We present a current and charge conserving theory for the low frequencyadmittance of a two-dimensional electron gas connected to ideal metalliccontacts and subject to a quantizing magnetic field. In the framework of theedge-channel picture, we calculate the admittance up to first order withrespect to frequency. The transport coefficients in first order with respect tofrequency, which are called emittances, determine the charge emitted into acontact of the sample or a gate in response to an oscillating voltage appliedto a contact of the sample or a nearby gate. The emittances depend on thepotential distribution inside the sample which is established in response tothe oscillation of the potential at a contact. We show that the emittances canbe related to the elements of an electro-chemical capacitance matrix whichdescribes a (fictitious) geometry in which each edge channel is coupled to itsown reservoir. The particular relation of the emittance matrix to thiselectro-chemical capacitance matrix depends strongly on the topology of theedge channels: We show that edge channels which connect different reservoirscontribute with a negative capacitance to the emittance. For example, while theemittance of a two-terminal Corbino disc is a capacitance, the emittance of atwo-terminal quantum Hall bar is a negative capacitance. The geometry of theedge-channel arrangement in a many-terminal setup is reflected by symmetryproperties of the emittance matrix. We investigate the effect of voltage probesand calculate the longitudinal and the Hall resistances of an idealfour-terminal Hall bar for low frequencies.
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