Hopf bifurcation, bistability, and onset of current-induced surface wave propagation on void surfaces in metallic thin films

Abstract

We analyze the electromigration-induced stable wave propagation on surfaces of voids in metallic thin films based on self-consistent numerical simulations of current-driven surface morphological evolution according to a fully nonlinear continuum model. The void surface morphological response is studied for single-crystalline films with twofold and fourfold symmetry of surface diffusional anisotropy, characteristic of <1 1 0>- and <1 0 0>-oriented film planes in face-centered cubic (fcc) metals, respectively, and the effects of the electric-field strength, the surface diffusional anisotropy strength, and the void size are examined. Variation of the above parameters past critical values is found to cause transitions from stable steady to stable oscillatory morphological responses on voids migrating along the films at constant speed; these transitions are the result of Hopf bifurcation at the corresponding critical points. For every bifurcation parameter, the nature of the Hopf bifurcation is determined by the symmetry of the surface diffusional anisotropy on the film plane: the bifurcation is supercritical for fourfold symmetry and subcritical for twofold symmetry. The amplitude and frequency of the void surface oscillations are calculated as a function of the bifurcation parameters and the corresponding bifurcation diagrams are constructed. Our simulations reveal hysteresis phenomena and bistability in both cases of Hopf bifurcation, i.e., for both twofold and fourfold symmetry. For twofold symmetry, i.e., for subcritical Hopf bifurcations, the transitions are between a stable steady state and a stable time-periodic state. For fourfold symmetry, i.e., for supercritical Hopf bifurcations, the transitions may also be between two stable steady states; one of the two steady states bifurcates (upon further variation of the bifurcation parameter) to a time-periodic state at the corresponding Hopf point. The film’s electrical resistance evolution reflecting the void’s morphological evolution is monitored in the regime of oscillatory void morphological response and it is found to be in good agreement with experimental measurements in accelerated electromigration testing.