Transmission Lines and Metamaterials Based on Quantum Hall Plasmonics

The characteristic impedance of a microwave transmission line is typically constrained to a value $Z_0=50\mathrm{\Omega }$, in part because of the low impedance of free space and the limited range of permittivity and permeability realizable with conventional materials. Here we suggest the possibility of constructing high-impedance transmission lines by exploiting the plasmonic response of edge states associated with the quantum Hall effect in gated devices. We analyze various implementations of quantum Hall transmission lines based on distributed networks and lumped-element circuits, including a detailed account of parasitic capacitance and Coulomb drag effects, which can modify device performance. We additionally conceive of a metamaterial structure comprising arrays of quantum Hall droplets and analyze its unusual properties. The realization of such structures holds promise for efficiently wiring-up quantum circuits on chip, as well as engineering strong coupling between semiconductor qubits and microwave photons.

Figure 1

Sketch of a quantum Hall transmission line. Four ideal electrodes of length $L_i$ are placed at a distance $d_i$ from the edge of a two-dimensional QH droplet; the capacitive coupling between the material and the $i\mathrm{th}$ lead is quantified by the velocities $v_i$. Each electrode can in principle be driven by a voltage $V_i$ applied with respect to ground or by a current $I_i$. The position in the material is parameterized by the coordinates $(x,y)$, which are normal and tangential to the perimeter, respectively. $l_i$ and $r_i$ indicate the left and right edges of the $i\mathrm{th}$ electrode, respectively. We refer to the pair of electrodes 1,2 (3,4) as driving (screening) electrodes.

Figure 2

Circuit equivalent of the QH droplet in Fig. 1. The circuit is composed of an ideal anticlockwise circulator with characteristic impedance $1/(2{\sigma }_{xy})$ and scattering matrix $S_{\circlearrowleft }$ [Eq. (8)] that connects in series four delay lines with frequency-dependent impedance $Z_i$ [Eq. (7)]. Each delay line is a transmission line with characteristic impedance $1/(2{\sigma }_{xy})$ and propagation time ${\tau }_i$ terminated by an open circuit, and it models the capacitive coupling between the QH material and the $i\mathrm{th}$ electrode.

Figure 3

Transmission line with a series double-stub tuner. This circuit is equivalent to a QH droplet with the screening electrodes characterized by the same propagation time ${\tau }_0$ and connected to ground; the transmission in this region is modeled by a conventional TL with a characteristic impedance $1/(2{\sigma }_{xy})$. The capacitive coupling to the driving electrodes is modeled by two stubs (TLs terminated with an open circuit) also having a characteristic impedance of $1/(2{\sigma }_{xy})$ and a propagation time ${\tau }_{1,2}$.

Figure 4

Scattering parameters of a QH droplet with grounded and identical screening electrodes. In the plots, we consider the device to be matched to the external circuitry, i.e., $\alpha =2{\sigma }_{xy}{Z}_0=1$ and we assume a symmetric configuration ${\tau }_1={\tau }_2$. The different plots are obtained by using different ratios of propagation times $p=2{\tau }_0/{\tau }_1$. The blue (red) solid line is the absolute value of the scattering parameter associated with reflection (transmission) $\left|S_{11}\right|(\left|S_{12}\right|)$; the dashed lines are the envelope functions that modulate the response obtained by the limit ${\tau }_0\to 0$; they capture the response of a configuration with floating screening electrodes, as described in Sec. 2b. The envelope functions of $\left|S_{11}\right|$ vanish at the central resonances ${\omega }_n$ (black crosses) defined in Eq. (10), and they attain the maximum value of one at the frequencies ${\omega }_m$ (black circles) defined in Eq. (11). The fast oscillations depend on the transmission line and they are associated with the resonances at the frequencies ${\omega }_l$ (orange dots) defined in Eq. (15).

Figure 5

Phase of the transmission parameter $S_{12}$. The device is matched to the external circuitry, i.e., $\alpha =1$, and the screening electrodes are identical. Also, we consider ${\tau }_1={\tau }_2$. The different plots are obtained by using different ratios of propagation times $p$. The solid red lines show the phase of the scattering parameter of a QH droplet with grounded screening electrodes, while the red dashed lines show the phases of the envelope function that modulates the response, and they are obtained by taking the limit ${\tau }_0\to 0$; these dashed lines capture the response of the setup described in Sec. 2b. The dotted black lines are obtained by using the linear expansion of the phase in Eq. (17).

Figure 6

Equivalent circuit model including parasitic capacitors. The circuit equivalent of the QH droplet in Fig. 2 is augmented by a parallel circuit that includes the possible capacitive coupling between external electrodes and ground; we do not include the coupling $C_{12}$ between the driving terminals because it is negligible for long TLs. Some of these additional capacitors modify the response in a qualitatively similar manner and they are thus drawn with the same color. Following the notation of the main text, the gray, red, green, and blue capacitors are associated with the charging times $T_0$, $T_1$, $T_2$, and $T_3$, respectively.

Figure 7

Changes in the scattering parameters of a QH droplet due to a small value of the parasitic capacitor $C_{34}\equiv 2{\sigma }_{xy}{T}_1$. In the plots, the screening gates are assumed to be floating, i.e., $T_0=0$, and we consider the device to be matched with the external circuitry $\alpha =1$ and symmetric, i.e., ${\tau }_1={\tau }_2$ and ${\tau }_3={\tau }_4={\tau }_{1}p/2$. The different plots show the scattering parameters for different values of $p$. The blue (red) solid line represents the absolute value of $\left|S_{11}\right|(\left|S_{12}\right|)$ when $T_1/{\tau }_1=0.05$, while the dashed lines show the response in the same configuration in the case without parasitics.

