# Edge Currents Induced by AC Electric Field in Two-Dimensional Dirac Structures
## 2 Microscopic Theory
The density of DC electric current \(j_{y}(x)\) is determined by the asymmetric part of the correction \(f_{2}\) and is given by  
\[j_{y}(x)=eg\sum_{\mathbf{p}}v_{y}f_{2}(\mathbf{p},x)\,, \tag{3}\]  
where \(g\) is the factor that takes into account possible spin and valley degeneracy (e.g., \(g=2\) for GaAs quantum wells and \(g=4\) for single-layer and bilayer graphene) and \(\sum_{\mathbf{p}}=(2\pi\hbar)^{-2}\int d^{2}\mathbf{p}\). Equation (1) with the perturbation term \(e\mathbf{\mathcal{E}}(x,t)\cdot\partial f/\partial\mathbf{p}\) yields the following differential equations for \(f_{1}\) and \(f_{2}\)  
\[-i\omega f_{1}+v_{x}\frac{\partial f_{1}}{\partial x}+e\mathbf{ \mathcal{E}}(x)\cdot\frac{\partial f_{0}}{\partial\mathbf{p}} =\text{I}\{f_{1}\}\,, \tag{4}\] \[v_{x}\frac{\partial f_{2}}{\partial x}+\left[e\mathbf{\mathcal{E}}( x)\cdot\frac{\partial f_{1}^{*}}{\partial\mathbf{p}}+\text{c.c}\right] =\text{I}\{f_{2}\}\,. \tag{5}\]  
We solve Equations (4) and (5) in the approximation of elastic electron scattering in the bulk of the 2D system and for specular reflection of electrons from the edge. The latter implies that \(f(p_{x},p_{y},0,t)=f(-p_{x},p_{y},0,t)\) which also ensures the lack of electric current across the edge. Multiplying Equation (5) by the velocity \(v_{y}\) and averaging the resulting equation over the directions of \(\mathbf{p}\) one obtains  
\[\left\langle v_{x}v_{y}\frac{\partial f_{2}}{\partial x}\right\rangle+\left\langle v _{y}\left(e\mathbf{\mathcal{E}}\cdot\frac{\partial f_{1}^{*}}{\partial\mathbf{p}}+ \text{c.c}\right)\right\rangle=-\frac{\langle v_{y}f_{2}\rangle}{\tau_{1}}\,, \tag{6}\]  
where the angular brackets \(\langle\ldots\rangle\) stand for the averaging and \(\tau_{1}\) is the momentum relaxation time (relaxation time of the first angular harmonic) defined as \(1/\tau_{1}=-\langle\mathbf{v}\,I\{f\}\rangle/\langle\mathbf{v}f\rangle\). Such a definition of \(\tau_{1}\) enables the consideration of its dependence on the electron energy \(\varepsilon(p)\). Equations (3) and (6) yield the equation for the current density  
\[j_{y}(x)=-eg\sum_{\mathbf{p}}\tau_{1}v_{x}v_{y}\frac{\partial f_{2}}{\partial x}- eg\sum_{\mathbf{p}}\tau_{1}v_{y}\bigg{(}e\mathbf{\mathcal{E}}\cdot\frac{\partial f _{1}^{*}}{\partial\mathbf{p}}+\text{c.c}\bigg{)}\,. \tag{7}\]  
After the integration of the second term by parts, Equation (7) assumes the form  
\[j_{y}(x) = -eg\sum_{\mathbf{p}}\tau_{1}v_{x}v_{y}\frac{\partial f_{2}}{\partial x }+e^{2}g\sum_{\mathbf{p}}\Bigl{(}\frac{\tau_{1}}{m}\Bigr{)}^{\prime}mv_{x}v_{y}( \mathcal{E}_{x}f_{1}^{*}+\text{c.c}) \tag{8}\] \[\frac{\partial f}{\partial t}+v_{x}\frac{\partial f}{\partial x}+e\mathbf{\mathcal{ E}}(x,t)\cdot\frac{\partial f}{\partial\mathbf{p}}=\text{I}\{f\}\;. \tag{1}\]0  
where \((\ldots)^{\prime}=d(\ldots)/d\varepsilon\) and \(m=p/v=p/(d\varepsilon/dp)\) is the (energy-dependent) effective mass. In the case of parabolic dispersion \(\varepsilon(p)=p^{2}/(2m^{*})\) the effective mass \(m=m^{*}\) is energy-independent whereas for linear dispersion \(\varepsilon(p)=v_{0}p\) the effective mass \(m=\varepsilon/v_{0}^{2}\) linearly depends on energy. Note that the mass \(m\) also determines the quasi-classical cyclotron motion. Equation (8) is valid for arbitrary dispersion and arbitrary boundary conditions at the edge. Obviously, the DC current vanishes for the AC field polarized along or perpendicularly to the edge and emerges only if the incident field \(E\) has both \(x\) and \(y\) components. Therefore, the corrections \(f_{1}\) in the second and third terms on the right-hand side of Equation (8) should be calculated for the \(y\) and \(x\) components of the field, respectively. For specular reflection of electrons from the edge, the second sum in Equation (8) vanishes since \(f_{1}\) in response to \(E_{y}\) is an even function of \(v_{x}\).