Graphene's Half-Integer Quantum Hall Effect

Field & Doping

Magnetic field B 10.0 T

Gate-tuned filling factor ν 0.0

Show orbiting carriers

Live Readouts

Filling factor ν

0.0

Landau spacing E₁

36.3 meV

Hall conductivity σ_xy

2 · e²/h (77.5 µS)

How it works

Electrons in graphene behave as massless Dirac fermions. In a perpendicular field B they still form discrete Landau levels, but because the low-energy dispersion is linear (a Dirac cone, not a parabola) the levels are spaced by a square root, not evenly:

E_n = sgn(n) · v_F · √(2ħeB|n|)  ,  n = 0, ±1, ±2, …
          ≈ sgn(n) · 36.3 meV · √(|n| · B[T])

Two features are unique to graphene here. First, there is an n = 0 level pinned exactly at E = 0, shared equally between the electron and hole side (an ordinary 2D electron gas has no such level — its ladder is E_n = ħω_c(n+½)). Second, each level carries a 4-fold degeneracy (2 spins × 2 valleys, K and K′). Filling levels one at a time as a gate voltage tunes the carrier density produces filling factors in steps of 4, but the shared n = 0 level shifts the whole sequence by half a step, giving the famous half-integer quantized Hall conductivity:

σ_xy = ±4(n + ½) e²/h = …, −6, −2, +2, +6, +10, … (× e²/h)

B slider — sets the magnetic field; the Landau ladder spreads apart as √B (levels shown at relative scale — the meV readout is the real value).
ν slider — models a back-gate sweeping the Dirac point through the Fermi level; the gold plane tracks the Fermi energy and the right-hand graph traces out the Hall-conductivity staircase live.
Note the plateau at the charge-neutrality point (ν = 0) sits at σ_xy = 2e²/h, not zero — a direct experimental fingerprint that distinguished graphene's Dirac electrons from a conventional 2D gas when first measured by Novoselov et al. and Zhang et al. in 2005.

Filled Landau level Empty Landau level Fermi level

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