## 5 Intra- and Inter-Layer Phonon Energy Renormalizations  
### Interlayer-Related Phonons in 2 LG  
One of the most important consequences of the interlayer (IL) interactions in 2 LG is the rise of IL vibrational modes. [PERSON] _et al._[14] were pioneers to studying the out-of-plane optical (ZO\({}^{\prime}\)) phonon mode (with frequency \(\omega_{\mathrm{ZO^{\prime}}}=90\) cm\({}^{-1}\) predicted at the \(\Gamma\)-point, as shown in Figure 2b. The ZO\({}^{\prime}\) mode is also known as the IL breathing mode-LBM. They also studied the combination mode LOZO\({}^{\prime}\) predicted at the \(\Gamma\)-point showing that this combination occurs in the range for \(\omega_{\mathrm{LOZO^{\prime}}}\) from 1600 to 1800 cm\({}^{-1}\) for multilayer graphene (MLG) with up to 6 layers [14]. The other IL-related combination modes present in 2 LG are the LOZA combination mode (ZA is the IL out-of-plane acoustic mode whose frequency \(\omega_{\mathrm{ZA}}\) is zero at the \(\Gamma\)-point, as shown in Figure 2b) and the 2 ZO overtone (ZO is the out-of-plane tangential optical mode with frequency \(\omega_{\mathrm{ZO}}=885\) cm\({}^{-1}\) predicted at the \(\Gamma\)-point, as shown in Figure 2b) demand a more detailed analysis, which is still elusive. All these features involve \(q\
eq 0\) intravalley (AV) processes, therefore occurring around the \(\Gamma\)-point in the Brillouin zone. However, only the 2 ZO overtone presents two possible forward (\(q\approx 0\)) and backward (\(q\approx 2k\)) scattering mechanisms [21]. Note that both the ZA and ZO modes are not Raman active at the \(\Gamma\)-point. In spite of recent advances in the study of these interlayer modes, their phonon self-energies and e-ph interactions for these IL-dependent modes have hardly been discussed. It is worth saying that, these modes ranging from 1600 to 1800 cm\({}^{-1}\) are spectroscopic signatures for 2 LG and by understanding them in detail, we can understand the vdW-related phonon-dependent phenomena associated with these systems. Figure 15f shows the phonon combination modes and overtones observed in 2 LG in the spectral range 1600 to 1800 cm\({}^{-1}\). The insets give the phonon vibration symmetries for the LOZA (P1) and the two LOZO\({}^{\prime}\) (P2) peaks (upper box in Figure 15f), and the two 2 ZO (P3) peaks (lower box in Figure 15f). The LOZA mode comes from a \(q\approx 2k\) intravalley phonon scattering process (\(q\approx 2k\) AV) showing a frequency dispersion \(\partial\omega_{\mathrm{LOZA}}/\partial E_{\mathrm{L}}=26.1\) cm\({}^{-1}\)/eV. By looking at the feature P2 in Figure 15f, we observe that the LOZO\({}^{\prime}\) mode (\(q\approx 2k\) AV process) splits into two peaks, LOZO\({}^{\prime}\)(+) and LOZO\({}^{\prime}(-)\), whose frequency dispersions are \(\partial\omega_{\mathrm{LOZO^{\prime}}(+)}/\partial E_{\mathrm{L}}=55.1\) cm\({}^{-1}\)/eV and \(\partial\omega_{\mathrm{LOZO^{\prime}}(-)}/\partial E_{\mathrm{L}}=34.2\) cm\({}^{-1}\)/eV, respectively [63]. As schematized in Figure 15c, the two P2 peaks do not arise from the phonon dispersion but rather, they come from different resonant regimes of the LOZO\({}^{\prime}\) combination mode with the two electronic valence bands (\(\pi_{1}\) and \(\pi_{2}\)) and the two electronic conduction bands (\(\pi_{1}^{*}\) and \(\pi_{2}^{*}\)) of 2 LG. The LOZO\({}^{\prime}\)(+) comes from a resonance process involving the \(\pi_{1}\)(\(\pi_{1}^{*}\)) bands, while the the LOZO\({}^{\prime}(-)\) comes from a resonance process involving the \(\pi_{2}\)(\(\pi_{2}^{*}\)) bands. These resonance conditions (see Figure 15c) require the phonon momentum \(\mathbf{q}\) for the LOZO\({}^{\prime}\)(+) mode to be larger than that for the LOZO\({}^{\prime}\)(\(-\)) mode (\(q_{\mathrm{LOZO^{\prime}}(+)}>q_{\mathrm{LOZO^{\prime}}(-)}\)). As a consequence the phonon energies are such that \(\hbar\omega_{\mathrm{LOZO^{\prime}}(+)}>\hbar\omega_{\mathrm{LOZO^{\prime}}( -)}\).