Frequency dependence single-crystal electrodes

The interfacial capacitance may also be measured at solid polarizable electrodes in an impedance experiment using phase-sensitive detection. Most experiments are carried out with single crystal electrodes at which the structure of the solid electrode remains constant from experiment to experiment. Nevertheless, capacity experiments with solid electrodes suffer from the problem of frequency dispersion. This means that the experimentally observed interfacial capacity depends to some extent on the frequency used in the a.c. impedance experiment. This observation is attributed to the fact that even a single crystal electrode is not smooth on the atomic scale but has on its surface atomic level steps and other imperfections. Using the theory of fractals, one can rationalize the frequency dependence of the interfacial properties [9]. The capacitance that one would observe at a perfect single crystal without imperfections is that obtained at infinite frequency. Details regarding the analysis of impedance data obtained at solid electrodes are given in [10]. 

In the last part of this chapter, our attention will be focused on the electrochemical properties of individual crystal faces of HTHP diamond single crystals, as well as single-crystal (homoepitaxial) CVD diamond films. Our preliminary studies showed that the HTHP single crystals, on the whole, are similar to the CVD polycrystalline films in terms of their electrode behavior. In particular, both the polycrystalline thin-film electrodes and the HTHP single crystal electrodes are equally characterized by the special type of frequency-dependent capacitance described by the CPE. 

Fig. 109. Frequency dependence of conductivity (G) for single crystals of MsNbjOFis and KsNb3OF,g (electrodes were deposited onto the (001) faces). Reproduced from [443], A. I. Agulyansky, J. Ravez, R Von Der Mtihll, A. Simon, Ferroelectrics 158 (1994) 139, Copyright 1994, with permission of Taylor Francis, Inc., http //www.routledge-ny.com.

The above-described situation is but an exception rather than the rule. Generally, the diamond electrode capacitance is frequency-dependent. In Fig. 12 we show a typical complex-plane plot of impedance for a single-crystal diamond electrode [69], At lower frequencies, the plot turns curved (Fig. 12a), due to a finite faradaic resistance Rp in the electrode s equivalent circuit (Fig. 10). And at an anodic or cathodic polarization, where Rf falls down, the curvature is still enhanced. At higher frequencies (1 to 100 kHz), the plot is a non-vertical line not crossing the origin (Fig. 12b). Complex-plane plots of this shape were often obtained with diamond electrodes [70-73],... 

In their chapter on time- and frequency-resolved studies of photoelectrochemical kinetics, Peter and Vanmaekelbergh give an extensive survey of how modulation techniques such as photoelectrochemical impedance spectroscopy or intensity-modulated photocurrent spectroscopy can yield valuable information on the time dependence of reactions at semiconducting surfaces over a broad range of time scales. Kinetic studies with single crystals as well as porous or nanocrystalline material reveal the important role that is played by the bulk structure of semiconductor electrodes. 

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