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Journal of the Physical Society of Japan

Landauer Type Approach to Electron Conduction in One-Dimensional Systems and Interaction Effects

Arisato Kawabata

Published Jul 15, 1998• 4 citations
Publisher

Abstract

Electron conduction in one-dimensional system is investigated explicitly taking into account the reservoirs to which 1D channel is connected at its ends. We calculate the current driven by the chemical potential difference of two reservoirs, using the linear response theory by Kubo. At 0K, the conductance of 1D channel is expressed in terms of one-electron Green's functions even in the presence of the interaction, in contrast to the theory based on Kubo formula, in which two-electron Green's functions are essentially involved. The formalism in this paper provides a new interpretation of Landauer formula. As regards the interaction effects, the conductance of Fermi liquid without potential scattering is 2 e 2 / h , while that of Tomonaga-Luttinger liquid deceases with power low as the length of 1D channel increases.

DOI
10.1143/jpsj.67.2430

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Electron conduction in one-dimensional system is investigated explicitly taking into account the reservoirs to which 1D channel is connected at its ends. We calculate the current driven by the chemical potential difference of two reservoirs, using the linear response theory by Kubo. At 0K, the conductance of 1D channel is expressed in terms of one-electron Green's functions even in the presence of the interaction, in contrast to the theory based on Kubo formula, in which two-electron Green's functions are essentially involved. The formalism in this paper provides a new interpretation of Landauer formula. As regards the interaction effects, the conductance of Fermi liquid without potential scattering is 2 e 2 / h , while that of Tomonaga-Luttinger liquid deceases with power low as the length of 1D channel increases.
The conductance of one-dimensional interacting electron systems is calculated in a manner similar to Landauer's argument for non-interacting systems.Unlike in previous studies in which the Kubo formula was used, the conductance is directly evaluated as the ratio of current J to the chemical potential difference ∆µ between right-going and left-going particles.It is shown that both J and ∆µ are renormalized by electron-electron (e-e) interactions, but their ratio, the conductance, is not renormalized at all if the e-e interactions are the only scattering mechanism.It is also shown that nonequilibrium current fluctuation at low frequency is absent in such a case.These conclusions are drawn for both Fermi liquids (in which quasi-particles are accompanied with the backflow) and Tomonaga-Luttinger liquids.
The conductance of one-dimensional interacting electron systems is calculated in a manner similar to Landauer's argument for non-interacting systems. Unlike in previous studies in which the Kubo formula was used, the conductance is directly evaluated as the ratio of current JJJ to the chemical potential difference ΔμΔμΔμ between right-going and left-going particles. It is shown that both JJJ and ΔμΔμΔμ are renormalized by electron-electron (e-e) interactions, but their ratio, the conductance, is not renormalized at all if the e-e interactions are the only scattering mechanism. It is also shown that nonequilibrium current fluctuation at low frequency is absent in such a case. These conclusions are drawn for both Fermi liquids (in which quasi-particles are accompanied with the backflow) and Tomonaga-Luttinger liquids.
The transport in a pure one-dimensional quantum wire is investigated for any range of interactions. First, the wire is connected to measuring leads. The transmission of an incident electron is found to be perfect, i.e., the conductance is not renormalized by the interactions. Either Landauer's approach or the Kubo formula can be used as long as the reservoirs impose the boundary conditions. Second, the Kubo formula as a response to the local field is reconsidered in a generic Luttinger liquid: the ``intrinsic'' conductance thus obtained is determined by the same combination of interaction parameters as that which renormalizes the current.
We describe microscopic theory for the quantum transport through finite interacting systems connected to noninteracting leads. It can be applied to small systems such as quantum dots, quantum wires, atomic chain, molecule, and so forth. The Keldysh formalism is introduced to study the nonlinear current-voltage characteristics, and general properties of the nonequilibrium Green's functions are provided. We apply the formulated to an out-of-equilibrium Anderson model that has been used widely for the Kondo effect in quantum dots. In the linear-response regime the Kubo formalism still has an advantage, because it relates the transport coefficients directly to the correlation functions defined with respect to the thermal equilibrium, and it has no ambiguities about a profile of the inner electric field. We introduce a many-body transmission coefficient, by which the dc conductance can be expressed in a Landauer-type form, quite generally. We also discuss transport properties of the Tomonaga-Luttinger liquid, which has also been an active field of research in this decade.
We review some of our recent results on the potential scattering in a weakly interacting one-dimensional(1D) electron gas. The technique we developed is a poor man's renormalization group procedure in the scattered wave basis. This technique can treat the renormalizations of the scattering on the barrier and the scattering between the electrons in a coherent way, and it allows us to find the scattering amplitudes on a localized potential of arbitrary strength for electrons at any energy. The obtained phase shifts are used to study the Fermi-edge singularity in an interacting 1D electron system, where anomalous exponent of the power-law singularity in the vicinity of the edge is found. The transmission coefficient is directly related to the conductance of a 1D channel by the Landauer formula. Simple formulas that describe the conductance at any temperature are derived. In spin-[Formula: see text] systems, the electron–electron backscattering induces renormalizations of the interaction constants, which causes the low-temperature conductance to deviate from the results of the Luttinger liquid theory. In particular, the temperature dependence of the conductance may become nonmonotonic. In the presence of a magnetic field, backscattering gives rise to a peak in the differential conductance at bias equal to the Zeeman splitting.
We investigate the effect of electron-electron interactions on the conductance of quasi one-dimensional systems without potential scattering. For a finite temperature or system length, the short-range interaction is not renormalized to 0, and it gives rise to a finite correction to the conductance if we calculate it using Kubo formula. We show that this correction can be absorbed into the renormalization of the chemical potential and that the properly defined conductance to be observed in the experiments is equal to that of non-interacting electrons.