Obvious answer: it is because the capacity C is changing with frequency. The question "why capacity has frequency dispersion?" is more fundamental. How do you extract capacity from overall impedance measured? Do you use any particular model, or just an out-phase impedance component? 

I'm currently using two equivalent circuit models (one for light and one for dark).

With regards to your answer of capacity dispersion due to frequency; why does this occur?

This is strange then: you are fitting EIS spectra for the whole frequency range, and finding 1 capacity value for a given potential. And this value does not depend on the frequency, because you've used all frequencies measured, haven't you?

Answering your question about the dispersion of capacity, there are 2 fundamental reasons (at least) to observe it for electrochemical systems.

1. Materials in electrochemical systems, like your semiconductor are often complex and non-homogeneous, contrary to usual capacitor. So, capacitance of different parts of your sample can be very different. Thus, modelling such a system by equivalent circuit with only 1 capacitor is an oversimplification, and one should use the system of capacitors in parallel. But, of course, it makes modelling more complicated and less reliable.  Otherwise, one is using CPE, as you did, and the value of a-parameter of CPE can be used as an indication of inhomogeniety of the material (for ideal capacitor a=1).  

You may want to look into: Z.Kerner, T.Pajkossy, Electrochim.Acta, 46(2000)207: On the origin of capacitance dispersion of rough electrodes.

2. Your experimental set-up may be not optimized for EIS measurements. In particular, relative position of WE and CE and RE is important. Conditions of polarization of WE should be the same for the whole electrode: it means that CE, ideally, should be larger than WE and placed around it. The tip of Luggin capillary should not be too far from WE also, to ensure correct and uniform probing of potential of the whole WE. It looks like you have rather high cell resistance (ca. 200-300 Ohm, according to your spectra), so, I am not quite sure if your setup is really optimized.

One way to check it it to measure an impedance of simplest system available. Take, for example, Au electrode, and measure it in perchloric acid. It should give something very close to ideal RC-behaviour (unless you have very rough Au). If not, you have a problem in your setup. 

The capacitance values were not calculated using the impedance measurements. Rather the Mott-Schottky experiment/analysis was used for the Mott-Schottky results. 

The Luggin capillary was not used for the experimental set up. A custom cell was used instead which placed the CE quite far from the WE. Unfortunately, the CE used was a Pt wire and thus was smaller than the WE. What effect would this have, however?

Quote: "The capacitance values were not calculated using the impedance measurements."

Then again, how the capacitance was measured, if not by impedance? MS analysis is just plotting the capacitance attributed to depletion layer in 1/C^2 - E coordinates.  I am sorry for insisting, but this is indeed the key question, because frequency dispersion of the parameters of MS plots is due to the frequency dispersion of capacitance measured. 

As for the setup, it might be OK, if your CE is far enough from WE. Then the ploarization conditions of WE should be similar in all points of the interface. 

I seem to understand now. Thank you Sergey!

Dear Jason,

when looking at your typical data, the following comes to my mind :

1. The data is incomplete. What is missing are the lowest frequencies until steady state is reached !!). In order to find the "right" frequency for your Schottky-Mott C-V analysis, you have to identify two (typically) CONSTANT capacitance levels in your EIS experiment, one at higher frequencies due to bulk response and one at lower frequencies, usually due to depletion region response. It is this capacitance level (the real part of the measured capacitance) that changes (approximately !!) with applied dc voltage according to Mott-Schottky formula. The measured capacitance in other frequency regions does also change a bit with the applied dc voltage bias, but NOT according Mott-Schottky formula. My guess is that in  your C-V analysis frequency range (between 1 Hz and 10 Hz), your measured capacitance DOES NOT HAVE a clear physical meaning. It is just a region between the low frequency large capacitance (depletion) and high frequency low capacitance  (bulk). What you call a "dispersion of capacitance", is just reflection of the fact that you measure and C-V analyze at "wrong" frequencies.

2. When you say that the Schottky-Mott analysis has not been done using your EIS data, then I am really confused as Sergey is. How did you get the C-V data shown in your first figure ?!! The only reasonable interpretation of the insert in this figure is that you measured capacitance of your system at 1 Hz, ... and 10 Hz ! Even if you used some kind of C-V meter, you must have chosen the frequencies of the measurement and these frequencies are NOT the frequencies where you measure depletion capacitance alone . This is , in my view, the reason that you do not get the same mobile charge densities with changing frequency of the measurement.

3. If universal response (your Constant Phase Element - CPE) is neglected, all the measured capacitance levels are geometrical capacitances of the respective regions (the same value of dielectric constant) within the system under test (bulk and depletion). The only reason that one "sees" these levels are the very different levels of the resistances within these regions, it is NOT because some mysterious depletion region polarisation sets in.

4. In electrochemical systems (blocking electrodes) , there is NO DEPLETION in a sense of an usual semiconductor (Schottky electrode(s)). There is though mobile charge polarisation leading to accumulation and depletion of mobile charges at the respective electrodes. This leads to a measurable capacitance level, but the width of this region is Debye screening length and not Schottky depletion length.

5. In your equivalent R,C,L fitting circuit(s), there is a missing capacitance level CR1 (in parallel with R1 - probably bulk) and capacitance level CR4 (in parallel with R4 - probably depletion ).

6. You use technical "jargon" that makes it difficult for non-experts to understand. What is WE, CE and RE ? Also, what is your system that you measure ? If you explain to me what these terms mean and describe your system, I might be able to help further,

With best regards

Petr

Dear Pete,

1. This makes sense and tells me I'll have to run the measurement at different frequencies. 

2. I was confused as to whether Sergey meant that I got the C-V data from potentiostatic electrochemical impedance spectroscopy. As you said, I measured the capacitance at the different frequencies (albeit incorrect ones)

3. I am not sure what you mean here. Can you offer more insight?

4. Does this mean I am effectively measuring the capacitance due to the Debye screening and not the Schottky depletion?

6. WE- Working electrode; CE- Counter electrode; RE- Reference electrode.

I'm currently investigating doped (in this case Ni) CuWO4 for water splitting applications. The counter electrode is a Pt wire. The ref. electrode is Ag/AgCl in 0.1M KCL. The electrolyte is a phosphate buffer solution at pH 7.4.