An Efficient Numerical Model for Solving the Single Electron Band Structure in Si Based on the Self-Consistent Pseudopotential Method

Abstract

The electronic band structure of a semiconductor is an essential property to determine most of its optical characteristics. The complexity of the energy band structure calculations makes analytical calculations impossible. Any calculation leading to electronic band structures has to utilize numerical methods. In this thesis, two solvers were developed to calculate the energy band structure of 1D Kronig-Penney lattice, 30 diamond lattice-structure and silicon lattice. In this thesis, many of the important methods of calculating the energy band structures were discussed. Through comparisons among different methods, we have determined that Self-Consistent Pseudopotential Method, SCPM, is the most suitable method for calculating the energy band structures when self-sufficiency and accuracy are of special importance. The SCPM is an iterative method which was utilized in this thesis by using efficient numerical methods. Instead of using conventional numerical methods such as Finite Difference Method or Finite Element Method which cause inefficiency, this thesis calculates the energy band structure by utilizing Orthogonal Plane-Wave expansion of the potentials. The 1D electronic band structure solver was developed as a foundation for the implementation of the 30 electronic band structure solver. It uses a minimal number of Fourier coefficients to calculate the energy band structure of the 1D Lattices without compromising accuracy. The 30 electronic band structure solver development needs multiple changes and modifications to the 1D solver. As the 30 solver is essentially made using the 10 solver platform, it is also efficient and needs a minimal number of Fourier coefficients for accurate results. The 30 solver can be used for either Nearly Free Electron Method, NFEM, or SCPM calculations. The NFEM calculations were done on the diamond lattice structure. The results were shown to be the same as the benchmarks of [28, 80]. The silicon lattice energy band structure was also calculated with the 30 solver using SCPM with LOA. The results were in the same range as the four sets of data gathered from three benchmarks [58, 81, 82], showing good agreement. Based on the two comparisons made for the 30 solver, it was shown that it is a reliable and efficient program to calculate energy band structures of the 30 lattices.