| This dissertation is organized as follows. Beginning with physical background discussions of many-body problems, Chapter 1 introduces the central Kohn-Sham equations of Density Functional Theory for electronic structure calculations. In order to discretize the system, the real-space mesh techniques are selected to apply to the Kohn-Sham equations because of their advantages over other discretization techniques. In addition, the high-order basis functions employed by real-space mesh technique are used to reduce the size of the system matrices and improve the simulation convergence. In the self-consistent simulations, the current challenge posed by the first-principle calculations is the high cost for computing the electron density, and this becomes a limiting factor for large-scale device simulations. Consequently, Chapter 2 investigates the relevance of mode decomposition techniques (i.e. mode approach) for solving the Schrodinger-type equation within a real-space mesh framework. It is shown how the full mode approach or its asymptotic counterpart can be of benefit to two distinct highly efficient numerical procedures for computing the electron density: (i) the CMB strategy and (ii) the FEAST algorithm. The numerical simulation examples of CNTs (carbon nanotubes) using empirical pseudopotential are also presented in this chapter to show the efficiency of the proposed techniques. In addition to the applications to empirical pseudopotential, Chapter 3 shows that these techniques are also successfully applied to all-electron calculations for systems of a single atom, molecules and polysparaphenylene, in which bare local core potential is taken. These simulations achieve high-accuracy using the 3rd order finite element method and non-uniform mesh. For large-scale simulation, it becomes necessary to implement parallelism on different levels of modeling process. Therefore, Chapter 4 proposes a domain decomposition technique to divide a large problem into small ones and then carry out calculations for subproblems simultaneously. In this dissertation, the linear system solver in FEAST is further customized to solve the solution on the coarse mesh level, while solution of fine mesh inside each independent subdomain can be retrieved simultaneously accounting for the information at the interfaces of fine/coarse mesh. In such way, the size of system can be reduced significantly, and the scalability of the all-electron calculations can be improved. |
