含时薛定谔方程的高阶辛算法研究
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摘要
纳米技术的发展之迅速异乎寻常,将成为二十一世纪的关键技术。纳米技术是电子器件小型化的必然趋势,也是现代理论研究和工程实践的前沿领域。纳米器件本身尺寸极小,传统半导体物理中的建模方法已经失效,采取实验手段直接测量器件的各种性能又比较困难。因此,研究精确和高效的数值方法是现代纳米器件建模和优化的重要课题。纳米器件的建模是一个极其复杂的多物理问题,描述电子传输中的电子与电子相互作用、电子和声子的散射、外界环境的影响(电极、电场、磁场)以及器件的局域化结构均需要采用不同方法。分析大部分纳米器件特性的切入点是确定器件结构电子的能量本征值和能量本征态,本征值和本征态的求解过程本质上也是表征器件格林函数的过程。一旦得到格林函数,器件的电荷密度、电流大小、和伏安曲线便可求出。因此,纳米器件本征值和本征态的快速和精确求解是一项极其重要的课题。
含时薛定谔(Schrodinger方程)可以用于求解纳米器件的本征值和本征态。FDTD方法——时域有限差分法,以其简单直观的特点,已成为求解含时薛定谔方程最常用的方法之一,它的优点包括形式简单、极易处理非均匀结构、方便得到目标的宽带信息、具有天然的并行性等等,但其缺点是无法克服在长时间仿真中产生的色散误差。
大量的物理现象可通过时间演变辛变换的哈密尔顿(Hamilton)微分方程描述。辛算法通过使用不同的时间差分方法保持了哈密尔顿系统向空间全局辛结构。大量实验已证明辛算法在哈密尔顿系统数值计算中的优势,该优势在长时间仿真中最为显著。辛算法已经成功用于求解薛定谔方程：对于含时薛定谔方程,一种方法是将其复波函数分成实数部分和虚数部分,另一种方法是将其哈密尔顿系统分成动能算子和势能算子。而对于稳态薛定谔方程,如果使用的是广义坐标(复波函数)和广义速度(复波函数的空间导数),那么,也可以使用辛算法。此外,辛算法亦可用于求解非线性薛定谔方程。
本文将辛算法和高阶时域有限差分算法结合构造SFDTD方法——高阶辛时域有限差分法,应用到量子力学的薛定谔方程的求解中,发展快速、高效、和精确的数值方案,解决任意结构纳米器件的本征问题。时间方向上,采用高阶辛积分,在长期仿真中保持薛定谔方程的辛结构；空间方向上,采用4阶交错差分,提高数值精度。
针对“含时薛定谔方程的高阶辛算法研究”这一课题,本文的创新工作主要包括以下几个方面的内容：
(1)研究了薛定谔方程的辛性质,探讨了薛定谔方程、离散方式、网格、空间拓扑之间的内在联系；
(2)研究SFDTD方法用于薛定谔方程的数值模拟。采用高阶辛时域有限差分法对含时薛定谔方程进行离散,对空间进行4阶交错差分,时间上引入高阶辛积分,给出了基本的迭代公式；分析算法的稳定性、色散性,研究本征值和本征态的有效提取方法,研究边界问题的处理方法。
(3)将SFDTD方法用于纳米器件能量本征值和能量本征态的分析与求解,研究算法的数值性态,发展相关的核心技术,对典型结构和复杂结构的纳米器件进行数值模拟。
(4)纳米材料量子传输模型的优化和设计。在构造麦克斯韦方程和薛定谔方程耦合方程的基础上,研究纳米材料的电学传输特性。比如,碳纳米管中电子传输模型的设计和优化。针对在碳纳米管中电子传输的量子特征,研究一种统一的多物理辛时域有限差分框架,从整体上统一求解薛定谔方程和麦克斯韦方程的耦合方程。
The advances in nanotechnology are unprecedentedly fast and nanotechnology will become one of key technologies in the21st century. Nanotechnology is driven by the miniaturization of electronic devices, and is also a cutting-edge field in theoretical research and engineering applications. The size of nanodevices is extremely small, and therefore the modeling approach adopted in classical semiconductor physics failed. Moreover, taking experimental tools to obtain various properties of nanodevices is difficult. Hence, studying an accurate and efficient numerical method is an important subject for modern nanodevice modeling and optimization. Nanoscale device modeling is a very complex multi-physics problem. Electron-electron interaction in the electron transport, the scattering between electrons and phonons, the influence of the external environment (the electrode, electric field and magnetic field) as well as the localized structure of the device should be described by different methods. Finding the eigenvalue and eigenstate of nanodevices is fundamentally important for capturing nanodevice characteristics, which strongly relates to the nonequilibrium Green's function. With the help of Green's function, charge density, current distribution, and current-voltage characteristics of the device will be obtained. Thus, an accurate and fast solution to eigenstates and eigenfrequencies is an important subject.
