Dynamical conductivity of gated AA-stacking multilayer graphene with spin-orbital coupling

An efficient method with no numerical diagonalization of a huge Hamiltonian matrix and calculation of a tedious Green's function is proposed to acquire the exact energy spectrum and dynamical conductivity in a gated AA-stacking $N$-layer Graphene (AANLG) with the intrinsic spin-orbital coupling (SOC). $2N \times 2N$ tight-binding Hamiltonian matrix, velocity operator and Green's function representation of an AANLG are simultaneously reduced to $N$ $2\times 2$ diagonal block matrices through a proper transformation matrix. A gated AANLG with intrinsic SOC is reduced to $N$ graphene-like layers. The energy spectrum of a graphene-like layer is $E= \varepsilon _{\bot}\pm \varepsilon_{||}$. $\varepsilon _{\bot}$ depends on the interlayer interaction, gated voltage and layer number. $\varepsilon_{||}=\sqrt{E_{MG}^2+ \Delta^2}$, where $E_{MG}$ is the energy spectrum of a monolayer graphene and $\Delta$ is the magnitude of intrinsic SOC. More importantly, by inserting the diagonal block velocity operator and Green's function representation in the Kubo formula, the exact dynamical conductivity of an AANLG is shown to be $\sigma = \Sigma_{j=1} ^N \sigma_j$, the sum of the dynamical conductivity of $N$ graphene-like layers. The analytical form of $\sigma_j$ is presented and the dependence of $\sigma_j$ on $\varepsilon_{\bot}$, $\Delta$, and chemical potential is clearly demonstrated. Moreover, the effect of Rashba SOC on the electronic properties of an AANLG is explored with the exact energy spectrum presented.