Quantum Anomalous Hall Effect

Film Geometry

Magnetization

Band Structure

Exchange gap Δ 28 meV

Fermi level E_F 0 meV

Transport

Gapped · C = +1 · one chiral edge channel

Chern number C

σ_xy (e²/h)

+1.00

σ_xx (longitudinal, schematic)

0.00 — dissipationless

How it works

A magnetic dopant (Cr, V) implanted in a thin topological-insulator film (Bi₂Se₃ family) polarizes and adds an exchange field to the surface Dirac Hamiltonian:

H(k) = ħv_F(k_x σ_y − k_y σ_x) + Δ σ_z
Δ ∝ exchange field from the ordered magnetic dopants

In a thin film the top and bottom surface Dirac cones hybridize strongly enough that a uniform out-of-plane magnetization gives both surfaces a mass term of the same sign, so their half-integer Berry-phase contributions add instead of cancel:

C = (1/2)[sign(Δ_top) + sign(Δ_bottom)] = sign(Δ)   (thin film, both surfaces gapped)

Whenever the Fermi level sits inside the gap (|E_F| < Δ/2), transport is carried entirely by one topologically protected chiral edge channel running around the sample boundary — no external magnetic field needed. The Hall conductance is exactly quantized and longitudinal transport vanishes:

σ_xy = C·e²/h   (exact plateau, C = ±1)
σ_xx = 0         (edge state cannot backscatter — dissipationless)

Thin / Thick film — only a thin film hybridizes enough to open the topological gap; a thick film has decoupled, ungapped surface Dirac cones and no quantization is possible.
↑M / ↓M — reversing the magnetization flips the sign of Δ, flips C, and reverses which way the edge channel circulates — with zero applied magnetic field.
Δ slider — the exchange-gap size; Δ = 0 removes the gap entirely (ordinary gapless TI surface).
E_F slider — once |E_F| exceeds Δ/2 the Fermi level enters the bulk bands: the edge channel mixes with diffusive bulk carriers and σ_xy drops off its plateau.

First observed in Cr-doped (Bi,Sb)₂Te₃ films (Chang et al., Science 2013) — the quantum anomalous Hall effect is a Hall-quantized state that needs no external magnetic field at all, unlike the ordinary integer quantum Hall effect.

Chiral edge channel Diffusive bulk carriers

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