Using gate-modulated Raman scattering and electron-phonon interactions to probe single-layer graphene: A different approach to assign phonon combination modes

Gate-modulated and laser-dependent Raman spectroscopy have been widely used to study $q=0$ zone center phonon modes, their self-energy, and their coupling to electrons in graphene systems. In this work we use gate-modulated Raman of $q\ne 0$ phonons as a technique to understand the nature of five second-order Raman combination modes observed in the frequency range of 1700–2300 cm${}^{-1}$ of single-layer graphene (SLG). Anomalous phonon self-energy renormalization phenomena are observed in all five combination modes within this intermediate frequency region, which can clearly be distinguished from one another. By combining the anomalous phonon renormalization effect with the double resonance Raman theory, which includes both phonon dispersion relations and angular dependence of the electron-phonon scattering matrix elements, and by comparing it to the experimentally obtained phonon dispersion, measured by using different laser excitation energies, we can assign each Raman peak to the proper phonon combination mode. This approach should also shed light on the understanding of more complex structures such as few-layer graphene (FLG) and its stacking orders as well as other two-dimensional (2D)-like materials.

Figure 1

(a) The Raman spectra for the combination modes in the frequency region from 1700 to 2300 cm${}^{-1}$ for four different laser lines: 488 nm (2.54 eV), 532 nm (2.33 eV), 575 nm (2.16 eV), and 590 nm (2.10 eV). In (b) and (c) we show the predicted phonon dispersion of single-layer graphene (SLG) (solid lines) adapted from Popov and Lambin,[27] showing the phonon branches near the $\mathit{\Gamma }$ point (left) and near the K point (right) for the combinations modes LO + iTA (black), iTO + LA (red), LO + LA (blue), oTO + iTO (purple), oTO + LO (gray), and iTO + iTA (green). (b) The phonon assignments of the present work for the experimental results obtained from the spectra in (a). Squares correspond to peak 1, circles to peak 2, triangles to peak 3, and diamonds and stars to peaks 4 and 5, respectively. Peaks 1, 2, and 3 come from an intravalley (AV) DRR process, while peaks 4 and 5 come from an intervalley (EV) DRR process. (c) The experimental data from this work are plotted considering the assignments proposed in the literature[7, 8]: Peaks 1, 3, and 4 coming from an intravalley DRR process and peaks 2 and 5 coming from an intervalley DRR process, which is different from the assignment in (b). The peak numbers are written below the corresponding phonon branch assigned to that peak. The absolute values for the angular dependence of the intravalley el-ph scattering matrix elements for the (d) LO + iTA (peak 1), (e) iTO + LA (peak 2), and (f) LO + LA (peak 3) phonon combination modes for $E_{\mathrm{L}}=2.54$ eV. ${\theta }_i$ and ${\theta }_f$ are, respectively, the initial and final scattering angles measured at the K point relative to the $k_x$ axis.

Figure 2

DRR processes involving phonons in the (a) KM or in the (b) $\mathbf{K}\mathit{\Gamma }$ direction, respectively, measured from the K point (red full arrows). (c) The phonon dispersion relation for the two intervalley (EV) combination modes: Peaks 4 (purple diamonds) and 5 (green stars). The half-colored symbols correspond to the DRR process in the KM direction and the open symbols correspond to the DRR process in the $\mathbf{K}\mathit{\Gamma }$ direction. The absolute value for the angular dependence of the EV el-ph scattering matrix elements for the (d) oTO + iTO, (e) iTO + iTA, and (f) oTO + LO phonon combination modes for $E_{\mathrm{L}}=2.54$ eV. ${\theta }_i$ and ${\theta }_f$ are, respectively, the initial and final scattering angles relative to the $k_x$ axis. The diagrams below (d), (e), and (f) show the scattering regions for which the el-ph matrix elements for the particular combination mode are maximum.

Figure 3

The dependence of the phonon frequency ${\omega }_q$ (black solid triangles) and phonon decay width ${\gamma }_q$ (open dots) on gate voltage ($V_{\mathrm{g}}$) for the (a) $G$ band and phonon combination modes (b) 1, (c) 2, (d) 3, (e) 4, and (f) 5. All the graphics, except for the $G$ band, are on the same scale for better comparison between the phonon renormalization results for the five combination modes assigned in this work. Notice that all five combination modes [(b) through (f)] show a decrease of ${\omega }_q$ and a broadening of ${\gamma }_q$ with increasing $V_{\mathrm{g}}$. The dashed lines are guides for the eyes and the error bars come from the fitting procedure.

Figure 4

Determination of the phonon wave vector q for (a) intravalley (AV) and (b) intervalley (EV) double resonance Raman processes. Namely, $k_0$, $k$, and $k^{\prime }$ stand for the electronic wave vectors and $E\left(k\right)$ is the electronic dispersion given by Eq. (A1). (c) ${\theta }_k$ is defined with relation to the $k_x$ axis.

Figure 5

All the possible final electronic states $k^{\prime }$ for a given initial electronic state $k$.