Flatband Potentials

When relation (28) is properly fitted, B,C, and the flatband potential Up can be determined. For a silicon electrode in contact with 0.6 M nh4f 

The Debye length of the electrode material can be determined from the constant B, and the sensitivity factor S from C, provided the diffusion length and the diffusion constant for minority carriers are known. 

In the experiment discussed (n-Si/0.6 M NH4F), the flatband potential (0.8 V vs. a saturated Hg-sulfate electrode) would have been immediately recognizable as the pronounced minimum between PMC and the photocurrent curve (Fig. 29). 

Another technique for flatband determination is based on the measurement of potential-modulated microwave conductivity signals and is described further in the next section. 

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Stationary microwave electrochemical measurements can be performed like stationary photoelectrochemical measurements simultaneously with the dynamic plot of photocurrents as a function of the voltage. The reflected photoinduced microwave power is recorded. A simultaneous plot of both photocurrents and microwave conductivity makes sense because the technique allows, as we will see, the determination of interfacial rate constants, flatband potential measurements, and the determination of a variety of interfacial and solid-state parameters. The accuracy increases when the photocurrent and the microwave conductivity are simultaneously determined for the same system. As in ordinary photoelectrochemistry, many parameters (light intensity, concentration of redox systems, temperature, the rotation speed of an electrode, or the pretreatment of an electrode) may be changed to obtain additional information. 

In the accumulation region, the situation is much more complicated, so that a reliable analytical expression is difficult to obtain. However, it can be shown17 that the PMC signals increase toward increased accumulation in a smooth, steplike function. The ratio between the PMC maximum and the PMC minimum (at the flatband potential) can be calculated... 

Figure 13. Numerically calculated PMC potential curves from transport equations (14)—(17) without simplifications for different interfacial reaction rate constants for minority carriers (holes in n-type semiconductor) (a) PMC peak in depletion region. Bulk lifetime 10" s, combined interfacial rate constants (sr = sr + kr) inserted in drawing. Dark points, calculation from analytical formula (18). (b) PMC peak in accumulation region. Bulk lifetime 10 5s. The combined interfacial charge-transfer and recombination rate ranges from 10 (1), 100 (2), 103 (3), 3 x 103 (4), 104 (5), 3 x 104 (6) to 106 (7) cm s"1. The flatband potential is indicated.

Figure 26. Dependence of function 4K.AU) [relation (19)] and of /

Figure 29. PMC potential and photocurrent-potential curves for n-Si in contact with 0.6 M NH4F. The flatband potential Up, is indicated.

For an electrode with high interfacial rate constants, for example, relation (28) can be plotted, which yields the flatband potential. It allows determination of the constant C, from which the sensitivity factor S can be calculated when the diffusion constant D, the absorption coefficient a, the diffusion length L, and the incident photon density I0 (corrected for reflection) are known ... 

Another way to determine the sensitivity factor consists in determining the difference between the PMC minimum (flatband potential) and the PMC maximum in the accumulation region (the infinite and negligible surface recombination rate). This difference can be calculated to be17... 

D for minority carriers in the material is known. The sensitivity factor can be determined from the maximum or minimum PMC signal. Using the minimum PMC signal at the flatband potential ( ps = 0, W=0),we derive from Equation (21). 

This is a relation in which PMCfl, the photoinduced microwave conductivity signal at the flatband potential, is measured and the rest of the constants are known. 

Otherwise, the effect of electrode potential and kinetic parameters as contained in the relevant expression for the PMC signal (21), which controls the lifetime of PMC transients (40), may lead to an erroneous interpretation of kinetic mechanisms. The fact that lifetime measurements of PMC transients largely match the pattern of PMC-potential curves, showing peaks in accumulation and depletion of the semiconductor electrode and a minimum at the flatband potential [Figs. 13, 16-18, 34, and 36(b)], demonstrates that kinetic constants are accessible via PMC transient measurements, as indicated by the simplified relation (40) derived for the depletion layer of an n-type electrode. 

The schemes in Figs. 44 and 45 may serve to summarize the main results on photoinduced microwave conductivity in a semiconductor electrode (an n-type material is used as an example). Before a limiting photocurrent at positive potentials is reached, minority carriers tend to accumulate in the space charge layer [Fig. 44(a)], producing a PMC peak [Fig. 45(a)], the shape and height of which are controlled by interfacial rate constants. Near the flatband potential, where surface recombination... 

Gerischer H (1989) Neglected problems in the pH dependence of the flatband potential of semiconducting oxides and semiconductors covered with oxide layers. Electrochim Acta 34 1005-1009... 

Ginley DS, Butler MA (1978) Flatband potential of cadmium sulfide (CdS) photoanodes and its dependence on surface ion effects. J Electrochem Soc 125 1968-1974... 

Roy AM, De GC, Sasmal N, Bhattacharyya SS (1995) Determination of the flatband potential of semiconductor particles in suspension by photovoltage measurement. J Hydrogen Energy 20 627-630... 

Frequently it has been observed with n-type as well as with p-type electrodes in aqueous solutions that the onset potential of the pure photocurrent differs considerably from the flatband potential. The latter can be determined by capacity measurements in the dark as illustrated by the dashed line in the ij — Ub curve in Fig. 8 a. This effect is usually explained by recombination and trapping of minority carriers created by light excitation at the surface. It is obvious that these effects have a negative effect... 

The magnitude of the photopotential is also related to light intensity and to the value of EDARK. The value of Elight cannot obviously exceed the flatband potential. At Dark = Efb, the photopotential drops to zero which can be used for a simple measurement of the flatband potential. 

The photocurrent density (/ph) is proportional to the light intensity, but almost independent of the electrode potential, provided that the band bending is sufficiently large to prevent recombination. At potentials close to the flatband potential, the photocurrent density again drops to zero. A typical current density-voltage characteristics of an n-semiconductor electrode in the dark and upon illumination is shown in Fig. 5.61. If the electrode reactions are slow, and/or if the e /h+ recombination via impurities or surface states takes place, more complicated curves for /light result. 

This value is maximized when EF reaches the flatband potential. Thus, a plot of incident light intensity versus open-circuit electrode potential is expected to saturate at the flatband potential, thereby allowing identification of this potential. A typical experiment of this sort is shown in Figure 28.3b for an n-CdSe electrode immersed in a ferri/ferrocyanide electrolyte. 

Figure 28.3 The flatband potential of a semiconductor can be established by measuring the photopotential of the semiconductor as a function of illumination intensity. In the dark (left), the semiconductor Fermi level and the redox potential of the electrolyte are equal, providing an equilibrium condition. However, illumination of the semiconductor (right) generates charge carriers that separate the Fermi level and the redox potential. The difference in these two parameters is the observed photovoltage as shown for an n-CdS electrode immersed in a ferri/ferrocyanide electrolyte (bottom). The measured photovoltage is observed to saturate at the flatband potential. In this case, a value of -0.2 V vs. SCE is obtained. Note that the photovoltage response yields a linear functionality at low light intensity with saturation behavior occurring as the flatband potential is approached.

In addition to the use of open-circuit photopotentials, the variation in interfacial capacitance with electrode potential can be utilized to determine the flatband potential as well as the semiconductor dopant concentration. A discussion of the capacitance-potential response of the semiconductor-electrolyte interface is beyond the scope of this text. The reader is referred to Reference 7 for a more complete discussion of this subject. 

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