Void electromigration as a moving free-boundary value problem

Abstract

The void electromigration process in the strip geometry is investigated analytically. The void is assumed to travel steadily along the axis of symmetry of the metal strip. When its length is large compared to the width of the metal ribbon, a strong similarity with the classical Saffman–Taylor viscous fingering is pointed out. It allows us to adapt the hodograph method to predict both the width and the growth velocity at fixed applied current. In the absence of surface tension, the void shape is unique and has a relative width close to 2/3, a result in strong contradiction with the viscous fingering in the same geometry. The propagation of symmetric bubbles is also investigated and shows that the set of solutions is less rich than the flow problem. The addition of surface tension appears non-standard in the area of pattern formation: it does not allow large void propagation, suppressing the unique infinite solution. As for the bubbles, the numerically computed shapes with surface tension appear rather circular. This paper shows that the modification of a single boundary condition: (the Dirichlet boundary condition) is enough to strongly modify our intuition about pattern formation.