Schottky-Mott relationship

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. 

Normally, for semiconductors, Csc < CH so CT Csc. Roat may be varied systematically and the decay of j can often be approximated by a single exponential form, i.e. kr 1/RtCi. or kT potentials well positive of V, the long-time transient time-constant t (Rm + Rout)Csc, and a plot of x vs. R]oad (sflin + Rout) is linear, as shown in Fig. 105. Confirmation of this is obtained from the fact that 1/t2 obeys the Mott-Schottky relationship. At potentials close to V, kec becomes much larger and the decay law more complex. 

If the contribution of a depletion layer in the semiconducting oxide to the interfacial potential is not negligible, the Mott-Schottky relationship holds between the interfacial capacitance and the electrode potential [13]. For an n-type oxide... 

Capacitance measurements are generally regarded as the most reliable method for determination of the band edge positions at a sensitized semiconductor-electrolyte interface [14]. The Mott-Schottky relationship, Eq. 7 ... 

Show that the capacity can be related to doping level (N — Na) and potential by the Mott-Schottky relationship... 

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]... 

The ratio g/I(0) defines the photocurrent efficiency . In the absence of surface recombination, qg corresponds to the photocurrent density Jphow measured in the external circuit. The Gartner equation has been used successfully to explain the photocurrent-potential characteristics of many semiconductor electrodes under conditions where surface recombination is absent. Plots of ln — O) against dsc (which according to the Mott-Schottky relationship is proportional to (1/ — have... 

The famous Mott-Schottky relationship [25,26] in Eq. 5-21 represents a different potential-dependent surface capacitive case. This relationship was derived to express the electronic properties of passive capacitive films of constant thickness formed on metals. The methods based on the Mott-Schottky equation have been widely used as a valid tool to determine semiconductive character and dopant density of the surface films in the semiconductor industry and in corrosion studies. The change of the space-charge layer capacitance of the passive film (or space charge distribution) depends on the difference between the applied DC potential V and flat band potential V g characteristic of the surface film, where Np = concentration of donors (or acceptors) or "doping density" ( 10 - lO cm" ), and Cg = 1.6 KT C electron charge ... 

The Mott-Schottky plot following from Eqs (4.5.12) and (4.5.14) is the relationship... 

The flat band potentials of a semiconductor can be determined from the photocurrent-potential relationship for small band bending [equation (4.2.1)], or derived from the intercept of Mott-Schottky plot [equation (4.2.2)] using following equations... 

A more reliable method for the determination of the fb potential can be drawn from a thorough investigation of the complete impedance diagram equivalent to the space charge layer. In fact, the main difficulty encountered in the Mott-Schottky plot is the rather wide range potential for the C extrapolation, which necessarily lead to values where electrochemical reactions contribute to changing the surface properties of the substrate. Moreover, the expected linear relationship shows a significant deviation, which is explained... 

Mott-Schottky plot — is a graphical representation of the relationship between the -> space charge layer - capacitance, and the potential of a semiconducting -> electrode (Mott-Schottky equation) ... 

A certain relationship, which exists between the bulk and surface properties of semiconducting materials and their electrochemical behavior, enables, in principle, electrochemical measurements to be used to characterize these materials. Since 1960, when Dewald was the first to determine the donor concentration in a zinc oxide electrode using Mott-Schottky plots, differential capacity measurements have frequently been used for this purpose in several materials. If possible sources of errors that were discussed in Section III.3 are taken into account correctly, the capacity method enables one to determine the distribution of the doping impurity concentration over the surface" and, in combination with the layer-by-layer etching method, also into the specimen depth. The impurity concentration profile can be constructed by this method. It has recently been developed in greatest detail as applied to gallium arsenide crystals and multilayer structures. 

Rearrangement yields a very useful relationship (first derived for the metal/semiconductor junction) called the Mott-Schottky equation (54, 55) ... 

Determining Eft, is based on the Mott-Schottky (M-S) relationship involves measuring the capacitance of the space charge layer (Csc) of the semiconductor electrode as a function of the applied potential (E) and applying the relationship according to Eq. (6.1) [9]. 

To determine the effect of oxidation, a Mott-Schottky plot of the space charge capacitance before and after oxidation was compared. In these plots, which were originally derived for a metal-semiconductor interface (Schottky [ 1939,1942], Mott [1939]) but hold equally well for the metal-electrolyte interface, a linear relationship is predicted between the applied potential and one over the square of the capacitance arising from the space charge layer in the saniconductor. The slope is inversely proportional to the effective donor or acceptor concentration in the semiconductor. For the semiconductor-electrolyte interface (Bard and Faulkner [1980]),... 

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. 

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