A very different picture of the redistribution of metal density and stress, caused by electric stressing, can be expected in multibranch interconnect structures formed by connected metal lines within the same metal layer. The absence of diffusion barriers in line junctions allows atoms to freely migrate between lines along the trajectories of the current carriers. When a multibranch structure includes metal lines that are connected in parallel, the creation of a void in one of the parallel branches does not necessarily result in a failure, which contrasts with what happens in a single line segment, because current can continue to flow in the unvoided parallel lines. The on-chip power/ground (p/g) grid is an example of such electrically redundant multibranch structures. In this chapter, we review a recently developed assessment methodology of the p/g grid MTTF and describe a novel experimental technique that could validate the proposed methodology. EM assessment performed on the grids with tens of millions of nodes has shown that the formation of the first void alone didn’t cause a grid failure. A failure criterion of 10% voltage drop increase was met due to cumulative effect of nucleation of several voids and their growth in the failed branches.

An accurate analysis of the stress evolution in a metal line loaded with an electric current requires solution of a number of coupled partial differential equations (PDEs). The continuity equations, describing the evolution of concentrations of vacancies and plated atoms along the line, are linked with the force balance equation yielding the elastic stress evolution due to interaction of the metal line volumetric deformation with the rigid confinement. The electric current density distribution is found by solving the corresponding Laplace equation. Accounting for the polycrystalline structure of the metals used as conductors in on-chip interconnects, and proper consideration of a variety of venues for diffusion of vacancies, such as grain boundaries and interfaces with liners and capping layers, requires a comprehensive 2D or 3D analysis. Following void nucleation, which happens when the tensile stress reaches a critical value, the void shape and size are described by a combination of the Cahn–Hilliard and Allen–Kahn equations with the phase-field formalism. Detailed description of these coupled PDEs and results of their solution for a number of cases using finite element analysis (FEA) are demonstrated in this chapter. A good fit between simulation results and measurements is demonstrated throughout the chapter.