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arXiv:cond-mat/9811011 · Physical review. B, Condensed matter

Transport in double-crossed Luttinger liquids

P. Durganandini, Sumathi Rao

Published May 15, 1999• 6 citations
arXivHTMLPDF

Abstract

We study transport through two Luttinger liquids (one-dimensional electrons interacting through a Coulomb repulsion in a metal) coupled together at two points. External voltage biases are incorporated through boundary conditions. We include density-density couplings as well as single-particle hops at the contacts. For weak repulsive interactions, transport through the wires remains undisturbed by the interwire couplings, which renormalize to zero. For strong repulsive interactions, the interwire couplings become strong. For symmetric barriers and no external voltage bias, a single gate voltage is sufficient to tune for resonance transmission in both wires. However, for asymmetric couplings or for finite external biases, the system is insulating.

DOI
10.1103/physrevb.59.13122

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We study transport through two Luttinger liquids (one-dimensional electrons interacting through a Coulomb repulsion in a metal) coupled together at two points. External voltage biases are incorporated through boundary conditions. We include density-density couplings as well as single-particle hops at the contacts. For weak repulsive interactions, transport through the wires remains undisturbed by the interwire couplings, which renormalize to zero. For strong repulsive interactions, the interwire couplings become strong. For symmetric barriers and no external voltage bias, a single gate voltage is sufficient to tune for resonance transmission in both wires. However, for asymmetric couplings or for finite external biases, the system is insulating.
Transport through two one-dimensional interacting metals (Luttinger liquids) coupled together at a single point is analyzed. The dominant coupling mechanism is shown to be of electrostatic nature. Describing the voltage sources by boundary conditions then allows for the full solution of the transport problem. For weak Coulomb interactions, transport is unperturbed by the coupling. In contrast, for strong interactions, unusual nonlinear conductance laws characteristic for the correlated system can be observed.
We study theoretically the transport of a one-channel Luttinger liquid through a weak link. For repulsive electron interactions, the electrons are completely reflected by even the smallest scatterer, leading to a truly insulating weak link, in striking contrast to that for noninteracting electrons. At finite temperature (T) the conductance is nonzero, and is predicted to vanish as a power of T. At T=0 power-law current-voltage characteristics are predicted. For attractive interactions, a Luttinger liquid is argued to be perfectly transmitted through even the largest of barriers. The role of Fermi-liquid leads is also explored.
Thermal transport through one-dimensional systems of strongly interacting electrons is studied in two cases: (i) a finite Luttinger liquid (LL) wire containing a strong impurity, and (ii) a LL ring connected by quantum wires to noninteracting electron reservoirs. It is shown that, for (i), a temperature-induced resonant transport takes place under certain conditions. In case (ii) the heat conductance under resonance conditions does not depend on the electron-electron interaction and the system thermal conductance coincides with the maximal Fermi-liquid thermal value.
We have calculated the two-probe Landauer conductance of a one-channel quantum wire containing a Luttinger liquid and connected to two noninteracting leads through tunnel barriers. The tunneling conductance shows broad resonances as a function of the bias voltage, this being a manifestation of the spin-charge separation. In the limit of zero bias and zero barrier, the tunneling conductance reduces to the ballistic contact value e2h\frac{{e}^{2}}{h}he2​ per channel.
We suggest an experiment to study Luttinger liquid behavior in a one-dimensional nanostructure, avoiding the usual complications associated with transport measurements. The proposed setup consists of a quantum box, biased by a gate voltage, and side coupled to a quantum wire by a point contact. Close to the degeneracy points of the Coulomb blockaded box, and in the presence of a magnetic field sufficiently strong to spin polarize the electrons, the setup can be described as a Luttinger liquid interacting with an effective Kondo impurity. Using exact nonperturbative techniques, we predict that the differential capacitance of the box will exhibit distinctive Luttinger liquid scaling with temperature and gate voltage.
We consider theoretically the transport in a one-channel spinless Luttinger liquid with two strong impurities in the presence of dissipation. As a difference with respect to the dissipation free case, where the two impurities fully transmit electrons at resonance points, the dissipation prevents complete transmission in the present situation. A rich crossover diagram for the conductance as a function of applied voltage, temperature, dissipation strength, Luttinger liquid parameter KKK, and the deviation from the resonance condition is obtained. For weak dissipation and 1/2<K<11/2<K<11/2<K<1, the conduction shows a nonmonotonic increase as a function of temperature or voltage. For strong dissipation the conduction increases monotonically but is exponentially small.