## 5 Intra- and Inter-Layer Phonon Energy Renormalizations
### Isotopes
As reported by [PERSON] _et al._[10] a signature of the IP and OP mixing is that the intensity ratio (\(I_{\rm HG}\)/\(I_{\rm LG}\)) between the HG (\(G^{+}\) in Section 4) (an anti-symmetric combination of IP and OP) and the LG (\(G^{-}\) in Section 4) modes (a symmetric combination of IP and OP) must increase with increasing \(|E_{\rm F}|\). This signature is confirmed by the experimental results, as shown in the insets of Figure 18c,d. Based on [PERSON] _et al._[10] and noting that due to conventions related to electrochemistry experiments, positive (negative) electrode potentials means that the \(E_{\rm F}\) is decreasing (increasing), [PERSON] _et al._[75] concluded that \(n_{bottom}>n_{top}\). Based on this conclusion, the authors [75] discuss the existence of an off-set in the change of the LG (\(G^{-}\) in Section 4) and HG (\(G^{+}\) in Section 4) mode frequencies in their dependence on electrode potential, as observed for the \({}^{12/13}\)C AB-2 LG. This is, in particular, clear for the case of positive doping. As one can see from Figure 18b, there is a small increase in the frequency of the LG (or \(G^{-}\)) mode and almost no changes in the frequency of the HG (or \(G^{+}\)) mode up to an electrode potential of \(+1\) V. Then, for the potentials above \(+1\) V, the G mode frequencies are gradually increased and seem to saturate from the potential \(+1.3\) V on. This behavior is in contrast to 1 LG or t-2 LG (Figure 18a) where the frequency starts to change even at very low positive and negative applied potentials (\(\pm 0.1\) V). Besides the competition for the corrections in the phonon frequency between both phenomena, phonon self-energy renormalizations and changes in the chemical C-C bonds (excess of holes hardens the C-C bonds while excess of electrons softens the C-C bonds) [30], in AB-2 LG, one should also consider subsequent filling (emptying) of the second electronic band in the conduction band \(\pi_{2}\)* (of the second electronic band in the valence band \(\pi_{2}\)) at higher electrode potentials. This band filling (emptying) is important to decide which phenomena will dominate and rule the frequency changes. In particular, it has been shown that the filling of \(\pi_{2}\)* may lead to a kink in the dependence of the Raman frequency on electrode potential [80] and this observation is consistent with a sudden change of the slope of the frequency/potential dependence (\(\delta\omega_{\rm G}/\delta V\)) during hole doping (positive potentials) as seen in Figure 18b[80]. Indeed, when \(E_{\rm F}\) is at zero, real e-h pairs involving the \(\pi_{1}\) and \(\pi_{1}\)* bands are being created all time, which contributes to decreasing the phonon frequency. At the same time, virtual e-h pairs are being created between \(\pi_{1}\) and \(\pi_{2}\) bands (in the valence band, which contributes to an increase in frequency) while no e-h pairs are being created between \(\pi_{1}\)* and \(\pi_{2}\)* in the conduction band. Note that, because it is a resonant effect, e-h pairs involving the \(\pi_{1}\) and \(\pi_{1}\)* bands will dominate the virtual processes. Nothing is happening to the bonds since the system is not doped yet. When the system is doped with holes, in a first stage (electrochemical potential between 0 and 1 V), the real e-h pair formation involving the bands \(\pi_{1}\)* and \(\pi_{1}\) will be halted (since \(E_{\rm F}\) is decreasing) and the e-h pairs involving the bands \(\pi_{1}\) and \(\pi_{2}\) will change their status from virtual to real e-h pairs. Therefore the phonon self-energy renormalizations will decrease the frequency (broaden the linewidth), since the real e-h pair formation will favor the frequency changes, while the C-C bonds will harden hence increasing the frequency. On average, no (or a small) change in frequency is expected, in accordance with the observations.