Two-Dimensional Analysis of the Double-Resonant 2D Raman Mode in Bilayer Graphene

By computing the double-resonant Raman scattering cross section completely from first principles and including the electron-electron interaction at the $GW$ level, we unravel the dominant contributions for the double-resonant 2D mode in bilayer graphene. We show that, in contrast to previous works, the so-called inner processes are dominant and that the 2D-mode line shape is described by three dominant resonances around the $K$ point. We show that the splitting of the transversal optical (TO) phonon branch in the $\mathrm{\Gamma }\text{−}K$ direction, as large as $12{\mathrm{cm}}^{-1}$ in the $GW$ approximation, is of great importance for a thorough description of the 2D-mode line shape. Finally, we present a method to extract the TO phonon splitting and the splitting of the electronic bands from experimental data.

Figure 1

Schematized illustration of the 2D-mode scattering processes along the $\mathrm{\Gamma }\text{−}K\text{−}M\text{−}{K}^{\prime }\text{−}\mathrm{\Gamma }$ high-symmetry direction for (a) symmetric and (b) antisymmetric processes. Inner and outer processes are marked with red and blue traces, respectively. (c) Goldstone diagram for a double-resonant $e\text{−}h$ scattering process $P_{mi}^{lj}$. (d) $GW$-corrected phonon dispersion of bilayer graphene close to $K$, showing the TO splitting in the $\mathrm{\Gamma }\text{−}K$ direction.

Figure 2

(a) Illustration of the phonon wave vector sectors for inner (red) and outer (blue) processes around the $K$ point. The solid and dashed white lines denote the $K\text{−}M$ and $K\text{−}\mathrm{\Gamma }$ high-symmetry lines, respectively. [(b)—(d)] Plots of the normalized 2D-mode scattering cross section ${\mathcal{I}}_{\mathbf{q}}$ around the $K$ point for different laser energies.

Figure 3

Comparison of calculated 2D-mode spectra with Raman spectra from freestanding bilayer graphene at different $\hslash {\omega }_L$ values. Calculations and experimental data are shown as red and black curves, respectively. Spectra are normalized and vertically offset.

Figure 4

Calculated 2D-mode spectra at 1.96 eV excitation energy. Decomposition of the calculated spectra into (a) $e\text{−}h$ scattering processes as well as inner and outer contributions, (b) the four different scattering processes $P_{mi}^{lj}$ (without interference between the different processes), and (c) into symmetric and antisymmetric processes (including interference). [(d) and (e)] Experimental values for the electronic and TO phonon splittings, respectively. The solid (red) and dashed (blue) lines denote the DFT calculated splittings in inner and outer directions, respectively. Open, green circles are data points from Ref. [11]. Filled, black circles represent data points from this work.