Relaxation of optically excited carriers in graphene

We explore the relaxation of photoexcited graphene by solving a transient Boltzmann transport equation with electron-phonon (e-ph) and electron-electron (e-e) scattering. Simulations show that when the excited carriers are relaxed by e-ph scattering only, a population inversion can be achieved at energies determined by the photon energy. However, e-e scattering quickly thermalizes the carrier energy distributions washing out the negative optical conductivity peaks. The relaxation rates and carrier multiplication effects are presented as a function of photon energy, graphene doping, and dielectric constant.

Figure 1

Simulation results for optically pumped graphene under steady state with e-ph scattering only. (a) $f_{+1}$ vs $E$ and (b) absorbance for $E_F$ $=$ 0 eV, $E_{\mathrm{opt}}$ $=$ 0.4, 0.6, and 0.8 eV, and $P_{\mathrm{opt}}$ $=$ 20, 30, and 40 $\mathrm{kW}{\mathrm{cm}}^{-2}$. (c) $f_{+1}$ vs $E$ and (d) absorbance for $E_F$ $=$ 0.05 and 0.1 eV, $E_{\mathrm{opt}}$ $=$ 0.8 eV, and $P_{\mathrm{opt}}$ $=$ 40 $\mathrm{kW}{\mathrm{cm}}^{-2}$. Population inversion is achieved and modulated by $E_{\mathrm{opt}}$.

Figure 2

Simulation results for optically pumped graphene under steady state with e-e scattering considered. (a) $f_{+1}$ vs $E$ for $E_F$ $=$ 0 eV, $\kappa$ $=$ 2.5, $E_{\mathrm{opt}}$ $=$ 0.4, 0.6, and 0.8 eV, and $P_{\mathrm{opt}}$ $=$ 20, 30, and 40 $\mathrm{kW}{\mathrm{cm}}^{-2}$. (b) $f_{+1}$ vs $E$ and (c) absorbance with increased pumping powers and $\kappa$ $=$ 10. (d) $f_{+1}$ vs $E$ for $E_F$ $=$ 0.05 and 0.1 eV, $\kappa$ $=$ 2.5, $E_{\mathrm{opt}}$ $=$ 0.8 eV, and $P_{\mathrm{opt}}$ $=$ 40 $\mathrm{kW}{\mathrm{cm}}^{-2}$. (e) $f_{+1}$ vs $E$ and (f) absorbance with increased pumping powers and $\kappa$ $=$ 10. Distribution functions are thermalized with elevated electron temperatures, and population inversion is not achieved.

Figure 3

Simulation results for ${\tau }^{-1}$ vs $E$ (conduction band, $\kappa$ $=$ 10) under equilibrium with (a) $E_F$ $=$ 0.1 eV and (b) $E_F$ $=$ 0.05 eV. Coulomb scattering rates are much larger than e-ph scattering rates and increase with decreasing $E_F$.

Figure 4

Simulation results for $\Delta n$ vs $t$ with optical pumping ($E_{\mathrm{opt}}$ $=$ 0.4, 0.6, and 0.8 eV, $P_{\mathrm{opt}}$ $=$ 20, 30, and 40 $\mathrm{kW}{\mathrm{cm}}^{-2}$) for $E_F$ $=$ 0 eV with (a) e-ph scattering only and (b) e-e scattering considered with $\kappa$ $=$ 2.5. (c) Relaxation time vs $E_{\mathrm{opt}}$ and (d) absorption efficiency vs $E_{\mathrm{opt}}$ for various $\kappa$ values for the same optical pumping conditions.