Measuring the absolute Raman cross section of nanographites as a function of laser energy and crystallite size

In this paper, the dependence of the differential Raman cross section $\beta$ of the $D$, $G$, $D^{\prime }$, and $G^{\prime }$ bands of nanographites on the excitation laser energy and also on the crystallite size is reported. We show that ${\beta }_G$ is proportional to the fourth power of the excitation laser energy $\left(E_l\right)$, as predicted by the Raman scattering theory. For the bands which arise from the double-resonance mechanism ($D$, $D^{\prime }$, and $G^{\prime }$), the differential cross section does not depend on $E_l$, explaining the strong dependence of the ratio $I_D∕I_G$ on the excitation laser energy $E_l$ used in the Raman experiment. The $L_a$ dependence of $D$ and $D^{\prime }$ band differential cross sections is measured, confirming that the proportionality $I_D∕I_G\propto L_a^{-1}$ originates from the strong dependence of ${\beta }_D$ on the inverse of the crystallite size. In the $G^{\prime }$ band case, the data show that its differential cross section increases with the increasing crystallite size $L_a$, following an opposite behavior when compared with the disorder induced $D$ and $D^{\prime }$ bands. An analysis on the dependence of the full width at half maximum $\left(\Gamma \right)$ of the $D$, $G$, $D^{\prime }$, and $G^{\prime }$ bands on the crystallite size $L_a$ of nanographites is performed, showing that the phonon lifetime is proportional to the crystallite size.

Figure 1

(Color online) The spectral differential cross section of the $D$, $G$, $D^{\prime }$, and $G^{\prime }$ bands for the nanographite sample with $L_a=35\mathrm{nm}$, using five different values of $E_l$ (indicated at the top of each spectrum). The same vertical scale was used for the five spectra for comparison.

Figure 2

The differential cross section of the $D$, $G$, and $D^{\prime }$ bands vs the excitation laser energy $E_l$, obtained from the samples with (a) $L_a=20\mathrm{nm}$, (b) $L_a=35\mathrm{nm}$, and (c) $L_a=65\mathrm{nm}$. (d) The differential cross section of the $G^{\prime }$ band vs the excitation laser energy, for the samples with crystallite sizes $L_a=20$, 35, and $65\mathrm{nm}$.

Figure 3

Plot of the $D$ (filled symbols) and $D^{\prime }$ (open symbols) band differential cross sections vs the inverse of the crystallite size $L_a$, obtained using the five values of $E_l$.

Figure 4

Plot of the ratios ${\beta }_D∕{\beta }_G$ (solid squares) and ${\beta }_{D^{\prime }}∕{\beta }_G$ (empty squares) vs the inverse of the product $E_l^4{L}_a$, for the whole data obtained in the experiment. The lines (solid and dashed lines for the $D$ and $D^{\prime }$ band data, respectively) are the linear fit originating Eqs. (1, 2).

Figure 5

(Color online) (a) Plot of the product ${\beta }_G{E}_l^{-4}$ vs the crystallite size $L_a$. (b) The differential cross section of the $G^{\prime }$ band vs the crystallite size $L_a$. (c) The ratio ${\beta }_{G^{\prime }}∕{\beta }_G$ vs the inverse of the crystallite size $L_a$, for the data obtained using five different values of $E_l$.

Figure 6

The full width at half maximum $\Gamma$ for the $D$, $D^{\prime }$, $G$, and $G^{\prime }$ bands [(a), (b), (c), and (d), respectively] vs the crystallite size observed in the Raman spectra taken from the samples with $L_a=20$, 35, and $65\mathrm{nm}$, using five different values for the excitation laser energy $E_l$.

Figure 7

Scanning tunneling microscopy (STM) images, with atomic resolution, obtained from the surface of a crystallite of the sample with $L_a=65\mathrm{nm}$. (a) A Moire pattern at the crystallite surface is observed. (b) Magnification of the region delimited by the white square in (a), where the carbon atom positions in the graphite lattice can be observed. The STM measurements were performed using a Nanoscope II MultiMode microscope from Digital Instruments. For more details, see Ref. 7.