Nonreciprocal quantum Hall devices with driven edge magnetoplasmons in two-dimensional materials

We develop a theory that describes the response of nonreciprocal devices employing two-dimensional materials in the quantum Hall regime capacitively coupled to external electrodes. As the conduction in these devices is understood to be associated to the edge magnetoplasmons (EMPs), we first investigate the EMP problem by using the linear response theory in the random phase approximation. Our model can incorporate several cases that were often treated on different grounds in literature. In particular, we analyze plasmonic excitations supported by a smooth and sharp confining potential in a two-dimensional electron gas, and in monolayer graphene, and we point out the similarities and differences in these materials. We also account for a general time-dependent external drive applied to the system. Finally, we describe the behavior of a nonreciprocal quantum Hall device: the response contains additional resonant features, which were not foreseen from previous models.

Figure 1

Geometry of the EMP model. A two-dimensional material (orange) subjected to a perpendicular magnetic field is situated in the $z=0$ plane; it extends infinitely in the $y$ direction, and it terminates at $x=0$. The material is capacitively coupled with an external electrode at $z=d$. To simplify the calculations, we will assume that the top gate is translational invariant in the $y$ direction, except that the drive changes the potential of a part of the gate only (dark grey region). For example, in the figure, we show a voltage drive of the form $V_e\left(t\right)\mathrm{\Theta }(-y)\mathrm{\Theta }(L_x-\left|x\right|)$.

Figure 2

Landau levels of a 2DEG terminated by a sharp edge potential as a function of the momentum $k_y$ (in units $1/l_B)$. The energy levels are computed within the WKB approximation (solid lines) and exactly, with the dispersion relation in Eq. (A4) (dashed lines). Static interactions have been neglected.

Figure 3

Velocity of EMPs supported by smooth edges in 2DEG as a function of the distance $d$ of the top gate. The velocities are expressed in units of $v_p\equiv {\sigma }_{xy}/\left(2\pi {\epsilon }_S\right)$, while $d$ is in units of the smoothness parameter $w$, defined in Eq. (30). In this calculation, we are neglecting the effects of the incompressible strips and of backscattering.

Figure 4

Motion of the $u_j(y,t)$ component of the EMP charge under the effect of the drive $V_e$ defined as the real part of Eq. (44). All the quantities are conveniently normalized to be dimensionless. In the plot, we are showing only the first two EMP modes. We used $d/w=0.5$ and $\eta /{\omega }_0=0.1$.

Figure 5

Velocity of the three fastest EMPs supported by a sharp edge in a 2DEG, sorted from (a) to (c) in decreasing order of velocity. The plots show the dependence on the magnetic field (in Tesla) and on the distance $d$ of the top gate, in units $l_1\equiv l_B(B=1\mathrm{T})\approx 26\mathrm{nm}$. For the plots, we used typical values of GaAs parameters, ${\epsilon }_S^{*}=8.7,m^{*}=0.063$, and $n_0={10}^{11}{\mathrm{cm}}^{-2}$. The steps as a function of $B$ occur when the Fermi energy crosses the bulk Landau levels: in these regions our model fails to account a finite real part of diagonal component of the conductivity and it is not applicable.

Figure 6

Velocity of the EMPs supported by sharp edge in a 2DEG, as a function of magnetic field (in Tesla) for two different values of $d$ in units $l_1\equiv l_B(B=1\mathrm{T})\approx 26\mathrm{nm}$. In (a), we used $d/l_1=1$ and in (b), we used $d/l_1=0.1$. For the plots, we used typical values of GaAs parameters, ${\epsilon }_S^{*}=8.7,m^{*}=0.063$, and $n_0={10}^{11}{\mathrm{cm}}^{-2}$. The solid lines represent the EMP velocities, while dashed lines represent the quantum contributions only. The quantum contributions are more relevant in (b). Our model is not applicable where the Fermi energy crosses bulk Landau levels, i.e., in the regions near the steep steps of the fastest EMP velocity in (a).

Figure 7

Landau levels of monolayer graphene terminated by a zigzag edge. Here, $\sigma$ is the valley (pseudospin) index. The energy levels are computed within the WKB approximation (solid line), and exactly (dashed lines). Static interactions have been neglected.

