| 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. |
