# An analytical study of electronic properties of ABC-stacking multilayer graphene

## TL;DR

This paper develops an analytical model for ABC-stacking multilayer graphene, revealing its electronic band structure and density of states through a simplified Hamiltonian approach, applicable to linear chain models.

## Contribution

The paper introduces a novel analytical framework that reduces the complex Hamiltonian of ABC-stacking multilayer graphene to simpler tridiagonal matrices, enabling easier analysis of its electronic properties.

## Key findings

- Derived the full band structure and density of states for ABC-stacking multilayer graphene.
- Established a relation between eigenvalues of symmetric and antisymmetric Hamiltonian matrices.
- Provided a method to analyze eigenvalue problems of tridiagonal matrices in layered materials.

## Abstract

We present an analytical model to study the electronic properties, including full band structure, low energy dispersions around the Dirac point and density of states of the ABC-stacking $N$-layer graphene (ABCNLG). An ABCNLG can be simulated by a linear atomic chain with $2N$ atoms. With only nearest-neighbor inter- and intra-layer hopping integrals taken into account, the Hamiltonian representation is a complex $2N \times 2N$ tridiagonal matrix $H_0$. Through a unitary transformation, we can reduce the $2N \times 2N$ Hamiltonian matrix into two real $N \times N$ tridiagonal matrices $\mathbb{H}_{s}$ and $\mathbb{H}_{a}$, i. e., $H_0=\mathbb{H}_{s} \oplus \mathbb{H}_{a} $. What's more, the two matrices satisfy the relation $\mathbb{H}_{a}=-\mathbb{H}_{s}$. As a result, energy spectrum associated with $\mathbb{H}_{s}$ and $\mathbb{H}_{s}$ have the relation $\lambda_{a}=-\lambda_{s}$. Such a characteristic is reflected on the energy dispersions and density of states. Our model can be applied to explore the basic properties of linear chain model and the eigenvalue problem of the tridiagonal matrices.