Electromigration-induced void drift and coalescence: Simulations and a dynamic scaling theory

Electromigration-induced failure of conductors in integrated circuitry is precipitated by the formation, drift, and coalescence of voids within the conductor. The kinetics of a collection of such voids is studied here through the development of a dynamic scaling theory, as well as through Monte Carlo simulations. In our model, voids originate at a source, drift with speeds inversely proportional to their radii, and coalesce according to a capture cross section which accounts for the effects of current crowding around the voids in an approximate fashion. The presence of a source leads to a spatially inhomogeneous distribution function n(Ω,x,t) for voids of volume {Ω} drifting in the x direction. We show that n(Ω,x,t) obeys the scaling form n(Ω,x,t)=Ω^-5/3f(Ωe^-ρx,Ωt^-3) where ρ0.054 and f is a scaling function. We also obtain scaling solutions for the void fraction φ(x,t) which agree well with the simulations. At a fixed time, {φ} grows monotonically with distance from the source, reaches a peak, and then decreases exponentially with distance. The position of the peak is shown to move with a velocity v∝1/t.