The finite electromigration boundary value problem
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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.
gradients were ignored. The diffusion equation
I. INTRODUCTION Since being recognized as a major failure mechanism in integrated circuit metallization over 25 years ago, electromigration failure modeling has been an important, often controversial, effort. The efforts have taken on a sometimes evangelical appearance which is not too surprising when one considers that the consequences of failure can be severe. The early attempts at modeling were plagued with a disturbing tendency to not agree with experiment, which have overwhelmingly produced a dependence of lifetime on the inverse square of the current density j , or to disagree with solid-state theory.1'2 Most models produced a l/j dependence except in the presence of Joule heating,3'4 or predicted that lifetime should vary as j ~ " , where n was to be determined by experiment.5 Others claimed that n was really dependent on j itself6-7 or produced a model by taking several distinct models and combining them into a computer program.8 Several years ago electromigration failure was modeled as the accumulation of vacancies at a flux divergence until a critical concentration was attained. It was postulated that failure would occur soon after this vacancy concentration was realized. For simplicity, a conductor stripe was modeled as a semi-infinite continuum with a perfectly blocking diffusion barrier and the effects of stress, stress gradients, and temperature
a
'Also with Max-Planck-Institut fiir Metallforschung, Seestrasse 75, Stuttgart, Germany. J. Mater. Res., Vol. 9, No. 3, Mar 1994
dC(x,t) dt
= D
d2C(x,t) [
2
dx
dC(x,t) — ~a- dx
(1)
where C(x, t) is the vacancy concentration, D is the vacancy diffusivity, a is the reduced electromigration driving force given by a = Z*epj/kT,
(2)
where Z* is the el
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