(iii) Pattern formation and selection in semiconductor superlattices

Figure 9: Schematic energy band structure of a superlattice.

Semiconductor superlattices consist of an alternating layer sequence of two different materials. At sufficient barrier thickness electrons are assumed to be localized in the individual wells. The resulting schematic energy band structure is shown in Fig. 9. Furthermore it is assumed, that electrons in one well are in local equilibrium with the majority of them occupying the lowest energy level. Electrons can tunnel from the ground state of a well to a free state of the next well, where a possible difference between the state energies can be compensated by the electric field acting between neighbouring wells. The current density $j_{m\to m+1}(F_m, n_m, n_{m+1})$ from well $m$ to $m+1$ is thus a nonlinear function of the electric field $F_m$ between the two wells as well as of the electron densities $n_m$ and $n_{m+1}$ in the involved wells. For concrete microscopic calculations of $j_{m \to m+1}$ we have used the sequential tunneling model which has been developed in our group, cf. review paper by A. Wacker [17]. For the contact currents at the emitter $j_{0\to 1}$ and the collector $j_{N\to N+1}$ we simply assume Ohmic boundary conditions, which are characterized by a contact conductivity $\sigma$ . $N$ is the number of quantum wells in the superlattice.

Therefore the following equations of motion for the electron densities arise:

$$e \dot{n}_m = j_{m-1 \to m} - j_{m\to m+1} m = 1, \ldots N,$$ (10)

$$\epsilon_r \epsilon_0 (F_m - F_{m-1}) = e(n_m -N_D) m = 1, \ldots N,$$ (11)

$$U_0 = - \sum_{m=0}^N F_m d,$$ (12)

with electron charge $e<0$ , $\epsilon_r$ and $\epsilon_0$ the relative and absolute permittivities respectively, $N_D$ the doping, $U_0$ the external voltage and $d$ the period of the superlattice. Eq. (12) describes a global constraint due to the total voltage. The total current through the superlattice is given by $j = \sum_m j_{m\to m+1} /(N+1)$ [34].

Figure 10: Chaotic scenario of the dynamics of the electron densities in a superlattice for various voltages (space-time plots). Light regions correspond to electron accumulation, dark regions to electron depletion. See [18].

Figure 11: Bifurcation diagram of the collision positions (quantum well index) of accumulation and depletion fronts for various voltage values $U$ . See [18].

Depending on the physical parameters (especially on $\sigma$ and $N_D$ ) the system of equations (10), (11) and (12) at a constant $U_0$ has either stationary or oscillatory spatially inhomogeneus solutions (field domains bounded by electron accumulation and depletion). In the stationary case the system is in general multistable, i.e. for one value of the voltage there are many stable branches, which differ for instance, in the resulting current. In collaboration with L. Bonilla (Madrid), the next question to be treated was which of these branches would be selected by the system after an abrupt or continuous change of the external voltage [9,36]. At this point we determined that the final state of the system can depend very sensitively on the difference between the initial and final voltage. This property could then be used to select operating points. The majority of the partially surprising effects could be explained by the fact that at the emitter, pairs of electron accumulation and depletion fronts (dipole) were generated. Our theoretical predictions on switching dynamics between multistable states were later quantitatively confirmed by experiments performed at the Paul-Drude-Institute in Berlin [5].

By closer investigations on the front generation process at the emitter, as well as of the motion of the fronts inside the device, we successfully generated complex self-oscillations like tripole modes [19]. It was shown that the front generation at the emitter depends substantially on the contact conductivity $\sigma$ and the total current $j$ . In particular, chaotic front dynamics in a non driven superlattice were proven for the first time [18]. A typical bifurcation scenario is shown in the electron density plots in Fig. 10. We can see that with increasing voltage the superlattice exhibits both periodic and chaotic behaviour. The full bifurcation diagram (Fig. 2.11) exhibits an alternating sequence of chaotic and periodic regions as well as a striking cobweb structure, whose center lies at $U_0=0.9V$ .

Our further objective was to reduce the front model to a simple elementary basis. In collaboration with the group of U. Parlitz (Göttingen) we found a surprising analogy to a tank model, which is normally used in a totally different context, for describing industrial production processes [28]. Consider a system of a given number of tanks. A swithching server fills one of the tanks and at the same time all nonempty tanks drain. The server then switches to a new tank, as soon as it is empty, under the condition that the tank which it is currently filling has reached the minimum filling height $p_h$ . The relation between the inflow- and outflow rates is chosen such, that the total amount of the water $L_h$ stays constant. The filling heights of the tanks correspond to the length of the high-field domain (in the superlattice system) between a depletion and an accumulation front or between the first depletion front and the emitter for the tank that is actually being filled. The switching of the server in the tank system coresponds to the generation of a dipole front at the emitter in the superlattice system.

Figure 12: Bifurcation scenario shown in the modified tent map in the inset. The parameters $x_1$ and $L_h$ correspond to the front position and the applied voltage respectively. See [28].

For three tanks the resulting dynamics is described by a one dimensional piecewise linear iterated map as in the inset of Fig. 12. This modified map has only one bifurcation parameter $L_h/p_h$ . The corresponding bifurcation diagram in Fig. 12 agrees in detail with the microscopically calculated bifurcation diagram in Fig. 11. In particular the cobweb structure is reproduced in detail. We can therefore show that the front dynamics in the superlattice can be explained on a very fundamental basis using iterated maps [28]. Because the microscopic properties of the superlattice do not come up, it must be assumed, that a similar reduction may also be possible for complex front systems with global coupling in many other disciplines, and that our reduced model may describe a universal bifurcation scenario.

From a technological point of view, oscillatory superlattices are interesting as GigaHertz-generators. In collaboration with the experimental group of E. Schomburg and K. Renk (Regensburg) we analyzed the high frequency impedance of the superlattices, as well as the behaviour of the superlattices in a resonator under the influence of an external AC voltage [20,21,38]. The front dynamics are controlled by a periodic AC voltage and exhibits typical behaviour like Arnold tongues, devil's staircase and phase synchronization. Apart from that we discovered that applying a suitable external circuit with capacitive and inductive elements to the superlattice, may change its oscillation mode in which front motion is supressed (quenched mode) and which leads to an eigenfrequency more than twice as big as the nominal frequency of the superlattice. In this context we developed together with the Regensburg group concrete proposals for experimental realization of electronic high frequency oscillators [22].

Figure 13: Successful stabilization of a chaotic front pattern (a) applying time-delayed feedback between voltage and current (b). See [29]

Furthermore, regarding applications of the superlattice as a high frequency oscillator, it is also important to create a stable periodic output signal and suppress potential chaotic oscillations. With that in mind we investigated the chaotic front dynamics under various feedback schemes. We could show for the first time that a control scheme with global time-delayed feedback, simple to realize, is successful. [29,30]. For this purpose we substitute in (12) $U_0$ by $U_0 +U_c(t)$ , with a control voltage

$$U_c(t) = - K\left(\overline{J}(t)-\overline{J}(t-\tau)\right) + R U_c(t-\tau),$$ (13)

with

$$\overline{J}(t) = \alpha A\int_0^t j(t') e^{-\alpha (t-t')} \mathrm{d}t'$$ (14)

where $A$ is the cross section of the device and $\alpha$ is a damping constant. We have shown that it is necessary to modify the conventional Pyragas method by a low-pass filter (14) due to the discrete structure of the superlattice. Successful control using this method is demonstrated in Fig. 13.