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. 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 — exactly why chip interconnects switched from aluminum (E_a ≈ 0.7 eV) to copper (E_a ≈ 0.9 eV) in the late 1990s. This 2D companion adds the other half of real interconnect reliability: the Blech short-length effect. In a short enough line, atoms piling up at the anode build up a mechanical back-stress gradient that pushes atoms back toward the cathode just as fast as the electron wind pushes them forward — net drift stops and the line becomes electromigration-immortal, no matter how long it is biased. This only happens below a critical product of current density and length:
(j·L) < (j·L)_crit → immortal (back-stress cancels the wind)
Copper's stronger diffusion barrier also raises its Blech threshold, so a copper line can be immortal at lengths and current densities that would still fail in aluminum — try shortening the Line length slider and watch the status chip flip to "IMMORTAL" first for copper, then for aluminum.
- Material buttons — switch E_a, the Black exponent n, prefactor A, and the Blech threshold between aluminum and copper.
- Current density / Temperature sliders — set J and T; MTTF updates live via Black's equation.
- Line length slider — sets L for the j·L Blech check; short lines at low J can stop failing entirely.
- 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.