Absolute Raman matrix elements of graphene and graphite

Abstract

Using sample substitution [Grimsditch et al., J. Raman Spectrosc. 10, 77 (1981)] we deconvolve the highly wavelength-dependent response of the spectrometer from the Raman spectra of graphene suspended on an ${\text{SiO}}_{2}\text{-Si}$ substrate and graphite for the $D$ and $G$ modes in the visible range. We derive a model that considers graphene suspended on an arbitrary stratified medium while sidestepping its problematic ascription as an object of finite thickness and calculate the absolute Raman response of graphene (and graphite) via its explicitly frequency-independent Raman matrix element [Falicov and Martin, Light Scattering in Solids I: Introductory Concepts (Springer-Verlag, Berlin, 1983), p. 1083] ${|{K}_{2f,10}^{\ensuremath{'}}|}^{2}$ vs laser frequency. For both graphene and graphite the ${|{K}_{2f,10}^{\ensuremath{'}}|}^{2}$ per graphene layer vs laser frequency rises rapidly for the $G$ mode and less so for the $D$ mode over the visible range. Although we find a dispersion of the $D$ mode position with laser frequency for both graphene and graphite of $41\text{ }{\text{cm}}^{\ensuremath{-}1}/\text{eV}$ and $35\text{ }{\text{cm}}^{\ensuremath{-}1}/\text{eV}$, respectively, in good agreement with Narula and Reich [Phys. Rev. B 78, 165422 (2008)] assuming constant matrix elements, the observed intensity dependence is in disagreement. Finally, we show the sensitivity of our calculation to the variation in thickness of the underlying ${\text{SiO}}_{2}$ layer for graphene. Our findings shall serve as an experimental verification of the behavior of the relevant matrix elements in graphene and its allotropes that may be calculated theoretically in the future.