Landauer formula for the current through an interacting electron region

Abstract

A Landauer formula for the current through a region of interacting electrons is derived using the nonequilibrium Keldysh formalism. The case of proportionate coupling to the left and right leads, where the formula takes an especially simple form, is studied in more detail. Two particular examples where interactions give rise to novel effects in the current are discussed: In the Kondo regime, an enhanced conductance is predicted, while a suppressed conductance is predicted for tunneling through a quantum dot in the fractional quantum Hall regime. The formulation by Landauer [1] and Biittiker [21 of the current through a finite, possibly disordered region of noninteracting electrons has tremendously enhanced the understanding of transport in mesoscopic systems [3]. The Landauer formula, which expresses the current in terms of local properties of the finite region (such as the transmission coefficient) and the distribution functions in connected reservoirs, has been used extensively and successfully in many areas, including the scaling theory of localization [4,5], universal conductance fluctuations [6], Aharonov-Bohm conductance oscillations [7], the integer quantum Hall efTect [8] and its quenching [91, the quantization of ballistic conductance [10], and recently in the field of quantum dynamics of driven systems (quantum chaos) [11].While both the derivation of the Landauer formula for noninteracting electrons [3] and its application are well established, an apt formulation of the current when interactions between electrons are involved has been lacking. In view of the recent technological progress in confinement of electrons into small regions, where the electron-electron interactions plays a major role in the transport [12], it is quite clear that a Landauer-type formula for the transport through such an interacting region is highly desirable. Several attempts have been made to deal with special cases [13]. Langreth [14] was able to express the linear conductance through a single site with an on-site interaction (the Anderson model) at zero temperature in terms of phase shifts and thus relate the conductance to a scattering matrix. Apel and Rice [151 approximated the interaction in one dimension by the values of the momentum transfer Bq at Bq =0 and Bq = 2k~and were able to derive a Landauer-like formula. Unfortunately, this approximation is unsuitable for electrons confined into a small region. More recently, Hershfield, Davis, and Wilkins [16] have been able to derive a formula for the current in the Anderson model.In this Letter (a) we derive an exact formula for the current through a region of interacting electrons coupled to two multichannel leads where the electrons are not interacting. The formula we derive [Eq. (6)] expresses the current, as in the noninteracting case, in terms of the Fermi functions in the leads and local properties of the interacting region. (b) We show how the noninteracting case and the results of Langreth and of Hershfield, Davis, and Wilkins follow as special cases. (c) The curren...

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