The project in the period 2001-2004 comprised two parts:
Activator-inhibitor kinetics in reaction-diffusion systems
Pattern formation in semiconductor superlattices
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 two-component reaction-diffusion system
with a non-cubic 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 reaction-diffusion
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 codimension-two bifurcation. The interaction of these two instabilities leads to highly interesting patterns, already in the one-dimensional 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 spatio-temporal 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 one-dimensional systems we find long chaotic transients and extensive spatio-temporal chaos, which may be characterized locally by a Karhunen-Loéve 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 spatio-temporal 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 reation-diffusion systems in one and two spatial dimension.
Another center of interest is chaos control by time-delayed feedback in globally coupled systems. The stabilisation of unstable spatio-temporal spiking orbits can be achieved by a method of time-delay autosynchronisation, which had previously been restricted essentially to purely temporal chaos (ordinary differential equations). Conditions for successful stabilisation and schemes are investigated.
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 N-shaped current-field 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 non-equivalent levels. Since in principle the domain wall may be localized in any of the quantum wells, there are as many stable branches in the current-voltage characteristic as there are superlattice periods. Thus a system with a high degree of multistability arises. If the charge carrier density is too low, self-generated 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 spatio-temporal dynamics. While the stationary multistable current-voltage characteristics as well as the self-generated domain oscillations are relatively well-understood, 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 spatio-temporal 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 S-shaped negative differential conductance (SNDC). The combination of SNDC and NNDC (or Z-shaped current-voltage characteristics in case of stationary domains) may lead to complex novel pattern formation effects. The self-stabilization of chaotic oscillations by time-delayed feedback control has been investigated.