## 5 Intra- and Inter-Layer Phonon Energy Renormalizations
### Isotopes
The diameter of the hexagon is about 10 \(\upmu\)m. Note that the center (small area delimited by the gray dotted lines) of the add-layer forms a \({}^{12/12}\)C AB-2 LG structure, while the remaining area is formed by \({}^{13/12}\)C AB-2 LG; **(b)** G\({}^{\prime}\) (2D) mode Full Width at Half Maximum (FWHM) identifying the \({}^{12}\)C 1 LG region (FWHM \(\sim 30\)  cm\({}^{-1}\)), the \({}^{12/12}\)C AB-2 LG region (FWHM \(\sim 45\)  cm\({}^{-1}\)) and the \({}^{12/13}\)C AB-2 LG region (FWHM \(\sim 75\)  cm\({}^{-1}\)) [75]. between the potential dependent behavior of the HG modes for the \({}^{12/13}\)C AB-2 LG and t-2 LG systems. However, the behavior of the LG mode for the \({}^{12/13}\)C AB-2 LG is completely different from both G mode lineshapes, that is, different from both the HG and LG modes of the \({}^{12/13}\)C t-2 LG. Namely, for the LG mode of the \({}^{12/13}\)C AB-2 LG, we observe small changes in the G mode frequency during negative doping, when compared to \({}^{12/13}\)C t-2 LG (and has an opposite direction). For positive electrochemical potentials, we observe a larger shift for the LG mode in AB-2 LG, as seen in Figure 17b, but this frequency change is still smaller than in the case of the LG mode in t-2 LG. In charged graphene, the frequency shift of the G mode, which is a first-order Raman mode around the \(\Gamma\)-point whose phonon momentum is null (\(q=0\)), is related to both, the change in the C-C bond strength and the renormalization of the phonon self-energy associated with many-body effects [80]. As discussed before, changes on the electrode potential lead to a renormalization of the phonon energy, the phonon lifetime and consequently the energy of the carriers, that is, for both electrons and holes [62, 83]. Observing Figure 18c,d, the different dependence on electrode potentials suggest that different mechanisms are ruling the doping of \({}^{12/13}\)C AB-2 LG in comparison to the \({}^{12/13}\)C t-2 LG at the same applied electrode potential. Indeed, the t-2 LG \(E_{\rm F}\) will depend on the charge concentration \(n\) as \(|E_{\rm F(1 LG)}|=\hbar|v_{\rm F}|(\pi n)^{1/2}\), since the weakly interacting layers composing the t-2 LG system keep the 1 LG properties, while \(E_{\rm F}\) for the AB-2 LG will obey \(|E_{\rm F(AB-2 LG)}|=[-\gamma_{1}+(4\pi\hbar^{2}v_{\rm F}^{2}n+\gamma_{1}^{2}) ^{1/2}]/2\), where \(\gamma_{1}=0.35\)  eV. By inspecting both, \(|E_{\rm F(1 LG)}|\) and \(|E_{\rm F(AB-2 LG)}|\) relations, we see that, for a given change  
Figure 18: (**a**) In-situ Raman spectro-electrochemistry of the \({}^{12/13}\)C t-2 LG G modes and (**b**) In-situ Raman spectro-electrochemistry of the \({}^{12/13}\)C AB-2 LG G modes. The spectra are excited by 2.33  eV laser excitation energy and the electrode potentials range from \(-1.5\) to \(1.5\)  V _vs._ Ag/Ag\({}^{+}\) (from bottom to top) in units of 0.1  V. The red dashed curves are guide to eyes. Note that, due to electrochemistry conventions, positive potentials are filling the system with holes while negative potential are filling the system with electrons; (**c**) and (**d**) Fitting analysis of the spectra showed in (**a**) and (**b**); (**c**) The frequency and linewidth behavior of the LG mode (left panel) and HG (right panel) in t-2 LG as a function of the electrode potential; (**d**) The frequency and linewidth behavior of the LG mode (left panel) and HG (right panel) in AB-2 LG as a function of the electrode potential. All insets present information on the intensity dependence with the electrode potential [75]. in \(n\), E\({}_{\rm F}\) for the t-2 LG changes more than the E\({}_{\rm F}\) for the AB-2 LG [65, 80]. Consequently, the frequency variations for the t-2 LG G modes are expected to be larger than the frequency variations for the AB-2 LG.