## 5 Intra- and Inter-Layer Phonon Energy Renormalizations
### Interlayer-Related Phonons in 2 LG
For the two 2 ZO features, [PERSON] _et al._[21] predicted through tight-binding calculations that, in 2 LG systems, the 2 ZO overtone should be observed for forward (\(q\approx 0\)) and backward (\(q\approx 2k\)) AV scattering, where the \(q\approx 2k\) mode presents a negative frequency dispersion. Although some information for the 2 ZO (\(q\approx 0\)) mode was reported by [PERSON] _et al._[14], the existence of the 2 ZO (\(q\approx 2k\)) mode remained elusive. Here we report the 2 ZO (\(q\approx 2k\)) mode which was found to show a negative frequency dispersion \(\partial\omega_{2 ZO(q\approx 2k)}/\partial E_{\mathrm{L}}=-48.1\) cm\({}^{-1}\)/eV [63]. The IL interactions in 2 LG rely on the interlayer hopping among equivalent and inequivalent carbon atoms and, therefore, will be directly related to the phonon self-energy and to the e-ph coupling regarding the ZO, ZO\({}^{\prime}\) and ZA modes [3, 40]. It is, however, important to note that the LO mode is dependent on the intralayer hopping between two inequivalent carbon atoms, and the LO mode remains essentially unchanged when changing the IL interactions [14, 19, 20, 21, 22, 23, 24, 25]. As depicted in the inset of Figure 15c, the LOZO\({}^{\prime}\) combination mode relates to a \(q\approx 2k\) AV process. Note that, Figure 15a,b show, respectively, the \(E_{\rm F}\) dependence of \(\omega_{\rm LOZO^{\prime}(+)}\) and \(\omega_{\rm LOZO^{\prime}(-)}\) when \(V_{\rm g}\) is varied. Both, \(\omega_{\rm LOZO^{\prime}(+)}\) and \(\omega_{\rm LOZO^{\prime}(-)}\) soften with increasing \(|E_{\rm F}|\). Correspondingly, as shown in the insets of Figure 15a,b, the phonon line widths \(\gamma_{\rm LOZO^{\prime}(+)}\) and \(\gamma_{\rm LOZO^{\prime}(-)}\) broaden with increasing \(|E_{\rm F}|\). Analogously, Figure16a shows that \(\omega_{\rm LOZA}\) (\(\gamma_{\rm LOZA}\)) softens (broadens) with increasing \(|E_{\rm F}|\), while for the 2 ZO overtone, a negligible dependence on \(V_{\rm g}\) is observed for both \(\omega_{\rm 2 ZO}\) and \(\gamma_{\rm 2 ZO}\). This behavior is opposite to what happens to the \(q=0\) phonons at the \(\Gamma\)-point [60], as is the case of the G-band feature. In order to quantify the ZO\({}^{\prime}\) phonon self-energy corrections, we measured the \(V_{\rm g}\) dependence of \(\omega_{\rm 2 LO}\) and \(\gamma_{\rm 2 LO}\) for the 2 LO overtone (see Figure 15d), which is known as the 2D\({}^{\prime}\) band around 3244  cm\({}^{-1}\). In analogy to the LOZO\({}^{\prime}\) combination mode about the \(\Gamma\)-point, the 2 LO overtone is a \(q\approx 2k\) AV double resonant process and is a fruitful choice for unraveling the two-phonon self-energy contributions that are merged in the LOZO\({}^{\prime}\) combination mode. The Raman scattering process involving overtone or combinations of phonon modes will conserve both the frequency and momentum, so that in our case \(\omega_{\rm s}=\omega_{\rm L}-(\omega_{\rm LO}+\omega_{\rm j})\) and \(\mathbf{k}_{\rm s}=\mathbf{k}_{\rm L}-(\mathbf{q}_{\rm LO}+\mathbf{q}_{\rm j})\), where \(\omega_{\rm s}(\mathbf{k}_{\rm s})\) is the frequency (momentum) of the stokes scattered light, \(\omega_{\rm L}(\mathbf{k}_{\rm L})\) is the frequency (momentum) of the incident light, \(\omega_{\rm LO}(\mathbf{q}_{\rm LO})\) is the frequency (momentum) of the LO phonon mode and \(\omega_{\rm j}(\mathbf{q}_{\rm j})\) is the frequency (momentum) of the ZO, ZO\({}^{\prime}\) or ZA phonon modes. Since the electron is vertically excited from the valence to the conduction band by the absorption of a photon, we have \(\mathbf{k}_{\rm s}=\mathbf{k}_{\rm L}\) and, therefore, the phonon momenta will be such that \(\mathbf{q}_{\rm LO}\approx\mathbf{q}_{\rm j}\). This scattering process is understood as follows: the electron is first scattered by one of the phonons, let us say the LO phonon, and then the electron is scattered again by a second phonon (LO, ZO, ZO\({}^{\prime}\) or ZA).