The time-dependent Schrodinger equation can be used to solve the eigenvalue and eigenstate of nanodevices. As the most standard algorithm, the traditional finite-difference time-domain (FDTD) method, which is simple and easy to implement, has been widely applied to solve the time-dependent Schrodinger equation. The main advantages of the FDTD-based techniques are computational simplicity and low operation count. Furthermore, it is very well suited to analyze transient problems and is very good at modeling inhomogeneous geometries. Most important of all, the method can readily be implemented on the massive computers. However, its drawback is unable to overcome the dispersion errors in the long simulation.
A large quantity of physical phenomena can be modeled by Hamiltonian differential equations whose time evolution is the symplectic transform and flow conserves the symplectic structure. The symplectic schemes include a variety of different temporal discretization strategies designed to preserve the global symplectic structure of the phase space for a Hamiltonian system. They have demonstrated their advantages in numerical computations for the Hamiltonian system, especially for a long-term simulation. The symplectic scheme has been successfully applied to solve Schrodinger equation with three different strategies. For the time-dependent Schrodinger equation, one scheme splits the complex wave function into real and imaginary parts, and another one decomposes the Hamiltonian into the kinetic and potential operators. For the time-independent Schrodinger equation, the symplectic scheme can also be employed if the generalized coordinate (complex wave function) and generalized velocity (spatial derivatives of complex wave function) are introduced. Moreover, the symplectic scheme has been extended to solve the nonlinear Schrodinger equation.
The high-order symplectic finite difference time domain is applied to the solution of the Schrodinger equation in quantum mechanics, developing rapid, efficient and accurate numerical scheme to solve the eigenvalue problem of arbitrarily structured nanodevices. In the time domain, high-order symplectic integration is adopted to preserve the symplectic structure of the Schrodinger equation in the long-term simulation. In the space domain,4th order staggered differences is taken to improve the numerical accuracy.
Focusing on the subject of "High-Order Symplectic FDTD Scheme for Solving Time-Dependent Schrodinger Equation", some novel contributions are made as follows
(1) The symplectiness of Schrodinger equations is discussed and its connections with Schrodinger equations, discretization method, grid, and spatial topology are established.
(2) The SFDTD scheme is employed in the numerical simulation to Schrodinger equation. Higher order symplectic finite difference time domain method is applied to discretize time-dependent Schrodinger equation. For time domain, the high-order symplectic integration scheme is used, and the staggered fourth-order difference scheme is adopted in spatial domain. At the same time, our work offers:(1) a rigorous numerical stability and dispersion analyses of the high-order SFDTD scheme;(2) the mathematical form and implementation for excitation source and the extraction method for eigenvalues and eigenstates;(3) boundary treatments
(3) The SFDTD method is introduced to analyze the energy eigenvalues and energy eigenstates of nanodevices. The numerical performance of algorithms is studied and core technologies are developed to simulate typical nanodevices with complex structures.
(4) Optimization and design of quantum transport model for nanomaterials. Based on the coupled form of Maxwell's equations and the Schrodinger equation, electron transport characteristics of nanomaterials and nanostructures is studied. For example, the design and optimization of the electron transport model in the carbon nanotubes. Regarding the quantum features of electron transport in carbon nanotubes, a unified and multi-physical symplectic framework is established to solve the coupled Schrodinger-Maxwell's equations globally.
含时薛定谔(Schrodinger方程)可以用于求解纳米器件的本征值和本征态。FDTD方法——时域有限差分法,以其简单直观的特点,已成为求解含时薛定谔方程最常用的方法之一,它的优点包括形式简单、极易处理非均匀结构、方便得到目标的宽带信息、具有天然的并行性等等,但其缺点是无法克服在长时间仿真中产生的色散误差。
大量的物理现象可通过时间演变辛变换的哈密尔顿(Hamilton)微分方程描述。辛算法通过使用不同的时间差分方法保持了哈密尔顿系统向空间全局辛结构。大量实验已证明辛算法在哈密尔顿系统数值计算中的优势,该优势在长时间仿真中最为显著。辛算法已经成功用于求解薛定谔方程：对于含时薛定谔方程,一种方法是将其复波函数分成实数部分和虚数部分,另一种方法是将其哈密尔顿系统分成动能算子和势能算子。而对于稳态薛定谔方程,如果使用的是广义坐标(复波函数)和广义速度(复波函数的空间导数),那么,也可以使用辛算法。此外,辛算法亦可用于求解非线性薛定谔方程。
本文将辛算法和高阶时域有限差分算法结合构造SFDTD方法——高阶辛时域有限差分法,应用到量子力学的薛定谔方程的求解中,发展快速、高效、和精确的数值方案,解决任意结构纳米器件的本征问题。时间方向上,采用高阶辛积分,在长期仿真中保持薛定谔方程的辛结构；空间方向上,采用4阶交错差分,提高数值精度。
针对“含时薛定谔方程的高阶辛算法研究”这一课题,本文的创新工作主要包括以下几个方面的内容：
(1)研究了薛定谔方程的辛性质,探讨了薛定谔方程、离散方式、网格、空间拓扑之间的内在联系；
(2)研究SFDTD方法用于薛定谔方程的数值模拟。采用高阶辛时域有限差分法对含时薛定谔方程进行离散,对空间进行4阶交错差分,时间上引入高阶辛积分,给出了基本的迭代公式；分析算法的稳定性、色散性,研究本征值和本征态的有效提取方法,研究边界问题的处理方法。
(3)将SFDTD方法用于纳米器件能量本征值和能量本征态的分析与求解,研究算法的数值性态,发展相关的核心技术,对典型结构和复杂结构的纳米器件进行数值模拟。
(4)纳米材料量子传输模型的优化和设计。在构造麦克斯韦方程和薛定谔方程耦合方程的基础上,研究纳米材料的电学传输特性。比如,碳纳米管中电子传输模型的设计和优化。针对在碳纳米管中电子传输的量子特征,研究一种统一的多物理辛时域有限差分框架,从整体上统一求解薛定谔方程和麦克斯韦方程的耦合方程。
