The Potential Dependence of the Rate Constant for Charge Transfer at the Semiconductor-Redox Electrolyte Interface

The kinetics of charge transfer at the semiconductor- redox electrolyte interface is described in terms of the Gurney- Gerischer -Marcus (GGM) model by using nuclear configuration potential energy diagrams, electronic configuration potential energy diagrams, density of state distributions and rate constant distributions. The model of identical parabolas for the nuclear configuration diagrams is used; this leads to Gaussian oxidant and reductant distribution functions, g(E), where E is the vertical transition (Franck-Condon) energy. The rate constant distribution, k(E), is obtained from the overlap between occupied and unoccupied state distribution functions of the semiconductor and redox electrolyte. Integration of k(E) gives the rate constant which is calculated as a function of the Helmholtz potential, VH, for various values of the reorganization energy, Ereorg. Three types of semiconductor are considered: intrinsic, doped and highly doped. For intrinsic semiconductors the charge transfer rate constant is relatively small and involves both the conduction and valence bands. For symmetric charge transfer (zero energy change, E0.0, for the reaction) both oxidation and reduction occur between the redox electrolyte and both bands of the semiconductor. For unsymmetrical reactions, charge transfer tends to involve only one of the bands; for net reduction, the valence band is involved, whereas for net oxidation the conduction band is involved. For doped semiconductors the rate constant is larger and only one band is involved; for n-type it is the conduction band, and for p-type it is the valence band. For highly doped semiconductors with the Fermi level in either the conduction or valence bands. the rate constant is even larger and only one band is involved. Changes in Helmholtz potential affect k(E) in a similar way to that for metals. However, unlike for metals, the calculated Tafel plots for highly doped n-type semiconductors are shown to exhibit a Marcus inversion region. This is a consequence of the energy gap between conduction and valence bands of the semiconductor. For doped semiconductors, changes in the Helmholtz potential also produce a maximum in the Tafel plot and because of the relatively low currents involved this maximum should be experimentally observable. For intrinsic semiconductors, variation of Helmholtz potential without inclusion of band bending in the semiconductor produces unexpectedly low Tafel slopes which are related to the ratio of the band gap to the reorganization energy, so that the larger the ratio the smaller the Tafel slope. This unexpected result, which amounts to an assumption of band edge unpinning, is shown to accurately account for the experimentally observed Tafel slopes for reduction at n-WSe2 of the dimethylferrocenium ion in acetonitrile.