Theory of the strongly nonlinear electrodynamic response of graphene: A hot electron model

An electrodynamic response of graphene to a strong electromagnetic radiation is considered. A hot electron model (HEM) is introduced and a corresponding system of nonlinear equations is formulated. Solutions of this system are found and discussed in detail for intrinsic and doped graphene: the hot electron temperature, nonequilibrium electron and hole densities, absorption coefficient, and other physical quantities are calculated as functions of the incident wave frequency $\omega$ and intensity $I$, of the equilibrium chemical potential ${\mu }_0$, and temperature $T_0$, scattering parameters, as well as of the ratio ${\tau }_{\epsilon }/{\tau }_{\mathrm{rec}}$ of the intraband energy relaxation time ${\tau }_{\epsilon }$ to the recombination time ${\tau }_{\mathrm{rec}}$. The influence of the radiation intensity on the absorption coefficient $A$ at low ($\hslash \omega \lesssim 2|{\mu }_0|, dA/dI>0$) and high ($\hslash \omega \gtrsim 2|{\mu }_0|, dA/dI<0$) frequencies is studied. The results are shown to be in good agreement with recent experimental data.

Figure 1

The time dependence of the normalized electron-hole pairs density $\mathrm{\Delta }\left(t\right)$, Eq. (32), at different values of ${\mathrm{\Delta }}_0$. The main plot and the inset show the same curves with linear and logarithmic scales of the $y$ axis, respectively. The decay of the electron-hole pairs density is exponential only at low excitation levels ${\mathrm{\Delta }}_0\ll 1$.

Figure 2

The relative change of the density $\mathrm{\Delta }=\delta n/(n_e^0+n_h^0)$ versus relative change of the temperature $(T-T_0)/T_0$ at different values of the input wave frequency $\hslash \omega$ (in eV). The ratio of the relaxation times is ${\tau }_{\epsilon }/{\tau }_{\mathrm{rec}}=0.1$, temperature $T_0=300\mathrm{K}$, and the energy $E_i=30\mathrm{meV}$. The equilibrium chemical potential is (a) ${\mu }_0=-0.2\mathrm{eV}$ and (b) ${\mu }_0=0\mathrm{eV}$. Arrows in (a) show the points where $I/I_0={10}^5$.

Figure 3

(a) The relative change of the density $\mathrm{\Delta }=\delta n/n_0=\delta n/(n_e^0+n_h^0)$ and temperature $(T-T_0)/T_0$, (b) the chemical potentials of electrons and holes ${\mu }_e$ and ${\mu }_h$, and (c) the absorption coefficient $A$, together with the intraband and interband contributions, as functions of the dimensionless power density $I/I_0$ at $\hslash \omega =0.6\mathrm{eV}, E_i=30\mathrm{meV}$, and ${\tau }_{\epsilon }/{\tau }_{\mathrm{rec}}=0.1$. The equilibrium chemical potential and temperature are ${\mu }_0=-0.2\mathrm{eV}$ and $T_0=300\mathrm{K}$. Left columns show the power dependencies at $I/I_0<{10}^4$, and the right columns the same dependencies in a broader power range. The horizontal axis scale is the same in all left and all right panels. The vertical axis scale is the same in both (c) panels.

Figure 4

(a) The relative change of the density $\mathrm{\Delta }=\delta n/n_0=\delta n/(n_e^0+n_h^0)$ and temperature $(T-T_0)/T_0$, (b) the chemical potentials of electrons and holes ${\mu }_e$ and ${\mu }_h$, and (c) the absorption coefficient $A$, together with the intraband and interband contributions, as functions of the dimensionless power density $I/I_0$ at $\hslash \omega =0.1\mathrm{eV}, E_i=30\mathrm{meV}$, and ${\tau }_{\epsilon }/{\tau }_{\mathrm{rec}}=0.1$. The equilibrium chemical potential and temperature are ${\mu }_0=-0.2\mathrm{eV}$ and $T_0=300\mathrm{K}$. Left columns show the power dependencies at $I/I_0<{10}^4$, and the right columns the same dependencies in a broader power range. The horizontal axis scale is the same in all left and all right panels. The vertical axis scale is the same in both (c) panels.

