## 3 Electron-Phonon Interactions: Theoretical Background
### Consequences of the Kohn Anomaly in Graphene
The inset in **(g)** shows illustrative predictions for the \(V_{\rm g}\)-dependence of the phonon frequency correction \(\omega_{\bf q}-\omega_{\bf q}^{0}\) (black solid line) and the corresponding decay width \(\gamma_{\bf q}\) (grey dashed line) when \(q\
eq 0\), both as a function of \(E_{\rm F}\). The \(\omega_{\bf q}-\omega_{\bf q}^{0}\) and \(\gamma_{\bf q}\) values in **(g)** were normalized to illustrate the concept of \(\omega_{\bf q}\) softening and \(\gamma_{\bf q}\) broadening [60]. Starting with the case \(q=0\), the insets of Figure 10e,f, give the results illustrated for the renormalization of the phonon frequencies (\(\omega_{\mathbf{q}}\)) and linewidth (\(\gamma_{\mathbf{q}}\)), respectively. From these insets, we observe that when \(2|E_{\mathrm{F}}|<\hbar\omega_{\mathbf{q}}\), real e-h pairs can be created (annihilated), which leads to a stronger electron-ion interaction screening. As a consequence, the phonon mode softens [60, 61, 62, 63, 64, 65, 66]. However, when \(2|E_{\mathrm{F}}|>\hbar\omega_{\mathbf{q}}\) the production of real e-h pairs becomes forbidden due to the Pauli principle. This leads to a phonon mode hardening where the phonons are not damped any more (they are now long lived) [60, 61, 62, 63, 64, 65, 66]. If \(2|E_{\mathrm{F}}|=\hbar\omega_{\mathbf{q}}\), the phonon mode softening shows its highest values, which represent two singularities, as shown by the black solid curve in the inset of Figure 10e. Due to a broadening (comparable to the phonon energy) in \(|E_{\mathrm{F}}|\) due to thermal excitations (relaxations) and non-uniformity in the density of carriers (due to foreign chemical species and charge traps in the substrate), these singularities are usually observed at low temperatures (77 K or less) [65, 66]. As an example of phonon renormalization when \(q=0\) for the AV process (see Figure 10a), the \(\omega_{G}\) and \(\gamma_{G}\) variations of the G band Raman feature are shown in Figure 10e,f, respectively, as \(|E_{\mathrm{F}}|\) is varied due to different \(V_{\mathrm{g}}\) values. The experimental results (Figure 10e,f) are in good agreement with theory [60, 61, 62, 63, 64, 65, 66], which shows a \(\omega_{G}\) hardening and \(\gamma_{G}\) narrowing when \(V_{\mathrm{g}}\) increases. eq 0\) AV/EV processes, in contrast to the observed for the G band (\(q=0\)), by observing Figure 10g, which brings the G\({}^{\prime}\) band as an example, it is seen that \(\omega_{q}\) is softened, and that the \(\omega_{q}\) softening is accompanied by a broadening of \(\gamma_{q}\) with increasing \(|E_{\mathrm{F}}|\) (increasing \(|V_{\mathrm{g}}|\)). Now, instead of the Pauli exclusion principle, the electronic and vibrational density of states (together with energy and momentum conservation requirements) are determining if the e-h pair creation (annihilation) due to a \(q\
eq 0\) phonon absorption (emission) will happen or not. In a previous work [60], the authors explained that the behavior observed is common to Raman modes that come from an AV or EV process with \(q\
eq 0\) and is opposite to the behavior observed for the \(\Gamma\) point \(q=0\) phonons, such as for \(\omega_{\mathrm{G}}\). Namely, for the \(q\
eq 0\) cases, when the Fermi level is around zero (see top diagrams in Figure 10b-d), either no e-h pair can be created by the absorption of a phonon (for an AV process) or the probability of an e-h pair creation is very small (for an AV process) because the density of states vanishes at the Dirac point. However, when \(|E_{\mathrm{F}}|\) increases, the renormalization of the phonon energy occurs by the creation of an e-h pair through the absorption of a phonon (bottom diagrams in Figure 10b through Figure 10d), then a softening of \(\omega_{q}\) and a broadening of \(\gamma_{q}\) are observed, in contrast to what happens for \(q=0\) phonons. The inset in Figure 10g brings an illustrative prediction for the \(q\
eq 0\) phonon renormalization case.