THE FINITE ELECTROMIGRATION BOUNDARY-VALUE PROBLEM

The electromigration boundary value problem is investigated for the three physically reasonable boundary conditions, assuming a perfectly blocking boundary on one side. The solution to this problem is believed to be that for nucleation dominated electromigration lifetime. The three boundary conditions investigated are the semi-infinite constant vacancy source of Shatzkes and Lloyd,9 the closed system of De Groot13 and Kirchheim and Kaber,19 and the heretofore unsolved constant vacancy source at a finite distance from the blocking boundary. It is argued that the first is unrealistic in that there is no length effect possible, which has been repeatedly observed experimentally. The second is argued to be too restrictive to account for failure, leaving the last as the most physically reasonable under most circumstances. The deceptively simple appearance of the boundary conditions belies a complex, double infinite series solution arrived at by a unique approach to inverting the Laplace transform of the solution. The solution correctly predicts the experimental observations of a length effect and, combined with the understanding provided by the solutions under the other two boundary conditions, the effect of a thick passivation layer on electromigration lifetime.