Three-dimensional nanoelectronic device simulation using spectral element methods
Authors: Qing Huo Liu
Venue:  (2007)

The purpose of this thesis is to develop an efficient 3-Dimensional (3-D) nanoelectronic device simulator. Specifically, the self-consistent Schr&ouml;dinger-Poisson model was implemented in this simulator to simulate band structures and quantum transport properties. Also, an efficient fast algorithm, spectral element method (SEM), was used in this simulator to achieve spectral accuracy where the error decreases exponentially with the increase of sampling densities and the basis order of the polynomial functions, thus significantly reducing the CPU time and memory usage. Moreover, within this simulator, a perfectly matched layer (PML) boundary condition method was used for the Schr&ouml;dinger solver, which significantly simplifies the problem and reduces the computational time. Furthermore, the effective mass in semiconductor devices was treated as a full anisotropic mass tensor, which provides an excellent tool to study the anisotropy characteristics along arbitrary orientation of the device. Nanoelectronic devices usually involve the simulations of energy band and quantum transport properties. One of the models to perform these simulations is by solving a self-consistent Schr&ouml;dinger-Poisson system. Two efficient fast algorithms, spectral grid method (SGM) and SEM, are investigated and implemented in this thesis. The spectral accuracy is achieved in both algorithms, whose errors decrease exponentially with the increase of the sampling density and basis orders. The spectral grid method is a pseudospectral method to achieve a high-accuracy result by choosing special nonuniform grid set and high-order Lagrange interpolants for a partial differential equation. Spectral element method is a high-order finite element method which uses the Gauss-Lobatto-Legendre (GLL) polynomials to represent the field variables in the Schr&ouml;dinger-Poisson system and, therefore, to achieve spectral accuracy. We have implemented the SGM in the Schr&ouml;dinger equation to solve the energy band structures, and found that for a typical quantum well, SGM is about 51 times faster for 1% and 295 times faster for 0.1% accuracy, than the second-order finite-difference method, respectively. It is, however, complex for SGM to deal with boundary conditions. We have also investigated and implemented the SEM in the self-consistent Schr&ouml;dinger-Poisson system to solve the band structure in a quantum well structure, and found that the error does converge exponentially with the increase of the sampling density and basis order. Therefore, we chose SEM to implement our 3-D self-consistent Schr&ouml;dinger-Poisson solver for quantum transport simulations. During our investigation of quantum transport simulations, we found that the traditional applied boundary condition for the Schr&ouml;dinger equation, the quantum transmitting boundary method (QTBM), is costly because it needs to independently consider each mode of incident waves, reflected waves, and transmitted waves from each contact region and take a summation up to a large number of modes. Therefore, we proposed and implemented a PML boundary method to simplify the implementation. The PML method is a method to extend contact regions to thin layers of perfectly absorbing material, which can treat the transmitted waves as a total wave and also achieve zero reflected waves, thus simplifying the problem. It can significantly reduce the computational cost without loss of accuracy. We have proved the validity of the PML method from theoretical proof and numerical results, and shown the exponential convergence of the SEM. Also, we have shown the utility of the Schr&ouml;dinger solver using the above SEM and PML method by a waveguide example, a carbon nanotube example and a multiple-terminal quantum dot device example. Moreover, in our implementation, we have used the effective mass in the semiconductor material as a full anisotropic mass tensor, which provides an excellent tool for studying the anisotropic characteristics along arbitrary orientation of the device. Finally, we have implemented the Poisson solver, and completed the implementation of the self-consistent Schr&ouml;dinger-Poisson solver for the quantum transport simulation. We have tested the validity and convergence of the Poisson solver by a parallel plate example, and a spherical doped carrier example. Also, we have tested the validity of the self-consistent Schr&ouml;dinger-Poisson solver by a carbon nanotube example. (Abstract shortened by UMI.)