## 2 Scattering Rate  
### Elastic Scattering  
Many scattering mechanisms have been suggested for the explanation of the experimental observation [1; 20; 21; 22; 23; 24; 25]. Among all possible scattering channels, phonon scattering is the intrinsic scattering mechanism that serves as the lower bound of the scattering rate that fundamentally limits the mobility of carriers in graphene. Two types of phonon scattering mechanisms are considered in this paper for carrier transport in graphene: elastic acoustic phonon scattering and inelastic optical phonon scattering; the former is discussed in this section and the latter is discussed in Subsection 2.2. The scattering rate arising from the longitudinal acoustic phonon scattering is given by [26]  
\[\tau_{\rm ac}^{-1}\left(E\right)=\frac{1}{\hbar^{3}}\frac{k_{\rm B}T}{4v_{\rm F }}^{2}\frac{D_{\rm A}^{2}}{\rho v_{\rm ph}^{2}}\,E \tag{1}\]  
where \(E\) is the energy of charged carriers, \(\hbar\) is the reduced Planck constant, \(k_{\rm B}\) is the Boltzmann constant, \(T\) is the temperature, \(v_{\rm F}\) (\(\equiv\) 10\({}^{6}\) m/s ) is the Fermi velocity of carriers in graphene, \(D_{\rm A}\) is the acoustic deformation potential, \(\rho\) = 7.6\(\times\)10\({}^{-7}\) kg/m\({}^{2}\) is the density of graphene per layer, and \(v_{\rm ph}\) = 2\(\times\)10\({}^{4}\) m/s is the phonon velocity of the longitudinal acoustic mode [27]. In this paper, \(D_{\rm A}\) = 18 eV is assumed, as measured in the experiments [22; 25; 26; 28].  
The impurity scattering due to the charged impurity is also considered in the literature for carrier transport in graphene [20; 21; 23]. For a monolayer graphene sandwiched between two media of different permittivities \(\varepsilon_{1}\) and \(\varepsilon_{2}\), the average dielectric permittivity of such a system is \(\varepsilon_{\text{avg}}=\left(\varepsilon_{1}+\varepsilon_{2}\right)/2\). The elastic scattering rate arising from the charged impurity scattering can then be written as [29]  
\[\tau_{\text{imp}}^{-1}\left(E\right)=\frac{n_{\text{i}}}{4\pi\hbar}\frac{E}{ \left(\hbar v_{\text{{}_{\text{F}}}}\right)^{2}}\int\limits_{0}^{2\pi}d\theta \frac{\tilde{V}^{2}}{\left(1+q_{\text{{}_{\text{s}}}}\left/\right.q\right)^{2} }\left(1-\cos^{2}\theta\right) \tag{2}\]  
Where \(n_{\text{i}}\) is the impurity density, \(q\) is the scattering wave number, \(\tilde{V}=e^{2}\) / \(2\varepsilon_{\text{avg}}q\) is the Fourier transform of the 2D potential energy, and \(q_{\text{{}_{\text{s}}}}\) is the screening wave number. With \(q=2k\sin\left(\theta/2\right)\), where \(k\) is the wave number of carriers and \(\theta\) is the scattering angle, we obtain the analytical expression for the impurity scattering rate in the limit of \(k>>q_{\text{{}_{\text{s}}}}\):  
\[\tau_{\text{imp}}^{-1}\left(E\right)=\frac{n_{\text{i}}}{4\hbar}\frac{1}{E}{ \left(\frac{e^{2}}{2\varepsilon_{\text{avg}}}\right)^{2}} \tag{3}\]  
In fact, \(q_{\text{{}_{\text{s}}}}\) is a function of both temperature \(T\) and wave number \(q\). In this paper, we consider the case of a Fermi energy \(E_{\text{f}}>>k_{\text{B}}T\) and a wave number \(q\leq 2k_{\text{{}_{\text{F}}}}\). Within this limit, \(q_{\text{{}_{\text{s}}}}\) is approximately the Thomas-Fermi screening wave number given by \(e^{2}\rho(E_{\text{{}_{\text{F}}}})\) / \(2\varepsilon_{\text{avg}}\), where \(\rho(E_{\text{{}_{\text{F}}}})=2E_{\text{{}_{\text{F}}}}\) / \(\pi\left(\hbar v_{\text{{}_{\text{F}}}}\right)^{2}\) is the density of states at the Fermi energy [29].