system can serve the detailed analysis of the observed external macroscopic oscillations.

As an indirect approach to gather additional information about the InP/electrolyte system can serve the detailed analysis of the observed external macroscopic oscillations. In what follows we will use the wavelet analysis, described in some more details in Appendix 2, to characterize the oscillations observed during the pore formation in n-InP. Such kind of analysis allows one to investigate the changes in oscillation frequency at different stages of pore formation.

The oscillation behavior of the etching system was investigated at different concentrations and temperatures of the electrolyte. Examples of voltage oscillations observed during the pore formation process at two electrolyte concentrations and their corresponding wavelet transformation maps are presented in Figure 3.29. The X and Y axis of the wavelet diagrams represent the (etching) time and frequency respectively. Note that the frequency on the Y-axis is inverted, i.e. low frequencies are up and high frequencies are down on the axis. Each point (t,f) in the diagram has a definite color showing the value of the amplitude of the wavelet transformation taken between the investigated signal in the neighborhood of the moment t, with a "mother" function of frequency f. For a normalized amplitude between 0 and 1, the red color means amplitude 1, whereas the black color means zero amplitude. The amplitude will tend to its maximum if the frequency of the investigated signal in the neighborhood of point t will match the frequency of the mother function.

In Figures 3.29b and d a continuous red region is easily observed. The red-color region represents the change in the dominant frequencies of the investigated signal along the time axis. From the diagrams it can be observed that in both cases the dominant frequency has a negative slope, or otherwise stated, the dominant frequency is decreasing in time. Thus, at the beginning the frequency is higher than at the end of the signal, which represent actually the beginning and end of the experiment. The experiments show that vanishing in time of the dominant frequency is always valid, independent of the electrolyte concentration, temperature or doping level of the samples.

wavelet transformation used to analyze the external voltage oscillations observed during the formation of porous n- InP." />

Figure 3.29: Wavelet transformation used to analyze the external voltage oscillations observed during the formation of porous n-InP. a,b) the oscillation in time at in 5% HCl aqueous electrolyte solution and the corresponding wavelet map; c,d) the oscillation in time at in 10% HCl aqueous electrolyte solution and the corresponding wavelet map. The red regions (see the white lines) represent the dominant frequency in the spectra.

The frequency dependence on temperature and external current density is shown in Figure 3.30. From Figures 3.30a-d it is obvious that the voltage oscillations are much more stable at lower temperatures. The amplitude as well as the stability of the signal is decreasing as the temperature is increased, e.g. from T=5^{o} C to T=20^{o} C. At j=26 mA/cm^{2} and T=20^{o}C, the oscillations nearly disappear.

In Figure 3.30e and f it is shown the dependence of the average frequency of oscillations on temperature and current density. As mentioned above, the dominant frequency is changing in time, therefore the average value in this case means 1/2× (f_{begin}+f_{end}), where f_{begin} and f_{end} are the oscillation frequencies at the beginning and at the end of the experiment respectively. The mean oscillation frequency increases with both increasing the current density and increasing the temperature of the electrolyte. In addition, the frequency increases linearly with the current density. This means that the charge consumed per oscillation cycle (see the slope of the curves in Figure 3.30f) is constant and does not depend upon the current density.

It is important to note that the observed data agree with qualitative predictions of the current burst model. As mentioned above, the dominant time constants of a current burst are the oxide formation time, oxide dissolution time and passivation time. Each parameter which will change the sum (t_{sum}, see Eq. 41) of these time constants will also affect the frequency of the external oscillations:

(41)

For example, the decrease of the frequency as the pores grow into the substrate can be directly related to oxide dissolution time constant. As the pores grow into the substrate the oxide is less effectively dissolved due to diffusion losses, as a consequence tsum increases, thus the frequency of the external oscillations decreases. On the other hand, for higher temperatures the oxide dissolution time constant decreases (tsum decreases), the oxide is faster dissolved and thus the frequency should increase.

Concerning the frequency increase as the current density increases, it is most probable that the passivation time constants is decreased in this case. Decrease of tsum results in an increase of the frequency. However, this can be true only if the passivation time constant has the same order of magnitude as the oxide dissolution time constant. This is not the case in Si, where the time for oxide dissolution is much higher than the time for passivation. In InP, however, all three constants can be expected to be of the same order of magnitude, because of the instability of InP-oxides in acidic solutions.

Figure 3.30: Temperature and current dependence of voltage oscillations. a,b,c,d) The voltage oscillations observed at different current densities and three different temperatures. The oscillations are more stable at low temperatures; e,f) Dependence of the average frequency on the current density and temperature.