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arXiv:cond-mat/0008188 · APS March Meeting Abstracts

Measuring Fractional Charge in Carbon Nanotubes

Cristina Bena B, Smitha Vishveshwara, Leon Balents, Matthew P. A. Fisher

Published Aug 11, 2000• 1 citation
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Abstract

The Luttinger model of the one-dimensional Fermi gas is the cornerstone of modern understanding of interacting electrons in one dimension. In fact, the enormous class of systems whose universal behavior is adiabatically connected to it are now deemed Luttinger liquids. Recently, it has been shown that metallic single-walled carbon nanotubes are almost perfectly described by the Luttinger Hamiltonian. Indeed, strongly non-Fermi liquid behavior has been observed in a variety of DC transport experiments, in very good agreement with theoretical predictions. Here, we describe how fractional quasiparticle charge, a fundamental property of Luttinger liquids, can be observed in impurity-induced shot noise.

DOI
10.48550/arxiv.cond-mat/0008188

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The Luttinger model of the one-dimensional Fermi gas is the cornerstone of modern understanding of interacting electrons in one dimension. In fact, the enormous class of systems whose universal behavior is adiabatically connected to it are now deemed Luttinger liquids. Recently, it has been shown that metallic single-walled carbon nanotubes are almost perfectly described by the Luttinger Hamiltonian. Indeed, strongly non-Fermi liquid behavior has been observed in a variety of DC transport experiments, in very good agreement with theoretical predictions. Here, we describe how fractional quasiparticle charge, a fundamental property of Luttinger liquids, can be observed in impurity-induced shot noise.
Theoretical studies suggest that an interacting one-dimensional electron system does not behave as a Fermi liquid. Instead, it is predicted to form a Luttinger liquid characterized e.g. by the absence of fermionic quasiparticles and interaction-dependent power-laws for transport quantities. So far, however, only a limited number of experiments have provided evidence for Luttinger liquid behavior. In this thesis we consider alternative methods to detect electronic correlations in one-dimensional electron systems based on our calculations of persistent currents and quantization of charge in ballistic single channel rings. The persistent current in an isolated impurity-free Luttinger liquid ring is found to be the same as for a system of non-interacting electrons. In contrast, a Luttinger liquid behavior is revealed if the ring is weakly connected to an electron reservoir and capacitively coupled to a gate electrode. The persistent current and the average charge in the ring are shown to change in a stepwise manner with the gate voltage at low temperatures. Both the step positions and the crossover temperatures are determined by the Luttinger liquid correlation parameter .alpha.(.alpha.c) which therefore can be measured. When a potential barrier is incorporated into the ring, the presence of strong interactions between electrons leads to a suppression of the persistent current and a change in its temperature dependence. A sharp maximum of the current is predicted to occur as a function of temperature. When two barriers are introduced into the ring, the possibility of resonant tunneling for the persistent current opens up. Two kinds of resonant behavior are discovered in the closed geometry: a current independent of the barrier transparency and a current analogous to the one with a single potential barrier. The persistent current in a ring containing a gas of fractionally charged excitations obeying fractional exclusion statistics is also studied. At finite temperatures, anomalous oscillations of the current are predicted as a manifestation of the statistics of the quasiparticles.
Charge excitations in a two dimensional electron gas, under a quantizing magnetic field and in the fractional quantum Hall effect regime, flow in one dimensional-like strips along the edges of the sample. These excitations (quasiparticles) may be independent or condense into an interacting chiral Luttinger liquid. Adding a backscattering potential, which reflects a forward propagating quasiparticle to a backward propagating one, partitions the stream of quasiparticles and induces quantum shot noise. The noise is proportional to quasiparticles charge and may be affected by their mutual interactions. The dependence of the determined charge on the temperature, excitation energy, and partitioning will be describes for a few fractional states, revealing in some cases a universal behavior.
The electronic transport properties of single-walled nanotubes (SWNTs) are characterized by the critical exponent $\ensuremath{\Theta}$, describing the behavior of the single-particle spectral function near the Fermi energy as $\ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\mid}\ensuremath{\omega}\ensuremath{\mid}}^{\ensuremath{\Theta}}$. Experimentally obtained $\ensuremath{\Theta}$ values have been reported in the range of 0.43--0.48 for metallic SWNTs. However, these values are much larger than the theoretical upper limit of the exponent, ${\ensuremath{\Theta}}_{\mathrm{max}}=1∕8$, for the Hubbard model. Here, we show that the observed electronic transport properties can be explained by taking into account a specific hard-core Coulomb interaction. For a many-body system with a hard-core potential, we analytically calculate the critical exponents using the Bethe ansatz and conformal field theory. We find strongly interacting Luttinger liquid states that are characterized by large $\ensuremath{\Theta}$ values, consistent with the empirical results, suggesting that these states are actually realized in metallic SWNTs and organic conductors.
The current noise of a voltage biased interacting quantum wire adiabatically connected to metallic leads is computed in presence of an impurity in the wire. We find that in the weak backscattering limit the Fano factor characterizing the ratio between shot noise and backscattering current crucially depends on the noise frequency relative to the ballistic frequency vF/gL, where vF is the Fermi velocity, g the Luttinger liquid interaction parameter, and L the length of the wire. In contrast to chiral Luttinger liquids, the noise is not only due to the Poissonian backscattering of fractionally charged quasiparticles at the impurity, but also depends on Andreev-type reflections of plasmons at the contacts, so that the frequency dependence of the noise needs to be analyzed to extract the fractional charge of the bulk excitations. We show that the frequencies needed to see interaction effects in the Fano factor are within experimental reach.
Using the appropriate fractionalization mechanism, we correctly derive the temperature (T) and interaction dependence of the electron lifetime ${\ensuremath{\tau}}_{F}$ in Luttinger liquids. For strong enough interactions, we report that $(T{\ensuremath{\tau}}_{F})\ensuremath{\propto}g,$ with $g\ensuremath{\ll}1$ being the standard Luttinger exponent; this reinforces that electrons are not good quasiparticles. We immediately emphasize that this is of importance for the detection of electronic interferences in ballistic one-dimensional rings and carbon nanotubes, inducing ``dephasing'' (strong reduction of Aharonov-Bohm oscillations).
With the surge of research in quantum information, the issue of producing entangled states has gained prominence. Here, we show that judiciously bringing together two systems of strongly interacting electrons with vastly differing ground states-the gapped BCS superconductor and the Luttinger liquid-can result in quantum entanglement. We propose three sets of measurements involving single-walled metallic carbon nanotubes which have been shown to exhibit Luttinger liquid physics, to test our claim and as nanoscience experiments of interest in and of themselves.