Calculation of refractive index of multilayer epitaxial graphene on C-face SiC measured by synchrotron using Kramers-Kronig and Newton-Raphson method

Abstract

Calculation of the refractive index of Multilayer Epitaxial Graphene (MEG) on C-face SiC using Kramers-Kronig and Newton-Raphson method has been done. Calculation of complex refractive index performed through the reflectance R data from Synchrotron radiation measurement with the incident angle of 17.5 °. The result of Kramers-Kronig transformation of R is phase shift (δ), and the equations that are used in the extracting refractive index of MEG on C-face SiC involve Fresnel and Snell’s equations. The problem in determining the root of Fresnel equation is resolved by using Newton-Raphson method. The results show that Kramers-Kronig and Newton-Raphson method can be used to extract the complex refractive index of nanostructure material, i.e., MEG on C-face SiC substrate. At lower wavenumber (up to 40000 cm−1), the complex refractive index resembles that of graphene on SiO2 and epitaxial graphene on Si-Face obtained using low energy spectroscopic ellipsometry and absorption spectroscopy, in which the refractive index is nearly constant at this particular wavenumber range indicating the transparency of MEG film on SiC substrate. we observed no difference in the value of the resulting complex refractive index of MEG films on SiC for difference Newton-Raphson tolerance, step size h, and initial guess used indicating that the resulting complex refractive index calculated using Newton-Raphson method converge very fast.