## 2 Scattering Rate
### Inelastic Scattering  
Using [PERSON]'s equations and appropriate boundary conditions, a surface optical phonon (SP) mode can be solved for a planar interface between two semi-infinite dielectrics with one dielectric characterized by a transverse optical (TO) frequency [30]. This SP mode has a frequency slightly higher than the associated TO mode; it can efficiently couple with electrons of graphene if the graphene layer is close enough to the polar substrate [31; 32]. The SP scattering rate \(\tau_{\text{sp}}^{-1}(E)\) contributed by various SP modes \(\
u\) with the phonon energy \(\hbar\omega_{\
u}\) is approximately given by [33]  
\[\tau_{\text{sp}}^{-1}\left(E\right)=\sum_{\
u}\tau_{\text{sp,\
u}}^{-1}\left( E\right)=\frac{e^{2}}{4\hbar^{2}v_{\text{{}_{\text{F}}}}^{2}}\sum_{\
u,\pm} \frac{F_{\
u}^{\ 2}\omega_{\
u}N_{\text{{}_{\text{F}}}}^{+}e^{-2\phi}}{q(1+q_{\text{{}_{\text{s }}}}\left/\right.q)^{2}}\frac{1-f_{0}(E\mp\hbar\omega_{\
u})}{1-f_{0}\left(E \right)}\left|\frac{E}{\hbar\omega_{\
u}}\mp 1\right| \tag{4}\]  
where the scattering wave number is \(q=\sqrt{\omega_{\
u}^{\ 2}/v_{\text{{}_{\text{F}}}}^{2}-2E\left(\pm\hbar \omega_{\
u}-E\right)/\hbar^{2}v_{\text{{}_{\text{F}}}}^{2}}\), with \(\pm\) standing for the phonon emission (plus sign) and absorption (minus sign) in the scattering process; \(e\) is the electronic charge; \(N_{\
u}^{\pm}=\left(\exp\left(\hbar\omega_{\
u}\ /\ k_{\text{{}_{\text{B}}}}T\right)-1 \right)^{-1}+0.5\pm 0.5\) is the equilibrium phonon occupation number of the surface phonon mode \(\
u\); \(d\) is the spacing between the graphene layer and the substrate; \(f_{0}\left(E\right)=\left\{\exp\left[\left(E-\mu\right)/\ k_{\text{{}_{\text{B} }}}T\right]+1\right\}^{-1}\) is the Fermi-Dirac function, with \(\mu\) being the chemical potential; \(F_{\
u}\) is the electron-phonon coupling parameter given by [31; 34]  
\[F_{\
u}^{\ 2}=\frac{\hbar\omega_{\
u}}{2}{\left(\frac{1}{\varepsilon_{\text{ high}}+\varepsilon_{0}}-\frac{1}{\varepsilon_{\text{low}}+\varepsilon_{0}}\right)} \tag{5}\]where \(\varepsilon_{\mathrm{high}}\) and \(\varepsilon_{\mathrm{low}}\) are the high-frequency and low-frequency dielectric permittivities of the polar substrate, respectively, and \(\varepsilon_{0}\) is the permittivity of free space. Equation (4) is a good approximation of \(\tau_{\mathrm{sp}}^{-1}(E)\) if the phonon energy \(\hbar\omega_{v}\) is small. For large phonon energies, each term in the summation of Equation (4) has to be multiplied by a correction factor \(c_{v}=1+0.0027\hbar\omega_{v}\), where \(\hbar\omega_{v}\) is in the unit of meV [33]. Besides the aforementioned extrinsic inelastic SP scattering, there is also intrinsic inelastic scattering, _i.e._, optical phonons scattering, in graphene. Among various optical phonon modes, the ZO mode at the Brillouin zone center (\(\Gamma-\mathrm{ZO}\)) has the lowest phonon energy (110 meV) and therefore the highest phonon occupation number [35]. Because the \(\Gamma-\mathrm{ZO}\) mode is out-of-plane vibrations, its coupling with in-plane conduction electrons is weak. By comparison, degenerated \(\Gamma-\mathrm{LO}\) and \(\Gamma-\mathrm{TO}\) modes can efficiently couple with conduction electrons [36], but at room temperature their contributions to the scattering rate is limited because these modes are energetic (200 meV). The TO mode at the zone boundary (K - TO ) has a lower energy around 160 meV; it is therefore suggested to have the highest efficiency in coupling with electrons [37]. However, as we previously showed theoretically [33], the SP modes have a higher coupling efficiency than the K - TO mode, which is also confirmed experimentally [22]. Therefore, in this paper, we only consider the surface optical phonon scattering as the dominant inelastic scattering mechanism. The K - TO phonon scattering becomes important when graphene is deposited on a substrate such as SiC or \(h\)-BN where the surface optical phonon scattering is inefficient.