| Summary: | 碩士 === 國立交通大學 === 電子物理系所 === 97 === In this thesis, we theoretically study the electronic structures of graphene nanostructures and the techniques of numerical diagonalization of large scale matrix, i.e., Lanczos and Arpack eigensolvers. In the first part of this thesis, the electronic structures of infinite two-dimensional(2D) graphene systems and one-dimensional(1D) graphene nano-ribbons are calculated by using one-orbital tight binding theory. The general Hamiltonian matrices for the 2D and 1D graphene systems are explicitly derived. The electronic structures of the graphene nano-systems are calculated by carrying out direct diagonalization. According to the study, we identify the localization of electron density at the zigzag edge in the zigzag ribbon. In the second part, we review the basic theory and algorithm of Lanczos and ARPACK eigensolvers for the diagonalization of large scale sparse matrix. The ARPACK package is applied to solve the problem of 1D simple harmonic oscillation. The maximum size of matrix diagonalized by the package is 8 million by 8 million on PC with 8G memory. Typically, the high accuracy 0.00002 % for the numerically calculated ground states can be achieved.
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