Stability of current filaments in a bistable semiconductor system with global coupling

We study the stability of steady current filaments in a bistable semiconductor system in the presence of global coupling given by an external circuit. The system is described by a reaction-diffusion model on a two-dimensional spatial domain with Neumann boundary conditions. We prove generally for the voltage-driven regime that in a convex domain any filament has at least one unstable linear eigenmode. Introducing a global coupling may either eliminate the unstable mode with the largest increment or induce oscillatory instabilities. Filaments with negative differential conductance can be stabilized by strong global coupling. Stabilization of filaments with positive differential conductance can be achieved only by an active external circuit with negative resistance and capacitance. We present analytical arguments and numerical simulations suggesting that the boundary of the domain always attracts current filaments. Our numerical results also show that seed inhomogeneities may pin current filaments in the center of sufficiently large domains. The competition between the attractive boundary and pinning by seed inhomogeneities is studied numerically.