Analysis of Stress-Driven Grain Boundary Diffusion. Part I

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 equations 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 study the properties of a type of Dirichlet-to-Neumann map that involves solving the Lamé equations with interesting interface boundary conditions. We identify a new class of degenerate grain boundary networks that lead to unsuppressed linear growth modes that are suggestive of continental drift in plate tectonics. 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 may or may not completely balance the electromigration force. In the latter (degenerate) case, the displacements continue to grow without bound along stress-free modes.