Surface evolution in bare bamboo-type metal lines under diffusion and electric field effects

• Authors: Amir Averbuch;Moshe Israeli;Menachem Nathan;Igor Ravve
• Venue: Journal of Computational Physics
• Year: 2003

Irregularities such as voids and cracks often occur in bamboo-type metal lines of microelectronic interconnects. They increase the resistance of the circuits, and may even lead to a fatal failure. In this work, we analyze numerically the electromigration of an unpassivated bamboo-type line with pre-existing irregularities in its top surface (also called a grain-void interface). The bamboo line is subjected to surface diffusion forces and external electric fields. Under these forces, initial defects may either heal or become worse. The grain-void interface is considered to be one-dimensional, and the physical formulation of an electromigration and diffusion model results in two coupled, fourth order, one-dimensional time-dependent PDEs, with the boundary conditions imposed at the electrode points and at the triple point, which belongs to two neighboring grains and the void. These equations are discretized by finite differences on a regular grid in space, and by a Runge-Kutta integration scheme in time, and solved simultaneously with a static Laplace equation describing the voltage distribution throughout each grain, when the substrate conductivity is neglected. Since the voltage distribution is required only along an interface line, the two-dimensional discretization of the grain interior is not needed, and the static problem is solved by the boundary element method at each time step. The motion of the interface line is studied for different ratios between diffusion and electric field forces, and for different initial configurations of the grain-void interface. We study plain and tilted contour lines, considering positive and negative tilts with respect to the external electric field, a stepped contour with field lines entering or exiting the 'step', and a number of modifications of the classical Mullins problem of thermal grooving. We also consider a two-grain Mullins problem with a normal and tilted boundary between the grains, examining positive and negative tilts.