Skeleton-Enhanced Discontinuous Galerkin Method for 3-D Nonlinear Semiconductor Modeling

Abstract

We present a mesh skeleton-enhanced discontinuous Galerkin (DG) method, i.e., hybridizable DG, to solve the 3-D highly nonlinear semiconductor drift-diffusion model. This skeleton-enhanced DG algorithm is a remedy but a significant extension of the classical DG method, where only the degrees of freedom on the skeleton are involved as a globally coupled problem, thus reducing the global dimension from 3-D to 2-D and 2-D to 1-D. Furthermore, high-order nodal basis functions over tetrahedra are easily obtained, and meticulous mesh designs are circumvented for complex semiconductor modeling. Rigorous analytical solutions demonstrate that the convergence rate achieves an optimal order of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\emph{p}+1$</tex-math> </inline-formula> in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\bm{L}^{\bm{2}}$</tex-math> </inline-formula> -norm. Then, we also apply our algorithm to solve semiconductor devices, including a bipolar transistor and a trigate fin-shaped field-effect transistor. Compared with the conventional finite volume and finite element solvers, the proposed algorithm exhibits superior stability, convergence, and efficiency.