Finite temperature inelastic mean free path and quasiparticle lifetime in graphene

We adopt the $GW$ and random phase approximations to study finite temperature effects on the inelastic mean free path and quasiparticle lifetime by directly calculating the imaginary part of the finite temperature self-energy induced by electron-electron interaction in extrinsic and intrinsic graphene. In particular, we provide the density-dependent leading order temperature correction to the inelastic scattering rate for both single-layer and double-layer graphene systems. We find that the inelastic mean free path is strongly influenced by finite-temperature effects. We present the similarity and the difference between graphene with linear chiral band dispersion and conventional two-dimensional electron systems with parabolic band dispersion. We also compare the calculated finite temperature inelastic scattering length with the elastic scattering length due to Coulomb disorder and comment on the prospects for quantum interference effects showing up in low-density graphene transport. We also carry out inelastic scattering calculation for electron-phonon interaction, which by itself gives rather long carrier mean free paths and lifetimes since the deformation potential coupling is weak in graphene, and therefore electron-phonon interaction contributes significantly to the inelastic scattering only at relatively high temperatures.

Figure 1

(a) Calculated ratio $\lambda$, which is the numerical result of $\text{Im}{\Sigma }_{+}^R({\mathbf{k}}_F,{\xi }_{{\mathbf{k}}_F})$ divided by the analytical asymptotic formula $-\frac{\pi {\left(k_{B}T\right)}^2}{8{\varepsilon }_F}[\ln \left(\frac{k_{B}T}{8{\varepsilon }_F}\right)+1.08387]$ [given in Eq. (10)], as a function of $T/T_F$. The inset presents $\lambda$ in the low-temperature regime. (b) Calculated ratio $\sigma$, the numerical result of $\text{Im}{\Sigma }_{+}^R(\mathbf{k},{\xi }_{\mathbf{k}})$ divided by $-\frac{{\xi }_k^2}{8\pi {\varepsilon }_F}[\ln \left(\frac{{\xi }_k}{8{\varepsilon }_F}\right)+\frac{1}{2}]$ [given in Eq. (8)] as a function of ${\xi }_{\mathbf{k}}/E_F$ at $T=0$. The inset presents $\sigma$ in the low-energy regime. The solid and dashed lines correspond to $\kappa =5$ ($r_s=0.44$) and $\kappa =1$ ($r_s=2.2$), respectively.

Figure 2

Calculated inelastic scattering length $l$ and the associated on-shell imaginary part of the self-energy $\text{Im}{\Sigma }_{+}^R(k,{\xi }_k)/E_F$ for extrinsic graphene (${\varepsilon }_F>0$) with dielectric constant $\kappa =5$ ($r_s=0.44$). (a) $l$ as a function of temperature $T$. The solid, dashed, and dot-dashed lines are for carrier density $n={10}^{10}$, ${10}^{11}$, and ${10}^{12}$ cm${}^{-2}$, respectively. The upper three lines are for ${\xi }_k={\xi }_{k_F}$, while the lower three lines are for ${\xi }_k=0.5{\epsilon }_F$. (b) $l$ as a function of ${\xi }_k/E_F$ for different temperatures. The solid and dashed lines are for carrier density $n={10}^{11}$ and ${10}^{10}$ cm${}^{-2}$, respectively. For the same carrier density, the lines from top to bottom correspond to $T=0$, 50, and 100 K. (c) and (d) Associated $\text{Im}{\Sigma }_{+}^R(k,{\xi }_{\mathbf{k}})/E_F$ corresponding to (a) and (b), respectively.

Figure 3

Calculated inelastic scattering length $l$ and the associated on-shell imaginary part of the self-energy $\text{Im}{\Sigma }_{+}^R(k,{\xi }_k)/E_F$ for extrinsic graphene (${\varepsilon }_F>0$) with dielectric constant $\kappa =1$ ($r_s=2.2$). (a) $l$ as a function of temperature $T$. The solid, dashed, and dot-dashed lines are for carrier density $n={10}^{10}$, ${10}^{11}$, and ${10}^{12}$ cm${}^{-2}$, respectively. The upper three lines are for ${\xi }_k={\xi }_{k_F}$, while the lower three lines are for ${\xi }_k=0.5{\epsilon }_F$. (b) $l$ as a function of ${\xi }_k/E_F$ for different temperatures. The solid and dashed lines are for carrier density $n={10}^{11}$ and ${10}^{10}$ cm${}^{-2}$, respectively. For the same carrier density, the lines from top to bottom correspond to $T=0$, 50, and 100 K. (c) and (d) Associated $\text{Im}{\Sigma }_{+}^R(k,{\xi }_{\mathbf{k}})/E_F$ corresponding to (a) and (b), respectively.

