# Edge Currents Induced by AC Electric Field in Two-Dimensional Dirac Structures
## 3 Results and Discussion  
Now, we discuss the spectral and polarization dependence of the edge current in 2D systems with parabolic and linear dispersions, also for different scattering mechanisms. First, we note that the current \(J_{y}\) is proportional to the square of the electric field amplitude (\(S_{2},S_{3}\propto E^{2}\)), i.e., to the intensity of the incident field. Therefore, it belongs to the class of photocurrents. Here, the photocurrent emerges due to the intraband (Drude-like) absorption of the electromagnetic field and is proportional to the real part of the high-frequency conductivity \(\sigma(\omega)\). The direction of the edge current (the polarity along the \(y\) axis) depends on the polarization state of the field via the Stokes parameters \(S_{2}=(E_{x}E_{y}^{*}+E_{y}E_{x}^{*})\) and \(S_{3}=i(E_{x}E_{y}^{*}-E_{y}E_{x}^{*})\). The contribution \(J_{y}\propto S_{2}\) is induced by linearly polarized radiation. This current is maximal if the radiation is polarized at the angle \(\pm\pi/4\) with respect to the edge and vanishes if the radiation is polarized along or perpendicular to the edge, Figure 1a,b. The contribution \(J_{y}\propto S_{3}\) describes the edge current induced by circularly polarized radiation and its direction is controlled by the radiation helicity, Figure 1c,d. Equation (21) is quite general and can be applied to a wide class of 2D systems, including conventional III-V and II-VI quantum wells, bilayer graphene, and transition metal dichalcogenide monolayers with the parabolic spectrum, graphene and HgTe/CdHgTe quantum wells of the critical thickness with the linear spectrum, and narrow-gap 2D systems with the Dirac-like spectrum \(\varepsilon(p)=v_{0}\sqrt{(m^{*}v_{0})^{2}+p^{2}}\). Below, we calculate \(J_{y}\) for single-layer and bilayer graphene. Figure 2 shows the frequency dependence of the edge photocurrent \(J_{y}\) in a 2D system with the linear dispersion law \(\varepsilon(p)=v_{0}p\). The parameters used for calculations are given in the caption of Figure 2 and correspond to high-quality graphene [25] with the electron density \(n=5\times 10^{11}\) cm\({}^{-2}\). The curves are calculated after Equation (23) for linearly polarized radiation with the electric field directed at the angle \(\pi/4\) with respect to the edge (\(S_{2}/E^{2}=1\), \(S_{3}=0\)) and for circularly polarized radiation (\(S_{3}/E^{2}=1\), \(S_{2}=0\)). We consider two model types of scattering potential: (i) short-range scatterers, resulting in the energy dependence of the relaxation times \(\tau_{1}=2\tau_{2}\propto\varepsilon^{-1}\), and (ii) charged scatterers with the Coulomb potential, resulting in \(\tau_{1}=3\tau_{2}\propto\varepsilon\)[7]. For charged scatterers, the ratio \(\tau_{1,2}/m\) does not depend on energy and the edge current is given by \(J_{y}=e\mathrm{Re}\sigma(\omega)v_{0}^{2}\tau_{1}^{2}(E_{F})/3E_{F}\) for linearly polarized radiation and \(J_{y}=-e\mathrm{Re}\sigma(\omega)v_{0}^{2}\tau_{1}(E_{F})/\omega E_{F}\) for circularly polarized radiation. Hence, the frequency dependence of the edge currents excited by linearly and circularly polarized radiation are determined by \(\mathrm{Re}\sigma(\omega)\propto 1/(1+\omega^{2}\tau_{1}^{2})\) and \(\mathrm{Re}\sigma(\omega)/\omega\), respectively. For short-range scatterers, all terms in Equation (21) contribute to the current, and the frequency dependence gets more complicated. In particular, the current induced by linearly polarized radiation is constant at low frequencies, changes its sign at an intermediate frequency, and decays as \(\propto\omega^{-2}\) at high frequencies. The circular contribution  
for short-range scatterers behaves as \(\propto\omega\) at low frequencies, in contrast to the diverging behavior \(\propto\omega^{-1}\) for Coulomb scatterers. The current magnitude is \(J_{y}\sim 10\) nA for 1 W/cm\({}^{2}\) of the radiation intensity. Figure 3 shows the frequency dependence of the edge photocurrent \(J_{y}\) in the 2D system with the parabolic dispersion law \(\varepsilon(p)=p^{2}/2m^{*}\).