Distributed electrochemical capacitance evidenced in high-frequency admittance measurements on a quantum Hall device

We describe high-frequency admittance measurements up to 3 GHz on a quantum point contact in a quantum Hall regime. When the excitation frequency is so high that the wavelength of the edge magnetoplasmons is comparable to the size of the system, the phase developing along the edge channels must be considered when analyzing electrochemical capacitance in admittance. The result cannot be accounted for by the local electrochemical capacitance near the quantum point contact but instead needs to be understood by considering the distributed capacitance within the sample as a whole. This implies that all electrons within the device are interacting because the screening in one-dimensional channels is extremely weak. A distributed-element model is essential for understanding the ac response in low-dimensional structures, which is analogous to that in high-frequency electrical circuits.

Figure 1

Schematic of the measurement setup and the device structure with the optical micrograph at the central part of the device. Also illustrated is the distribution of the capacitive coupling. We measured the dc and rf signals traveling from the upper-left Ohmic contact ${\Omega }_1$ to the upper right contact ${\Omega }_2$. The QPC was formed by applying $-$1.2 V to the upper gate electrode and $V_{\mathrm{g}}$ to the lower gate electrode.

Figure 2

(a) Absolute admittance $|Y_{21}\left(\omega \right)|$ shown with dc conductance as a function of $V_{\mathrm{g}}$ measured at $\nu$ $=$ 12 ($B$ $=$ 0.70 T). (Inset) Magnetic field dependence of $|S_{21,\mathrm{meas}}|$ measured for a 500-MHz rf signal. The QPC was tuned to $T$ $=$ 1 ($V_{\mathrm{g}}$ $=$ 0 V) during this measurement. (b) The admittance phase arg$(Y_{21}$($\omega ))$ as a function of $V_{\mathrm{g}}$. Each arg$(Y_{21}$$\left(\omega \right))$ shifts 90° accompanied by the closing of the QPC.

Figure 3

(a) Frequency dependence of $|Y_{21}\left(\omega \right)|$ at $T$ $=$ 0 ($V_{\mathrm{g}}$ $=$ $-$1.2 V) obtained at several bulk filling factors with the fittings calculated from the model in Fig. 4 (thick solid lines). The dashed line shows the calculated $|Y_{21}\left(\omega \right)|$, assuming a lumped electrochemical capacitance at the QPC as $C_{\mu }$ $=$ 3.1 fF. (b) and (c) Frequency dependence of arg$(G_{21}$$\left(\omega \right))$ measured at (b) $T$ $=$ 0 ($V_{\mathrm{g}}$ $=$ $-$1.2 V) and (c) $T$ $=$ 1 ($V_{\mathrm{g}}$ $=$ $-$0 V), respectively. We observed a phase difference of 90° between the data at $T$ $=$ 0 and $T$ $=$ 1, which corresponds to the expectation for capacitive and transmissive transport.

Figure 4

(a) A simple example of the capacitive circuit to show the interference of the EMPs. The edge channels (length $L$) are connected to each other by the electrochemical capacitance $c_{//}$ per unit length distributed in length λ) $|Y_{21}\left(\omega \right)|$ of the circuit in (a) [red (dark gray) solid line] shown with that of a single-capacitance model (black dashed line). (c) The admittance phase in the model.