Current amplification and relaxation in Dirac systems

Recent experiments provide evidence for photocurrent generation in Dirac systems such as topological-insulator surface states and graphene. Within the simplest picture, the magnitude of the photocurrents is governed by the competition between photoexcitation of particle-hole pairs and current relaxation by scattering. Here, we study the relaxation of photocurrents by electron-electron $(e-e)$ collisions, which should dominate in clean systems. We compute the current relaxation rate as a function of the initial energies of the photoexcited carriers and the Fermi energy. For a positive Fermi energy, we find that collisions of a single excited electron with the Fermi sea can substantially increase the current, while for a single excited hole the current initially decreases. Together these processes partially cancel leading to a relative suppression of the relaxation of the total photocurrent carried by an electron-hole pair. We also analyze the limit of many scattering events and find that while $e-e$ collisions initially reduce the current associated with a single hole, the current eventually reverses sign and becomes as large in magnitude as in the electron case. Thus, for photoexcited electron-hole pairs, the current ultimately relaxes to zero. We discuss schemes which may allow one to probe the nontrivial current amplification physics for individual carriers in experiment.

Figure 1

Mean change of the current per electron scattering event $\langle \Delta j\rangle /j_0$ relative to the initial current $j_0$ vs ${\epsilon }_F$ for fixed initial energy ${\epsilon }_{{\mathit{k}}_1}\approx 0.15$ eV. The blue and red shaded areas indicate current amplification and relaxation, respectively. The rate of change of the current is normalized by the total scattering rate $\Gamma$. The results are obtained for realistic parameters for ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, including particle-hole asymmetry $\xi =23.7$ eV ${\text{Å}}^2,v_F=5\times {10}^5$ m/s [37], and $\alpha =0.1$ (see text for definitions), where we assumed an average dielectric constant of air and $B{i}_{2}S{e}_3$ of $\varepsilon \sim 50$ [38, 39]. (Inset) Schematic of the excitation of an electron-hole pair by the initial photoexcited electron.

Figure 2

The photoexcitation of an electron-hole pair within the Dirac cone and the relaxation process of the hot electron by excitation of an electron-hole pair, for (a) ${\epsilon }_F=0$ and (b) ${\epsilon }_F>0$. (c) A possible relaxation process of the excited hole.

Figure 3

Kinematic ellipse for allowed electron scattering processes for ${\epsilon }_i>0$. ${\mathit{k}}_1$ and ${\mathit{k}}_2$ are drawn head to tail starting and ending at the left and right focal points (red dots). ${\mathit{k}}_1^{\prime }$ and ${\mathit{k}}_2^{\prime }$ are drawn in a similar fashion and touch at points that lie on the ellipse. $(k_1+k_2)/2$ and $|{\mathit{k}}_1+{\mathit{k}}_2|$ are the lengths of the semimajor axis and the distance between focal points, respectively. The green dashed circle indicates the Fermi momentum. Because of Pauli's principle we have $k_2\le k_F<k_2^{\prime },k_1^{\prime }$ and the point where ${\mathit{k}}_1^{\prime }$ is connected to ${\mathit{k}}_2^{\prime }$ must lie on the ellipse outside the green dashed circles, while the point of connection of ${\mathit{k}}_1$ and ${\mathit{k}}_2$ has to lie inside the green dashed circle.

Figure 4

Rate of change of the total current (blue circles) [from Eq. (10)] and the asymptotic behavior $\sim {\left({\epsilon }_F/{\epsilon }_1\right)}^{3/2}$ given by Eq. (14) (red). (Inset) Rates of change of the electron (red squares), hole (green diamonds), and total (blue circles) currents. Parameters as in Fig. 1. Relaxation of the total current is strongly suppressed due to cancellations between the electron and hole contributions. Results for ${\epsilon }_F<0$ follow by electron-hole symmetry.

Figure 5

(a) Schematic illustration of the setup of a nanotube array on top of a graphene sheet. Using an array instead of a single nanotube might be advantageous in terms of gating, but, in principle, this effect also works for a single nanotube. The graphene sheet and the nanotube array will be gated independently with $V_{G,\mathrm{graphene}}$ and $V_{G,NT}$ respectively, such that we have two shifted Dirac cones in momentum space as illustrated in part (b). The source-drain voltage $V_{sd}$ leads to a bias between right movers and left movers in the nanotubes. (b) Two shifted Dirac cones for the nanotubes (top cone, 1D) and graphene (bottom, 2D), assuming that the nanotubes and graphene have the same Fermi velocity. The gate and source-drain voltages applied to the nanotubes should be adjusted such that ${\mu }_L>{\epsilon }_c>{\mu }_R$ so only, e.g., left-moving electrons can tunnel into graphene. The red color indicates occupied states in the nanotubes.

Figure 6

When averaging over all allowed ${\mathit{k}}_2$, each ${\mathit{k}}_2$ parametrizes a different ellipse. For each ${\mathit{k}}_2$ and the resulting ellipse (red) there is a mirror image with respect to ${\mathit{k}}_1,{\tilde{\mathit{k}}}_2$ (blue), such that an increase in the current along ${\widehat{\mu }}_{\parallel }\left({\mathit{k}}_2\right)$ averages to an increase along ${\widehat{\mathit{k}}}_1$. In general, there can also be a change of the current in direction ${\widehat{\mu }}_{\perp }$. The change in the component parallel to ${\mathit{k}}_1$, however, averages to zero when averaging over ${\mathit{k}}_2$. By symmetry there can be no change in current perpendicular to ${\mathit{k}}_1$. Changes of the current in the ${\widehat{\mu }}_{\perp }$ direction are therefore not important for the average change in current. The green dashed circle indicates the Fermi momentum. The allowed states ${\mathit{k}}_2$ must lie within this circle.

Figure 7

(a) Electron and (b) hole process for the case of ${\epsilon }_F>0$. (a) Scattering processes where $|{\mathit{k}}_2\rangle$ is in the lower band are collinear and will not change the current. (b) Processes where $|{\mathit{k}}_1^{\prime }\rangle ,|{\mathit{k}}_2^{\prime }\rangle$ are in the same band are also collinear.