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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.