The Gerischer Model

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

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 ... 

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... 

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