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arXiv:cond-mat/9408094 · Physical review. B, Condensed matter

Transport properties of a quantum wire in the presence of impurities and long-range Coulomb forces

H. Maurey, Thierry Giamarchi

Published Apr 15, 1995• 47 citations
arXivHTMLPDF

Abstract

One-dimensional electron systems interacting with long-range Coulomb forces (quantum wires) show a Wigner crystal structure. We investigate in this paper the transport properties of such a Wigner crystal in the presence of impurities. Contrary to what happens when only short-range interactions are included, the system is dominated by 4${\mathit{k}}_{\mathit{F}}$ scattering on the impurities. There are two important length scales in such a problem: one is the pinning length above which the (quasi-)long-range order of the Wigner crystal is destroyed by disorder. The other length, ${\ensuremath{\xi}}_{\mathrm{cr}}$, is the length below which Coulomb interactions are not important and the system is behaving as a standard Luttinger liquid with short-range interactions. We obtain the frequency and temperature dependence of the conductivity. We show that such a system is very similar to a classical charge density wave pinned by impurities, but with important differences due to quantum fluctuations and long-range Coulomb interactions. Finally, we discuss our results in comparison with experimental systems.

DOI
10.1103/physrevb.51.10833

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One-dimensional electron systems interacting with long-range Coulomb forces (quantum wires) show a Wigner crystal structure. We investigate in this paper the transport properties of such a Wigner crystal in the presence of impurities. Contrary to what happens when only short-range interactions are included, the system is dominated by 4kF{\mathit{k}}_{\mathit{F}}kF​ scattering on the impurities. There are two important length scales in such a problem: one is the pinning length above which the (quasi-)long-range order of the Wigner crystal is destroyed by disorder. The other length, \ensuremathξcr{\ensuremath{\xi}}_{\mathrm{cr}}\ensuremathξcr​, is the length below which Coulomb interactions are not important and the system is behaving as a standard Luttinger liquid with short-range interactions. We obtain the frequency and temperature dependence of the conductivity. We show that such a system is very similar to a classical charge density wave pinned by impurities, but with important differences due to quantum fluctuations and long-range Coulomb interactions. Finally, we discuss our results in comparison with experimental systems.
A one-dimensional gas of electrons interacting with long-range Coulomb forces [V(r)\ensuremath{\approxeq}1/r] is investigated. The excitation spectrum consists of separate collective charge and spin modes. For arbitrarily weak Coulomb repulsion density correlations at wave vector 4kF{\mathit{k}}_{\mathit{F}}kF​ decay extremely slowly and are best described as those of a one-dimensional Wigner crystal. Pinning of the Wigner crystal then leads to the nonlinear transport properties characteristic of charge density waves. The results allow a consistent interpretation of the plasmon and spin excitations observed in one-dimensional semiconductor structures, and suggest an interpretation of some of the observed features in terms of ``spinons.''
The effects of a long-range electronic potential on a one-dimensional chain of spinless fermions are investigated by numerical techniques (exact diagonalization of rings with up to 30 sites complemented by finite size analysis) and analytic calculations. Due to a competition between the 2kF{2k}_{F}2kF​ oscillations in the density and the (very slow) log divergence of the long wavelength part of the Coulomb potential, the metallic character of the system is enhanced at intermediate (and up to quite large) length scales. Despite some similarities, we found that this quasimetallic regime is not of the Luttinger liquid type, as evidenced by strong deviations from predictions of conformal field theory and in agreement with the picture of a (very weakly pinned) Wigner crystal. When the strength of the Coulomb interaction is substantially increased, we observe a smooth crossover to a strongly localized charge density wave.
The effects of a long range electronic potential on a one dimensional chain of spinless fermions are investigated by numerical techniques and analytic calculations. Due to a competition between the 2kF oscillations in the density and the log divergence of the long wavelength part of the Coulomb potential, the metallic character of the system is enhanced at intermediate (and up to quite large) lengthscales. We found that this quasi-metallic regime is not of the Luttinger Liquid type, as evidenced by strong deviations from predictions of conformal field theory and in agreement with the picture of a (very weakly pinned) Wigner crystal. When the strength of the Coulomb interaction is substantially increased, we observe a smooth cross-over to a strongly localized charge density wave.
A model is found whereby the Coulomb interaction during electron transport in quantum wires of finite length with current-lead contacts can be taken into account exactly (within the framework of the bosonization method). It is established that the charge density distribution along a wire in the case of the real Coulomb interaction is strongly different from the Luttinger liquid model, where the interaction is assumed to be short-ranged. The charge density near the contacts is found to be much higher than in the model with a short-range interaction, but away from the contacts the two densities are closer to each other.
We examine the effects of disorder in one-dimensional systems. We link the case of a few impurities, typical of a short quantum wire, to that of a finite density of scatterers more appropriate for a long wire or a macroscopic system. Finally we investigate the effects of long-range interactions on the transport in 1D systems. We predict in that case a conductivity behaving as σ(T)∼T2σ(T) \sim T^2σ(T)∼T2.}
Two Luttinger liquids, with an equal density and opposite sign of charge carriers, may exhibit enhanced excitonic correlations. We term such a system an exciton quasicondensate, with a possible realization being two parallel oppositely doped quantum wires, coupled by repulsive Coulomb interactions. We show that this quasiexciton condensate can be stabilized in an extended range of parameters, in both spinless and spinful systems. We calculate the interwire tunneling current-voltage characteristic, and find that a negative differential conductance is a signature of excitonic correlations. For spinful electrons, the excitonic regime is shown to be distinct from the usual quasi-long-range ordered Wigner crystal phase characterized by power-law density wave correlations. The two phases can be clearly distinguished through their interwire tunneling current-voltage characteristics. In the quasiexciton condensate regime the tunneling conductivity diverges at low temperatures and voltages, whereas in the Wigner crystal it is strongly suppressed. Both the Wigner crystal and the excitonic regime are characterized by a divergent Coulomb drag at low temperature. Finally, metallic carbon nanotubes are considered as a special case of such a one-dimensional setup, and it is shown that exciton condensation is favorable due to the additional valley degree of freedom.