Numerical simulation of electrochemical migration of Cu based on the Nernst-Plank equation

Abstract

Electrochemical Migration (ECM) is getting more attention in the microelectronics industry due to the continuing miniaturization, which increases the possibility of short circuit formation caused by the ECM-induced dendrites. This work presents a 2D numerical model of the ECM based on the Nernst-Plank equation. The model contains the deterministic description of the metal dissolution, the changes of electrolyte properties, and the ion transport in the electrolyte. However, the reduction of the ions and the dendrite growth is described stochastically. The capability of the model was tested in the case of pure copper electrodes with a gap distance 200 µm, 10 VDC bias, 20 °C temperature, and a contaminant-free electrolyte. The results of the model were validated by experimental water-drop tests. The results showed very good agreement between the experimental and the calculated mean time to failure values, dendrite morphologies, and the kinetics of the dendrite formation. The model showed that the developing dendrites consume most of the Cu2+ ions around them, which answers why only some dominant dendrites can develop in a given area. The model proved that not only the electric field but the diffusion of the ions is also dominant in given phases of the ECM process. Our model could be a useful tool for ECM failure prediction and for further ECM researches as the digital twin of the ECM process. Also the approach can be applied in various aspects of failure-prediction in modern reflow-soldering.