the Landau levels.

*Eₙ = ħω (n + ½) · eB/h per level and spin · 10 T: 2.42 × 10¹¹ per cm² · orbits drawn about 250,000 times larger*

## 05 Steps

The stronger the field, the more electrons each level holds, and the fewer levels are filled. The number filled is counted by the filling factor, ν = nh/(eB). Yet the measured Hall resistance does not climb along a straight line: wherever ν nears a whole number it rests on a step, R = h/(νe²) with ν that whole number. On each step the resistance along the current, R\_xx, drops to zero.

In the early hours of 5 February 1980, at the high-field laboratory in Grenoble, France, Klaus von Klitzing was measuring silicon transistors and found that the Hall resistance on the steps is h/e² divided by a whole number. It depends only on fundamental constants, not on the device’s shape; he measured h/4e² = 6453.17 ± 0.02 Ω, with an uncertainty of about three parts per million. Steps had already appeared in a 1978 curve of his with Th. Englert; that paper passed over them. He received the 1985 Nobel Prize in Physics.

*R\_xy = h/(νe²) · ν = 2: 12,906.4037 Ω · traces computed by this page’s model*

## 06 One-Way Edges

Why are the steps so flat? On a step, as the field changes, the electrons gained or lost go only into states trapped around impurities, where they circle in place and carry no current; the states that do carry current stay as they were, and so does the Hall resistance. Along the edges, electrons can travel on: they hit the boundary, bounce, circle again, and skip forward. Along one edge they all go one way; along the opposite edge, the other.

An electron that meets an impurity cannot turn back, because there is no backward lane on its edge; it can only go round. So the current runs along the sample without losing energy (heat appears only where it enters and leaves), R\_xx is zero, and the Hall resistance is set by the number of lanes alone, whatever the impurities or the shape. Bertrand Halperin pointed out these edge states in 1982; Markus Büttiker used them to explain the effect in samples with many contacts in 1988.

*ν one-way lanes along each edge (here ν = 2) · orbits drawn about 250,000 times larger*

## 07 A Whole Number

In 1982 David Thouless and colleagues showed that for electrons in a crystal lattice, whenever the Fermi level lies in a gap, the Hall conductance is a whole number times e²/h. Later work identified that number as a topological invariant, the Chern number: as long as the gap stays open, smooth changes to the sample leave it alone; it can change only by a whole jump, so impurities and shape cannot alter it. Each filled Landau level adds one.

The scientific background to the 2016 prize calls the integer quantum Hall effect the first topological quantum liquid to be discovered. That year’s Nobel Prize in Physics went to Thouless, F. Duncan M. Haldane and J. Michael Kosterlitz for theoretical discoveries of topological phase transitions and topological phases of matter. Related states that need no external field have since been found: a 2024 experiment at about 34 mK found one differs from h/e² by (4.4 ± 8.7) × 10⁻⁹. Reaching that cold takes a dilution refrigerator, the plate “The Dilution Refrigerator”.

*σ\_xy = C · e²/h · each filled level adds 1 to C*

## 08 One Third

In October 1981, Daniel Tsui, Horst Störmer and Arthur Gossard took a cleaner gallium arsenide sample below 1 K in a field of about 15 tesla and found a new step: a Hall resistance of 3h/e², with the lowest level only a third full. At 0.42 K it was within 1% of 3h/e².

Single electrons circling cannot explain it. In 1983 Robert Laughlin proposed that the electrons condense together into a new quantum liquid whose excitations carry a fraction of the electron’s charge, such as e/3. Laughlin, Störmer and Tsui shared the 1998 Nobel Prize in Physics. Fractional steps need colder, cleaner samples; resistance standards use the whole-number step ν = 2.

*ν = 1/3 · R\_xy = 3h/e² · within 1% (0.42 K, 1982)*

## 09 Cut Open: the Cryostat

Outermost is a stainless-steel vacuum can that stops heat carried by air; inside it an aluminium shield, wrapped in many layers of aluminised film, stops heat carried by radiation; inside that, the vessel of liquid helium. The superconducting magnet sits in the liquid, hung from the top plate on three rods, and two copper leads bring its current in.

The sample probe drops through the middle of the top plate; the sample space at its end sits at the centre of the field and can be cooled further by pumping. The transportable system the BIPM takes to other laboratories is also a liquid-helium cryostat, with an 11.3 tesla magnet and a sample space cooled to 1.4 K.

