Studies On 1-D Quasineutral Drift Diffusion Model For Semiconductors

There are two big class of mathematical models for semiconductor devices. One are the microscopic models, namely the kinetic models based on semiconductor Boltzmann equation. Then second class are the macroscopic models describing directly macroscopic quantities such as charge density, charge current density, energy density and temperature etc.. The most representative macroscopic model for semiconductors is hydrodynamic model . A simplification for hydrodynamic model through relaxation limit leads to drift diffusion model. Drift diffusion model is the first model which is widely used in semiconductor simulation and real application. For the scaled drift diffusion model, by quasineutral limit leads to the quasineutral drift diffusion model. In the literature there are very few results about quasineutral drift diffusion model even though it is a simplest model in the modelling sets of semiconductors. The main difficulty to be met is that the third equation of it is algebraic equation, which is very different from the classical drift-diffusion model. This is the reason why we can not use the methods which have successfully used in classical drift diffusion model system . In this thesis, we mainly study the 1-d quasineutral drift diffusion model.There are altogether two parts in this thesis. In the first part, we study the local and global wellposedness of the 1-d quasineutral drift diffusion model. We will use regular-ization method and upper and lower solution technique to give the local existence, global existence and uniqueness results. Some singular stationary solutions are also presented in this part. In the second part, we mainly investigate asymptotic behavior of globally smooth solution of this model. Using a different method, fixed point argument, from the first part, we give local existence and uniqueness of smooth solution. Then we apply energy method to get a priori estimate which yields the the global existence and asymptotic result with the help of the local existence result.