具有非紧间断势的Wigner输运方程的数值模拟

项目名称： 具有非紧间断势的Wigner输运方程的数值模拟

项目编号： No.91230107

项目类型： 重大研究计划

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 卢朓

作者单位： 北京大学

项目金额： 60万元

中文摘要： 纳米科技已经成为国际上研究的重要课题，在我国，也是有望取得跨越式发展的重要研究领域。随着器件尺寸的缩小进入纳米尺度， 量子效应在器件设计中的作用已不容忽视。本课题主要研究最精确的量子输运模型之一—Wigner方程的算法设计和数值模拟。截至目前，Wigner方程的研究仍存在一些瓶颈，使得该方程并没有得到真正广泛的应用。我们认为原因在于它的非紧间断势积分项的离散和不相容的入射边界条件的设置。如何攻克这两个极具挑战性的困难，得到高效收敛的算法是我们这个项目要解决的核心问题。我们拟利用广义Fourier变化和Gibbs现象消除法研究势积分项的离散，利用非平衡Green函数法研究边界条件的设置。最终研究目标是应用我们的数值算法开发出一套纳米器件模拟软件，为下一代半导体器件设计提供高精度的数值模拟工具，从而为我国半导体器件的发展做出贡献。

中文关键词： 量子动理学理论；入射边界条件；香农采样定理；适定性；

英文摘要： Nanotechnology research has become one of the most important topics in the world, and is also one of the fields in which our country could gain a leap-forward development. As the device scaling goes into the nano scale, the role of the quantum effects in the design of semiconductor devices can not be ignored. This project is mainly concerned with the algorithm design and numerical simulation of the Wigner equation, one of the most accurate models for the investigation of the quantum transport. Until now, there are some bottlenecks in the study of Wigner equation which prevents its more wide application. We think they are related to two aspects: one is the discretization of the pseudo-differential operator of the Wigner equation with a non-compactly supported and discontinuous potential, and the other is the inflow boundary condition which is not fully consistent with the quantum mechanics. How to solve the challenging issues and obtain a highly efficient and convergent numerical algorithm is the kernel problem of this project. We will make use of the generalized Fourier transform and the Gibbs phenomenon removal method to study the potential integral term, and the non-equilibrium Green function method to study the boundary conditions. The final goal of this project is to develop a nano-device simulation software

英文关键词： quantum Kinetic thoery；inflow boundary conditions；Shannon sampling thoery；well-posedness；