Dynamic admittance of mesoscopic conductors: Discrete-potential model

We present a discussion of the low-frequency admittance of mesoscopic conductors in close analogy with the scattering approach to do conductance. The mesoscopic conductor is coupled via contacts and gates to a macroscopic circuit which contains ac-current sources or ac-voltage sources. We find the admittance matrix which relates the currents at the contacts of the mesoscopic sample and of nearby gates to the voltages at these contacts. The problem is solved in two steps: we first evaluate the currents at the sample contacts in response to the oscillating voltages at the contacts, keeping the internal electrostatic potential fixed. In a second stage an internal response due to the potential induced by the injected charges is evaluated. The self-consistent calculation is carried out for the simple limit in which each conductor is characterized by a single induced potential. Our discussion treats the conductors and gates on equal footing. Since our approach includes all conductors on which induced fields can change the charge distribution, the admittance of the total response is current conserving, and the current response depends only on ac-voltage differences. We apply our approach to a mesoscopic capacitor for which each capacitor plate is coupled via a lead to an electron reservoir. We find an electrochemical capacitance with density-of-state contributions in series with the geometrical capacitance. The dissipative part of the admittance is governed by a charge-relaxation resistance which is a consequence of the dynamics of the charge pileup one the capacitor plates. We specialize on a geometry displaying an Aharonov-Bohm effect only at nonzero frequencies. For a double barrier with a well coupled capacitively to a gate the low-frequency admittance terms may have either sign, reflecting either a capacitive or a kinetic-inductive behavior. The validity of a second-quantization-current-operator expression which neglects spatial information is examined for perfect leads in both the frequency and the magnetic-field domain.