## 5 Intra- and Inter-Layer Phonon Energy Renormalizations
### Interlayer-Related Phonons in 2 LG
On top of this, the renormalization will happen independently for each phonon, so that, as a first approximation for a combination mode or overtone, the renormalization will be \(\Pi(\omega_{\rm LO+j},E_{\rm F})=\Pi(\omega_{\rm LO},E_{\rm F})+\Pi(\omega_{ \rm j},E_{\rm F})\), where j can be a LO, ZO, ZO\({}^{\prime}\) or ZA phonon mode. Moreover, the IL interactions governing the ZO, ZO\({}^{\prime}\) and ZA modes will not change the intralayer dependent LO mode [14, 19, 20, 21]. Therefore, what we are observing in the gate-modulated Raman experiment for overtone and phonon combination modes is, indeed, the summation of the individual phonon self-energy corrections of each phonon participating of the scattering process [1]. The phonon self-energy renormalization strengths will be quantified by the difference between the frequencies \(\omega_{\mathbf{q}}^{0}\) at \(V_{\rm g}=0\) and \(\omega_{\mathbf{q}}\) for \(V_{\rm g}\
eq 0\). As explained above, \(\omega_{\mathbf{q}}^{0}\) at \(V_{\rm g}=0\) describes the system with no renormalizations that are associated with e-h pair formation. By inspecting Figure 15a,b, we find that \(\Delta\omega_{\rm LOZO^{\prime}(+)}=9\) cm\({}^{-1}\) and \(\Delta\omega_{\rm LOZO^{\prime}(-)}=7\) cm\({}^{-1}\), respectively. On the other hand, by looking at Figure 15d we see that \(\Delta\omega_{\rm 2 LO}=5\) cm\({}^{-1}\), which means that the LO frequency renormalization for this AV process is \(\Delta\omega_{\rm LO}\approx 2.5\) cm\({}^{-1}\). The self-energy corrections regarding the LO mode will be the same for the LO contribution for both the LOZO\({}^{\prime}\)(+) and LOZO\({}^{\prime}\)(\(-\)) features. Therefore the phonon self-energy correction \(\Delta\omega_{\rm ZO^{\prime(+)}}\) for the ZO\({}^{\prime}\)(+) mode will be given by \(\Delta\omega_{\rm LOZO^{\prime}(+)}-\Delta\omega_{\rm LO}=6.5\) cm\({}^{-1}\) while the phonon self-energy correction \(\Delta\omega_{\rm ZO^{\prime}(-)}\) for the ZO\({}^{\prime}\)(\(-\)) mode will be given by \(\Delta\omega_{\rm LOZO^{\prime}(-)}-\Delta\omega_{\rm LO}=4.5\) cm\({}^{-1}\). From the analysis above and remembering that, the larger the self-energy corrections \(\Delta\omega_{\mathbf{q}}\), the stronger are the e-ph couplings, we deduce that the IL e-ph coupling mediating the renormalizations for the ZO\({}^{\prime}\) mode is stronger than the renormalization for the LO mode. It is noteworthy that the self-energy renormalizations for LOZO\({}^{\prime}\)(+) and LOZO\({}^{\prime}\)(\(-\)) are different even though they involve the same LO phonon. We understand these differences as follows: the phonon self-energy corrections for \(q\
eq 0\) phonons rely on the density of electron and phonon states [40]. The density of phonons states will be the same because the same phonon is involved. However, as shown in Figure 15e, for energies smaller than \(|2|\) eV, the density of electronic states for \(\pi_{2}\)(\(\pi_{2}^{*}\)) is always smaller than that for \(\pi_{1}\)(\(\pi_{1}^{*}\)). This means that the phonon self-energy corrections are weaker for the ZO\({}^{\prime}\)(\(-\)) in comparison to that for the ZO\({}^{\prime}\)(+) not due to a different e-ph coupling symmetry, but because the density of electronic states for \(\pi_{2}\)(\(\pi_{2}^{*}\)) is smaller in comparison to that for \(\pi_{1}\)(\(\pi_{1}^{*}\)). By following the same strategy, we could also estimate the phonon self-energy corrections for the ZA mode, whose LOZA combination mode frequency (linewidth) also hardens (broadens) as expected for \(q\
eq 0\) AV processes. As shown in Figure 16a, \(\Delta\omega_{\mathrm{LOZA}}=8\)  cm\({}^{-1}\). Therefore, the ZA mode self-energy corrections \(\Delta\omega_{\mathrm{ZA}}=5.5\)  cm\({}^{-1}\). It is interesting to note that, the renormalization for the ZA mode is similar to that ruling the ZO\({}^{\prime}\) mode [63].