Variation of current with potential

The expressions for the components of the total faradaic current at the semiconductor surface as given in eqns. (173) (176) show that this current is given as the product of factors intrinsic to the electron transfer process taking place, to the concentration and thermal energy distribution of the redox couple, and to the concentration of carriers or the density of states. If we restrict attention to an n-type semiconductor and assume that only electron transfer to and from the conduction band is significant, then the nett current can be written 

The potential distribution at the surface of the semiconductor is such that the bulk of the potential change is accommodated within the depletion layer. It follows, as discussed in Sect. 4, that ns will be a strong function of the applied potential. However, the corollary of this is that the matrix element V and the thermal distribution parameters ox(Ec) and Qrei(Ec) will be much weaker functions of potential. Although, therefore, we would expect to find an exponential or Tafel-like variation of current with potential for a faradaic reaction on a semiconductor, the underlying situation is quite different from that of a metal. In the latter case, the exponential behaviour arises from the nature of the thermal distribution function Q and the concentration of carriers at the surface of the metal varies little with potential. To see this more clearly, we may expand eqn. (179) assuming that the reverse process of electron injection into the CB can be neglected eqn. (179) then reduces to 

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In the limit of large anodic polarization, n, relative to the open-circuit potential Ecoa< the variation of current with potential will be log-linear and vice versa for large cathodic potentials (relative to open-circuit potential). This behavior can be modeled by the semilo-garithmic Tafel equations ... 

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