Fractal Generalization of the Scher-Montroll Model for Anomalous Transit-Time Dispersion in Disordered Solids

The Scher-Montroll model successfully describes subdiffusive photocurrents in homogeneously disordered semiconductors. The present paper generalizes this model to the case of fractal spatial disorder (self-similar random distribution of localized states) under the conditions of the time-of-flight experiment. Within the fractal model, we calculate charge carrier densities and transient current for different cases, solving the corresponding fractional-order equations of dispersive transport. Photocurrent response after injection of non-equilibrium carriers by the short laser pulse is expressed via fractional stable distributions. For the simplest case of one-sided instantaneous jumps (tunneling) between neighboring localized states, the dispersive transport equation contains fractional Riemann-Liouville derivatives on time and longitudinal coordinate. We discuss the role of back-scattering, spatial correlations induced by quenching of disorder, and spatiotemporal non-locality produced by the fractal trap distribution and the finite velocity of motion between localized states. We derive expressions for the photocurrent and transit time that allow us to determine the fractal dimension of the distribution of traps and the dispersion parameter from the time-of-flight measurements.