Mott-Schottky plots, frequency dependence

Occasionally, the impedance spectra of diamond electrodes are well described by the Randles equivalent circuit with a frequency-independent capacitance (in the 1 to 105 Hz range) [66], Shown in Fig. 11 is the potential dependence of the reciprocal of capacitance squared, a well-known Mott-Schottky plot. Physically, the plot reflects the potential dependence of the space charge region thickness in a semiconductor [6], The intercept on the potential axis is the flat-band potential E whereas the slope of the line gives the uncompensated acceptor concentration NA - Nd in what follows, we shall for brevity denote it as Na ... 

It goes without saying that the frequency dependence of capacitance, which follows from the complex-plane plots of the type shown in Fig. 12, manifests itself in a frequency-dependent slope of Mott Schottky plots (Fig. 13) [78], The complications in calculating Na thus involved will be discussed at length in Section 5.3 below. 

The problem of the frequency dependence of the differential capacitance of diamond electrodes, which manifests itself in the frequency dependence of the slope of Mott-Schottky plots, can be subdivided into two aspects (1) by the process(es) causing the frequency dependence and (2) the most convenient format for the presentation of this dependence. 

The nature of the frequency dependence of Mott-Schottky plots for semiconductor electrodes has been discussed in the electrochemical literature for more than three decades (see e.g. reviews [6, 84]). It has been speculated that it can be caused by the following factors (1) frequency dependence of dielectric relaxation of the space charge region [85], (2) roughness of the electrode surface [84], (3) slow ionization of deep donors (acceptors) in the space charge region in the semiconductor [86], and (4) effect of surface states. 

Thus, we have to conclude that, without knowing the physical nature of the frequency dependence of the differential capacitance of a semiconductor electrode, the donor (or acceptor) concentration in the electrode cannot be reliably determined on the basis of the Schottky theory, irrespective of the Mott-Schottky plot presentation format. Therefore, the reported in literature acceptor concentrations in diamond, determined by the Schottky theory disregarding the frequency effect under discussion, must be taken as an approximation only. However, we believe that the o 2 vs. E plot (the more so, when the exponent a approaches 1), or the Ccaic 2 vs. E plot, are more convenient for a qualitative comparison of electrodes made of the same semiconductor material. 

FRA systems are versatile, and they can be controlled to acquire and analyse the data required to construct Mott-Schottky plots, for example. Unfortunately, the ease of use of FRA-fitting software can lead to errors of interpretation that arise from a failure to relate fitting elements to the physical system. Several equivalent circuits may give the same frequency-dependent impedance response. No a priori distinction between degenerate circuits is possible, ft is necessary to study the system response as a function of additional experimental variables (DC voltage, concentration, mass transport conditions etc.) in order to establish whether the circuit elements are related in a predictable way to a model of the physical system. 

Finally, it should be noted that in many cases where < 0, is determined by the capacity method uncertainty arises, which is related to the frequency dependence of Mott-Schottky plots. (In particular, the frequency of the measuring current is increased in order to reduce the contribution of surface states to the capacity measured.) As the frequency varies, these plots, as well as the plots of the squared leakage resistance R vs. the potential (in the electrode equivalent circuit, R and C are connected in parallel), are deformed in either of two ways (see Figs. 6a and 6b). In most of the cases, only the slopes of these plots change but their intercepts on the potential axis remain unchanged and are the same for capacity and resistance plots (Fig. 6b). Sometimes, however, not only does the slope vary but the straight line shifts, as a whole, with respect to the potential axis, so that the intercept on this axis depends upon the frequency (Fig. 6a). 

We consider first the results on p-GaP. The impedance data for p-GaP has been a fruitful source of controversy, though not of comprehension. If a sample of p-GaP is held at a negative potential for a considerable period and then slowly ramped towards positive potentials, the a.c. impedance data cannot be analysed within the framework of the two-component model. Attempts to do so lead to Mott-Schottky plots whose slopes and intercepts are both frequency-dependent as shown in Fig. 25. If the data are analysed according to the more complex five-component equivalent circuit shown above, then a much better fit is obtained for the potential region more than about 0.6 V negative of the predicted flat-band potential. In this region, the Mott-Schottky plot is linear with a slope that corresponds reasonably well... 

Finally, it should be mentioned that various peculiarities in the capacity measurements have been observed. In many cases a certain frequency dependence of the slope of the Mott-Schottky plots [Eq. (10)] and of the series was found. Gomes and co-workers have analyzed these effects in detail. 

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