Schottky-Mott model

The general method for the determination of the flat band potential is based on the Mott-Schottky linear plot based on ca-pacitance/voltage relation. Starting from Eq. (9) the space charge distribution was calculated, and its potential dependence lead to the derivation of a model equivalent to a capacitance, given by ... 

An example of the type of behaviour encountered for exponential surface-state distributions is provided by n-Fe203 in 1M NaOH [69]. The equivalent conductance and susceptance of the circuit comprising Zss in parallel with C8c clearly show a power law dependence on values obtained from this model again obeyed the Mott-Schottky relationship, although the donor density of 8 x 1018 cm 3 and dielectric constant of 25 suggest that the true flat-band potential may lie rather positive of the value given. 

As already mentioned in Sect. 3.3, frequently a shift of the Mott-Schottky curves has been observed upon illumination, which has been interpreted as a corresponding shift of the flatband potential. In a recent study, Allongue et al. [119] have correlated this effect with the formation of surface radicals by using the same model proposed in Ref. [112, 114]. The shift of flatband is then given by... 

D. B. Bonham and M. E. Orazem, "A Mathematical Model for the Influence of Deep-Level Electronic States on Photoelectrochemical Impedance Spectroscopy 2. Assessment of Characterization Methods Based on Mott-Schottky Theory," Journal of The Electrochemical Society, 139 (1992) 126-131. 

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. 

Although this model is a natural extension of that derived for metal/semi-conductor or p-n junctions, it has proved remarkably difficult to verify it for semiconductors in contact with those electrolytes normally employed by electrochemists. As an example, the electrochemistry of germanium initially proved very difficult to understand in aqueous solution [2] and it was only with DeWaid s studies of n-ZnO [3] that a paradigmatic example of the classical model was discovered. The data found by DeWaid in his study of the ZnO electrolyte interface confirmed quantitatively the behaviour of the a.c. response of the semiconductor/electrolyte as predicted by the classical model. In particular, DeWald confirmed that the series capacitance of the interface obeyed the Mott-Schottky relationship [1]... 

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... 

Fig. 3.17 Simulated Mott-Schottky plots for an n-type semiconductor. The same model is used as in Fig. 3.16, except that the depletion layer thickness is now calculated from (2.41) using a donor density of lO cm. Left. Calculated curves using parameters that correspond to the Si = 10 kQ curve in Fig. 3.16a, showing the deviations that can occur at high frequencies when a (too) low current range limits the bandwidth of the potentiostat. Right. Deviations at low frequencies due to the presence of Ssc parallel to Csc. based on the parameters used in Fig. 3.16d...

Fig. 3.18 Mott-Schottky plots of Si-doped (curves a-c) and undoped (curve d) mesoporous hematite photoanode. The capacitances for curves a, b, and c are obtained from an Si-doped sample and models a, b, and c, respectively, shown in the inset of the left-hand plot. Curve d is obtained from the undoped film and model a (series RC). The dashed lines connecting the data points represent the variable active surface area fit. Sketches e-g depict the development of the space-charge layer in a mesoporous semiconductor as function of applied potential, illustrating a decrease in active surface area at advancing space-charge layer width in two dimensions, (e) Near flat band potential with maximum surface area, (f) Total depletion of smaller feature at increased bias potential, (g) Decreased active surface area in concave curved surface. Reprinted with permission from ref. [57], copyright, 2009 American Chemical Society...

Figure 17.4 Charge profiles of acceptor dopants, oxygen vacancies, and electrons near a grain boundary interface with a space charge potential of + 0.44 V, according to both the Gouy-Chapman (dotted lines) and Mott-Schottky (solid lines) models.

Mott-Schottky model, with a modification to include an impurity gradient near the layer interface. 

Fig. 10.3 Mott-Schottky plots at Cr electrode using capacitance measured at one frequency (indicated), Eq. (10.3), and using Brag et al. [305] and Hsu and Mansfeld [306] models (From Ref.

Perhaps the simplest of these techniques are the potentiostatic photocurrent transients (79) that were shown to be sensitive to the semiconductor electrodes down to 1 ns. (80) and below (81). Often the time resolution is limited by the RC of the system and the technique is most valuable in the longer time scales for identification of intermediates and products of photo redox reactions (79). The interpretation of the data follows the routine in some of the methods that we have explored to interpret impedance data, i.e., assume an equivalent circuit and analyze the decay as a superposition of exponential decays where the time constants are correlated with the elements of the equivalent circuit (79)(80)(82). The time constant that was associated with the space charge layer was in reasonable agreement with the Mott-Schottky data (79)(80). The time-scale of the predicted response (83) is much faster than the one observed by the authors of Ref. 79, but the much faster resolution reported in Ref. 81 was in agreement with the time-dependent version of Gartner s model. Etching wasfoundtohavealargeeffecton the amplitude and decay time of the transients (82). This method was also applied to the study of dye sensitization and the role of a super sensitizers in these systems (84). 

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