Transcritical Hopf bifurcation and breathing of limit cycles in sequential tunnelling of superlattices

Within a discrete drift model, we study the evolution of the self-sustained current oscillation (SSCO) solutions ( limit cycles) in sequential tunnelling of superlattices under dc bias. We propose two possible modes: one is co-existence of both fixed point and limit cycle solutions, exchanging stabilities at the bifurcation point, termed the transcritical Hopf bifurcation and the other is that, at high doping densities and inside SSCO regime, the breathing motion of the limit cycles, in which the amplitude and frequency of SSCOs oscillate as a function of an applied dc bias, in contrast with a square-root dependence expected in a conventional Hopf bifurcation.