Modelling charge carrier transport with anomalous diffusion and heat conduction in amorphous semiconductors using fractional calculus

Abstract

An ordered semiconductor has a crystalline lattice in which charge carriers move around by the Gaussian process of normal diffusion. The mean square displacement (MSD) of these charge carriers is proportional to time. On the contrary, the movement of carriers in a material with a non-crystalline structure such as amorphous semiconductors is considered to be non-Gaussian in nature. In this case, MSD is proportional to some power of time. Diffusion in this type of transport mechanism is classified as anomalous diffusion. The usual drift-diffusion equation (DDE) cannot adequately describe this process because it has non-Gaussian and dispersive transport mechanisms. Fractional calculus has been used to generalize the standard DDE to a time fractional equation in order to include the hereditary effects of the carrier transport. For power devices, the distribution and conduction of heat is the primary criteria considered when making a device. Therefore, an equation for heat conduction is added to the model for inclusion of variable temperature. The coupled system is solved using a Numerical scheme wherein Finite Difference method has been employed to discretize the Riemann - Liouville time derivative of order α and the space variable. The effects of different physical factors such as light intensity, heat and applied electric field are discussed with the help of graphical illustrations.