Continuous-time random-walk model of electron transport in nanocrystallineTiO2electrodes

Electronic junctions made from porous, nanocrystalline ${\mathrm{TiO}}_2$ films in contact with an electrolyte are important for applications such as dye-sensitized solar cells. They exhibit anomalous electron transport properties: extremely slow, nonexponential current and charge recombination transients, and intensity-dependent response times. These features are attributed to a high density of intraband-gap trap states. Most available models of the electron transport are based on the diffusion equation and predict transient and intensity-dependent behavior which is not observed. In this paper, a preliminary model of dispersive transport based on the continuous-time random walk is applied to nanocrystalline ${\mathrm{TiO}}_2$ electrodes. Electrons perform a random walk on a lattice of trap states, each electron moving after a waiting time which is determined by the activation energy of the trap currently occupied. An exponential density of trap states $g\left(E\right)\mathrm{}\sim e^{\alpha {(E}_C-E)/kT}$ is used giving rise to a power-law waiting-time distribution, $\psi {\mathrm{}\left(t\right)=At}^{-1-\alpha }.$ Occupancy of traps is limited to simulate trap filling. The model predicts photocurrents that vary like $t^{-1-\alpha }$ at long time, and charge recombination transients that are approximately stretched exponential in form. Monte Carlo simulations of photocurrent and charge recombination transients reproduce many of the features that have been observed in practice. Using $\alpha =0.37,$ good quantitative agreement is obtained with measurements of charge recombination kinetics in dye-sensitized ${\mathrm{TiO}}_2$ electrodes under applied bias. The intensity dependence of photocurrent transients can be reproduced. It is also shown that normal diffusive transport, which is represented by $\psi \mathrm{}\left(t\right)=\lambda e^{-\lambda t}$ fails to explain the observed kinetic behavior. The model is proposed as a starting point for a more refined microscopic treatment in which an experimentally determined density of states can be easily incorporated.