Stress distribution and mass transport along grain boundaries during steady-state electromigration

Abstract

The distribution of normal stress and mass flux in the plane of each grain boundary of a polycrystalline thin-film conductor during electromigration has been calculated, assuming zero grain boundary flux divergence (steady state), for various boundary conditions by standard boundary value methods of diffusion theory. Steady state, representing a balance between the (applied) electric and (induced) stress driving forces, develops after a transient time, which is estimated to be short for the cases considered. Continuity of stress and flux is assumed at the intersection of grain boundaries with each other (triple junctions). Intersections of grain boundaries with the film surfaces are assumed to be of two types: either open (passes flux freely) or closed (zero flux). The bottom film surface (substrate) is assumed closed and the top surface (bare metal) open; grain boundary intersections with the film edges (edge junctions) can be of either type. Several grain boundary configurations and combinations of boundary conditions are considered. The results clearly show accumulation or depletion of matter at open edge junctions and formation of incipient holes and hillocks near the intersection of triple junctions with the top surface. Maximum tensile or compressive stress occurs at the intersections of triple junctions and closed edge junctions with the bottom surface.