The advances in nanotechnology are unprecedentedly fast and nanotechnology will become one of key technologies in the21st century. Nanotechnology is driven by the miniaturization of electronic devices, and is also a cutting-edge field in theoretical research and engineering applications. The size of nanodevices is extremely small, and therefore the modeling approach adopted in classical semiconductor physics failed. Moreover, taking experimental tools to obtain various properties of nanodevices is difficult. Hence, studying an accurate and efficient numerical method is an important subject for modern nanodevice modeling and optimization. Nanoscale device modeling is a very complex multi-physics problem. Electron-electron interaction in the electron transport, the scattering between electrons and phonons, the influence of the external environment (the electrode, electric field and magnetic field) as well as the localized structure of the device should be described by different methods. Finding the eigenvalue and eigenstate of nanodevices is fundamentally important for capturing nanodevice characteristics, which strongly relates to the nonequilibrium Green's function. With the help of Green's function, charge density, current distribution, and current-voltage characteristics of the device will be obtained. Thus, an accurate and fast solution to eigenstates and eigenfrequencies is an important subject.
The time-dependent Schrodinger equation can be used to solve the eigenvalue and eigenstate of nanodevices. As the most standard algorithm, the traditional finite-difference time-domain (FDTD) method, which is simple and easy to implement, has been widely applied to solve the time-dependent Schrodinger equation. The main advantages of the FDTD-based techniques are computational simplicity and low operation count. Furthermore, it is very well suited to analyze transient problems and is very good at modeling inhomogeneous geometries. Most important of all, the method can readily be implemented on the massive computers. However, its drawback is unable to overcome the dispersion errors in the long simulation.
A large quantity of physical phenomena can be modeled by Hamiltonian differential equations whose time evolution is the symplectic transform and flow conserves the symplectic structure. The symplectic schemes include a variety of different temporal discretization strategies designed to preserve the global symplectic structure of the phase space for a Hamiltonian system. They have demonstrated their advantages in numerical computations for the Hamiltonian system, especially for a long-term simulation. The symplectic scheme has been successfully applied to solve Schrodinger equation with three different strategies. For the time-dependent Schrodinger equation, one scheme splits the complex wave function into real and imaginary parts, and another one decomposes the Hamiltonian into the kinetic and potential operators. For the time-independent Schrodinger equation, the symplectic scheme can also be employed if the generalized coordinate (complex wave function) and generalized velocity (spatial derivatives of complex wave function) are introduced. Moreover, the symplectic scheme has been extended to solve the nonlinear Schrodinger equation.
The high-order symplectic finite difference time domain is applied to the solution of the Schrodinger equation in quantum mechanics, developing rapid, efficient and accurate numerical scheme to solve the eigenvalue problem of arbitrarily structured nanodevices. In the time domain, high-order symplectic integration is adopted to preserve the symplectic structure of the Schrodinger equation in the long-term simulation. In the space domain,4th order staggered differences is taken to improve the numerical accuracy.
Focusing on the subject of "High-Order Symplectic FDTD Scheme for Solving Time-Dependent Schrodinger Equation", some novel contributions are made as follows
(1) The symplectiness of Schrodinger equations is discussed and its connections with Schrodinger equations, discretization method, grid, and spatial topology are established.
(2) The SFDTD scheme is employed in the numerical simulation to Schrodinger equation. Higher order symplectic finite difference time domain method is applied to discretize time-dependent Schrodinger equation. For time domain, the high-order symplectic integration scheme is used, and the staggered fourth-order difference scheme is adopted in spatial domain. At the same time, our work offers:(1) a rigorous numerical stability and dispersion analyses of the high-order SFDTD scheme;(2) the mathematical form and implementation for excitation source and the extraction method for eigenvalues and eigenstates;(3) boundary treatments
(3) The SFDTD method is introduced to analyze the energy eigenvalues and energy eigenstates of nanodevices. The numerical performance of algorithms is studied and core technologies are developed to simulate typical nanodevices with complex structures.
(4) Optimization and design of quantum transport model for nanomaterials. Based on the coupled form of Maxwell's equations and the Schrodinger equation, electron transport characteristics of nanomaterials and nanostructures is studied. For example, the design and optimization of the electron transport model in the carbon nanotubes. Regarding the quantum features of electron transport in carbon nanotubes, a unified and multi-physical symplectic framework is established to solve the coupled Schrodinger-Maxwell's equations globally.
引文
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