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 σ₁(ω) of graphene layers. We present an accurate expression for σ₁(ω) 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 σ₁(ω) 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 σ₁(ω_p) value of N-layer graphene by the corresponding σ₁(ω_p) of monolayer graphene, where ω_p is the peak frequency of the ordinary dielectric function's imaginary part ε₂(ω) of graphene.