Dependence of electromigration lifetime on the square of current density

A model to explain the conductor lifetime dependence on the square of the current density (Black's relationship) is proposed. It is based on both experimental and computer-simulated results. The number of voids and hillocks formed per conductor length by electromigration are proportional to the current density. The relationship between the growth rate of voids (and hillocks) and the current density is derived from Fick's diffusion equation. That relationship is shown by numerically solving a one-dimensional diffusion equation with nonuniform diffusivity and a critical concentration for void/hillock formation. The average growth rate of each void is observed to be proportional to the current density. The relationships of the number of voids and hillocks and the average void growth rate to the current density are simultaneously incorporated during electromigration resulting in the inverse square relationship between lifetime and current density.<>