General coupling model for electromigration and one-dimensional numerical solutions

Abstract

A three-dimensional (3D) general coupling model for electromigration has been developed with the use of the mass conservation equation. The flux terms that include concentration gradient, electron wind force, stress migration, and thermal migration are considered. The constitutive equation for the electromigration strain has been derived. Then, the governing equations for one-dimensional (1D) metal lines are obtained for both totally fixed and stress-free mechanical boundary conditions. The numerical results reveal that the hydrostatic stress is significantly lower than the predicted results in the existing literature for the totally fixed configuration. Extensive discussions are presented to provide the explanations of such difference. The vacancy concentration gradient plays an important role in formulating electromigration problems. The current-driven flux can be entirely balanced by the concentration gradient that acts as an opposing force during electromigration under a stress-free condition in steady-state. The new solutions of the critical threshold jL, the product of current density, and metal line length are obtained in terms of vacancy concentration. As electromigration is eventually determined by the void growth, the critical vacancy concentration is used to reanalyze Blech's experiment data. The theoretical predictions are consistent with the experimental observations.