Determination of the In-Plane Optical Conductivity of Multilayer Graphene Supported on a Transparent Substrate of Finite Thickness from Normal-Incidence Transmission Spectra

Normal-incidence transmission measurements are commonly used for determining the real part of the in-plane optical conductivities sigma(1)(omega) of graphene layers. We present an accurate expression for sigma(1)(omega) in a closed form for a multilayer graphene film supported on a finite-thickness transparent substrate. This form takes into account the coherent and incoherent multiple reflections of the system, whereas the traditional method assumes a semi-infinite substrate. The simulated results for graphene sheets with a layer number N <= 10 show that no matter what the transparent substrate is, the accuracy to which sigma(1)(omega) is determined by applying this expression is improved with no systematic error. Moreover, the layer number N can be exactly determined by simply dividing the sigma(1)(omega(p)) value of N-layer graphene by the corresponding sigma(1)(omega(p)) of monolayer graphene, where omega(p) is the peak frequency of the ordinary dielectric function's imaginary part epsilon(2)(omega) of graphene.