Drift velocity of edge magnetoplasmons due to magnetic edge channels

Edge magnetoplasmons arise on a boundary of conducting layers in perpendicular magnetic field due to an interplay of electron cyclotron motion and Coulomb repulsion. Lateral electric field, which confines electrons inside the sample, drives their spiraling motion in magnetic field along the edge with the average drift velocity contributing to the total magnetoplasmon velocity. We revisit this classical picture by developing fully quantum theory of drift velocity starting from analysis of magnetic edge channels and their electrodynamic response. We derive the quantum-mechanical expression for the drift velocity, which arises in our theory as a characteristic of such response and can be calculated as the harmonic mean of group velocities of edge channels crossing the Fermi level. Using the Wiener-Hopf method to analytically solve the edge mode electrodynamic problem, we demonstrate that the edge channel response effectively enhances the bulk Hall response of the conducting layer and thus increases the edge magnetoplasmon velocity. In the long-wavelength limit of our model, the drift velocity is simply added to the total magnetoplasmon velocity, in agreement with the classical picture.

Figure 1

Illustration of the origin of edge magnetoplasmon drift velocity. Wavelike oscillations of the local Fermi level $E_{\mathrm{F}}^{\mathrm{e}}$ change the filling of magnetic edge channels, i.e., Landau levels with the energies $E_{n{k}_y}$, which are bent upward near the edge $x=0$. Since the edge channels are unidirectional, oscillations of their occupation give rise to oscillations of edge current, shown by red cones. The edge current appears in phase with the bulk current response of the 2DEG (blue cones) and thus effectively enhances its Hall conductivity ${\sigma }_{xy}$. The edge magnetoplasmon velocity, which is roughly proportional to $|{\sigma }_{xy}|$, is increased too by the quantity $v_{\mathrm{dr}}$, which has the meaning of drift velocity.

Figure 2

Energies of the edge states $E_{n{k}_y}$ in magnetic field as functions of the wave vector $k_y$ along the edge. The curves from bottom to top correspond to the levels $n=0...5$. Dashed line shows the example of Fermi level location $E_{\mathrm{F}}^{\mathrm{e}}$ between the bulk Landau levels $n=2$ and 3, and circles demonstrate its intersections with the edge state energies at momenta $k_n^{\max }, n=0,1,2$.

Figure 3

Wave function square moduli ${\left|\psi \right|}^2$ for (a)–(c) edge states (2), (d)–(f) bulk Landau levels (4) at different wave vectors $k_y$ along the edge. The columns correspond to the level numbers $n=0,1,2$.

Figure 4

Contributions (9) and (10) of individual edge channels $n=0,1,2$ to (a) charge density ${\rho }_{\mathrm{e}}\left(x\right)$ and (b) current density $j_y\left(x\right)$ as functions of the distance $x$ from the edge. The Fermi level $E_{\mathrm{F}}^{\mathrm{e}}$ is located in the midgap between the bulk Landau levels $n=2$ and 3, as shown in Fig. 2.

Figure 5

Drift velocity $v_{\mathrm{dr}}$ (solid lines) as function of (a) Fermi level $E_{\mathrm{F}}$ and (b) Landau-level filling factor ${\nu }_{\mathrm{L}}$. Dashed lines show the quasiclassical approximation (44) which coincides (circles) with the exact quantum-mechanical $v_{\mathrm{dr}}$ when the Fermi level is located in the midgap between Landau levels, i.e., when $E_{\mathrm{F}}/\hslash {\omega }_{\mathrm{c}}$ and $\frac{1}{2}{\nu }_{\mathrm{L}}$ are equal integers.

Figure 6

Edge magnetoplasmon dispersions found numerically in the presence (solid lines) and absence (dashed lines) of the drift velocity $v_{\mathrm{dr}}$. Black circles show the analytical approximation (43) for the long-wavelength limit. The panels correspond to different Landau level filling factors ${\nu }_{\mathrm{L}}$ and Coulomb interaction strengths $r_{\mathrm{s}}$.

Figure 7

Exact energies of the edge states $E_{n{k}_y}$ in magnetic field (solid lines) as functions of $k_y$, and their quasiclassical approximations (circles) given by Eq. (A6). Insets show quadratic confining potential $V\left(x\right)$ and wave function ${\psi }_{n{k}_y}\left(x\right)$ for $n=2$ in different regions of quasiclassical quantization.