The space–time-fractional derivatives order effect of Caputo–Fabrizio on the doping profiles for formation a p-n junction

In this study, we treated the space–time-fractional diffusion equation in a semi-infinite medium using a recently developed fractional derivative introduced by Caputo and Fabrizio. Our main focus was on simulating the diffusion profiles during the creation of a p-n junction according to the obtained solution. We made an interesting observation regarding the influence of the fractional-order derivatives on the depth estimation of the p-n junction. Increasing the order of the time-fractional derivative, denoted as $$\alpha $$ α , resulted in faster diffusion and deeper p-n junctions. On the other hand, increasing the order of the space fractional derivative, denoted as $$\beta $$ β , led to slower diffusion and shallower p-n junctions. These findings demonstrate the significant impact of the fractional derivative orders on the diffusion behavior and depth characteristics of the p-n junction in the studied system. Graphical abstract