Fractional multiple trapping model of time-of-flight transient photocurrents in amorphous semiconductors

The use of the multiple-trapping (MT) model to comprehend the transport of nonequilibrium charge carriers in amorphous semiconductors has proven highly effective. Under specific conditions, this model generates anomalous diffusion equations characterized by fractional time derivatives. This underscores the utility of the MT model in interpreting fractional transport equations, establishing initial and boundary conditions, and developing numerical methods for solving fractional kinetic equations. Also, this work provides a concise overview of applying fractional MT equations to address challenges in time-of-flight (TOF) experiments. Furthermore, it delves into the connection between the MT model and generalized fractional kinetic equations. In addition, the study introduces analytic approximate solutions of the fractional diffusion equation, incorporating MT phenomena and employing Laplace transforms. This approach is suitable for analyzing both the pre- and post-regimes of TOF transient current, applicable to amorphous semiconductors that display either nondispersive or dispersive transport characteristics. The effectiveness of this method is illustrated through numerical simulations of TOF transient current using the inverse Laplace transform technique with the Padé approximation. The practicality of the method is confronted with the experimental data obtained from thin films of amorphous selenium (a-Se), and the results of this confrontation are deemed satisfactory. The results of this study offer a new promising perspective for the two following reasons. First, employing fractional calculus to address the MT equations introduces a distinct approach compared to methodologies in the existing literature. This is substantiated by the inclusion of memory effects in fractional calculus, implying that the present solution is influenced by preceding time steps. Second, the numerical results demonstrate good agreement with experimental data. Consequently, the introduction of fractional calculus has the potential to offer fresh insights into the behavior of charge carriers in amorphous semiconductors.