Nanoscopic Tunneling Contacts on Mesoscopic Multiprobe Conductors

TL;DR

This paper develops a theoretical framework for understanding tunneling contacts between mesoscopic conductors, highlighting their dual role as current sources and sinks, and analyzing magnetic field effects and local density of states.

Contribution

It introduces Bardeen-like expressions for transmission probabilities in multi-probe mesoscopic systems with weak tunneling contacts, including new formulas for partial density of states.

Findings

Transmission probability equals the product of local densities of states in the point contact limit.

The theory describes voltage and resistance measurements in mesoscopic conductors.

Application to a simple scatterer illustrates the development of Hall resistance with weak probes.

Abstract

We derive Bardeen-like expressions for the transmission probabilities between two multi-probe mesoscopic conductors coupled by a weak tunneling contact. We emphasize especially the dual role of a weak coupling contact as a current source and sink and analyze the magnetic field symmetry. In the limit of a point-like tunneling contact the transmission probability becomes a product of local, partial density of states of the two mesoscopic conductors. We present expressions for the partial density of states in terms of functional derivatives of the scattering matrix with respect to the local potential and in terms of wave functions. We discuss voltage measurements and resistance measurements in the transport state of conductors. We illustrate the theory for the simple case of a scatterer in an otherwise perfect wire. In particular, we investigate the development of the Hall-resistance as measured with weak coupling probes.