Dispersive Transport Described by the Generalized Fick Law with Different Fractional Operators

Abstract

The approach based on fractional advection–diffusion equations provides an effective and meaningful tool to describe the dispersive transport of charge carriers in disordered semiconductors. A fractional generalization of Fick’s law containing the Riemann–Liouville fractional derivative is related to the well-known fractional Fokker–Planck equation, and it is consistent with the universal characteristics of dispersive transport observed in the time-of-flight experiment (ToF). In the present paper, we consider the generalized Fick laws containing other forms of fractional time operators with singular and non-singular kernels and find out features of ToF transient currents that can indicate the presence of such fractional dynamics. Solutions of the corresponding fractional Fokker–Planck equations are expressed through solutions of integer-order equation in terms of an integral with the subordinating function. This representation is used to calculate the ToF transient current curves. The physical reasons leading to the considered fractional generalizations are elucidated and discussed.

Keywords

• anomalous diffusion
• fractional equation
• dispersive transport
• time-of-flight experiment