Figure 5

The electron distribution function (6) in the valence ($E<0$) and conduction ($E>0$) bands at different values of the radiation intensity. The radiation frequency is (a) $\hslash \omega =0.6\mathrm{eV}$ and (b) $\hslash \omega =0.1\mathrm{eV}$, other parameters are ${\mu }_0=-0.2\mathrm{eV}, T_0=300\mathrm{K}, E_i=30\mathrm{meV}$, and ${\tau }_{\epsilon }/{\tau }_{\mathrm{rec}}=0.1$.

Figure 6

(a) The relative change of the density $\mathrm{\Delta }=\delta n/n_0=\delta n/(n_e^0+n_h^0)$ and temperature $(T-T_0)/T_0$, (b) the chemical potentials of electrons and holes ${\mu }_e$ and ${\mu }_h$, and (c) the absorption coefficient $A$, together with the intraband and interband contributions, as functions of the dimensionless power density $I/I_0$ at $\hslash \omega =0.4\mathrm{eV}, E_i=30\mathrm{meV}$, and ${\tau }_{\epsilon }/{\tau }_{\mathrm{rec}}=0.1$. The equilibrium chemical potential and temperature are ${\mu }_0=0\mathrm{eV}$ (intrinsic graphene) and $T_0=300\mathrm{K}$. Left columns show the power dependencies at $I/I_0<{10}^4$, and the right columns the same dependencies in a broader power range. The horizontal axis scale is the same in all left and all right panels. The vertical axis scale is the same in left and right (b) and (c) panels.

Figure 7

The electron distribution function (6) in the valence ($E<0$) and conduction ($E>0$) bands of intrinsic graphene (${\mu }_0=0\mathrm{eV}$) at different values of the radiation intensity. Other parameters are $\hslash \omega =0.4\mathrm{eV}, T_0=300\mathrm{K}, E_i=30\mathrm{meV}$, and ${\tau }_{\epsilon }/{\tau }_{\mathrm{rec}}=0.1$.

Figure 8

The absorption coefficient vs frequency at different values of the radiation intensity, for (a) $|{\mu }_0|=0.2\mathrm{eV}$ and (b) ${\mu }_0=0\mathrm{eV}$. Other parameters are ${\tau }_{\epsilon }/{\tau }_{\mathrm{rec}}=0.1, E_i=30\mathrm{meV}$, and $T_0=300\mathrm{K}$.

Figure 9

Interband transitions in graphene (a) in equilibrium and under (b) low-frequency and (c) high-frequency powerful irradiation at a negative chemical potential ${\mu }_0<0$ and low temperature $T_0$. (a) In equilibrium, the high-frequency photons are absorbed (left arrow: electrons jump from occupied initial to the empty final states) and the low-frequency photons are not (right arrow: initial states are empty). (b) Under the low-frequency powerful irradiation, the electron gas gets heated due to the intraband absorption $T_0\to T$, the initial states in the valence band get partly occupied, and the interband absorption increases (induced absorption). (c) Under the high-frequency powerful irradiation, the final states in the conduction band get partly occupied and the interband absorption decreases (absorption saturation).

Figure 10

The hot electron temperature $T$ as a function of the photon energy. Four curves in the upper right corner are plotted for parameters corresponding to Fig. 8. The curves in the lower left corner are plotted for parameters of the experiment [48] (see discussion in the text). Arrows indicate the positions of the double chemical potential $2|{\mu }_0|$.

Figure 11

(a) The absorption coefficient $A$ as a function of the radiation frequency at ${\tau }_{\epsilon }/{\tau }_{\mathrm{rec}}=0.01, |{\mu }_0|=16\mathrm{meV}, T_0=10\mathrm{K}$, and (a) ${\tau }_p=300$ fs and (b) ${\tau }_p=100$ fs.

Figure 12

The Coulomb impurity energy $E_i$ (in meV) and the mobility $\mu$ as a function of the impurity density $N_i$. Graphene is assumed to lie on a ${\mathrm{SiO}}_2$ substrate so that $\kappa =2.45, \alpha =0.36$, and $\beta =9.61$.

Figure 13

Experimental data from Ref. [91] for two different potassium atom densities, corresponding to 0-s and 12-s doping times (see Fig. 2 in Ref. [91]). Symbols correspond to the experimental data, solid curves correspond to Eq. (A6) for parameters $\zeta$ and $N_i$ indicated on the plot. The density of charge carriers $n_s$ and of impurities $N_i$ are in units ${10}^{12}{\mathrm{cm}}^{-2}$, the conductivity is in units $e^2/h$.