Electron injection in a nanotube: Noise correlations and entanglement

Abstract

Transport through a metallic carbon nanotube is considered, where electrons are injected in the bulk by a scanning tunneling microscope tip. The charge current and noise are computed both in the absence and in the presence of one dimensional Fermi liquid leads. For an infinite homogeneous nanotube, the shot noise exhibits effective charges different from the electron charge. Noise correlations between both ends of the nanotube are positive, and occur to second order only in the tunneling amplitude. The positive correlations are symptomatic of an entanglement phenomenon between quasiparticles moving right and left from the tip. This entanglement involves many body states of the boson operators which describe the collective excitations of the Luttinger liquid. I. INTRODUCTIONOver the years, the study of current noise and noise correlations has become a respected and useful diagnosis for transport measurements on mesoscopic conductors. Theoretically, noise was first computed mostly for non-interacting systems 1 . However, it soon became clear that low frequency noise could be used to isolate the quasiparticle charge 2,3 and to study the statistical correlations 4,5 in specific quasi one-dimensional correlated electron systems, such as the edge waves in the quantum Hall effect. In these chiral Luttinger liquids, the charge of the collective excitations along the edges corresponds to the electron charge multiplied by the filling factor.Attention is now turning towards conductors -individual nano-objects -which occur naturally, and which can be connected to current/voltage probes in order to perform a transport experiment. The crucial advantage of such nanoobjects is that they are essentially free of defects and in some circumstances they have an inherent one dimensional character. Carbon nanotubes constitute the archetype of such 1D nano-objects: single wall armchair nanotubes have metallic behavior, with two propagating modes at the Fermi level. Incidentally, electronic correlations are known to play an important role in such systems. Carbon nanotubes seem to constitute good candidates to study Luttinger liquid behavior. In particular, their tunneling density of states -and thus the tunneling I(V ) characteristics is known to have a power law behavior 6,7,8 in accordance with Luttinger liquid theory.Luttinger models for nanotubes differ significantly from their quantum Hall effect counterpart, because of their nonchiral character. Forward and backward fields describing collective excitations effectively mix, because the interactions between electrons are spread along the whole length of the nanotube. For this reason, a straightforward transposition of the results obtained for chiral edge system proves difficult. Nevertheless, non-chiral Luttinger liquids can be described with chiral fields 9,10 . Such chiral fields correspond to excitations with anomalous (non-integer) charge, which has eluded detection so far.In the present work, we propose an experimental geometry which allows to probe directly the underlying charg...

Junctions of multiple quantum wires with different Luttinger parameters

“…Moreover, a single TLL can have inhomogeneities: e.g., a contact between an interacting TLL and a Fermi-liquid lead, a key ingredient of most transport measurements, is often studied as an inhomogeneous TLL wire smoothly interpolating between interacting (TLL) and noninteracting (Fermi-liquid) regions or as a two-wire junction with the Luttinger parameter abruptly changing at the junction. [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56] A junction of three quantum wires with different Luttinger parameters has been studied in the weak coupling regime. 21,[57][58][59] The experimental importance of junctions of TLL wires with generally unequal Luttinger parameters motivates an in-depth study of their properties, which is the main objective of the present paper.…”

Charge fractionalization in nonchiral Luttinger systems

“…When an electron is injected in the bulk of a quantum wire at a given Fermi point, say, −k F , this gives rise to two counterpropagating pieces which carry charge f e and (1 − f )e, respectively, where f = (1 + g)/2 [8,9,10,11,12]. We have shown that the ratio between the asymmetry A S = (I L − I R )/I S and the two-terminal conductance G 2 in the almost equilibrium regime, where the applied bias voltages are small compared to k B T /e, allows to construct a novel dimensionless ratio reflecting the charge fractionalization phenomenon,…”

“…The difference with the case of a point-like tunnel coupling should be noted; in that case, the terms G θφ in Eq. (A.5) would not be there and as a result | I(x → 0) | = I S /2 and therefore on gets I(x → L) = I S /2 [12].…”

“…[12] and extend the analysis to our context where an electron is injected in the lower wire at a single Fermi point, say, −k F . The current takes the general form,…”

“…This fractional charge is not just a statistical average, but is predicted to be a definite property of the quantum mechanical state, subject in principle to verification by repeated measurements after a single tunneling event [8,9,10,11,12].…”