Numerical simulation of nonlinear processes in semiconductor devices with the application of the Newton's method for linearization

This article relates to the use of Newton's method and Scharfetter–Gummel scheme, to linearize and discretize the equations, for numerical modeling of nonlinear processes in semiconductor devices. The mathematical model of the problem represents a system of nonlinear differential equations, in the unknowns $\varphi$–electrostatic potential, $~n, p $–the concentrations of electrons and holes, respectively. The problem is further complicated by the fact that the boundary conditions are of two types: the Dirichlet conditions and the Neumman conditions, which act on different portions of the boundary. The subproblems that were solved in this paper: linearization of nonlinear differential equations, using Newton's method; discretization of equations, using Scharfetter–Gummel scheme. The obtained systems have five diagonal and nonsymmetrical matrices. The numerical method of Bi–Conjugate Gradients was used to solve the systems.