Numerical methods for solving the 3D Neumann problem of laser‐induced plasma evolution in a semiconductor: Direct and iteration methods

Abstract

The article is devoted to an effective numerical method for solving the 3D problem of laser‐induced plasma evolution in a semiconductor. The physical processes under consideration are accompanied by the existence of the nonlinear feedback between semiconductor characteristics. Mathematically the problem is described by the set of time‐dependent nonlinear PDEs, which involves the Poisson equation with Neumann boundary conditions. One of the problem complexities correlates with providing of well‐known solvability condition for the Neumann problem at its numerical solution. This condition for the problem under consideration coincides with the fulfillment of the charge conservation law, and it is necessary to take into account at developing the numerical method. We apply and compare two numerical approaches: the direct method and the iteration method for a numerical solution of the Poisson equation together with the solution of other PDEs. The computer simulation results are shown that using the direct method leads to the solution distortion because of the presence of the problem nonlinear feedback between functions containing in the equations, and this feedback leads to the conservation law violation. Opposite, using the iteration method for the Poisson equation solution allows us to fulfill the conservation law of the equations set, whose numerical solution is based on developed iteration process.

Violation of Neumann Problem’s Solvability Condition for Poisson Equation, Involved in the Non-Linear PDEs, Containing the Reaction-Diffusion-Convection-Type Equation, at Numerical Solution by Direct Method

“…Evidently, this non-linear feedback is "switched off" at zero-value electron mobility. In [30], we demonstrated that the presence of such a term plays a crucial role in the Neumann problem's solvability condition violation using DM.…”

“…It should be emphasized that the obtained results and conclusions are regardless of the particular method's implementation and have a general character because they arise from the solvability condition's violation caused by the round-off errors accumulation at using DM. It should be emphasized that in [30], the crucial influence of a convection-type term related to electron mobility on the computation's effectiveness was demonstrated, and at zero-value electron mobility, both methods (DM and IM) possessed the same high accuracy. However, in [30], the problem with homogeneous BCs or a particular case of the external electric field (EEF) action along the direction that is transverse to the optical pulse propagation direction was only considered.…”

“…It should be emphasized that in [30], the crucial influence of a convection-type term related to electron mobility on the computation's effectiveness was demonstrated, and at zero-value electron mobility, both methods (DM and IM) possessed the same high accuracy. However, in [30], the problem with homogeneous BCs or a particular case of the external electric field (EEF) action along the direction that is transverse to the optical pulse propagation direction was only considered. On the contrary, below, we study the problem with inhomogeneous BCs corresponding to the action of EEF in one or two directions, including the optical pulse propagation direction.…”