Band Structures of One-Dimensional Periodic Materials with Graph Theory

TL;DR

This paper introduces a graph-theoretic approach to model and analyze the electronic band structures of one-dimensional periodic materials, enabling flexible representation and analysis of various unit cell configurations.

Contribution

It presents a novel method to represent periodic materials as graphs and compute their band structures, including analysis of connectivity and band gap relationships.

Findings

Connectivity of the unit cell is not correlated with its band gap at half filling.

The method applies to both regular and randomly-generated structures.

Provides a new computational tool for solid-state physics analysis.

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.