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

Nonlinear Dynamics of Surface Steps

“…In the presence of crystalline anisotropy in the mobility of adatoms along the inner void surface, a transition from stationary to oscillatory behavior occurs with increasing electromigration force or void area. Subsequent detailed analysis has shown that this transition has the character of a Hopf bifurcation [22]. The experimental signature of oscillatory void evolution are rapid oscillations in the resistance of the conductor, which have indeed been reported in the literature [23].…”

“…To check the consistency of this assumption, we consider the outflow region of the bunch, where the spacing between steps leaving the bunch becomes large and hence the nonlinear interaction terms on the right hand side of (1.14) can be neglected. We are thus left with the linear system 22) which can be solved by the exponential traveling wave ansatz…”

Electromechanically driven chaotic dynamics of voids in metallic thin films

“…22 The transition onset corresponds to a Hopf bifurcation that may be either supercritical or subcritical, depending on the symmetry of the surface diffusional anisotropy as determined ͑through m͒ by the crystallographic orientation of the film plane. 22 In this section, the focus is on m = 1, where the ⌺ = 0 Hopf bifurcation is subcritical.…”

“…These range from current-induced solitary waves and nonlinear surface wave trains propagating on an infinite metal surface in the direction of the electric field, 12,16 to soliton-like features that travel on large-size void surfaces preceding the failure of metallic thin films, 8,21 and to stable wave propagation on smaller-size void surfaces in films driven by a strongerthan-critical electric field. 14,22 Recent theoretical studies also have predicted electromigration-induced complex shape evolution of homoepitaxial islands on electrically conducting substrates 18 and step meandering on vicinal surfaces. 20 Fundamental understanding of the nature and origin of such surface wave phenomena and mapping of the surface morphological stability domains associated with the propagation of such wave patterns are crucial for identifying the conditions under which electromechanically driven oscillatory surface dynamics can be generated and stabilized.…”

“…In self-consistent modeling studies of electromigration-driven dynamics of void surfaces in metallic thin films, over a broad range of electric-field strengths, void sizes, and surface transport anisotropy parameters, the most complex asymptotic states that have been stabilized are time-periodic states that consist of surface waves traveling on voids; 14,22 these morphologically stable voids are solitons ͑driven by the electric field͒ that migrate at a constant speed along the metallic film in the appliedfield direction. 17 In addition to these studies, models of electromigration-driven homoepitaxial islands on conducting solid substrate surfaces have been shown to exhibit oscillatory and chaotic dynamics; 18 these are the most complex electromigration-driven stable surface states that have been reported to date.…”