Mott-Schottky line

From the slope of the Mott-Schottky line one can determine the charge career density (donor or acceptor density) if the relative dielectric permittivity is known. 

Erne et al. [240] have recently shown that the potential dependence of the DA under the depletion conditions is similar to the Mott-Schottky line, which allows determination of the flat-band potential of the electrode. Other examples of employing DA in spectroelectrochemical studies are analysis of the relaxation process of free carriers in semiconductor electrodes [241] and elucidating the mechanisms of interfacial processes (Section 7.5). 

Fig. 10.2 Mott-Schottky plots of an n-type and a p-type silicon electrode in an electrolyte composed of 0.5 mol I"1 HF and 0.5 mol I-1 NH4CI. Dotted lines correspond to ND = l.l xlO15 cm" and NA=3.4xl015 cm-3.

Where Vs is the potential value at the surface of the electrode. Then plotting the value of 1/Csc versus the applied potential E should yield a straight line whose intercept with the E axis represents the flat band potential, and the slope is used for the calculation of N, the charge carrier density in the semiconductor. A typical example of Mott-Schottky plot is given in Fig. 2 [7] in this graph, the extrapolated values of the fb potential are -1-0.8 V and —0.6 V vs. SCE for p-Si and n-Si respectively. 

It follows from Eq. (19) that the dependence Csc(0sc) becomes a straight line in the coordinates (C 2, 5 ) the line thus obtained is called the Mott-Schottky plot. 

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. 

Figure 5. Variation of Mott-Schottky intercept with pCl for (100) orientation n-GaAs, 40°C. Circles denote intercept values from automated admittance measurements. Bars signify standard deviation of least-squares straight line.

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 ... 

Fig. 11. Mott-Schottky plot for a polycrystalline diamond electrode with tance in 0.5 M H2S04. Solid line shows a linear fitting [66], Reproduced chemical Society, Inc.

Fig. 7. Mott -Schottky plots for two different n-ZnO crystals. The experimental points form two lines intercepting the potential axis at -0.4 0.05 V vs. SCE and the behaviour calculated from eqn. (40) is shown as the dotted lines.

Fig. 98. (A) Current-voltage curves for n-GaAs/selenide junction under (a) 1.5 mW cm 2, (b) 9mWera"2, (c) 22raWcm"2, and (d) 50mWcm"2. (B) Mott-Schottky plots using data from the equivalent circuit of Fig. 97(a) at light intensities as in (A). The line (e) was obtained in the dark and gives Vn, - 2.06 V/SCE.

Fig. 3. Mott-Schottky plots of capacitance vs bias for two typical zinc oxide crystals. The dotted lines represent absolute theoretical predictions in the absence of surface states.

Fig. 4. The Mott-Schottky plot for the a = 0.025 crystal of Fig. 1. The dotted line shows the theoretical slope predicted from the room temperature Hall effect and conductivity measurements. The slope at strongly anodic bias (V positive) is a measure of the total donor density in this crystal.

Deviations from straight lines in Mott-Schottky plots can be attributed to the influence of potential-dependent charging of surface or bulk states. This interpretation is supported by analytic calculations of the contribution of defects to the space charge as a function of applied potential. In principle, the... 

Fig. 5.14 Space charge capacity vs. electrode potential for an n type silicon electrode (1 Q cm material) in 10 M HF open circles experimental values solid line theoretical curve, a) Linear plot of Csc b) Mott-Schottky type of plot. (After ref. [26])...

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). 

FIGURE 2.4.2 (a) Simple band line-up diagram for a metal-organic semiconductor interface assuming that the Mott-Schottky rule holds and that the vacuum levels for the metal and semiconductor are aligned, (b) Application of a positive bias to the metal can result in hole injection into the semiconductor by thermionic emission over the barrier, (c) Band line-up diagram in the case where an interface dipole is present, causing a shift (A) in the vacuum levels across the junction. 

Since —A(f> = E — Ef, a plot of HC q vs. E should be linear. The potential where the line intersects the potential axis yields the value of Ef, and the slope can be used to obtain the doping level Ni. While such Mott-Schottky plots have been useful in characterizing the semiconductor-solution interface, they must be used with caution, because perturbing effects, such as those attributable to surface states can cause... 

