Low-frequency admittance of quantized Hall conductors

TL;DR

This paper develops a theory for the low-frequency admittance of quantum Hall conductors, linking emittance to electro-chemical capacitance and revealing how edge channel topology affects charge response.

Contribution

It introduces a first-order frequency expansion of admittance in quantum Hall systems, relating emittance to electro-chemical capacitance and analyzing the impact of edge channel topology.

Findings

Emittance can be related to electro-chemical capacitance matrix elements.

Edge channels connecting different reservoirs contribute negative capacitance.

The geometry influences the symmetry and properties of the emittance matrix.

Abstract

We present a current and charge conserving theory for the low frequency admittance of a two-dimensional electron gas connected to ideal metallic contacts and subject to a quantizing magnetic field. In the framework of the edge-channel picture, we calculate the admittance up to first order with respect to frequency. The transport coefficients in first order with respect to frequency, which are called emittances, determine the charge emitted into a contact of the sample or a gate in response to an oscillating voltage applied to a contact of the sample or a nearby gate. The emittances depend on the potential distribution inside the sample which is established in response to the oscillation of the potential at a contact. We show that the emittances can be related to the elements of an electro-chemical capacitance matrix which describes a (fictitious) geometry in which each edge channel is coupled to its own reservoir. The particular relation of the emittance matrix to this electro-chemical capacitance matrix depends strongly on the topology of the edge channels: We show that edge channels which connect different reservoirs contribute with a negative capacitance to the emittance. For example, while the emittance of a two-terminal Corbino disc is a capacitance, the emittance of a two-terminal quantum Hall bar is a negative capacitance. The geometry of the edge-channel arrangement in a many-terminal setup is reflected by symmetry properties of the emittance matrix. We investigate the effect of voltage probes and calculate the longitudinal and the Hall resistances of an ideal four-terminal Hall bar for low frequencies.