Local Discontinuous Galerkin Method For Drift-Diffusion Model Of Semiconductor Problem

This paper considers the Drift-Diffusion(DD)model of the semiconductor problem.DD model is described by an equation system consisting of a concentration equation and an electron potential equation.The concentration equation includes not only the first-order derivative convection term,but also the second-order derivative diffusion term.This paper presents the local discontinuous Galerkin(LDG)method for the one-dimensional and two-dimensional problems of the DD model,and carries out the numerical simulation.The main technical difficulties include cell boundary processing caused by the discontinuity of the LDG method,and the nonlinear coupling of concentration and electric field in the concentration equation.When simulating a one-dimensional problem,fine meshes are used in the parts where the concentration changes sharply,and coarse meshes are used in the places where the concentration changes gently,and compared with the numerical simulation of uniform meshes,it realizes the purpose of saving space and dividing the number of elements and speeding up the running speed under non-uniform division.When simulating two-dimensional problems,a combination of Dirichlet and Neumann boundaries is used.Numerical results verify the stability of the LDG method.This article is divided into four chapters.Chapter 1 is the introduction.It briefly introduces the origin and development of the semiconductor DD model,as well as the advantages of the LDG method.Chapter 2 is the preparation.In order to facilitate the theoretical analysis of the chapter 4,some basic notation,projection,interpolation properties,and inverse properties are mainly proposed.Chapter 3 uses the LDG method to solve the following semiconductor problem 1D DD model:n1-(μEn)x=τθnxx,φxx=e/ε(n-nd),in which,x∈(0,1),the first concentration equation is a periodic boundary condition,and the second electron potential equation is Dirichlet boundary condition:φ(0,t)=0,φ(1,t)=vhias,and the unknown functions are n and φ,E=-φx represents the electric field.The LDG format is constructed for the 1D DD model,and specific calculation examples under uniform and non-uniform division are given,which saves the number of computing space nodes and speeds up the operation speed under non-uniform division.Chapter 4 uses the LDG method to solve the following semiconductor problem 2D DD model:nt-▽·(μEn)=τθΔn,Δφ=e/ε(n-nd),in which,x×y∈Ω,Ω=(0,1)×(0,1),the first concentration equation is a periodic boundary condition,the second electron potential equation is Dirichlet boundary condition,and the unknown functions are n and φ,E=-▽φ(x，y)represents the electric field.The LDG format is constructed for the 2D DD model,and the error analysis is given.Finally,specific numerical examples are given for the 2D DD model.Finally,the conclusion and prospect of the full text are given.