Influence of out-of-plane response on optical properties of two-dimensional materials: First principles approach

The ab initio calculation of optical spectra of sheet crystals usually arranges them in a three-dimensional superlattice with a sufficiently large interlayer distance. We show how the resulting frequency-dependent dielectric tensor is related to the anisotropic optical conductivity of an individual sheet or to the dielectric tensor of a corresponding film with thickness $d$. Their out-of-plane component is taken into account, in contrast to usual treatments. We demonstrate that the generalized transfer-matrix method to model the optical properties of a layer system containing a sheet crystal accounts for all tensor components. As long as $d\ensuremath{\ll}\ensuremath{\lambda}$ ($\ensuremath{\lambda}$-wavelength of light) this generalized formulation of the optical properties for anisotropic two-dimensional (2D) systems of arbitrary thickness reproduces the limits found in literature that are based either on electromagnetic boundary conditions for a conducting surface or on an isotropic dielectric tensor. For $s$-polarized light, the results are independent of the sheet description. For oblique incidence of $p$-polarized light, the tensor nature of the optical conductivity (or the dielectric function) of the sheet crystal strongly impacts on reflectance, transmittance, and absorbance due to the out-of-plane optical conductivity. The limit $d\ensuremath{\rightarrow}0$ should be taken in the final expressions. Example spectra are given for the group-IV honeycomb 2D crystals graphene and silicene.