*The BIPM’s transportable system: 11.3 T, 1.4 K · the cryostat drawn about half its real height*

## 10 Ten Tesla

A gallium arsenide sample needs about 10 tesla to sit on the ν = 2 step, between 150,000 and 400,000 times the Earth’s field. Fields like that come from a coil wound from superconducting wire: it sits in liquid helium, has no resistance, and carries a large current without heating up.

In the middle of the coil the field is strong and even; towards the ends it weakens. The sample sits at the centre. The curve is the field along the axis, computed for this coil’s dimensions. The magnet PTB used in the 2025 comparison goes up to 12 tesla.

*Field along the axis (computed for the coil shown) · the sample at the centre*

## 11 Two Materials

The sample is a chip with a Hall bar etched into it, a strip a few tenths of a millimetre wide with side arms: current flows from end to end, and the arms pick up the voltages. A gallium arsenide sample sits on the ν = 2 step at about 1.4 K and 10 tesla, with measuring currents of a few tens of microamperes.

The graphene used for standards is a single layer of carbon atoms, grown on a silicon carbide wafer. Its Landau levels are much further apart: at 10 tesla, about 17 meV in gallium arsenide but about 115 meV between graphene’s zero-energy level and the next one up. So a graphene standard works at 5 tesla and 4.2 K, takes ten times the current or more, and fits in a small refrigerator that needs no liquid helium.

*Gallium arsenide ħω ≈ 1.7 · B meV · graphene E₁ ≈ 36 · √B meV (B in tesla)*

## 12 Counting Turns

At ν = 2 the quantum Hall resistance is about 12,906.4 Ω, but everyday standards are round values such as 1 Ω or 100 Ω. The step between them is made with a cryogenic current comparator: two windings sit inside a ring-shaped superconducting shield, one of N₁ turns carrying a current I₁, the other of N₂ turns carrying I₂. The superconductor keeps magnetic flux out, so the field left outside the shield depends only on N₁I₁ − N₂I₂, not on where the wires lie inside. A superconducting quantum interference device (SQUID) watches it and holds it at zero.

The ratio of the currents is then the ratio of the turns, to better than 10⁻¹⁰ in a well-built comparator. Pass the two currents through the quantum Hall resistance and a 100 Ω standard, compare the voltages, and the ratio of the resistances follows from the turns. With 2065 turns against 16, for instance, the ratio is 129.0625, only about 1.2 × 10⁻⁵ from 12,906.4037 ÷ 100; the bridge measures that remainder.

*N₁I₁ = N₂I₂ · 2065 : 16 = 129.0625*

## 13 Where an Ohm Comes From

From 1990, laboratories agreed to convert quantum Hall resistance with R\_K-90 = 25,812.807 Ω. Since 20 May 2019 the SI fixes Planck’s constant h and the elementary charge e exactly, and the ohm is realised with R\_K = h/e² = 25,812.807 459 3… Ω. That is 1.78 × 10⁻⁸ larger than the old conventional value: from that day, the same resistor reads that much higher. A new definition of the second is being prepared; see the plate “The Optical Lattice Clock”.

An institute first calibrates a group of 100 Ω standards against its quantum Hall resistance, then steps from them to values from 1 Ω to 1 MΩ, and these calibrate other resistors. Since 1993 the BIPM has carried its own transportable standard to institutes for on-site comparisons: the same 100 Ω resistor is calibrated against both quantum Hall standards.

*R\_K = h/e² = 25,812.807 459 3… Ω (since 2019) · 1.78 × 10⁻⁸ above R\_K-90*

## 14 Taken Apart

The cryostat’s outer can, shield and helium vessel, the magnet, the top plate and the sample probe, the comparator’s dewar and probe, the magnet supply and the bridge, the standard resistor, the recorder and the enlarged Hall bar come apart, piece by piece.

Whichever piece the pointer rests on is lifted out and the rest fade; left alone, the pieces are all named.

## 15 Sweep the Field

The lever is the magnetic field. The recorder’s black pen draws the Hall resistance and its red pen the longitudinal resistance R\_xx, both computed by this page’s toy model; the readouts follow the lever.

As the temperature rises the steps narrow and R\_xx on them is no longer zero. With graphene, the same ν = 2 step starts at a much lower field and stays flat even at 10 K.

