# Edge Currents Induced by AC Electric Field in Two-Dimensional Dirac Structures
## 2 Microscopic Theory  
Consider an electromagnetic wave incident on the structure hosting a 2D electron gas occupying the \((xy)\) half-plane at \(x\geq 0\), see Figure 1. The AC electric field of the incident wave has the form \(\mathbf{E}(t)=\mathbf{E}\exp(-i\omega t)+\text{c.c. }\), where \(\mathbf{E}\) and \(\omega\) are the field amplitude and frequency, respectively, and the abbreviation \"c.c.\" denotes the complex conjugation. The AC electric field causes the back-and-forth in-plane motion of electrons. At the edge of the structure (here, at \(x=0\)), the AC motion of electrons gets distorted due to electron reflection from the edge and dynamic charge accumulation, which leads to an asymmetry of the high-frequency electron transport. This asymmetry results in the rectification of the AC current and, hence, the emergence of a DC current \(J_{y}\) flowing along the edge. As Figure 1 illustrates, the DC edge current can be excited by both linearly polarized and circularly polarized electromagnetic waves and the current direction is controlled by the wave polarization. Now we present the microscopic theory of high-frequency non-linear electron transport and calculate the edge currently. We consider the classical range of the electromagnetic wave frequencies, i.e., \(\hbar\omega\ll E_{F}\), where \(E_{F}\) is the Fermi energy of the 2D electron gas, and describe the electron kinetics by the Boltzmann equation  
\[\frac{\partial f}{\partial t}+v_{x}\frac{\partial f}{\partial x}+e\mathbf{\mathcal{ E}}(x,t)\cdot\frac{\partial f}{\partial\mathbf{p}}=\text{I}\{f\}\;. \tag{1}\]  
Here, \(f=f(\mathbf{p},x,t)\) is the electron distribution function, \(\mathbf{p}\) and \(\varepsilon(p)\) are the electron momentum and energy, respectively, \(\mathbf{v}=d\varepsilon/d\mathbf{p}=vp/p\) is the velocity, \(v=d\varepsilon/dp\), \(e\) is the electron charge, \(\mathbf{\mathcal{E}}(x,t)=\mathbf{\mathcal{E}}(x)\exp(-i\omega t)+\text{c.c. }\) is the local electric field acting upon the electrons, and \(\text{I}\{f\}\) is the collision integral. At this stage, we assume that the electron spectrum is isotropic in the 2D plane but do not specify the exact form of the dispersion \(\varepsilon(p)\). The field \(\mathbf{\mathcal{E}}(x)\) near the edge differs from the incident field \(\mathbf{E}\) by the correction \(\delta\mathbf{E}(x)\parallel x\) due to the screening produced by dynamical charge accumulation [6; 23; 24]. Therefore, \(\mathcal{E}_{y}=E_{y}\) whereas \(\mathcal{E}_{x}(x)\propto E_{x}\). Figure 1: Illustration of the edge current formation. The back-and-forth motion of 2D carriers occupying the \(x\geq 0\) half-plane by linearly polarized (**a,b**) or circularly polarized (**c,d**) AC electric field results, due to electron scattering from the edge and dynamical charge accumulation, in the DC current \(J_{y}\) flowing along the edge. The edge current direction is controlled by the field polarization. At equilibrium, the electron distribution is isotropic and homogenous at \(x\geq 0\) and is described by the Fermi-Dirac function \(f_{0}(\varepsilon)\). In the presence of an AC electric field, the distribution function acquires corrections. We expand the resulting distribution function \(f(\mathbf{p},x,t)\) in the series in the electric field amplitude as follows  
\[f(\mathbf{p},x,t)=f_{0}+\big{[}f_{1}(\mathbf{p},x)\exp(-i\omega t)+\text{c.c.}\big{]}+f_ {2}(\mathbf{p},x)+\ldots\,, \tag{2}\]  
where \(f_{1}\) is the first-order correction, which determines the linear (Drude) conductivity, and \(f_{2}\) is the time-independent second-order correction. The second-order correction oscillating at \(2\omega\) is not considered here since it does not contribute to DC electric current.