  Semiconductor $p^+ - p^- - n - p^+ - n^{++}$ structures with large junction
and contact areas are treated as 1 \times 2-dimensional active media, in which
self-organized pattern formation can be expected. The local bistable behavior
of the structures may emanate from two different mechanisms both governed by a
nonlinear current feedback-loop between the electrons and holes injected from
the outer layers. By considering the device to be composed of an active
subsystem with negative differential resistance and a passive resistive layer
with positive differential resistance an analytical approach is suggested to
understand and describe the corresponding physical mechanisms in a
self-consistent way. Analytical solutions of the derived model equations allow
a description of homogeneous stationary states and yield explicit expressions
of the current-density vs. voltage characteristics of the whole structure and
its subsystems. A stability analysis of the homogeneous states with respect to
two-dimensional transversal harmonic fluctuations is performed for the two
cases under study. The resulting dispersion relations allow two different types
of instability. While the first one is of Ridley's type which is characteristic
for any spatially extended electrical system with negative differential
resistance, the second type can be considered as a solid-state analogue of
Turing's instability known as a generic instability mechanism which may lead,
e. g., to the formation of periodic patterns.