Figure 8

Changes in the scattering parameters of a QH droplet due to a large value of the parasitic capacitor $C_{34}\equiv 2{\sigma }_{xy}{T}_1$. In the plots we used the same conventions as in Fig. 7, but the solid lines are now obtained by using a higher value of the parasitic capacitor, $T_1/{\tau }_1=10$.

Figure 9

Changes in the scattering parameters of a QH droplet due to the parasitic capacitors connecting adjacent electrodes $C_{13}=C_{14}=C_{24}=C_{24}\equiv 2{\sigma }_{xy}{T}_3$. In these plots, we assume $\alpha \equiv 2{\sigma }_{xy}{Z}_0=1$ and a symmetric configuration with ${\tau }_1={\tau }_2$ and ${\tau }_3={\tau }_4={\tau }_{1}p/2$. Here, we set $p=1$. In (a) and (b) we consider screening gates that are floating ($T_0\to 0$) and connected to ground ($T_0\to \mathrm{\infty }$), respectively. The blue (red) solid line represents the absolute value of $\left|S_{11}\right|(\left|S_{12}\right|)$ when $T_3/{\tau }_1=1$, while the dotted lines are obtained for $T_3/{\tau }_1=0.1$.

Figure 10

Metamaterial transmission lines. The metamaterials are composed of a chain of identical QH droplets coupled in different ways. The electrostatic coupling to the different metal electrodes is quantified by the corresponding propagation time ${\tau }_i$. Here, we focus on symmetric droplets characterized by only two propagation times ${\tau }_{0,1}$ and whose screening electrodes are connected to ground. In (a) the coupling between adjacent droplets is mediated by a metal electrode of negligible length, while in (b) the droplets are coupled via unscreened Coulomb interactions. In the latter case, the coupling region is characterized by its length $a$ and by the intra- and interedge velocities, $v_F$ and $v_1$, respectively.

Figure 11

Absolute values of the $S$ parameters and dispersion relation for the metamaterial transmission line in Fig. 10. The different plots are obtained with different values of $p=2{\tau }_0/{\tau }_1$ and assuming a matched circuit $\alpha =1$. The blue (red) solid lines in the top subplots are the reflection (transmission) parameters $\left|S_{11}\right|(\left|S_{12}\right|)$ obtained for a chain comprising $N=10$ unit cells. The thin dashed lines represent the response for a single droplet and the thick dashed lines in darker colors are the smooth envelope functions modulating the fast resonances. In the subplots at the bottom, we show the band dispersion $k(\omega )$ [Eq. (32)] of the corresponding periodic chain.

Figure 12

Phase of the transmission coefficient $S_{12}$ of the metamaterial in Fig. 10. The red lines are the phases of $S_{12}$, the black lines are obtained by using the linear approximation in Eq. (35). In the plot, we use a different number $N$ of unit cells; in particular, $N=1,5,10$ for dotted, dashed, and solid lines, respectively. The thick vertical lines at $\omega {\tau }_1=2\pi /3$ and $\omega {\tau }_1=4\pi /3$ represent the edges of the transmission band.

Figure 13

Equivalent circuit model of the metamaterial shown in Fig. 10. The Coulomb interactions between adjacent droplets are modeled by the transfer matrix ${\mathcal{T}}_{\mathrm{int}}$ in Eq. (37). This is represented by a $T$ junction composed of the delay lines $Z_I$ and $Z_G$, characterized by the propagation time ${\tau }_I\equiv a/v$ and by the characteristic impedances $(v_F-v_1)/(2{\sigma }_{xy}v)$ and $v_1/(2{\sigma }_{xy}v)$, respectively. Here, $v=\sqrt{v_F^2-v_1^2}$. This $T$ junction is repeated $N-1$ times, where $N$ is the number of QH droplets. Each junction is connected to the next one by an ideal transmission line with characteristic impedance $1/(2{\sigma }_{xy})$ and propagation time ${\tau }_0$. The capacitive coupling to the external electrodes is assumed to be the same for the initial and final droplets and is modeled by the delay lines $Z_1$ [Eq. (7)] with a propagation time ${\tau }_1$ and a characteristic impedance $1/(2{\sigma }_{xy})$.

Figure 14

Absolute values of the $S$ parameters and dispersion relation for the metamaterial transmission line shown in Fig. 10. The single droplets are assumed to be symmetric and matched with the external microwave circuit, i.e., ${\tau }_I={\tau }_1$ and $\alpha =2{\sigma }_{xy}{Z}_0=1$. The chain is composed of $N=10$ unit cells and the Coulomb coupling between adjacent droplets is parameterized by $v/v_F=0.15$. The two plots are given for different values of $p\equiv 2{\tau }_0/{\tau }_1$; in particular, in (a) we used $p=1$ and in (b) $p=2$. In the top subplots, the blue (red) lines are $\left|S_{11}\right|(\left|S_{12}\right|)$. In the bottom subplots, the band dispersions [Eq. (39)] obtained with $v/v_F=0.15$ are plotted as solid lines; the dashed line is the linear dispersion obtained in the strong-coupling limit $v_1\to v_F$.