Finite element analysis of the effects of geometry and microstructure on electromigration in confined metal lines

Abstract

A simple mathematical model based on atomic drift and diffusion is advanced to describe electromigration-induced stress in confined metal lines. Using a finite element approach, a MATLAB program was developed to simulate stress evolution in lines with variable geometry and microstructure under different boundary conditions. The simulation results show that the contact pads connected to the line end in the National Institute of Standards and Technology (NIST) test structure postpone the stress buildup in a microstructurally homogeneous line and the time to reach a certain stress is proportional to the pad sizes when the pads are much wider than the line. Subtractive defects are not the preferred failure sites when the defects fall on a wide polycrystalline line or a bamboo structure line. In a narrow polycrystalline conductor with a distribution of grain size, a subtractive defect may result in an abrupt change in the effective diffusion coefficient (blocking effect) along the line and quick buildup of the stress at the defect site. The stress at a center blocking site in a 190 μm line reaches maximum in 146 h, under 1 MA/cm2 and at 200 °C. Electromigration lifetime is most sensitive to distribution of grain size and variation of the linewidth when the linewidth is a few times average grain size. For a near bamboo structure line, the evolution of the stress after local quasisteady state is affected by line end conditions as well as by distribution of the polygranular clusters. Under a constant source boundary condition, the maximum stress at final steady state depends on the length and distribution of individual polygranular clusters in a line. Under blocking end conditions, the stress distribution in final global steady state (if it exists) is determined by electrical current density and is independent of microstructure of a line. The cluster/bamboo juncture is the most severely stressed site in the early stages, and the blocking line end will finally become the most severely stressed site if the line does not fail at an early time. The magnitude of the maximum stress at the cluster/bamboo juncture at the local steady state depends on the ratio of the effective diffusivity of the cluster to that of the bamboo segment, Dg/Db, as well as on the cluster length and current density.