Level set modeling of transient electromigration grooving

Abstract

A numerical investigation of grain-boundary (GB) grooving by means of the level set (LS) method is carried out. GB grooving is emerging as a key element of electromigration (EM) drift in polycrystalline microelectronic (ME) interconnects, as evidenced by a number of recent studies. The purpose of the present study is to provide an efficient numerical simulation, allowing a parametric study of the effect of key physical parameters (GB and surface diffusivities, grain size, current density, etc.) on the EM drift velocity as well as on the morphology of the affected regions. An idealized polycrystalline interconnect which consists of grains separated by parallel GBs aligned normal to the average orientation of interconnect's surface is considered. Surface and GB diffusions are the only diffusion mechanisms assumed. The diffusion is driven by surface curvature gradients and by an externally applied electric field. The corresponding mathematical system is an initial boundary value problem for a two-dimensional Hamilton–Jacobi type equation. To solve the electrostatic problem at a given time step, a full model based on the solution of Laplace's equation for the electric potential is employed. The resulting set of linear algebraic equations (from the finite difference discretization of the equation) is solved with an effective multigrid iterative procedure. The details of transient slit and ridge formation processes are presented and compared with theoretical predictions on steady-state grooving [E. Glickman, M. Nathan, J. Appl. Phys. 80 (1996) 3782; L. Klinger, E. Glickman, V. Fradkov, W. Mullins, C. Bauer, J. Appl. Phys. 78(6) (1995) 3833; L. Klinger, X. Chu, W. Mullins, C. Bauer, J. Appl. Phys. 80(12) (1996) 6670]