Absolute Raman matrix elements of graphene and graphite

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] ${\left|K_{2f,10}^{\prime }\right|}^2$ vs laser frequency. For both graphene and graphite the ${\left|K_{2f,10}^{\prime }\right|}^2$ 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 $41{\text{cm}}^{-1}/\text{eV}$ and $35{\text{cm}}^{-1}/\text{eV}$, 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.

Figure 1

(Color online) The model dipole layer (graphene) located at position $z_d$ is taken to be 0.335 nm above the ambient-${\text{SiO}}_2$ interface denoted by $z_1$. The ${\text{Si-SiO}}_2$ is denoted by $z_2$. The contribution of the total enhancement factor ${\left|F\left[z_d,{\omega }_l\right]\right|}^2{\left|F\left[z_d,{\omega }_s\right]\right|}^2$ is illustrated while comparing the configurations (a) bare graphene and (b) graphene suspended over the ${\text{SiO}}_2\text{-Si}$ strata.

Figure 2

(Color online) (a) Experimental configuration for measuring the $D$ mode at the graphene edge which is located at a distance $z$ above the ambient-${\text{SiO}}_2$ interface located at $z_1$. $z_2$ locates the ${\text{Si-SiO}}_2$ interface. The ${\text{SiO}}_2$ layer thickness is $\left(293.7\pm 0.5\right)\text{nm}$ while the Si layer is assumed semi-infinite. (b) Optical image of the graphene flake under consideration. The laser spot was scanned along the line indicated. (c) The pencil lead upon which the measurement for graphite was carried out. (d) Microstructure of pencil graphite observed under $100\times$ optical magnification indicating a highly defected structure.

Figure 3

(Color online) (a) Representative Raman spectra of graphene obtained for laser excitation wavelengths 632 nm (red), 514 nm (green), and 472 nm (blue). (b) ${\left|K_{2f,10}^{\prime }\right|}^2$ for the graphene monolayer vs laser energy (eV) for the $G$ (black circles) and $D$ modes (red squares). The corresponding lines are a guide to the eye.

Figure 4

(Color online) Variation in the total enhancement factor of Eq. (8), ${\left|F\left[z_d,{\omega }_s\right]\right|}^2{\left|F\left[z_d,{\omega }_l\right]\right|}^2$ for graphene suspended on a ${\text{SiO}}_2\text{-Si}$ layer as a function of laser energy for a $\left(293.7\pm 0.5\right)\text{nm}$ variation in ${\text{SiO}}_2$ thickness for (a) the $D$ mode and (b) the $G$ mode. The solid black lines indicate the variation in the total enhancement factor ${\left|F\left[z_d,{\omega }_s\right]\right|}^2{\left|F\left[z_d,{\omega }_l\right]\right|}^2$ for the incident laser energies employed.

Figure 5

(Color online) (a) Representative Raman spectra of graphite obtained for laser excitation wavelengths 632 nm (red), 514 nm (green), and 472 nm (blue) with indicated $D$, $G$, $G^{-}$, and $G^{+}$ modes. (b) ${\left|K_{2f,10}^{\prime }\right|}^2$ for graphite per unit area per graphene layer vs laser energy (eV) for the $G$ (solid black circles), $D$ (red squares), $G^{-}$ (blue diamonds), and the $G^{+}$ modes (green squares). The corresponding lines are a guide to the eye.