單一原子石墨層電子結構的計算模擬及大型矩陣對角化技術研究

Abstract

在這份論文中，我們將計算奈米尺度的單一原子石墨層(graphene)的電子結構並研究大規模矩陣(Large scale matrix)對角化的技術，首先我們介紹Tight-Binding的基本理論，建立K空間與實空間矩陣之後，我們建構出單一原子石墨層的奈米結構(graphene & graphene nanoribbon)的Tight-Binding矩陣，並只計算出單一原子石墨層與nanoribbon的能帶結構。接著再介紹大規模稀疏矩陣(Large scale sparse matrix)對角化技術中的LANCZOS以及使用最多時間研究的ARPACK演算法，應用在簡單的一維簡諧振子問題並利用有限差分法(finite difference)計算其能階，檢驗其準確度，會隨著矩陣增大而增加其準確度，使用一般配置8G記憶體的電腦，目前ARPACK可對角化約十五萬乘十五萬的大矩陣，未來可以計算更大的矩陣。

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.