## 3 Electron-Phonon Interactions: Theoretical Background
### Consequences of the Kohn Anomaly in Graphene  
Hereafter, this review discusses phonons in 2 LG that occur either, around the \(\Gamma\) point or around the K point. As a reminder, there are two types of e-ph interactions, namely intra-valley (AV) (Figure 10a,c) and inter-valley (EV) (Figure 10b,d) processes [60, 61, 62, 63, 64]. For an AV process, the initial and final states both occur within the region close to a K [K\({}^{\prime}\)] point, while for inter-valley processes, K is connected to K\({}^{\prime}\) in a different valley by a \(q\
eq 0\) phonon. Thus the AV (EV) process corresponds to \(\Gamma\) (K) point phonons. The phonon wavevector \(\mathbf{q}\) for AV (EV) processes are measured from the \(\mathbf{\Gamma}\) (K) points with both \(q=0\) and \(q\
eq 0\) possible [60, 61, 62, 63, 64]. The phonon self-energy renormalization will happen any time the matrix element \(M_{\mathbf{k},\mathbf{k}^{\prime}}\) is different of zero. Gate-modulated Raman and electrochemistry Raman are the most used techniques to observe such renormalization effects. In gate-modulated Raman, the 2 LG flake is deposited over a silicon (Si) substrate with a 300 nm thick layer of SiO\({}_{2}\). Next, lithography followed by liftoff is required to place contacts on the flake so that gate voltages (\(V_{\text{g}}\)) can be applied to the device in a backgate configuration, as illustrated in Figure 11a. Then, for each \(V_{\text{g}}\) value, Raman spectrum is taken and analyzed using Lorentzian line-shapes from which frequencies and decay widths are extracted [60, 61, 62, 63, 64]. Note that resonance Raman spectroscopy provides information about both the electronic and vibrational structures, while the \(V_{\text{g}}\) variation allows for control of the Fermi level energy (\(E_{\text{F}}\)) of the 2 LG. In electrochemistry Raman, the device preparation is done in the very same way as described above. However, instead of applying \(V_{\text{g}}\) in a backgate configuration, an electrochemical polymer is utilized to topgating the devices. Then, instead of varying \(V_{\text{g}}\), the electrode potential is varied (see Figure 11b). Likewise \(V_{\text{g}}\), the electrode potential also allows for control of \(E_{\text{F}}\). Figure 9: The two possible vector (**a**) \(q=\Gamma\) and (**b**) \(q\) = K that can connect two electronic states in the Fermi surface (red dots) in graphene. Figure 11: **(a)** Schematic view of the backgate graphene device. A variable voltage \(V_{\rm g}\) can be applied between the graphene and the silicon substrate. The device is treated as a parallel plate capacitor where the silicon oxide layer is the dielectric medium; **(b)** Schematic view of topgate graphene devices using electrochemical polymers. Figure 10: **(a)** Possible (\(E_{\rm F}=0\)) and not possible (\(E_{\rm F}\
eq 0\)) AV \(q=0\) processes for e-h pair creation (annihilation) due to phonon (with energy \(\hbar\omega_{\bf q}\)) absorption (emission); **(b)** possible (\(E_{\rm F}=0\) and \(E_{\rm F}\
eq 0\)) \(q=0\) (measured from the \(\bf K\) point) EV processes; **(c)** not possible (\(E_{\rm F}=0\)) and possible (\(E_{\rm F}\
eq 0\)) AV processes and **(d)** possible EV processes for electron-hole pair creation (annihilation) due to phonon (with energy \(\hbar\omega_{\bf q}\)) absorption (emission) when the phonon wave-vector is not zero (\(q\
eq 0\)). \(E^{eh}\) is the e-h pair energy and \(E_{\bf K\to K^{\prime}}\) is the energy required to translate an electron from K to K\({}^{\prime}\); **(e)** The frequency \(\omega_{G}\) hardening and **(f)** decay width \(\gamma_{G}\) narrowing for the G band Raman feature as a function of gate voltage \(V_{\rm g}\). The insets in **(e)** and **(f)** are theoretical predictions of the \(E_{\rm F}\) dependence of \(\hbar\omega_{\bf q}-\hbar\omega_{\bf q}^{0}\) and \(\gamma_{\bf q}\) for an AV \(q=0\) process; **(g)** The \(V_{\rm g}\) dependence of the 2 iTO (\(q=2k\)) \(\omega_{\rm G^{\prime}}\) and \(\gamma_{\rm G^{\prime}}\).