Analysis of Stress-Driven Grain Boundary Diffusion. Part I

Abstract

The stress-driven grain boundary diffusion problem is a continuum model of mass transport phenomena in microelectronic circuits due to high current densities (electromigration) and gradients in normal stress along grain boundaries. The model involves coupling many different equa- tions and phenomena, and difficulties such as nonlocality, complex geometry, and singularities in the stress tensor have left open such mathematical questions as existence of solutions and compatibility of boundary conditions. In this paper and its companion, we address these issues and establish a firm mathematical foundation for this problem. We use techniques from semigroup theory to prove that the problem is well posed and that the stress field relaxes to a steady state distribution which, in the nondegenerate case, balances the elec- tromigration force along grain boundaries. Our analysis shows that while the role of electromigration is important, it is the interplay among grain growth, stress generation, and mass transport that is responsible for the diffusive nature of the problem. Electromigration acts as a passive driving force that determines the steady state stress distribution, but it is not responsible for the dynamics that drive the system to steady state. We also show that stress singularities may develop near grain boundary junctions; however, stress components directly involved in the diffusion process remain finite for all time. Thus, we have identified a mechanism by which large hidden stresses may develop that are not directly involved in the diffusion process but may play a role in void nucleation and stress-induced damage.