In a ferromagnetic metal, spin-orbit coupling makes an electron's scattering cross-section off d-band states depend on the angle θ between the local magnetization M and the current direction I, not just their magnitudes. That gives an angle-dependent sheet resistance:
R(θ) = R⊥ + (R∥ − R⊥)·cos²θ
θ = angle between magnetization M and current I
R∥ = resistance when M ∥ I (θ = 0°)
R⊥ = resistance when M ⊥ I (θ = 90°)
ΔR/R = (R∥ − R⊥) / R⊥ — the AMR ratio, a few % for 3d ferromagnets
The magnetization does not track the external field instantly — shape anisotropy along the strip pins it, so M relaxes toward the field direction at a rate set by the field strength:
dφ_M/dt = (φ_H − φ_M) / τ(H) τ(H) shrinks as field strength grows
At low field strength the magnetization lags visibly behind a rotating field — exactly the nonlinearity that real AMR sensors fight with a "barber-pole" bias structure. The lower panel traces R(θ) live as the field sweeps, revealing the real cos² curve and its period-π (180°) symmetry — a signature that distinguishes AMR from GMR/TMR, which are period-2π.