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Physical Review Letters

Mesoscopic Voltage Fluctuations in Luttinger Liquids

Scot R. Renn, Daniel P. Arovas

Published May 26, 1997• 3 citations
Publisher

Abstract

In this Letter, the mesoscopic voltage and conductance fluctuations of a Luttinger liquid are examined. For quasiballistic wires, one finds that the amplitude of conductance fluctuations, i.e., $\mathrm{Var}(G)$, is proportional to ${T}^{4g\ensuremath{-}5}$ where $g\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\pi}\ensuremath{\Elzxh}v(\ensuremath{\partial}n/\ensuremath{\partial}\ensuremath{\mu})$ is a dimensionless measure of compressibility. For macroscopic wires, one finds resistance fluctuations of the form $\mathrm{Var}(R)\ensuremath{\sim}{T}^{4g\ensuremath{-}5}$. We also investigate the voltage distribution function and the length scale over which self-averaging occurs.

DOI
10.1103/physrevlett.78.4091

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In this Letter, the mesoscopic voltage and conductance fluctuations of a Luttinger liquid are examined. For quasiballistic wires, one finds that the amplitude of conductance fluctuations, i.e., Var(G)\mathrm{Var}(G)Var(G), is proportional to T4g\ensuremath−5{T}^{4g\ensuremath{-}5}T4g\ensuremath−5 where g=\ensuremathπ\ensuremath\Elzxhv(\ensuremath∂n/\ensuremath∂\ensuremathμ)g\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\pi}\ensuremath{\Elzxh}v(\ensuremath{\partial}n/\ensuremath{\partial}\ensuremath{\mu})g=\ensuremathπ\ensuremath\Elzxhv(\ensuremath∂n/\ensuremath∂\ensuremathμ) is a dimensionless measure of compressibility. For macroscopic wires, one finds resistance fluctuations of the form Var(R)\ensuremath∼T4g\ensuremath−5\mathrm{Var}(R)\ensuremath{\sim}{T}^{4g\ensuremath{-}5}Var(R)\ensuremath∼T4g\ensuremath−5. We also investigate the voltage distribution function and the length scale over which self-averaging occurs.
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.
Recent experiments have revealed that the temperature dependence of the conductance of quasiballistic quantum wires bears characteristic features of the Luttinger-liquid state. In this paper, the conductance of an N-channel quantum wire is calculated within the model of N coupled Luttinger liquids and under the assumption of weak disorder. It is shown that as the number of channels increases, a crossover from the Luttinger-liquid to the Fermi-liquid behavior occurs. This crossover manifests itself in the 1/N decrease of the scaling exponent of the temperature dependence. An exact expression for the scaling exponent for the case of N coupled Luttinger chains is obtained, and the large N limit is studied for the case of a quantum wire. The case of N=2 for electrons with spin is analyzed in detail, and a qualitative agreement with the experiment is achieved.
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.
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.
Nonlinear current voltage characteristics of a disordered Luttinger liquid are calculated using a perturbative formalism. One finds the nonuniversal power-law characteristics of the form I(V)\ensuremath{\sim}${\mathit{V}}^{1/(2\mathit{g}\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}\mathrm{\ensuremath{-}}1)}$, which is valid both in the superfluid phase when I is small and also in the insulator phase when I is large. Mesoscopic voltage fluctuations are also calculated. One finds Var(\ensuremath{\Delta}V)\ensuremath{\sim}${\mathit{I}}^{4\mathit{g}\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}\mathrm{\ensuremath{-}}3}$. Both the I(V) characteristic and the voltage fluctuations exhibit universal power-law behavior at the superfluid insulator transition where g\ifmmode \tilde{}\else \~{}\fi{}=3/2. The possible application of these results to the nonlinear transport properties of gated Hall bars is discussed.
In a first part, I study the electronic transport through a one-dimensional, finite-length quantum wire of correlated electrons (Luttinger liquid) coupled at arbitrary position via tunnel barriers to two semi-infinite, one-dimensional as well as stripe-like (two-dimensional) leads, thereby bringing theory closer towards systems resembling setups realized in experiments. In particular, I compute the temperature dependence of the linear conductance of a system without bulk impurities. The appearance of new temperature scales introduced by the lengths of the overhanging parts of the leads and the wire implies a conductance function which is much more complex than the simple power-law behavior obtained in earlier approaches. My results can be used to optimize the experimental setups designed for a verification of Luttinger-liquid scaling. In a second part, I suggest a setup to study Luttinger-liquid behavior in quantum wires which allows to determine the Luttinger-liquid parameter from two independent measurements: transport through a quantum dot embedded in the wire and the charge on the dot. To this end, I identified novel Luttinger-liquid power-laws in the charging of the dot. In a more technical part, I extended an adaption of the functional renormalization group such that it can be used on correlated electrons in complex geometries like the wire-lead structures described above and arbitrary dot structures coupled to Luttinger-liquid leads.