explaining the macroscopic oscillations in InP will be discussed.

In what follows a simplified model, based on the current burst model, explaining the macroscopic oscillations in InP will be discussed.

In contrast to current oscillations which may occur locally under constant voltage conditions at pore tips (or, more generally, in arbitrarily large domains) but add up to a constant external current if the phases are distributed randomly, local voltage oscillations are not possible - the voltage along any path between the electrodes must be the same. Therefore, the voltage oscillation under constant external current conditions may be understood if we assume that the current at the pore tip generally oscillates while the diameter stays nearly constant, i.e. the current density oscillates, too. Both assumptions which we adopt, in general, without knowing the exact current oscillation mechanism at this point [10] are general properties of the current burst model. In terms of an equivalent circuit each pore may then be described by an oscillating resistor R(t) with the average value

Figure 3.31: A schematic representation of pores as oscillating resistors, where R_{i}(t), I^{i}_{por}, j^{i}_{por}- are resistance, current and current density respectively of a pore; U_{ex}, I_{ex} - are externally implied current (constant) and measured voltage respectively. a) uncorrelated pores, j^{i}_{por}, I^{i}_{por}, R_{i}(t) - oscillate, U_{ex} constant; b) synchronized pores, j^{i}_{por}, I^{i}_{por}, R_{i}(t) - oscillate, U_{ex} and pore diameter are oscillating.

A schematic representation of pores as oscillating resistors:

(42)

where ρ is the resistivity of the oxide layer formed at the tip, L is the thickness of the oxide layer, and A is the surface of one pore tip. In a general case at least one on these parameters should oscillate in order to obtain an oscillating "pore-resistor". However, we will consider only the most obvious case. Namely, A oscillates and L and r are constant. The oscillation of A is proved by the observed modulation of pores (see Figure 3.26 and 3.28).

The system is varying this parameter in time aiming to minimize the resistance of the entire sample. The pores are closed packed and begin to interact via their space charge regions. The interaction between pores may influence the whole system, and a correlation (synchronization) between a certain number of pores can be achieved. Synchronization of pores in some parts of the sample (domains) or on the whole sample can only be achieved by correlating the phases of the oscillating resistors (see Figure 3.54b). The total current through a sample now will no longer average out to a constant value, but will also oscillate and the constant current condition enforced by the external current source now can only be maintained if the voltage oscillates so that

< I > = constant = U_{oscillates}/< R >_{oscillates} (43)

This simple model can explain the observation. Of course, in a better approximation one would have to describe a pore by a more complex equivalent circuit containing capacitors. The displacement currents (Eq. 44) have also to be compensated by the voltage adjustments, causing some degree of feedback in the system:

(44)

Nevertheless, the ultimate causes of the voltage oscillation are most probably intrinsic current oscillations together with some phase coupling or correlation between pores. This consideration, if turned around, gives a clue to the interaction mechanism between neighboring pores. If, by random fluctuation of pore diameters, pores come close enough to experience some kind of interaction, a feedback mechanism may start that leads to phase coupling of the pore growth states and by percolation to the formation of a synchronized domain.

This domain, however, may cover only a part of the specimen surface, i.e. regions with uncorrelated pores may also be found. Moreover, since percolation does not have to take place at every cycle of the oscillation - especially if the general conditions are just near the percolation point of the system - somewhat irregular voltage oscillations as shown in Figure 3.28a are possible. The frequency of the voltage oscillations in this model can be determined by the frequency of the current oscillations inherent to the current burst model.

From the observation described above, it can be concluded that the voltage oscillation can be considered as an emergent property of strongly interacting pores, which means that the oscillation will immediately disappear if the pores stop interacting.

If the pores do not interact, they have the freedom (more or less) to choose their own mode of growth: the frequency of current (not voltage) oscillation at the pore tip, diameter, rate of growth etc. Actually, these parameters do not differ extremely from pore to pore, but nevertheless can be different and randomly distributed in the huge "sea" of pores. If the pores begin to interact, a phase coupling can occur. The interaction of pores restricts the available options to a small "volume" of its parameters space, i.e.,all the pores in the system will tend to have the same oscillation frequency, diameter etc..

In essence the system moves to an "attractor" that covers only a small volume of the parameter space. An attractor is a stable state for the system and if the system starts from another state it will evolve until it reaches the attractor, and will then stay there in the absence of other effects. An attractor can be a point, a path, a complex series of states etc. All of them specify a restricted volume in parameter space. In our case, the Current-line Oriented Pores can be considered to be an attractor and the point at which the macroscopic voltage oscillations start can be considered as an evidence that the system has reached or is on the way to reach the attractor.

Actually, this point can be considered also to be a critical point of the system, where the system properties change suddenly, e.g. the pore matrix can go from a non-percolating to a percolating state and vice versa. The two states of the system on both sides of the critical point are usually defined as two different phases, pretty much in the same way as the 0_{o}C is considered the critical point between the solid and liquid state of water. Percolation in our system can be regarded as the arrangement of pores in such a way that one property, in our system the phase of current oscillation, connects the opposite sides of the structure. This can be regarded as making a path in a "disconnected" pore array. The boundary at which the system goes from a "disconnected" array to a "connected" array is a sudden one and its main feature is that at this point the system has a correlation length that just spans over the entire system.

It is known that self-organization of a system will occur if the system is neither too sparsely connected (so most of units are independent) nor too strongly connected (so that every unit affects every other). An autocorrelation-analysis of the pore positions shows an interaction between the pores of up to the sixth neighbor. This means that one pore can 'feel' or is connected up to six neighbors aligned along one straight line.

Thus, it can be concluded that in order to initiate voltage oscillations it is not necessary that each pore in the sample interacts directly with the rest of the growing pores. It is sufficient that each pore interacts directly with a limited number of its neighbors - making a domain. The domains will interact at their turn between them making bigger domains and so on until self-organization will appear along the whole sample.