For various illumination intensities, the diameter of the semicircle fitting the data at high frequencies equals approximately kT/ely pHl [45-47, 49]. In addition, it was shown that upon illumination, a capacitive peak appears in the C versus V plot of the n-GaAsjO.l M H2SO4 interface [45,46, 51], The peak value proved to be a function of the frequency and the photocurrent density as measured in region G [51]. This behavior is markedly different from the purely capacitive impedance (vertical line in the Nyquist plane and straight Mott-Schottky plot) expected for a blocking s/e interface (see Sect. 2.1.3.1). 

Equation 6 indicates that a plot of l/C against U gives a straight line with a slope of l/qsQSsNi)), which is termed the Mott-Schottky plot, as mentioned earlier. The extrapolation of the straight line to 1/C = 0 gives ((7fi, 4- 7 / ). Therefore, the plot can be used to determine the flat band potential t/ft. The donor density Vd (or the acceptor density Va) can also be determined from the slopes of the plots. Figure 4 shows examples of Mott-Schottky plots, obtained for n-Si(lll) and n-Si(lOO) electrodes in 7.1 M... 

An interesting feature of the Mott-Schottky curves in Fig. 3.14 is the capacitance plateau found at potentials positive of —0.7 V vs. SCE. At these potentials, the depletion layer width actually exceeds the Ti02 film thickness, and further extends into the underlying (n-type) ITO film. Clearly, such a plateau will only be observed for low donor densities and/or very thin films. The slope of the curve at the plateau is determined by the donor density of the ITO, for which a value of 1 x 10 cm is found in this particular case [50, 51]. Moreover, from the capacitance Cl at the intersection of both curves (dashed lines in Fig. 3.14) and the known thickness (L) of the films, it is possible to determine the dielectric constant of the Ti02 using the expression for a parallel plate capacitor (Cl = sqe IL). A value of 55 was found for the dielectric constant of polycrystalline anatase Ti02 (this value was also used to calculate the donor densities mentioned in the previous paragraph). 

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 7.31. Free-carrier absorption at p-Si-fluoride electrolyte interface as function of potential determined as IR absorption in 1000-2000-cm range, induced by potential modulation (50 mV, 1 kFIz). Fluoride concentration 0.1 M, phi 4.5. Inset shows variation of A/// on expanded scale in depletion region. Dashed and solid straight lines are Mott-Schottky fits for A/// and (A///)", respectively l/ft, shows calculated value of flat-band potential. Reprinted, by permission, from F. Ozanam, C. da Fonseca, A. V. Rao, and J.-N Chazalviel, Appl. Spectrosc. 51,519 (1997), p. 523, Fig. 3. Copyright 1997 Society for Applied Spectroscopy.

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.

Figure 17.14 Schematic diagram (top) ofthe buik (center), space charge iayers (frame), and grain boundaries (outer dark lines) in a poiycrystaiiine materiai. Shown beiow are Mott-Schottky oxygen vacancy profiles for AO values of +0.5, +0.3, 0, and —0.1 V in the space charge layer.

Figure 4.3.15. Mott-Schottky plots of the space charge capacitance (curve 1) as derived from data like those shown in Figure 4.3.9a and the capacitance associated with the high-frequency response, Q (curve 2) derived from data like those shown in Figure 4.3.96. The flat-band potential is the same in both cases (0.69 V), but the doping level, as calculated from the slope of the lines, is an order of magnitude lower for curve 2 (polished + etched + oxidized sample) than for curve 1 (polished + etched sample). (Shen et al. [1986]). Reprinted by permission of the publisher, The Electrochemical Society, Inc.

Easily measurable, the capacity of a semiconducting electrode provides direct information about the space-charge layer. Figure 3.56 shows the experimental values for the capacitance of a ZnO electrode, an n-type semiconductor [21], The data are presented in the form of a Mott-Schottky diagram. From the slope of the straight line, the concentration of the majority carriers can be determined. In this case, we have... 

Combining (20) and (21) leads to a dependence of capacitance on the square root of the concentration of charged defects. To be able to determine the defect concentration by fitting a straight line to the data, the Mott-Schottky analysis involves the plot of C vs applied d.c. voltage. The intersect of C with the voltage axis yields... 

If the device thickness is sufficiently high, the Mott-Schottky plot [i.e. C vs voltage see (22)] yields a straight line over a considerable voltage range. At larger reverse bias, C may saturate to a constant value. This indicates that the device is... 

---