Modeling anomalous charge carrier transport in disordered organic semiconductors using the fractional drift-diffusion equation

Anomalous transport is ever-present in many disordered organic semiconductor materials. The long-tail behavior observed in the transient photocurrent is a manifestation of anomalous transport. Owing to the fact that anomalous transport has dispersive and non-Gaussian transport dynamics, thus anomalous transport cannot be adequately described by the standard drift-diffusion equation which is a framework commonly used to model normal diffusive transport. In this work, we generalized the standard drift-diffusion equation to time fractional drift-diffusion equation (TFDDE) using the fractional calculus approach to model the anomalous transport in the regio-random poly(3-hexylthiophene) (RRa-P3HT) and regio-regular poly(3-hexylthiophene) (RR-P3HT). Physical elucidation of TFDDE is given by stressing how the influence of the multiple-trapping mechanisms and energy disorder lead to the long-tail behavior in the transient photocurrent curves. TFDDE is solved numerically using finite difference scheme to obtain the profiles of charge carriers density evolution and hence to reproduce the corresponding transient photocurrents of RRa-P3HT and RR-P3HT. Poisson solver is also included in the model to account for the fluctuation of localized electric field due to the evolution of charge carriers. It is found that charge carriers acquire additional energy from high electric field that helps them to escape from the trap centers more easily and then propagating at higher velocity, which yields higher transient current. Higher concentration of charge carriers can be generated at higher light intensity and they can occupy energy levels close to the mobility edge, where charge carriers will encounter smaller capturing rate and hop at a longer length in each hopping event. Thus, the transport dynamic of charge carriers at high light intensity is less dispersive than that of the low light intensity. Besides, the transport dynamic of charge carriers in RR-P3HT is relatively less dispersive and has higher mobility than that of the RRa-P3HT since RR-P3HT has lower capturing rate and is less energy disordered.