Limit cycle theory of temporal current self-oscillations in sequential tunneling of superlattices

A unified theory of the temporal current self-oscillations is presented. We establish these oscillations as the manifestations of limit cycles, around unstable steady-state solutions caused by the negative differential conductance. This theory implies that both the generation and the motion of an electric-field domain boundary are universal in the sense that they do not depend on the initial conditions. Under an extra weak ac bias with a frequency $\omega_{ac}$, the frequency must be either $\omega_{ac}$ or an integer fractional of $\omega_{ac}$ if the tunneling current oscillates periodically in time, indicating the periodic doubling for this non-linear dynamical system