Trap-limited recombination in dye-sensitized nanocrystalline metal oxide electrodes

Abstract

We use transient and steady-state optical spectroscopies to study the recombination reaction between electrons and dye cations in a dye-sensitized nanocrystalline ${\mathrm{TiO}}_{2}$ electrode in several different chemical environments. Kinetic decay curves are approximately stretched exponential, and the cation half-life, ${t}_{50%}$ varies with electron density n as ${t}_{50%}\ensuremath{\propto}{n}^{\ensuremath{-}1/\ensuremath{\alpha}},$ where \ensuremath{\alpha} is a constant in the range 0.2--0.5. We have developed a model of electron transport in the presence of an energetic distribution of trap states and consider two regimes. In the first, the continuous-time random-walk (CTRW) electrons are free to diffuse through the lattice, by means of multiple trapping events mediated by the conduction band. In the second, the hopping regime, trapped electrons are allowed to tunnel to other, vacant trap sites, or to the dye cation, according to a Miller-Abrahams model for the transition rate. We carry out Monte Carlo simulations of the recombination kinetics as a function of electron density, trap state distributions and other parameters. The CTRW reproduces both the dependence of ${t}_{50%}$ on n and the shape of the kinetic curves with only one free fitting parameter, for the case of an exponential density of trap states. The hopping model is ruled out by subnanosecond measurements. We conclude that multiple trapping with a broad energetic distribution of electron traps is responsible for the slow recombination kinetics. When applied to recombination in a nanocrystalline photovoltaic junction at open circuit, the model predicts a sublinear power-law variation of electron density with light intensity G, $n\ensuremath{\propto}{G}^{\ensuremath{\alpha}},$ compatible with the observed behavior.