Modelling the electric potential distribution in the dark in nanoporous semiconductor electrodes

This study concerns the electric potential distribution in the dark in nanocrystalline porous semiconductor electrodes, in full depletion conditions. Since band bending in a single colloidal particle is small, the idea is to develop a model that accounts for the total potential drop resulting from the equilibration between the Fermi level and the redox potential in the solution. As preliminary steps, the band bending and potential distribution in a planar electrode and also in a colloidal semiconductor particle are reviewed. In order to overcome the limitations of results based on these geometries, a model based on a columnar shape is developed. The Poisson equation is solved in the columnar electrode. with careful consideration of the boundary conditions. A large potential drop is shown to take place at the back contact. To complete the study, the effect of the depletion zone in the transparent conducting oxide is analysed. Simple expressions are derived that permit evaluation of how the total potential drop is distributed between the electrode and the substrate. From this, the strength and spatial range of the electric field in the electrode can be estimated.