The Quantum Hall Effect Explained: Resistance in Perfect Steps

The quantum Hall effect is what happens when the Hall effect, a sideways voltage in a conductor sitting in a magnetic field, meets quantum physics. In a thin enough layer of electrons, very cold and in a very strong field, the Hall resistance stops changing smoothly and climbs in flat steps. Our character Stairohm is literally built from those steps. This post builds on our explainer How Are Electricity and Magnetism Connected?.

First, the ordinary Hall effect

In 1879 Edwin H. Hall, then a young student, found that if a thin gold plate is placed in a magnetic field at right angles to its surface, a current flowing along the plate produces a voltage across it (NobelPrize.org, 1998 press release^[1]). The magnetic force pushes moving charges toward one edge, so charge piles up there and a voltage appears sideways. That sideways voltage divided by the current is the Hall resistance. The Nobel committee’s 1998 release notes that the effect is used to measure the density of charge carriers in conductors and semiconductors and has become a standard tool in physics labs. Hall worked at room temperature with fields of less than one tesla.

What changes in the quantum Hall effect?

By the end of the 1970s researchers were using temperatures only a few degrees above absolute zero (about −272 °C) and very strong magnetic fields, up to about 30 T, on the kind of semiconductor structures used in electronics (NobelPrize.org, 1998 press release^[1]). In those samples the electrons can only move in a flat, two-dimensional layer. The 1985 Nobel press release explains that the experiments used a specially designed transistor called a MOSFET, and that at such low temperatures and strong fields collisions no longer break up the electrons’ ordered motion, so classical theory no longer applies and the motion becomes quantised (NobelPrize.org, 1985 press release^[2]).

In the spring of 1980 Klaus von Klitzing showed that the Hall conductivity forms step-like plateaux that follow a simple rule with exceptional accuracy, deviating from a whole number by less than 0.000 000 1 (NobelPrize.org, 1985 press release^[2]). He published the result with G. Dorda and M. Pepper in Physical Review Letters (von Klitzing, Dorda & Pepper, 1980^[3]) and received the 1985 Nobel Prize in Physics “for the discovery of the quantized Hall effect” (NobelPrize.org, Physics 1985^[4]).

The formula for the steps

On each plateau the Hall resistance takes the value

R_H = h / (ν e²) = R_K / ν

where h is the Planck constant, e is the elementary charge and ν (the Greek letter nu) is a whole number, 1, 2, 3 and so on, called the filling factor. The combination h/e² is named the von Klitzing constant:

R_K = h / e² = 25 812.807 45… Ω

The value comes from the NIST list of CODATA recommended constants, where it is marked exact (NIST, R_K^[5]). It is exact because both h = 6.626 070 15 × 10⁻³⁴ J s and e = 1.602 176 634 × 10⁻¹⁹ C now have fixed values (NIST, h^[6]; NIST, e^[7]). Both are among the seven defining constants of the SI (BIPM, SI defining constants^[8]).

Worked numbers

• ν = 1: R_H = 25 812.807 Ω (about 25.8 kΩ)
• ν = 2: R_H = 12 906.404 Ω
• ν = 3: R_H = 8 604.269 Ω
• ν = 4: R_H = 6 453.202 Ω

As the magnetic field rises, ν drops, so the resistance climbs from step to step, just like Stairohm’s staircase. The 1998 press release adds that on the plateaux the ordinary resistance along the sample disappears, so in that direction the material behaves in a sense like a superconductor (NobelPrize.org, 1998 press release^[1]).

Why the steps do not care about the material

The striking thing is that the step values depend only on fundamental constants, not on the size, shape or impurities of the sample. The 1998 release states this directly: the steps occur at resistance values that do not depend on the properties of the material (NobelPrize.org, 1998 press release^[1]). Why a messy real-world device gives such clean numbers took further theory to explain, and the 1985 committee called it a new phenomenon whose characteristics were then still only partially understood (NobelPrize.org, 1985 press release^[2]).

A resistance standard and a constant of nature

Because of its extreme precision, the 1985 press release says the effect can be used as a standard of electrical resistance and also gives a new way to measure a fundamental constant important in atomic and particle physics (NobelPrize.org, 1985 press release^[2]). That second use is the fine-structure constant, which is in the title of the 1980 paper. National measurement labs in Germany, the USA, Canada, Australia, France, Japan and other countries tested the precision in detail so the effect could serve as a standard.

The fractional surprise

At the start of the 1980s a team at Bell Laboratories found plateaux at fractions such as 1/3, 2/3, 2/5 and 3/5 multiplied by e²/h (NobelPrize.org, 1985 press release^[2]). Horst Störmer and Daniel Tsui made the discovery in 1982, and Robert Laughlin explained it within a year as a new quantum fluid whose excitations carry fractions of the electron’s charge. All three shared the 1998 Nobel Prize (NobelPrize.org, 1998 press release^[1]).

For more quantum characters, see the sticker collection. For the magnet basics behind the Hall effect, read why a magnet falls slowly through a copper pipe.

Quick FAQ

Is the quantum Hall effect the same as superconductivity?

No. On a plateau the resistance along the sample vanishes, but the sideways Hall resistance is not zero; it is locked to h/(νe²).

Why does it need such cold temperatures?

Heat causes collisions that scramble the electrons’ ordered motion. Only when collisions are suppressed does the quantum behaviour show.

What is the von Klitzing constant?

It is h/e², exactly 25 812.807 45… ohms in today’s SI.

References

1. NobelPrize.org, Press release: The 1998 Nobel Prize in Physics (Laughlin, Störmer, Tsui). https://www.nobelprize.org/prizes/physics/1998/press-release/
2. NobelPrize.org, Press release: The 1985 Nobel Prize in Physics. https://www.nobelprize.org/prizes/physics/1985/press-release/
3. K. v. Klitzing, G. Dorda & M. Pepper (1980), “New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance,” Phys. Rev. Lett. 45, 494–497 (DOI 10.1103/PhysRevLett.45.494; metadata verified via Crossref). https://doi.org/10.1103/PhysRevLett.45.494
4. NobelPrize.org, “The Nobel Prize in Physics 1985” (K. von Klitzing). https://www.nobelprize.org/prizes/physics/1985/summary/
5. NIST CODATA 2022 recommended values: von Klitzing constant R_K (exact). https://physics.nist.gov/cgi-bin/cuu/Value?rk
6. NIST CODATA 2022 recommended values: Planck constant (exact). https://physics.nist.gov/cgi-bin/cuu/Value?h
7. NIST CODATA 2022 recommended values: elementary charge (exact). https://physics.nist.gov/cgi-bin/cuu/Value?e
8. BIPM, “SI defining constants”. https://www.bipm.org/en/measurement-units/si-defining-constants

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.