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

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.