Figure 4

Calculated Im${\Sigma }^R(k,\omega )$ for extrinsic graphene (the chemical potential $\mu >0$) with $\kappa =5$. (a) and (b) correspond to conduction band, while (c) and (d) correspond to valence band. In (a), the solid, dashed and dot-dashed lines correspond to $\omega /E_F=0.5$, $1.0$ and $2.0$, respectively. (b) The solid, dashed, and dot-dashed lines correspond to $T/T_F=0$, $0.2$ and $0.4$, respectively. In (c) and (d), the solid and dashed lines correspond to $k=0$ and $k=1.5k_F$. (c) For the same momentum, the lines from bottom to top correspond to $\omega /E_F=0.5$, $1.0$ and $2.0$, respectively. (d) For the same momentum, the lines from bottom to top correspond to $T/T_F=0$, $0.2$, and $0.4$, respectively. Note that Im${\Sigma }_{+}^R(k=0,\omega )=\text{Im}{\Sigma }_{-}^R(k=0,\omega )$.

Figure 5

Calculated ratio $\zeta =l/l_e$, the inelastic mean free path divided by the elastic mean free path, as a function of temperature for carrier density $n={10}^9$ cm${}^{-2}$, $\kappa =5$, and charged impurity density $9\times {10}^{10}$ cm${}^{-2}$. The solid, dotted, dashed, and dot-dashed lines correspond to $s=1.0$, $5.0$, $15.0$, and $25.0$ meV, respectively, indicating increasing effects of Coulomb disorder induced inhomogeneous puddles. The horizontal line indicates where (above the line) quantum interference could play a role.

Figure 6

Calculated inelastic scattering length $l$ and the associated on-shell imaginary part of the self-energy $\text{Im}{\Sigma }_{+}^R(k,{\epsilon }_k)$ for intrinsic graphene (${\varepsilon }_F\equiv 0$) with dielectric constant $\kappa =5$. (a) $l$ as a function of temperature $T$ for different energy ${\varepsilon }_{\mathbf{k}}$. The solid, dotted, dashed, and dot-dashed lines are for energy ${\varepsilon }_{\mathbf{k}}=1$, 5, 10, and 20 meV. (b) $l$ as a function of energy $\omega$ for different temperatures. The solid, dashed, and dot-dashed lines are for temperature $T=10$, 50, and 100 K. (c) and (d) Associated $\text{Im}{\Sigma }_{+}^R(k,{\epsilon }_k)$ corresponding to (a) and (b), respectively.

Figure 7

Calculated imaginary part of the self-energy Im${\Sigma }^R(k,\omega )$ for intrinsic graphene (the chemical potential $\mu \equiv 0$) with $\kappa =5$. (a) and (b) Conduction band Im${\Sigma }_{+}^R(k=0,\omega )$ as a function of temperature $T$ and energy $\omega$, respectively. (c) and (d) Conduction band Im${\Sigma }_{+}^R(k,\omega )$ with $k=10$ meV$/\hslash v_F$ as a function of temperature $T$ and energy $\omega$, respectively. (e) and (f) Valence band Im${\Sigma }_{-}^R(k,\omega )$ with $k=10$ meV$/\hslash v_F$ as a function of temperature $T$ and energy $\omega$, respectively. The solid, dotted, dashed, and dot-dashed lines in (a), (c), and (e) are for energy $\omega =1$, 5, 10, and 20 meV. While the solid, dotted, dashed, and dot-dashed lines in (b), (d), and (f) are for temperature $T=0$, 10, 50, and 100 K. Note that $\text{Im}{\Sigma }_{+}^R(k=0,\omega )=\text{Im}{\Sigma }_{-}^R(k=0,\omega )$.

Figure 8

Calculated mean free path $l_{\mathrm{ep}}$ induced by e-p interaction. We use[37] $D=25$ eV, $v_l=2.6\times {10}^6$ cm/s, and ${\rho }_m=7.6\times {10}^{-8}$ g/cm${}^2$. (a) $l_{\mathrm{ep}}$ for ${\xi }_k={\xi }_{k_F}$ as a function of temperature $T$ for different carrier densities. (b) $l_{\mathrm{ep}}$ as a function of energy ${\xi }_k/E_F$ for different carrier densities and temperatures. (c) $l_{\mathrm{ep}}$ for ${\xi }_k={\xi }_{k_F}$ as a function of carrier density for different temperatures.