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arXiv:cond-mat/9411130 · Europhysics Letters (EPL)

Asymptotic Tunnelling Conductance in Luttinger Liquids

F. Guinea, G. Gómez-Santos, Maura Sassetti, Masahito Ueda

Published Jun 20, 1995• 39 citations
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Abstract

Conductance through weak constrictions in Luttinger liquids is shown to vanish with frequency ω as c 1 ω 2 + c 2 ω 2/ g -2 , where g is a dimensionless parameter characterizing the Luttinger-liquid phase, and c 1 and c 2 are non-universal constants. The first term arises from the “Coulomb blockade” effect and dominates for g < 1/2, whereas the second results from eliminating high-energy modes and dominates for g > 1/2.

DOI
10.1209/0295-5075/30/9/010

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Conductance through weak constrictions in Luttinger liquids is shown to vanish with frequency ω as c 1 ω 2 + c 2 ω 2/ g -2 , where g is a dimensionless parameter characterizing the Luttinger-liquid phase, and c 1 and c 2 are non-universal constants. The first term arises from the “Coulomb blockade” effect and dominates for g < 1/2, whereas the second results from eliminating high-energy modes and dominates for g > 1/2.
The temperature-dependent nonlinear conductance for transport of a Luttinger liquid through a barrier is calculated in the nonperturbative regime for g=1/2-\ensuremath{\epsilon}, where g is the dimensionless interaction constant. To describe the low-energy behavior, we perform a leading-logarithmic summation of all diagrams contributing to the conductance which is valid for \ensuremath{\Vert}\ensuremath{\epsilon}\ensuremath{\Vert}\ensuremath{\ll}1. With increasing external voltage, the asymptotic low-temperature behavior displays a turnover from the ${\mathit{T}}^{2/\mathit{g}\mathrm{\ensuremath{-}}2}$ to a universal T2{\mathit{T}}^{2}T2 law.
The nonlinear charge transfer through a tunnel junction between two Luttinger systems is studied for repulsive, finite-range interaction between electrons on the same side, V11{\mathrm{V}}_{11}V11​, and on different sides, V12{\mathrm{V}}_{12}V12​, of a junction. Features of the Coulomb blockade effect are observed if V12{\mathrm{V}}_{12}V12​=0. We predict an interaction-induced enhancement of the current if V12{\mathrm{V}}_{12}V12​>0. When V12{\mathrm{V}}_{12}V12​=V11{\mathrm{V}}_{11}V11​, the current is suppressed at small bias, but the ``charging energy,'' obtained from the asymptotic behavior at high-bias voltage, vanishes.
The theoretical model of the short-range interacting Luttinger liquid predicts a power-law scaling of the density of states and the momentum distribution function around the Fermi surface, which can be readily tested through tunneling experiments. However, some physical systems have long-range interaction, most notably the Coulomb interaction, leading to significantly different behaviors from the short-range interacting system. In this paper, we revisit the tunneling theory for the one-dimensional electrons interacting via the long-range Coulomb force. We show that even though in a small dynamic range of temperature and bias voltage, the tunneling conductance may appear to have a power-law decay similar to short-range interacting systems, the effective exponent is scale-dependent and slowly increases with decreasing energy. This factor may lead to the sample-to-sample variation in the measured tunneling exponents. We also discuss the crossover to a free Fermi gas at high energy and the effect of the finite size. Our work demonstrates that experimental tunneling measurements in one-dimensional electron systems should be interpreted with great caution when the system is a Coulomb Luttinger liquid.
Luttinger liquids occupy a notable place in physics as one of the most understood classes of quantum many-body systems. The experimental mission of measuring its main prediction, power laws in observable quantities, has already produced a body of exponents in different semiconductor and metallic structures. Here, we combine tunneling spectroscopy with density-dependent transport measurements in the same quantum wires over more than two orders of magnitude in temperature to very low electron temperatures down to ∼40 mK. This reveals that, when the second 1D subband becomes populated, the temperature dependence splits into two ranges with different exponents in the power-law dependence of the conductance, both dominated by the finite-size effect of the end-tunneling process. This result demonstrates the importance of measuring the Luttinger parameters as well as the number of modes independently through spectroscopy in addition to the transport exponent in the characterization of Luttinger liquids. This opens a pathway to unambiguous interpretation of the exponents observed in quantum wires.
Several recent results concerning frequency dependent and nonlinear transport of one-dimensional correlated electrons are reviewed. The frequency dependent conductance of a Luttinger liquid is discussed. For zero-range interaction the conductance is similar to that of a non-interacting ideal quantum wire, with a renormalized zero-frequency limit. For a screened Coulomb potential, the conductance as a function of the frequency shows features of the elementary excitations. They can be simulated in terms of a capacitance, an inductance and a resistance. The non-linear DC-conductance of the Luttinger liquid in presence of a tunnel barrier shows Coulomb blockade if the interaction is of finite range. The capacitance scales in the same way as for the system without the tunnel barrier. The linear DC-response of a tunnel barrier as a function of the frequency and the voltage of a monochromatic driving field shows oscillations which can be interpreted as signature of a frequency-induced depinning of a charge density wave.