Layered Dynamical Conductivity for a Transfer Matrix Method -- Application to an N-layer Graphene

We calculated the optical properties of an $N$-layer graphene by formulating the dynamical conductivity of each layer. This is the conductivity when an electromagnetic field is localized at a particular layer and differs from the standard conductivity calculated assuming a uniform field throughout all layers. By combining these conductivities with a transfer matrix method, we took into account the spatial variation of the electromagnetic field caused by internal reflections. The results obtained from the two conductivities show that similar peak structures originating from the interlayer electronic interaction appear in reflectance of an $N$-layer graphene at any $N$. The peak is inherent to the AB stacking and is not seen for the AA stacking, and the peak corresponding to a sufficiently large $N$ is considered to the one observed for natural graphite. We also gave physical explanations of the existing experimental results on highly oriented pyrolytic graphite and natural graphite under high pressure. Although a layered conductivity underestimates the reflectance of graphite at photon energies below the peak, we will show that the disagreement is attributed to a nonlocal conductivity caused by interlayer interaction. The calculations with layered conductivity are useful in knowing the local response to light and may be further validated by an observation of a correction by interlayer electronic interaction to the universal layer number that we have discovered recently.