**Try this**: Set the temperature to 10 K and sweep the field again: is R\_xx still zero on gallium arsenide’s ν = 2 plateau? And with graphene? At what field does the plateau begin?

## What the model leaves out

The bench follows the layout metrology institutes describe in their papers: a liquid-helium cryostat with a superconducting magnet and a sample probe, a cryogenic current comparator in a dewar of its own, the bridge electronics and a temperature-controlled standard resistor. It is not any laboratory’s plans. The cryostat is drawn about half its real height and the sample space larger than life; the enlarged Hall bar is schematic: its width over a hundred times real, its layers enlarged far more, the electrons’ orbits about 250,000 times. The X–Y recorder is how field sweeps were once recorded; today a computer logs them. The traces on the recorder and in the laboratory come from a toy model: impurities broaden each Landau level, only states near a level’s centre carry current, and temperature smears the filling; the conductances that follow are turned into resistances. It gives the steps, the peaks in R\_xx and how they change with temperature, but the widths of the steps and the heights of the peaks are not a real sample’s. Gallium arsenide’s spin splitting is left out, so the odd-numbered steps do not appear, though real samples show them. In graphene on silicon carbide, charge moving between substrate and graphene keeps the ν = 2 step going to very high fields; the model imitates this with a fixed position for the Fermi level.

## Some common misreadings

Zero R\_xx on a step is not superconductivity: the current runs along the edges and cannot scatter backwards, so it loses no energy along the way, and this holds only on a step, at low enough temperature and modest current. Nor are the steps exact because the sample is perfect: it is the impurities, trapping some of the electrons, that give the steps their width. Since 2019 h/e² is an exact number, but that the quantum Hall resistance equals h/e² is a premise backed by a large body of experiment and theory; institutes check every sample against technical guidelines, with low contact resistances and an R\_xx too small to measure, before using it. The circling “orbits” are a classical picture; in quantum mechanics an electron has no definite orbit. The Chern number is a topological property of the electrons’ states; it is often likened to counting the holes in a doughnut, but it does not count holes in the sample.

## Common questions

### Who discovered the quantum Hall effect?

In the early hours of 5 February 1980, at the high-field laboratory in Grenoble, France, Klaus von Klitzing was measuring silicon transistors and found that the Hall resistance on the steps is h/e² divided by a whole number.

See: [05 Steps](https://world.lapla.si/en/quantumhall/#plateaus)

### Why is the Hall resistance on a step so exact?

An electron that meets an impurity cannot turn back, because there is no backward lane on its edge; it can only go round.

See: [06 One-Way Edges](https://world.lapla.si/en/quantumhall/#edges)

### Why do graphene standards need a weaker field?

Its Landau levels are much further apart: at 10 tesla, about 17 meV in gallium arsenide but about 115 meV between graphene’s zero-energy level and the next one up.

See: [11 Two Materials](https://world.lapla.si/en/quantumhall/#sample)

### How has the ohm been realised since 2019?

Since 20 May 2019 the SI fixes Planck’s constant h and the elementary charge e exactly, and the ohm is realised with R\_K = h/e² = 25,812.807 459 3… Ω.

See: [13 Where an Ohm Comes From](https://world.lapla.si/en/quantumhall/#ohm)

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## Data

The bench follows the layout that national metrology institutes describe in their papers (a liquid-helium cryostat with a superconducting magnet and a sample probe, a cryogenic current comparator in its own dewar, the bridge electronics, a temperature-controlled 100 Ω standard resistor); it is not drawn from any one laboratory’s plans. The Hall and longitudinal resistances on the recorder and in the laboratory come from a toy model computed by this page; they are not measured data. Material presets, parts and the softbox studio of the site’s model kit are ported from AirsupHQ/machine-skill (MIT, © 2026 Air Supply Technologies Inc.).

## Licence CC BY-NC 4.0

The text, pictures and models on this page are licensed CC BY-NC 4.0: free to use and adapt for teaching and other non-commercial purposes, with credit to world.lapla.si. Third-party material listed above keeps its own licence.

https://creativecommons.org/licenses/by-nc/4.0/deed.en

## Cite this plate

Laplace’s Mini World, Plate F36, “The Quantum Hall Effect”, https://world.lapla.si/en/quantumhall/, CC BY-NC 4.0, updated 2026-10-22.