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arXiv:cond-mat/9604193 · Physical review. B, Condensed matter

Tunneling through a barrier in a Tomonaga-Luttinger liquid connected to reservoirs

Akira Furusaki, Naoto Nagaosa

Published Aug 15, 1996• 71 citations
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Abstract

The transport through a barrier is studied for a Tomonaga-Luttinger liquid of finite length connected to reservoirs. An effective action for the phase variable at the barrier is derived for spatially varying electric field. In the dc limit only the total voltage drop between the left and right reservoirs appears in the action. We discuss crossovers of the renormalization group flow and hence of the temperature and wire length dependence of the conductance, taking into account the location of the barrier.

DOI
10.1103/physrevb.54.r5239

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The transport through a barrier is studied for a Tomonaga-Luttinger liquid of finite length connected to reservoirs. An effective action for the phase variable at the barrier is derived for spatially varying electric field. In the dc limit only the total voltage drop between the left and right reservoirs appears in the action. We discuss crossovers of the renormalization group flow and hence of the temperature and wire length dependence of the conductance, taking into account the location of the barrier.
The finiteness of the length of a one-channel wire results in a crossover from Tomonaga-Luttinger to Fermi-liquid behavior with lowering energy scale. Under the condition that the voltage drop (V)(V)(V) mostly occurs across a tunnel barrier inside the wire, we determined the coefficients of temperature and voltage expansion for low-energy conductance as a function of the interaction strength and the right and left traversal times. At higher voltage, the finite-length contribution exhibits oscillations related to both traversal times and becomes a slowly decaying correction to the scale-invariant ${V}^{1/g\ensuremath{-}1}$ dependence of the conductance.
As voltage decreases d.c. condctivity of a Tomonaga-Luttinger liquid wire collapses to a small value determined by the length of the wire and its contacts with the leads. In condition that voltage drop (V) mostly occurs across a tunnel barrier inside the wire the tunneling density of states and, hence, the differential conductivity are shown to exhibit an interference structure resulted from the transition of the Luttinger liquid quasiparticles into free electrons at the exits from the wire. The finite length correction to the scale-invariant V2/g−2V^{2/g-2}V2/g−2 dependence of the conductivity oscillates as a function of voltage with periodicities related to both rigth and left traversal times.
We investigate the conductivity using Green's functions in the Tomonaga-Luttinger liquid. It is shown that the resistivity has a linear temperature dependence if we take account of the physical process that the accelerated electron decays into spinon and holon. The effect of impurity scattering becomes visible only below a crossover temperature which strongly depends on the impurity strength, energy cutoff, and spin and charge velocities. The optical conductivity is also studied and it is shown that the relaxation rate has a linear frequency dependence.
We use the technique of bosonization to understand a variety of recent experimental results on the conductivity of a quantum wire. The quantum wire is taken to be a finite-length Luttinger liquid connected on two sides to semi-infinite Fermi liquids through contacts. The contacts are modeled as (short) Luttinger liquids bounded by localized one-body potentials. We use effective actions and the renormalization group to study the effects of electronic interactions within the wire, the length of the wire, finite temperature, and a magnetic field, on the conductivity. We explain the deviations of the conductivity from 2Ne2/h{2Ne}^{2}/h2Ne2/h in wires that are not too short as arising from renormalization effects caused by the repulsive interactions. We also explain the universal conductance corrections observed in different channels at higher temperatures. We study the effects of an external magnetic field on electronic transport through this system and explain why odd and even spin-split bands show different renormalizations from the universal conductance values. We discuss the case of resonant transmission and of the possibility of producing a spin valve that only allows electrons of one value of the spin to go through. We compare our results for the conductance corrections with experimental observations. We also propose an experimental test of our model of the contact regions.
We study the effect that resistive regions have on the conductance of a quantum wire with interacting electrons which is connected to Fermi liquid leads. Using the bosonization formalism and a Rayleigh dissipation function to model the power dissipation, we use both scattering theory and Green's function techniques to derive the DC conductance. The resistive regions are generally found to lead to incoherent transport. For a single wire, we find that the resistance adds in series to the contact resistance of h/e(2) for spinless electrons, and the total resistance is independent of the Luttinger parameter K-W of the wire. We numerically solve the bosonic equations to illustrate what happens when a charge density pulse is incident on the wire; the results depend on the parameters of the resistive and interacting regions in interesting ways. For a junction of Tomonaga-Luttinger liquid wires, we use a dissipationless current splitting matrix to model the junction. For a junction of three wires connected to Fermi liquid leads, there are two families of such matrices; we find that the conductance matrix generally depends on K-W for one family but is independent of K-W for the other family, regardless of the resistances present in the system. Copyright (c) EPLA, 2011
The electrical current through an arbitrary junction connecting quantum wires of spinless interacting fermions is calculated in fermionic representation. The wires are adiabatically attached to two reservoirs at chemical potentials differing by the applied voltage bias. The relevant scale-dependent contributions in perturbation theory in the interaction up to infinite order are evaluated and summed up. The result allows one to construct renormalization group equations for the conductance as a function of voltage (or temperature, wire length). There are two fixed points at which the conductance follows a power law in terms of a scaling variable $\ensuremath{\Lambda}$, which equals the bias voltage VVV, if VVV is the largest energy scale compared to temperature TTT and inverse wire length ${L}^{\ensuremath{-}1}$, and interpolates between these quantities in the crossover regimes.