### MATHEMATICAL MODELLING OF TECHNOLOGICAL PROCESSES AND SYSTEMS

## MODELING OF ELECTRONIC STATES OF A SINGLE DONOR IN MIS-STRUCTURE USING THE FINITE DIFFERENCE METHOD

Numerical modeling of electronic state evolution due to external electric field in the structure metal-insulator-semiconductor with solitary donor center is carried out. Considering a nanometer disc-shaped gate as a source of the electric field, the problem for the Laplace equation in infinite multilayered medium is solved to determine the gate potential. The energy spectrum of a bound electron is calculated from the problem for the stationary SchrÃ¶dinger equation. Finite difference schemes are constructed to solve both the problems. Difference scheme for the SchrÃ¶dinger equation takes into account cusp condition for the wave function at the donor location. To solve the problem for the Laplace equation, asymptotic boundary conditions for approximating the potential at large distances from the gate are proposed. On the basis of calculation results, a controlling parameter is suggested, which allows to determine the localization of electron wave function regardless of insulator thickness and permittivity.