Quantum Hall Edge States | mysimulator.uk

Magnetic Field

Field strength B 3.0 T

2D Electron Gas

Areal density n 3.0 ×10¹¹ cm⁻²

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Cyclotron freq f_c

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Landau gap ħω_c

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Filling factor ν = n·h/(eB)

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Localized (bulk)

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Chiral (edge)

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Sweep B — every time ν passes through an integer, the bulk states have just finished localizing and the Hall resistance sits on a quantized plateau R_xy = h/(νe²).

Bulk cyclotron orbit (closed loop, no net current) Edge skipping orbit (chiral, carries current) Localized on disorder (pinned, insulating)

How it works

In a strong perpendicular magnetic field B, a 2D electron gas (e.g. the interface of a GaAs/AlGaAs heterostructure) doesn't conduct like an ordinary metal. Each electron of effective mass m* moves on a circular cyclotron orbit at angular frequency:

ω_c = eB / m*      (all orbits share the same rotation sense)
E_n = ħω_c (n + 1/2)      — quantized Landau levels

An orbit whose guiding centre sits deep in the bulk closes on itself and carries zero net current — it just sits there. An orbit whose guiding centre is close to the sample boundary gets clipped by the hard wall: the electron specularly reflects and re-enters its circular motion, producing a skipping orbit that hops persistently along the edge, one direction only (chirality set by sign(B)). Stacked around the whole perimeter, these skipping orbits form a one-way, backscattering-free edge current — the real charge carrier in the integer quantum Hall effect.

Real samples have disorder (impurities, lattice defects). Bulk cyclotron orbits near an impurity get trapped in a localized loop around it and contribute nothing to transport — this is what makes the Hall conductance sit on a flat, precisely quantized plateau over a range of B instead of jumping discontinuously:

ν = n·h / (eB)        — Landau level filling factor
R_xy = h / (ν e²)     — quantized only when ν is (near) an integer

Field strength B — tighter B means tighter, faster orbits (r ∝ 1/B, ω_c ∝ B) and a lower filling factor ν for the same density.
Areal density n — more electrons raises ν at fixed B; both together set which Landau level is at the Fermi energy.
Disorder — toggles impurity sites that pin nearby bulk orbits (grey), leaving only true edge states (gold) to conduct.
Flip field — reverses sign(B), which reverses the chirality of the edge current, exactly as in a real Hall bar.

This is the same physics measured with von Klitzing's 1980 experiment that gave the quantum Hall effect its 1985 Nobel Prize, and it defines the SI ohm today via R_K = h/e² ≈ 25 812.807 Ω.

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Bulk orbit Edge current Localized

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