Fractional differential approach to dispersive transport in semiconductors

A novel approach using equations with fractional order derivatives to describe dispersive transport in disordered semiconductors is described. A relationship between the self-similarity of dispersive transport, stable limiting distributions and kinetic equations with fractional derivatives is established. It is shown that unlike the well-known Scher–Montroll and Arkhipov–Rudenko models, which are in a sense alternative to the normal transport model, fractional differential equations provide a unified mathematical framework for describing normal and dispersive transport. The fractional differential formalism allows the equations of bipolar transport to be written down and transport in distributed dispersion systems to be described. The relationship between fractional transport equations and the generalized limit theorem reveals the probabilistic aspects of the phenomenon in which a dispersive to Gaussian transport transition occurs in a time-of-flight experiment as the applied voltage is decreased and/or the sample thickness increased.