Hybrid Model for Chaotic Front Dynamics: From Semiconductors to Water Tanks

We present a general method for studying front propagation in nonlinear systems with a global constraint in the language of hybrid tank models. The method is illustrated in the case of semiconductor superlattices, where the dynamics of the electron accumulation and depletion fronts shows complex spatiotemporal patterns, including chaos. We show that this behavior may be elegantly explained by a tank model, for which analytical results on the emergence of chaos are available. In particular, for the case of three tanks the bifurcation scenario is characterized by a modified version of the one-dimensional iterated tent map.

Figure 1

Space-time plot of the electron density evolution in a semiconductor superlattice. Electron accumulation and depletion fronts are indicated by light and dark shading, respectively. The full microscopic sequential tunneling model of Ref. 12 for $N=100$ quantum wells and contact conductivity $\sigma =0.5{\Omega }^{-1}{\mathrm{m}}^{-1}$ is used for a bias $U=0.95\mathrm{V}$. The inset shows a simplified view of the pattern, which is adopted in the front and the tank model. The dashed lines denote the switching times $t_m$.

Figure 2

Scheme of an $n$-tank switched arrival system with minimal filling height $x_{\mathrm{m}\mathrm{i}\mathrm{n}}$.

Figure 3

Current density vs electric field characteristic at the emitter barrier (straight line, Ohmic conductivity $\sigma =0.5{\Omega }^{-1}{\mathrm{m}}^{-1}$) and between two neutral wells (N-shaped curve) for the microscopic superlattice model [12]. $J_c$ denotes the intersection point of the two characteristics. $F^l$ and $F^h$ are the stable low and high field states, respectively, for the current density $J_c$. Inset: Positions where accumulation and depletion fronts annihilate vs voltage.

Figure 4

Bifurcation diagram of $x_1$ vs $L_h$ in units of $p_h$. Inset: Graph of the return map for the $n=3$ tank model Eq. (5). In the shaded region the map is not defined.