Mott-Schottky case

Approximate solutions for the Gouy-Chapman case were presented by Maier [21], while more recently analytical solutions for the Gouy-Chapman as well as the Mott-Schottky cases were presented by litzehnan et al. [22, 23]. The relevant expressions for each partial conductivity are summarized for the Gouy-Chapman case ... 

Table 5.3 Effective values for the Gouy-Chapman and the Mott-Schottky cases ( < — 1).

The relationships are different in the Mott-Schottky case. There 91nco"(x=0)oc 5In E does not apply. Rather, because of E a [D ]A and Eqs. (5.215), (5.231), it follows that 51nco (x=0)oc 5 E, i.e. the absolute change is important (more than that The concentration effect is proportional to Ej9 El, in complete contrast to 5 S / E in the Gouy-Chapman case). A detailed example will be discussed in the next section. 

When doping is 10% the calculated Det e length is less than the nearest neighbour distance. However, the values may be larger because of nonidealities. In the Mott-Schottky case (see Section 5.8 constant dopant concentration, depletion of Vq) the thickness of the zone increases with increasing space charge p>otential. 

Let us briefly consider the temperature dependence of Csc for the simple Gouy-Chapman and Mott-Schottky cases. In the first case at small space charge potentials the temperature dependence follows from A(T) and, hence, from c (T). In pure materials Cbc can thus increase considerably with temperature, while in highly doped materials Csc will remain almost constant over a large temperature range. In the case of large effects the temperature dependence of the interfacial concentration, as discussed in Chapter 5, also becomes important. 

Straight lines plotted in the coordinates iph — < sc or iph —

Mott-Schottky plots C 2 —

depletion layer thickness Lsc on the potential. The straight line segments intersect at a single point. For materials studied in the works cited below, this point coincides, within 0.2 V, with the value of q>tb measured independently by the differential capacity technique. 

Flat-band potential — In the energy barrier formed for example at metal-semiconductor junctions (- Schottky barrier), metal-insulator-semiconductor junctions, and solution-semiconductor interfaces the flat-band potential corresponds to the potential at which the electric field equals zero at the semiconductor interface, i.e., there is no -+ band bending. In case of solution-semiconductor interfaces, the flat-band potential corresponds to the condition of absence of excess charge and consequently, depletion layer, in the semiconductor. See also -> Mott-Schottky plot, and -> semiconductor. 

It can be seen that, both for the case of inhomogeneous doping and for the case of deep traps, the intercept of the Mott-Schottky plots at high reverse bias do not yield the flat-band potential or any simple function thereof. 

Finally, it should be mentioned that frequently, as in the case of Ti02, a frequency dispersion of the slope of the Mott-Schottky curves has been observed (see e.g. [67,68]), although the Hatband potential was not affected. Modern methods, such as impedance spectroscopy, have shown, however, that this frequency dispersion is an artifact [59]. 

In the simplest case, as more fully discussed elsewhere [14, 15, 29], one obtains the Mott-Schottky relation (for the specific instance of an n-type semiconductor)... 

The Mott-Schottky regime spans about 1 V in applied bias potential for most semiconductor-electrolyte interfaces (i.e., in the region of depletion layer formation of the semiconductor space-charge layer, see above) [15]. The simple case considered here involves no mediator trap states or surface states at the interface such that the equivalent circuit of the interface essentially collapses to its most rudimentary form of Csc in series with the bulk resistance of the semiconductor. Further, in all the discussions above, it is reiterated that the redox electrolyte is sufficiently concentrated that the potential drop across the Gouy layer can be neglected. Specific adsorption and other processes at the semiconductor-electrolyte interface will influence Ffb these are discussed elsewhere [29, 30], as are anomalies related to the measurement process itself [31]. Figure 7 contains representative Mott-Schottky... 

In an idealized case when the effect of the Helmholtz layer can be neglected, i.e., when Ch Csc, there is a negligible amount of surface states, that is. Css Qc. The total capacitance of the semiconductor/electrolyte interface described by Eq. (1.45) becomes C Csc. The interface capacitance as a function of the electrode potential then follows the Mott-Schottky equation ... 

In the potential range where the hydroxyl surface was converted to a hydride surface, a high density of surface-states was found which was related to the radical or to a dangling bond formed as an intermediate (Memming and Neumann, 1968). Additional capacitance has also been observed in Mott-Schottky plots for doped semiconductors. In most cases, however, correlation to surface-states was not unambiguously possible. 

In the case of GaAs a change of the potential across the Helmholtz layer was observed upon anodic and cathodic prepolarization, which was interpreted in terms of hydroxyl and hydride surface layers, as for Ge (see Section 5.3.1) A linear Mott-Schottky dependence for an n-GaAs electrode was only found at sufficiently high scan rates after anodic or cathodic prepolarization as shown in Fig. 5.17 [40], It is worth mentioning that all reliable capacity measurements could be interpreted in terms of space charge capacities, i.e. additional capacities due to surface states were not found. 

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). 

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