Quantum Hall Effect | mysimulator.uk

The classical Hall effect, and where it breaks down

In 1879 Edwin Hall discovered that passing a current through a conductor in a perpendicular magnetic field produces a transverse voltage — the Hall voltage — and the resulting Hall resistance grows smoothly and linearly with the field strength, exactly as classical electromagnetism predicts. The quantum version needs three special ingredients that Hall's setup never had: a genuinely two-dimensional electron gas (first realized in silicon MOSFETs, later in much cleaner GaAs/AlGaAs heterostructures), very low temperature, and a very strong magnetic field.

Klaus von Klitzing's 1980 discovery

Under those conditions, Klaus von Klitzing found in 1980 that instead of rising smoothly, the Hall resistance climbs in a series of extremely flat plateaus, while at the same time the ordinary longitudinal resistance drops to essentially zero on each plateau — current flows through the sample with no dissipation at all. The plateau values turn out to be exact to roughly one part in a billion, and remarkably, they are completely independent of the specific sample, its exact geometry, or the amount of disorder in the material — a universality that is almost unheard of in condensed matter physics, where measured properties usually depend heavily on messy material details. Von Klitzing was awarded the 1985 Nobel Prize in Physics for the discovery.

R_xy = h / (n · e²)        n = 1, 2, 3, …     (integer quantum Hall effect)

h/e² ≈ 25 812.807 Ω        the "von Klitzing constant," R_K

live demo · flat resistance plateaus as the field is swept● LIVE

Landau levels explain the plateaus

In a strong perpendicular magnetic field, the kinetic energy of a two-dimensional electron gas is quantized into discrete Landau levels rather than forming a continuous band. Crucially, the number of electron states that fit into each Landau level per unit area — its degeneracy — grows in direct proportion to the magnetic field. As the field increases, fewer and fewer Landau levels are needed to hold all the electrons in the sample, and every time a Landau level empties out completely, the Hall resistance jumps cleanly to the next quantized plateau. Between those special field values, disorder in the sample localizes the extra electrons in states that don't carry any current, which is exactly why the plateaus are flat and robust rather than an instantaneous jump — the localized states pin the electrons' collective behavior in place over a whole range of magnetic field.

Why it defines a metrology standard

Because the quantization is tied only to the fundamental constants h and e rather than to any material detail, it is so precise and so universal that in 1990 the international metrology community adopted the quantum Hall effect as the practical realization of the ohm. It was used to fix the value of e²/h with extraordinary precision ahead of the 2019 redefinition of the SI, which fixed both h and e as exact defined quantities rather than measured ones — making the quantum Hall resistance now exactly defined by definition, not merely measured to very high precision.

Fractional quantum Hall: a second, stranger discovery

In 1982, Daniel Tsui, Horst Störmer and Arthur Gossard discovered that plateaus also appear at fractional values — most famously at one-third — which cannot possibly be explained by single-particle electrons simply filling Landau levels one at a time. Robert Laughlin's 1983 theory explained the 1/3 plateau as an entirely new, strongly correlated quantum liquid state arising from electron-electron interactions, whose excitations carry a fraction of the electron's charge — genuinely fractional-charge quasiparticles with even more exotic "anyonic" quantum statistics. This fractional quantum Hall effect earned the 1998 Nobel Prize and remains one of the leading experimental platforms for proposals of topological quantum computing.

Frequently asked questions

Why is the quantum Hall resistance so precisely reproducible across different samples?

Because the plateau values depend only on fundamental constants — Planck's constant and the electron charge — not on the sample's geometry, material purity, or exact electron density. That's a hallmark of topological quantization: what's being counted is an integer, the number of filled Landau levels, and integers can't drift continuously the way ordinary material properties do.

What causes the resistance plateaus to be flat instead of a sharp staircase?

Real samples always have some disorder, which creates localized electron states between the mobile, extended states of each Landau level. Electrons trapped in these localized states don't contribute to current, so as you sweep the magnetic field, extra electrons pile into localized states over a whole range of field values before the next Landau level starts to empty — that range is exactly the width of the plateau.

What's the difference between the integer and fractional quantum Hall effects?

The integer effect is explained by non-interacting electrons filling discrete Landau levels one at a time. The fractional effect appears at certain fractional filling factors and can only be explained by strong electron-electron interactions forming an entirely new correlated quantum liquid, first described by Robert Laughlin, whose excitations carry fractional electric charge.

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The Quantum Hall Effect: Conductance in Exact, Uncrossable Steps