A Deterministic Limit for Electromigration: Deriving the Critical Void Radius via Lattice Diffusion

The prediction of electromigration-induced failure in semiconductor interconnects relies heavily on Black's Equation, a classical empirical model that estimates median time-to-failure (MTTF) based on statistical current density and temperature parameters. While this framework provides aggregate lifespan estimates for integrated circuits, it fails to deterministically define the exact spatial boundary where localized momentum transfer from the electron wind triggers irreversible atomic voiding. This paper introduces a strict continuum framework for solid-state electromigration. By modeling the interconnect lattice as a dynamic kinematic balance between the spatial capacity for atomic healing (lattice diffusion) and the localized rate of atomic excavation (electron wind force), we derive a universal critical void radius (Rvoid). We demonstrate that interconnect failure is not a probabilistic function of time, but an exact deterministic limit where localized momentum transfer strictly overpowers the advective vacancy-diffusion capacity of the surrounding metallic crystal. We propose a blueprint for Absolute Interconnect Scaling and real-time electromigration telemetry using embedded ring-oscillator delay sensors to provide spatial prediction prior to catastrophic severing.