Figure 8

Velocity of the three fastest EMPs supported by a zigzag edge of monolayer graphene, sorted from (a) to (c) in decreasing order of velocity. The plots show the dependence on the Fermi energy in units of the cyclotron energy ${\epsilon }_F/\left(\hslash {\omega }_c\right)$ and on the distance of the electrode in units of the magnetic length $d/l_B$. For the plot, we used the typical value of the dielectric constant of silicon dioxide ${\epsilon }_S^{*}=3.9$. When the Fermi energy crosses the bulk Landau levels, the fastest plasmon velocity has a step: in these regions our model fails to account for a finite real part of the diagonal component of the conductivity and it is not applicable.

Figure 9

Velocity of the EMPs supported supported by a zigzag edge of monolayer graphene, as a function of the Fermi energy (in units of the cyclotron energy) ${\epsilon }_F/\left(\hslash {\omega }_c\right)$ for two different values of $d/l_B$. In (a), we used $d/l_B=1$ and in (b), we used $d/l_B=0.1$. For the plots, we used the typical value of the dielectric constant of silicon dioxide ${\epsilon }_S^{*}=3.9$. The solid lines represent the EMP velocities, while dashed lines represent the quantum contributions only. Our model is not applicable where the Fermi energy crosses bulk Landau levels, i.e., in the regions in the vicinity of the steep steps of the fastest plasmon velocity in (a).

Figure 10

Three-terminal gyrator. Three electrodes of length $L_n$ are capacitively coupled to a Hall bar of perimeter $L_y$, whose boundary is parametrized by $y$. The starting position of the electrodes in the $y$ direction is labeled by $y_n$ and they are separated by gaps of length $l$ with the nearest ones. These gaps are assumed to be equal and small compared to $L_n$. The device behaves as a two-port gyrator when the voltage of two electrodes is measured with respect to the last one, which acts as a common ground. The convention of currents and voltages is shown in the plot.

Figure 11

Single mode response. We plot the dependence of the $\mathrm{\Delta }$ parameter, defined in Eq. (68), on the dimensionless frequency $\mathrm{\Omega }$, defined in Eq. (69a). We normalize $\mathrm{\Omega }$ over the first gyration frequency ${\mathrm{\Omega }}_0=\pi L_1/L_3$ and consider different values of the gap $l$ between electrodes. For the plot, we used $\alpha =0.2$ and $L_3/L_1=2$.

Figure 12

Response including multiple modes. We plot the dependence of $\mathrm{\Delta }$, defined in Eq. (68), on the dimensionless frequency $\mathrm{\Omega }$, defined in Eq. (69a). We normalize $\mathrm{\Omega }$ over the first gyration frequency ${\mathrm{\Omega }}_0=\pi L_1/L_3$ and we include up to three modes in the response. For the plot, we used $\alpha =0.2,L_3/L_1=2$ and $l/L_1=0.05$. In (a) and (b), we consider the EMPs in a 2DEG with sharp edges (with the same parameters used in Fig. 5), at $B=0.5\text{T}$ and with $d/l_1=1$ and $d/l_1=0.1$ respectively. In (c), we consider the EMPs in a 2DEG with smooth edges, with $d/w=0.1$. All the resonant structures go up to a value very close to 1 (or to 0), but the plots do not have enough resolution to show this behavior.

Figure 13

Effect of a damping rate. We plot the dependence of $\mathrm{\Delta }$, defined in Eq. (68), on the dimensionless frequency $\mathrm{\Omega }$, defined in Eq. (69a). We normalize $\mathrm{\Omega }$ the first gyration frequency ${\mathrm{\Omega }}_0=\pi L_1/L_3$ and use different damping rates ${\mathrm{\Omega }}_{\eta }\equiv \eta L_y/v_0$. Here, the plots are obtained in the same way as the ones in Fig. 12, and we zoomed in a narrower frequency range. In (a) and (b), we consider the EMPs in a 2DEG with sharp edges (with the same parameters used in Fig. 5), at $B=0.5\text{T}$ and with $d/l_1=1$ and $d/l_1=0.1$ respectively. In (c), we consider the EMPs in a 2DEG with smooth edges, with $d/w=0.1$.