Magnetic-field symmetry of dynamic capacitances

Abstract

While the static-capacitance matrix is always symmetric and thus an even function of the magnetic field, the dynamic-capacitance matrix of multiterminal samples obeys in general only the weaker Onsager-Casimir symmetry relations. Our results are in accordance with recent experimental observations of asymmetric dynamic capacitances in quantized Hall systems. We predict quantization of the four-terminal resistances of an insulating Hall sample. ͓S0163-1829͑97͒51204-5͔ Symmetry relations play an important role in both theoretical and experimental physics. In mesoscopic conductors, microreversibility coupled with the ͑irreversible͒ thermodynamics of electron reservoirs leads to conductances which obey the Onsager-Casimir 1 reciprocity relations under magnetic-field reversal. A manifestly symmetric formulation of the dc-transmission approach to electrical conductance 2-4 proved to be close to experimental reality 5 and has been applied to a wide range of problems 6 including the quantized Hall effect. 7 In this paper we are concerned not with stationary dc-transport but with the magnetic-field symmetry of the dynamic-capacitance matrix for arrangements of conductors which couple purely capacitively. In contrast to the dcconductances, the static-capacitance matrix can be obtained from thermodynamic potentials. As a consequence, the static-capacitance matrix is symmetric and all elements of the capacitance matrix are even functions of the magnetic field. Clearly, as we increase the frequency the capacitances themselves become frequency dependent. A thermodynamic description becomes invalid and the symmetry under magnetic-field reversal might be weakened. Below we show that the dynamic-capacitance matrix is in general not an even function of the magnetic field but obeys only the weaker Onsager-Casimir reciprocity symmetry. This result is in agreement with a recent experiment by Sommerfeld et al. 8 Linear electrical transport is characterized by a quadraticadmittance matrix G ␣␤ with a dimension equal to the number of contacts which are connected to the sample under investigation. The Fourier amplitudes, ␦I ␣ , of the current through contact ␣ are then related to the Fourier amplitudes of the voltage oscillation, ␦V ␤ exp(Ϫit), at contact ␤ byThe Onsager-Casimir symmetry relations require that the admittance matrix at a given magnetic field is equal to the transposed-admittance matrix at the reversed magnetic field:The requirements of current conservation and of invariance against a global voltage shift ͑gauge invariance͒ imply the additional sum rules ͚ ␣ G ␣␤ ϭ͚ ␤ G ␣␤ ϭ0. 9 As a consequence, the admittance of a two-terminal sample is determined by a single-scalar admittance and is thus an even function of the magnetic field B. An asymmetric conductance, on the other hand, requires at least a three-terminal sample. While asymmetric dc conductances are well established, 5,6 the investigation of an asymmetric ac response is relatively new. Two years ago, Chen et al. 10 reported on an asymmetric capacitance of a gate clo...

Friedel phases and phases of transmission amplitudes in quantum scattering systems

“…A rigorous procedure to obtain local densities is via derivatives of phases and scattering matrices with respect to local potentials. 8 The discussion given here (in terms of energy derivatives) is correct only up to WKB like corrections. The last term in Eq.…”

“…[2][3][4][5] Since the Friedel phase is related to the density of states it is also connected to thermodynamic statistical mechanics quantities 5,6 like the persistent current. 7 In addition to the total density of states the scattering phases also play an important role in the partial density of states 8 (density of states with a preselection or postselection of the incident or exiting quantum channel) and in transport coefficients like capacitances 9 and charge relaxation resistances. 10 The principal aim of this work is to investigate the behavior of the phase in simple scattering problems as they frequently occur in mesoscopic physics.…”

Untitled

“…The conductance matrix, Eqs. (23,24) is current conserving if the gate contact is included. The elements of the conductance matrix relating to the gate (contact 3) can be obtained by using the sum rules µ G µν = ν G µν = 0.…”

Interaction‐induced magnetic field asymmetry of nonlinear mesoscopic electrical transport

“…It is well known that the magnetic field symmetry of linear conductance played an important role in the development of a scattering theory that is applicable to a wide range of experimental configurations 14. The magnetic field symmetry of ac transport conductances has similarly been discussed 15, 16, but only a few experimental works are available 17, 18. The self‐consistent, nonlinear scattering approach to dc transport, which we will use here, has been developed in Ref.…”