Oscillations at the Si/electrolyte contact: Discretization of phase oscillators

The origin of sustained current oscillations at the Si/electrolyte contact is not fully understood. Oscillatory functions are regarded which describe the oscillating oxide thickness at the silicon electrode. We consider an initially vanishing two-dimensional time dependent function which oscillates between a minimum and a maximum oxide thickness at each location of the electrode. The function is continuous except at single points of the electrode at which the oxide thickness drops deeply due to the formation of nanopores in the oxide. The oscillatory function is represented by a set of infinite (infinitesimal) oscillators. The mathematical model is based on the fact that it is sufficient to register the oscillators only one time per i-th cycle at their minimum or when the phase of the oscillator equals i 2π. In phase-space representation, the passing of the phase trough the i 2π planes leads to oscillator density functions pi(t) which define the (differential) number of oscillators passing their minimum at the i-th time at the time t. Two consecutive oscillator density functions are connected by an integral equation representing a Markov process. Together with a local model for the oxide microstructure, a fit of the model parameter to the measured current is possible. The result is that the existence of two types of oxides (with different nanopore densities) can explain sustained current oscillations and, further, it is possible to calculate the mean nanopore distance in both types of oxide.