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Feedback controlled pattern formation in globally coupled semiconductor systems (Sfb 555)
The project in the period 20012004 comprised two parts:

Activatorinhibitor kinetics in reactiondiffusion systems

Pattern formation in semiconductor superlattices
(1) Activatorinhibitor kinetics in reactiondiffusion systems
The vertical charge transport in layered semiconductor structures, such as the heterostructure hot electron diode, resonant tunneling structures, thyristors, can be described under appropiate approximations by means of a twocomponent reactiondiffusion system with a noncubic nonlinearity. The activator variable is given by an internal degree of freedom, for instance the charge density at the interface, while the long range inhibitor is represented by the sample voltage. Depending on the assumptions of the model and the external circuit, the sample voltage is either globally coupled to the activator or is subject to an additional local diffusive coupling. We study both the globally and the locally coupled reactiondiffusion system on one and two dimensional spatial domains.
In the locally coupled system a Hopf bifurcation or a Turing bifurcation can occur for suitable values of the control parameter. They may coincide at a codimensiontwo bifurcation. The interaction of these two instabilities leads to highly interesting patterns, already in the onedimensional case under current controlled conditions. Besides pure Hopf and Turing modes, there also exist localized bistable patterns and mixed modes, which are characterized by a Hopf frequency and a Turing wavelength. There exist also subharmonic resonances which contain two frequencies and two wavelengths resulting in a characteristic spatiotemporal spiking pattern, where current filaments are formed and subsequently vanish. The extension of this investigations to two spatial dimensions yields various Turing patterns of symmetry and spiral waves. In onedimensional systems we find long chaotic transients and extensive spatiotemporal chaos, which may be characterized locally by a KarhunenLoéve correlation length.
In the globally coupled system different competing instabilities may occur: a Hopf bifurcation and a filamentary instability of the homogeneous state, an oscillatory instability of the stationary filament, and a spatial instability of the homogeneous relaxation oscillations. Depending upon the position of the instability thresholds relative to each other, simple or complex spatiotemporal dynamics may occur. A variety of scenarios with periodically or chaotically breathing or spiking filaments, and accelerated or decelerated fronts may be generated by variation of the system parameters, the global coupling or the system size. These studies are of basic interest not only in the context of special semiconductor models but more generally for globally coupled reationdiffusion systems in one and two spatial dimension.
Another center of interest is chaos control by timedelayed feedback in globally coupled systems. The stabilisation of unstable spatiotemporal spiking orbits can be achieved by a method of timedelay autosynchronisation, which had previously been restricted essentially to purely temporal chaos (ordinary differential equations). Conditions for successful stabilisation and schemes are investigated.
(2) Pattern formation in semiconductor superlattices
Semiconductor superlattices are composed of alternating layers of two materials; they may be conceived as a periodic sequence of potential barriers and quantum wells. At sufficiently high electric fields resonant tunneling between the ground state in a quantum well and the excited state in the adjacent well results in an Nshaped currentfield characteristic (NNDC). If the sample contains a sufficient number of charge carriers (generated optically or by doping), a high and a low field domain may arise in the growth direction, with the interface being formed by a charge accumulation. The low field domain corresponds to sequential tunneling between equivalent levels in adjacent wells, while the high field domain corresponds to resonant tunneling between nonequivalent levels. Since in principle the domain wall may be localized in any of the quantum wells, there are as many stable branches in the currentvoltage characteristic as there are superlattice periods. Thus a system with a high degree of multistability arises. If the charge carrier density is too low, selfgenerated periodic or chaotic oscillations of the domains may occur. The stationary and oscillating domains depend sensitively upon structural imperfections of the periodicity of the superlattice or doping fluctuations. We have extensively studied the formation of field domains and oscillations of the domain walls, and are focussing on the complex nonlinear spatiotemporal dynamics. While the stationary multistable currentvoltage characteristics as well as the selfgenerated domain oscillations are relatively wellunderstood, more work is needed on the following aspects: The switching behavior between the various multistable states can be controlled by different scenarios of the motion of the domain walls. Monopole, dipole, and tripole oscillations, and chaos are possible. The operation of the superlattice in a load circuit leads to a global coupling which can sensitively affect the stability and the oscillations, and the spatiotemporal patterns, in particular under ac drive. Previous theoretical and experimental work has often neglected the effect of the load and of parasitic capacitances. As we have shown numerically, electron heating leads to an additional regime of Sshaped negative differential conductance (SNDC). The combination of SNDC and NNDC (or Zshaped currentvoltage characteristics in case of stationary domains) may lead to complex novel pattern formation effects. The selfstabilization of chaotic oscillations by timedelayed feedback control has been investigated.