Band Structures of One-Dimensional Periodic Materials with Graph Theory

Abstract

We show how arbitrary unit cells of periodic materials can be represented as graphs whose nodes represent atoms and whose weighted edges represent tunneling connections between atoms. Further, we present methods to calculate the band structure of a material with an arbitrary graphical representation, which allows one to study the Fermi level of the material as well as conductivity at zero temperature. We present results for both circular chains as well as randomly-generated unit cell structures, and also use this representation to show that the connectivity of the unit cell is not correlated to its band gap at half filling. This paper provides an introductory insight into the utilization of graph theory for computational solid-state physics.