## 4 Inversion Symmetry in 2 LG Systems
exhibit a logarithmic singularity at \(E_{\rm F}=\hbar\omega/2\), while a narrowing of the line width is observed. The renormalization for this phonon mode is analogous to the case of 1 LG [30]. For the OP phonons, with the increase of \(E_{\rm F}\), the intraband transitions contributes now to the phonon self-energy, and the frequency of this mode gradually decreases, and a small broadening of the line width can be observed. Special care must be taken when analyzing the results for 2 LG e-ph coupling. The results showed in Figure 13a,b are valid only if you change the Fermi level position but keep the inversion symmetry between the two layers. When there is an asymmetry between the two layers (for example, an electric field perpendicular to the layers induced by the application of an external gate voltage), the inversion symmetry of bilayer graphene is broken, lowering the symmetry of the system, that now belongs to the C\({}_{3v}\) point group [70], and opening an electronic band gap at the K point. As a consequence, the in-phase and the out-of-phase lattice vibrations are no longer eigenstates os the system, but the resulting eigenstates can be regarded as superpositions of the IP and OP displacements [71,72]. Since the Raman active mode (in-phase) is now present in both resulting modes, there will be two peaks in the Raman spectrum: one with lower frequency G\({}^{-}\) and another with higher frequency G\({}^{+}\). The Raman spectrum can be quantitatively analyzed using a simple coupled-mode description [72]  
\[\left| \begin{array}{cc}E-E_{\rm IP}&g\\ g&E-E_{\rm OP}\end{array} \right|=0 \tag{20}\]  
where \(E_{\rm IP}=\hbar\omega_{\rm IP}-i\gamma_{\rm IP}\), \(E_{\rm OP}=\hbar\omega_{\rm OP}-i\gamma_{\rm OP}\) (\(\gamma\) is the line width), and \(g\) is the coupling between the IP-OP modes. Solutions to Equation (20) are given by [72]  
\[E^{\pm}=\frac{E_{\rm IP}+E_{\rm OP}}{2}\pm\sqrt{\left(\frac{E_{\rm IP}-E_{\rm OP }}{2}\right)^{2}+g^{2}} \tag{21}\]  
so that the real and imaginary parts of \(E^{\pm}\), respectively, describe the energy and broadening of the G\({}^{+}\) and G\({}^{-}\) modes. The behavior of the relative intensities, the width broadening and the frequency  
Figure 13: Calculated frequency shift (full lines) and broadening (dashed lines) for the (**a**) in-phase and (**b**) out-of-phase lattice vibrations of bilayer graphene as a function of the Fermi energy for two different values of crystal disorder \(\delta\)[69]. shift of the G\({}^{+}\) and G\({}^{-}\) peaks as a function of the charge concentration and Fermi energy is shown in Figure 14a-c, respectively [72]. The peak intensity is determined by the size of \(E_{\rm IP}\) content within each mode. At \(E_{\rm F}=\pm 200\) meV, the two peaks in the Raman spectrum have the same intensity as shown in Figure 14a because the coupling partitions of the Raman active E\({}_{g}\) mode is equally distributed to the G\({}^{+}\) and G\({}^{-}\) peaks. Away from \(E_{\rm F}=\pm 200\) meV, the relative intensities of the G\({}^{+}\) and G\({}^{-}\) reverse, reflecting the fact that G\({}^{-}\) (G\({}^{+}\)) is dominated by the IP vibration at low (high) charge concentration. It is important to note that the \({}^{12/12}\)C AB-2 LG systems, without any electric field applied to them, have the inversion symmetry present naturally for both electrons and phonons. However, the \({}^{12/13}\)C AB-2 LG systems have the inversion symmetry only for electrons. Phonons do not have the inversion symmetry because the atoms in the top layer have different masses from those in the bottom layer. In other words, the unit cell in \({}^{12/13}\)C AB-2 LG systems do not have inversion symmetry for phonons and what we naturally see in the G band of such systems (see Figure 3f) are two peaks active in Raman resulting from the combination between the IP and OP phonons (\(G^{+}\) and \(G^{-}\)).