Bifurcation Behavior of a Superlattice Model

We present a complete description of the stationary and dynamical behavior of semiconductor superlattices in the framework of a discrete drift model by means of numerical continuation, singular perturbation analysis, and bifurcation techniques. The control parameters are the applied DC voltage (ϕ) and the doping (ν) in nondimensional units. We show that the organizing centers for the long time dynamics are Takens-Bogdanov bifurcation points in a broad range of parameters and we cast our results in a ϕ-ν phase diagram. For small values of the doping, the system has only one uniform solution where all the variables are almost equal. For high doping we find multistability corresponding to domain solutions and the stationary solutions may exhibit chaotic spatial behavior. In the intermediate regime of ν the solution can be time-periodic depending on the bias. The oscillatory regions are related to the appearance and disappearance of Hopf bifurcation tongues which can be sub- or supercritical. These results are in good agreement with most of the experimental observations and also predict new interesting dynamical behavior.