On a Mathematical Model of the Diffusion of Excitons in a Semiconductor Taking into Account Their Variable Lifetime

The time dependence of the concentration of nonequilibrium minority charge carriers generated in a homogeneous semiconductor material by a sharply focused electron beam, an electron probe, after the target irradiation is described. A mathematical model is constructed for the nonstationary diffusion of charged particles generated by a low-energy electron probe in a semiconductor target in the presence of two independent recombination channels of generated charge carriers in the target material. The use of a low-energy electron probe made it possible to use a two-dimensional mathematical model of diffusion in the simulation. As an initial condition in this model, the distribution of the concentration of nonequilibrium minority charge carriers under quasi-equilibrium conditions, before turning off the electron irradiation, is used. In the presence of two independent recombination channels, the profile of the decline in the concentration of nonequilibrium minority charge carriers in the target after the electron irradiation is turned off is described by the sum of two time-dependent exponents. This approach made it possible to obtain a solution to the differential equation for two-dimensional diffusion of charge carriers, taking into account their variable effective lifetime. The practical application of the developed mathematical model is implemented for the method of time-of-flight cathodoluminescence in describing the diffusion of excitons in single-crystal gallium nitride, taking into account the dependence of the concentration of generated excitons on time.