On the hydrodynamic model for semiconductor devices
Abstract
The hydrodynamic model for semiconductor devices plays an important role in simulating the behavior of the charge carrier in submicron semiconductor devices. This model consists of a set of nonlinear conservation laws for the particle density, current density, and energy density. The Poisson equation for electrostatic potential is also used. The hydrodynamic model P.D.Es. have hyperbolic, parabolic, and elliptic modes. There have recently been some of the computational and physical aspects of this model. A mathematical analysis of the time-dependent hydrodynamic model has not been presented yet; only results exist for steady-state hydrodynamic model. This work emphasizes a mathematical analysis of the time-dependent hydrodynamic model for semiconductor devices. In Chapters 2 through 4, a simplified one-dimensional hydrodynamic model for semiconductor devices, where the energy equation is replaced by a pressure-density relationship, is studied. The local existence of a smooth solution of the system of equations is obtained by using Lagrangian mass coordinates. Physical global weak solutions of the simplified one-dimensional hydrodynamic model are also investigated. There is an essential difficulty which is caused by the vacuum state. To overcome this difficulty, the theory of compensated compactness is applied to this model. First, a sequence of approximate solutions is constructed by the Godunov scheme. Secondly, it is shown that the sequence of approximate solutions satisfies the following: (1) The sequence of approximate solutions is uniformly bounded. (2) The sequence of weak entropy dissipation measures corresponding to the uniformly bounded approximate solutions is compact in $H\sbsp{loc}{-1}.$ Then by applying the theory of compensated compactness, there exists a convergent subsequence. The limit function is a physical weak solution. With this approach, not only is a global existence theorem shown, but also a numerical method is provided for the computation of this model. A regularized system of equations of the simplified one-dimensional hydrodynamic model is studied. The existence and uniqueness of the global smooth solution of the regularized system for this model are obtained by using the characteristic method and the Leray-Schauder theorem. The asymptotic stability of the solution is also shown by the energy method. Finally, the full one-dimensional hydrodynamic model with small viscosity is studied with an emphasis on the system of the compressible Navier-Stokes equations coupled to the Poisson equation. As in Chapter 4, this system is reduced to coupled differential-integral equations by using the characteristic method. Similar results are obtained by using the Leray-Schauder theorem and the energy method.
Degree
Ph.D.
Advisors
Douglas, Purdue University.
Subject Area
Mathematics
Off-Campus Purdue Users:
To access this dissertation, please log in to our
proxy server.