Gerischers model

In this section, we first consider a general model of the faradaic processes occurring at the semiconductor-electrolyte interface due to Gerischer [11]. From Gerischer s model, using the potential distribution at the interface, we may derive a Tafel-type description of the variation of electron transfer with potential and we will then consider the transport limitations discussed above. We then turn to the case of intermediate interactions, in which the electron transfer process is mediated by surface states on the semiconductor and, finally, we consider situations in which the simple Gerischer model breaks down. 

An example will clarify this if °ER is the redox energy for the process Ox+ + e ac - Red, °E1 the redox energy for Ox + + e - Red, where 0x + is the oxidised complex, distorted to have the same structure and solvation geometry as Red, and °E2 the redox energy for 0x+ + evac - Red, we have, from 

Clearly, °E1 corresponds to the most probable energy of the reduced form and °E2 to the most probable energy of the oxidised form. The midpoint 

Using the harmonic approximation, the distribution of energies is given by 

For the weak interaction case, where the redox ion in solution remains coordinated only to solvent molecules and at no time becomes adsorbed on to the surface, we have, for the cathodic and anodic currents from one band of the semiconductor 

Gumey treated the neutralization of ions (H + c H and Cl e + Cl), which is a rather complicated process that involves the strong interaction between an electrode and an reaction intermediate such as an adsorbed hydrogen for the hydrogen evolution reaction. Gerischer extended Gurney s treatment to simple redox reactions such as 

In this case the potential energy diagram is something like the one shown in Fig. 16, in which I is the ionization energy of the reduced species in vacuum. The corresponding reaction can be written as 

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With reference to the Gerischer model [11,76,77], the charge transfer reaction of a semiconductor electrode in contact with a... 

Salvador [100] introduced a non-equilibrium thermodynamic approach taking entropy into account, which is not present in the conventional Gerischer model, formulating a dependence between the charge transfer mechanism at a semiconductor-electrolyte interface under illumination and the physical properties thermodynamically defining the irreversible photoelectrochemical system properties. The force of the resulting photoelectrochemical reactions are described in terms of photocurrent intensity, photoelectochemical activity, and interfacial charge transfer... 

Fig. 43. Current voltage curves for the partial currents in valence and conduction bands in the Gerischer model.

The exponential dependence of the current on applied potential for p-type silicon and highly doped n-type silicon in the pore formation regime can be analyzed using the Gerischer model of the semiconductor/electrolyte interface [77]. In the absence of surface states, the hole current for a p-type semiconductor is given by ... 

Willig and co-workers used near-infrared spectroscopy to measure excited-state interfacial electron transfer rates after pulsed light excitation of cis-Ru(dcb)2(NCS)2-Ti02 in vacuum from 20 to 295 K [208]. They reported that excited-state electron injection occurred in less than 25 fs, prior to the redistribution of the excited-state vibrational energy, and that the classical Gerischer model for electron injection was inappropriate for this process. They concluded that the injection reaction is controlled by the electronic tunneling barrier and by the escape of the initially prepared wave packet describing the hot electron from the reaction distance of the oxidized dye molecule. It appears that some sensitizer decomposition occurred in these studies as the transient spectrum was reported to be similar to that of the thermal oxidation product of m-Ru(dcb)2(NCS)2. 

It is surprising that Kamat, O Regan and co-workers found a decreased injection yield at potentials near the flat-band condition. In the standard Gerischer model for sensitized planar electrodes, the low photocurrent near the flat band results because the injected carriers rapidly recombine with the oxidized sensitizer owing to the lack of a substantial depletion layer. Gerischer theory would not predict a decreased injection yield near the flat band, but this behavior can clearly be realized at sensitized nanocrystalline semiconductor films. 

Standard driving force for each couple is nominally the same (Hamann et al., 2005a). These collective measurements demonstrate that simple one-electron charge-transfer processes at semiconductor/liquid junctions are experimentally in accord with the Marcus-Gerischer model of interfacial charge transfer. 

In the Gerischer model, an electron transfer occurs from an occupied state in the metal or the semiconductor to an empty state in the redox system, as illustrated in Fig. 6.10. The reverse process occurs then from an occupied state in the redox system to an empty state of the solid (not shown). The electron transfer takes place at a certain and constant energy as indicated by arrows in Fig. 6.10. This means that the electron transfer is faster than any rearrangement of the solvent molecules, i.e. the Frank-Condon principle is valid. In this approach, the rate of an electron transfer depends on the density of energy states on both sides of the interface. For instance, in the case of an electron transfer from the electrode to the redox system the rate is given by... 

If such a multiorbital system were to be used in the Gerischer model then the corresponding distribution of VTox and Wred or Dqx and Dred would look more complicated. 

The redox process at metal electrodes described above, should also be briefly discussed in terms of the Gerischer model (see Section 6.2). Assuming equal concentrations for the reduced and oxidized species of the redox system then the energetics of the metal liquid interface are given in Fig. 7.5 for equilibrium, cathodic and anodic polarization. The anodic and cathodic currents are then given by (see Eq. 6.42) ... 

As already mentioned in the previous section, any electron transfer across the semiconductor-liquid interface occurs via the energy bands. There may also be an electron transfer via surface states at the interface the electrons or holes, however, must finally be transported via one of the energy bands. This is possible by capturing an electron from the conduction band or a hole from the valence band in the surface states. In the present section the basic rules for the charge transfer will be given, in particular, physical factors which determine whether an electron transfer occurs via the conduction or the valence band, will be derived. For illustration, the Gerischer model will be used here because it best shows the energetic conditions. 

In the preceding derivations we have used the Gerischer model, i.e. we have described the currents in terms of a charge transfer between occupied states on one side of the interface and empty states on the other. In principle one obtains the same equations when using one of the other theories described in Chapter 6. The reason is that in all theories the same exponential term occurs which originates from the assumption that the fluctuation of the solvent molecules or dipoles is assumed to behave like an harmonic oscillator. Quantitatively speaking, the use of the different theories mainly leads to different pre-exponentials [19]. Since the exponential terms are independent of potential, it is useful to include them in the rate constant [19]. The conduction band processes can then be described by... 

The ultrashort time constants of <25 fs for electron transfer, found by Hannappel et al., indicate a different reaction mechanism for the electron transfer process [53]. The electron transfer occurs more quickly than the vibrational relaxation within the dye molecule, i.e. the electron is transferred from any excitation level directly into a corresponding energy level in the conduction band. Because of this strong coupling, this electron transfer cannot be described by the simple Marcus-Gerischer model which is only valid for comparably weak interactions. 

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