Relaxation of optically excited carriers in graphene: Anomalous diffusion and Lévy flights

We present a theoretical analysis of the relaxation cascade of a photoexcited electron in graphene in the presence of screened electron-electron interaction in the random phase approximation. We calculate the relaxation rate of high energy electrons and the jump-size distribution of the random walk constituting the cascade which exhibits fat tails. We find that the statistics of the entire cascade are described by Lévy flights with constant drift instead of standard drift diffusion in energy space. The Lévy flight manifests nontrivial scaling relations of the fluctuations in the cascade time, which is related to the problem of the first passage time of Lévy processes. Furthermore we determine the transient differential transmission of graphene after an excitation by a laser pulse taking into account the fractional kinetics of the relaxation dynamics.

Figure 1

The kernel $\mathcal{K}(\omega ,q)$, Eqs. (9) and (C1), determining the phase space of scattering for thermal electrons for different frequencies $\omega$ and (a) $\mu =0$, (b) $\mu /T=10$. The regions of intraband ($q>\left|\omega \right|$) and interband ($q<\left|\omega \right|$) scattering are separated by the dashed line.

Figure 2

The jump-size distribution (7) for $T\gg \left|\mu \right|$. The inset (b) shows the contributions of $q>2T$ (dashed line) and $q<2T$ (dash-dotted line) to $P\left(\omega \right)$ (solid line) for $\left|\omega \right|<2T$. Both curves are calculated for ${\alpha }_g=0.75$.

Figure 3

The JSD (7) for $\mu /T=10$. In the region $\left|\omega \right|<2\left|\mu \right|$ processes with $q<2\left|\mu \right|$ are dominant. For $\left|\omega \right|>2\left|\mu \right|$ processes with $q>2\left|\mu \right|$ determine the fat tail of the JSD. The inset (b) illustrates the evolution of the forward scattering resonance with lowering temperature. Both curves are calculated for ${\alpha }_g=0.75$. For details of the calculation, see Appendix pp1.

Figure 4

(a) Sample of the JSD for $T\gg \left|\mu \right|$ (see Fig. 2). (b) Sample of the cascade variable $S_n={\omega }_1+\cdots +{\omega }_n$ from the JSD for $n=4$ with a high energy cutoff for the JSD given by the particle energy ${\varepsilon }_p/T=100$. The solid line is the stable distribution with $\alpha =3/2$ and $\beta =1$. (c) The average number of steps sampled from the JSD as a function of the cascade length $\Delta \varepsilon$. The error bars show the typical fluctuations ${\sigma }_n$ of the number of cascade steps. (d) The fluctuation ${\sigma }_n$ as a function of the cascade length $\Delta \varepsilon$. The solid line is the $\Delta {\varepsilon }^{2/3}$ law (19). The dashed line illustrates Gaussian fluctuations for comparison. The inset shows a typical distribution of cascade steps for $\Delta \varepsilon /2T=50$.

Figure 5

Pump-probe setup for (a) ${\omega }_{\mathrm{probe}}<{\omega }_0$. The probe measures mostly the secondary particles which are created with the probability $P\left(\omega \right)\propto {\omega }^{-5/2}$. (b) ${\omega }_{\mathrm{probe}}>{\omega }_0$. The density of secondary particles is negligible and the situation is suitable for studying the cascade time and its fluctuations depending on the length of the cascade $\Delta \varepsilon =({\omega }_{\mathrm{pump}}-{\omega }_{\mathrm{probe}})/2$. (c) The fluctuations (19) determine the width of the rise time in the measured change of the transmission [see also the inset of Fig. 4].

Figure 6

The solution [see Eqs. (28, 29, 30)] of the FFPE (21) (solid line) as a function of energy in comparison to the result obtained for Gaussian diffusion (dashed line) for different times.

Figure 7

The normalized differential transmission $\Delta T/\Delta T_{\max }$ as a function of the dimensionless time $tT$. Here $\Delta T_{\max }$ denotes the maximum value of $\Delta T$. (b) Shows the results on a logarithmic scale. The solid curves are calculated according to Eq. (34) and Eqs. (28) and (30), for $\Delta \varepsilon /T=({\omega }_{\mathrm{pump}}-{\omega }_{\mathrm{probe}})/T=25$ and $\Gamma \langle \omega \rangle /T^2=20$ as well as $D/T^{\alpha +1}$ from Eq. (23). The dashed lines in (a) and (b) illustrate the result for usual diffusion in comparison to the fractional kinetics (solid line).

Figure 8

(a) Auger process and (b) phase space for low energy electrons in the $p_x$-$p_y$ plane; (c) phase space for high energy electron.