Photocurrent calculation

Let us calculate the correction to the proton polaron direct current density conditioned by light-induced transitions between the sites. This photocurrent calculation is analogous to that of the correction to the drift activation current carried out in the previous subsection [the analogy lies in the fact that operator (276) is similar to the correction to the Hamiltonian if the electric field is taken into account, with the correction being nondiagonal on the operator of the coordinate see expression (236)]. 

It is interesting to note that independent, direct calculations of the PMC transients by Ramakrishna and Rangarajan (the time-dependent generation term considered in the transport equation and solved by Laplace transformation) have yielded an analogous inverse root dependence of the PMC transient lifetime on the electrode potential.37 This shows that our simple derivation from stationary equations is sufficiently reliable. It is interesting that these authors do not discuss a lifetime maximum for their formula, such as that observed near the onset of photocurrents (Fig. 22). Their complicated formula may still contain this information for certain parameter constellations, but it is applicable only for moderate flash intensities. 

The first estimations of for photoinduced processes were reported by Dvorak et al. for the photoreaction in Eq. (40) [157,158]. In this work, the authors proposed that the impedance under illumination could be estimated from the ratio between the AC photopotential under chopped illumination and the AC photocurrent responses. Subsequently, the faradaic impedance was calculated following a treatment similar to that described in Eqs. (22) to (26), i.e., subtracting the impedance under illumination and in the dark. From this analysis, a pseudo-first-order photoinduced ET rate constant of the order of 10 to 10 ms was estimated, corresponding to a rather unrealistic ket > 10 M cms . Considering the nonactivated limit for adiabatic outer sphere heterogeneous ET at liquid-liquid interfaces given by Eq. (17) [5], the maximum bimolecular rate constant is approximately 1000 smaller than the values reported by these authors. 

FIGURE 7.16 (A) Photocurrents of salamander rod cells following light flashes giving between 10 and 2000 rhodopsin molecule isomerizations. (B) Calculated increments in phosphodiesterase hydrolytic rate constant. (From Lamb, T. D. and Pugh, Jr., E. N., Trends Neurosci., 15, 291-299, 1992. With permission.)... 

Halmann reported in 1978 the first example of the reduction of carbon dioxide at a p-GaP electrode in an aqueous solution (0.05 M phosphate buffer, pH 6.8).95 At -1.0 V versus SCE, the initial photocurrent under C02 was 6 mA/ cm2, decreasing to 1 mA/cm2 after 24 h, while the dark current was 0.1 mA/cm2. In contrast to the electrochemical reduction of C02 on metal electrodes, formic acid, which is a main product at metal electrodes, was further reduced to formaldehyde and methanol at an illuminated p-GaP. Analysis of the solution after photoassisted electrolysis for 18 and 90 h showed that the products were 1.2 x 10-2 and 5 x 10 2 M formic acid, 3.2 x 10 4 and 2.8 x 10-4 M formaldehyde, and 1.1 x 10-4 and 8.1xlO 4M methanol, respectively. The maximum optical conversion efficiency calculated from Eq. (23) for production of formaldehyde and methanol (assuming 100% current efficiency) was 5.6 and 3.6%, respectively, where the bias voltage against a carbon anode was -0.8 to -0.9 V and 365-nm monochromatic light was used. In a later publication,4 these values were given as ca. 1% or less, where actual current efficiencies were taken into account [Eq. (24)]. 

Gartner s equation can be derived by calculating that part of the photocurrent which comes from the bulk. The concentration p(x) of holes obeys the following equation, which combines the familiar diffusion equation with a source and a loss term ... 

This fast removal of Si-F species can be ascribed to the weakening of the Si backbonds induced by the strong polarizing effect of F [Ubl], The weak back-bonds are then attacked by HF or H20. This reaction scheme for the dissolution process is supported by quantum-chemical calculations [Trl]. The observed dissolution valence of two for Jelectron injection current and Si-F bond density [Be22] are experimental findings that are in support of the divalent dissolution mechanism, as shown in Fig. 4.3 [Lei, Ge7, Ho6]. 

Fig. 3.22 (a) Current - voltage characteristics of a p-type photocathode and that when it is replaced by a platinum shown to illustrate the efficiency calculation using the power saving approach. The shaded area represents the maximum power saving as a result of photoelectrolysis, (b) The efficiency at various photocurrent densities obtained useing the base graph. 

For example, the solar photocurrent spectrum of a titania nanotube array photoanode calculated using relahon (3.6.23) is given in Fig. 3.25. The solar irradiance P (k) for the calculation is taken from Fig. 3.19. The total photocurrent that can be obtained from this electrode when exposed to global sunlight is... 

Curve (b) of Fig. 3.25 is obtained by the integral of Curve (a). As can be seen from Curve (b) of Fig. 3.25, Ip= 847 uA/cml Outdoor measurements yielded values agreeing well with this calculated photocurrent value (State college, Pennsylvania latitude 40.79°N, longitude 77.86°W, on April 3, 2007 at 3 00 PM, clear sky, incident irradiance 950 W/m ). 

The effect of nanotube-array length on the photoresponse, with all samples annealed at 530°C was also studied both photocurrent magnitude and photoconversion efficiency are seen to increase with length [5]. On exposing 6 pm nanotube-array samples annealed at 600°C to individual wavelengths of 337 nm (3.1 mW/cm ) and 365 nm (89mW/cm ), the quantum efficiency was calculated as 81% and 80% respectively. The high quantum... 

Steady-state wavelength-specific photocurrents were measured for the Ti-Fe-0 films in a two-electrode arrangement at different applied voltages. Incident photon-to-current efficiencies (IPCE) are calculated using the following equation ... 

Photocurrent measurements permit the determination of the hole diffusion length Lp. As was already noted, comparison of measured and calculated polarization curves allows Lp to be determined by a fitting procedure. For example, Butler (1977) and Wilson (1977) obtained for W03 and TiOz the values of Lp equal to 0.5 x 10-4 and 4 x 10-4 cm, respectively. 

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