Origin of CPE Dispersion

Because of the presence of the unknown parameter/g in Eq. (8.18), it impossible to determine the value of the factor b, and the values of the charge transfer resistance and double-layer capacitance remain unknown. From the experiment it is possible to determine only these parameters multiplied or divided by the factor 

The presence of the CPE in experimentally measured impedances generated a lot of discussion and confusion in the literature [334]. Of course, the fractal model described earlier also leads to CPE behavior for blocking electrodes however, in the presence of faradaic reactions it leads to skewed semicircles instead of a decrease in the center of the semicircle below the real axis, without the further deformation typically observed in experimental conditions (CPE behavior). 

In general, the appearance of CPE-like behavior can be attributed to two phenomena  

Dispersion due to surface adsorption/diffusion processes (the so-called kinetic dispersion effect). 

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It should be noted, though that defective semiconductors - for instance oxide layers on metals - typically do not show an ideal capacitive behavior which often leads to a strong frequency-dependence of the capacitance. There are many possible reasons for non-ideal behavior such as ionic participation , a frequency-dependent dielectric constant, contributions from the Helmoltz-layer or from surface states, non-ideal structure or nonideal donor distribution, as well as inhomogeneous depth distribution in the composition or structure of the oxide layer. Independent of the origin of the non-ideal behavior, the frequency dispersion can partially be corrected by replacing the capacitance in impedance fits by a so-called constant phase element (CPE), which takes into consideration the non-ideal nature of a capacitance. While the introduction of CPE may eliminate the... 

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