Paper
Solution Procedure for a Class of Band Structures –Application of the Finite Element Method to Schrödinger’s Equation
- Authors:
- Najib A. Kasti
- Abstract
- Several publications used the finite element method to determine the band structures of periodic solids by solving the Schrödinger equation (for example, Pask et al., Sukumar et al. [1-4]). The approaches used by these publications could basically be divided into two. The first approach (Pask et al. [1-3]) expresses the wave function ψ as the product of a harmonic function and a periodic function using Bloch’s theorem. The periodic function is then discretized over the domain. In contrast, the second approach (Sukumar et al. [4]) discretizes the wave function over the domain with its nodal values being complex in this case. This paper discusses a solution procedure for determining the band structures for a class of materials starting from the approach followed by Sukumar [4]. It assumes that one can obtain the discrete Hamiltonian and overlap matrices from a conventional finite element analysis program without reverting to a special program. The application of the boundary conditions and the solution of the band structures, for the defined class of material, are performed through matrix operations of well defined steps. The final complex eigenvalue problem, to determine the band energies of the system, is then solved by conventional methods. When solving the resulting system, two representations of the overlap matrix were tested in this work, namely, the consistent and lumped representations. Each of these representations displayed a different response when compared to the exact solution. The results from the lumped and consistent formulations as well as those from a simple averaging process are discussed in this paper.
- Keywords
- Quantum Mechanics; Finite Element Method; Schrödinger Equation; Periodic Solids
- StartPage
- 129
- EndPage
- 138
- Doi