Diffuse interface model for electromigration and stress voiding

Abstract

We report a diffuse interface or phase field model for simulating electromigration and stress-induced void evolution in interconnect lines. Our approach is based on the introduction of an order parameter field to characterize the damaged state of the interconnect. The order parameter takes on distinct uniform values within the material and the void, varying rapidly from one to the other over narrow interfacial layers associated with the void surface. The evolution of this order parameter is shown to be governed by a form of the Cahn–Hilliard equation. An asymptotic analysis of the equation demonstrates, as intended, that the zero contour of the order parameter tracks the motion of a void evolving by diffusion under the coupled effects of stresses and the “electron wind” force. An implicit finite element scheme is used to solve the modified Cahn–Hilliard equation, together with equations associated with the accompanying mechanical and electrical problems. The diffuse interface model is applied to simulate a range of problems involving void evolution in interconnect lines. Results obtained are shown to be in excellent agreement with those computed previously using equivalent sharp interface models.