The Study On Asymptotic Limit Of Drift-Diffusion Model For Semiconductors And Related Models

In this dissertation, we study the drift-diffusion model for semiconductors and related models. We will study the well-posedness and asymptotic limit of drift-diffusion model for semiconductors and related models by using the asymp-totic expansion method of the singular perturbation, classical energy methods and some important inequalities, such as Cauchy-Schwarz inequality, Holder inequality, Hardy-Littlewood inequality, Sobolev's lemma.In Chapter 1, first we briefly introduce the history of semiconductors. Then we show the model and its research progress. Finally the structure of this dissertation and the main research contents are introduced.In Chapter 2, we mainly study the quasi-neutral limit of the initial-boundary value problem for the one-dimensional drift-diffusion model for semiconductors with mixed boundary conditions. Here the mixed boundary conditions are that contacts (Dirichlet boundary conditions) are assumed on the left boundary while insulating (Neumann) boundary conditions are assumed on the right boundary. The quasi-neutral limit in this case is different from the known results in the case of Neumann boundary conditions. The research emphasis of Chapter 2 is the differences between the structure of the boundary layers in Dirichlet boundary conditions and that in Neumann boundary conditions. For simplicity, we assume that the doping profile and the initial data satisfy some conditions, which can avoid the occurrence of the right boundary layer and the initial layer. That is, in this chapter, we study the case in which only the left boundary layer exists. First, by matched asymptotic analysis, we construct directly an accurate approximate solution for the electron density, the hole density and the electric potential. Then we construct the error function and perform the energy estimate. In the estimate, we find that, to deal with the singular terms and to obtain uniform estimates, a condition must be satisfied in the left boundary, that is, the total density of electron and hole after doping is not very big in the left boundary (for example, smaller than some constantη). Finally, we obtain the conclusion:for the well-prepared boundary conditions, the quasi-neutral limit holds; for the ill-prepared boundary conditions, if the above condition is satisfied, then the quasi-neutral limit holds globally in time up to the maximal existence time of the limit equations; otherwise, we can only get the local convergence result.In Chapter 3, we continue to study the quasi-neutral limit of the initial-boundary value problem for the one-dimensional drift-diffusion models for semi-conductors in the case of mixed boundary conditions. On the basis of Chapter 2, we remove the assumptions imposed on the doping profile and the initial data there and study the case in which the left boundary layer, the right boundary layer and the initial layer all exist. Introducing the density transformation and using the method of formal asymptotic expansions, we construct a more accurate approximate solution (constituted by the inner solution, the left boundary layer function, the right boundary layer function and the initial layer function) and prove the quasi-neutral limit. The technique here is that the left boundary layer function is taken asφ+0 and the right boundary layer function is taken as E-0, where E0=-φx0. It is because of the difference of boundary conditions (the left Dirichlet boundary condition and the right Neumann boundary condition). In Chapter 4, we study the mixed layer problem and the quasi-neutral limit of the drift-diffusion model for semiconductors in the case of Neumann boundary conditions. In 2006, Wang, Xin and Markowich studied the quasi-neutral limit of this model. There a compatibility assumption was imposed to avoid the occurrence of the mixed layers. Here we remove that compatibility assumption and study the case in which the mixed layers exist. By constructing a upper solution and a lower solution, we deduce the exponential decay of the mixed layer functions and prove the quasi-neutral limit by the method of formal asymptotic expansions and the classical energy estimate.In Chapter 5, the initial layer problem of the electro-diffusion model arising in electro-hydrodynamics is studied. The electro-diffusion model is a coupling between the Poisson-Nernst-Planck system and the incompressible Navier-Stokes equations. For the general initial data, an accurate approximate solution involving the effect of initial layer is constructed and the quasi-neutral limit is performed rigorously by using multiple scaling asymptotic analysis. Hardy-Littlewood in-equality is used to deal with the singular terms caused by the initial layer function.The study on drift-diffusion model for semiconductors and related models has not only important theoretical significance, but also extensive application value.