Mathematical Sciences: Analytical and Numerical Studies of Partial Differential Equations of Semiconductor Devices

主要研究人员将研究用解析和数值方法模拟半导体器件的偏微分方程和积分微分方程。他们的工作将涵盖从基本方程的实际推导到具体边值问题的解的整个范围。特别是，研究人员将使用渐近方法研究玻尔兹曼方程及其衍生系统，如水动力模型和漂移-扩散模型。描述半导体器件行为的微分方程组很难解析求解。即使是大型计算机也不能准确地解决解决方案的各个方面。因此，我们非常需要一种近似理论，这种近似理论既要包含足够的设备物理特性，既要切合实际，又要在数学上易于处理。主要研究人员将推导半导体的近似模型，并用渐近和数值方法研究它们，以便了解实际设备的工作原理。

The principal investigators will study the partial differential equations and integro-differential equations that model semiconductor devices by using analytical and numerical methods. Their work will span the entire range from the actual derivation of the basic equations to the solution of concrete boundary value problems. In particular, the investigators will study the Boltzmann equation and systems derived from it such as the hydrodynamic model and the drift-diffusion model using asymptotic methods. The systems of differential equations that describe the behavior of semiconductor devices are difficult to solve analytically. Even large computers are unable to resolve aspects of the solutions accurately. Thus there is a great need for approximate theories that contain enough of the physics of the devices to be realistic and yet are tractable mathematically. The principal investigators will derive approximate models of semiconductors and study them with asymptotic and numerical methods, in order to understand the workings of actual devices.