Current flowing through a metal line means electrons stream from the cathode end toward the anode end. Each electron occasionally collides with a lattice metal ion and transfers momentum — the "electron wind." At high enough current density this wind force exceeds the field force pulling ions back, so ions slowly drift in the electron-flow direction:
v_drift ∝ D(T) · Z*·e·ρ·J / (k_B·T)
D(T) = D₀·exp(−E_a / k_B·T) (Arrhenius diffusivity)
Net atom transport empties material at the cathode end — nucleating a void that can grow until it severs the line (open circuit) — while atoms pile up at the anode end, extruding a hillock that can breach neighboring layers and short the die. Time-to-failure follows Black's equation:
MTTF = A · J^(−n) · exp(E_a / k_B·T)
Higher current density or higher temperature both shrink MTTF; a larger activation energy E_a makes it exponentially longer. This is exactly why chip interconnects switched from aluminum (E_a ≈ 0.7 eV) to copper (E_a ≈ 0.9 eV) in the late 1990s — copper's stronger self-diffusion barrier buys roughly two orders of magnitude more electromigration lifetime at the same current density.
- Material buttons — switch the activation energy E_a, Black exponent n and prefactor A between aluminum and copper.
- Current density / Temperature sliders — set J and T; the void-growth rate and MTTF update live, following the same Black's-equation scaling.
- Speed slider — the real MTTF is months to decades, so playback compresses it; relative timing between settings stays physically consistent.
Constants used here are literature-representative but calibrated for a readable demo timescale, not a specific foundry's qualified process.