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arXiv:cond-mat/0508752 · Physical Review B

Multiple-particle scattering in quantum point contacts

Dganit Meidan, Yuval Oreg

Published Sep 30, 2005• 18 citations
arXivHTMLPDF

Abstract

Recent experiments performed on weakly pinched quantum point contacts have shown a resistance that tends to decrease at low source drain voltage. We show that enhanced Coulomb interactions, prompted by the presence of the point contact, may lead to anomalously large multiple-particle scattering at finite bias voltage. These processes tend to decrease at low voltage, and thus may account for the observed reduction of the resistance. We concentrate on the case of a normal point contact, and model it by a spinful interacting Tomonaga-Luttinger liquid, with a single impurity, connected to noninteracting leads. We find that sufficiently strong Coulomb interactions enhance two-electron scatterings, which dominate the conductance. Our calculation shows that the effective charge, probed by the shot noise of such a system, approaches a value proportional to ${e}^{*}=2e$ at a sufficiently large backscattering current. This distinctive hallmark may be tested experimentally. We discuss possible applications of this model to experiments conducted on Hall bars.

DOI
10.1103/physrevb.72.121312

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•Jan 17, 2013•0 citations
Recent experiments performed on weakly pinched quantum point contacts have shown a resistance that tends to decrease at low source drain voltage. We show that enhanced Coulomb interactions, prompted by the presence of the point contact, may lead to anomalously large multiple-particle scattering at finite bias voltage. These processes tend to decrease at low voltage, and thus may account for the observed reduction of the resistance. We concentrate on the case of a normal point contact, and model it by a spinful interacting Tomonaga-Luttinger liquid, with a single impurity, connected to noninteracting leads. We find that sufficiently strong Coulomb interactions enhance two-electron scatterings, which dominate the conductance. Our calculation shows that the effective charge, probed by the shot noise of such a system, approaches a value proportional to e∗=2e{e}^{*}=2ee∗=2e at a sufficiently large backscattering current. This distinctive hallmark may be tested experimentally. We discuss possible applications of this model to experiments conducted on Hall bars.
We study a quantum point contact in a quantum spin Hall insulator. It has recently been shown that the Luttinger liquid theory of such a structure maps to the theory of a weak link in a Luttinger liquid with spin with Luttinger liquid parameters ${g}_{\ensuremath{\rho}}=1/{g}_{\ensuremath{\sigma}}=g<1$. We show that for weak interactions, 1/2<g<11/2<g<11/2<g<1, the pinch-off of the point contact as a function of gate voltage is controlled by a novel quantum critical point, which is a realization of a nontrivial intermediate fixed point found previously in the Luttinger liquid model with spin. We predict that the dependence of the conductance on gate voltage near the pinch-off transition for different temperatures collapses onto a universal curve described by a crossover scaling function associated with that fixed point. We compute the conductance and critical exponents of the critical point as well as the universal scaling function in solvable limits, which include $g=1\ensuremath{-}ϵ$, g=1/2+ϵg=1/2+ϵg=1/2+ϵ, and g=1/3g=1/\sqrt{3}g=1/3​. These results, along with a general scaling analysis, provide an overall picture of the critical behavior as a function of ggg. In addition, we analyze the structure of the four-terminal conductance of the point contact in the weak tunneling and weak backscattering limits. We find that different components of the conductance can have different temperature dependences. In particular, we identify a skew conductance GXY{G}_{XY}GXY​, which we predict vanishes as ${T}^{\ensuremath{\gamma}}$ with $\ensuremath{\gamma}\ensuremath{\ge}2$. This behavior is a direct consequence of the unique edge state structure of the quantum spin Hall insulator. Finally, we show that for strong interactions, g<1/2g<1/2g<1/2, the presence of spin nonconserving spin-orbit interactions leads to a novel time-reversal-symmetry breaking insulating phase. In this phase, the transport is carried by spinless chargons and chargeless spinons. These lead to nontrivial correlations in the low frequency shot noise. Implications for experiments on HgCdTe quantum well structures will be discussed.
It is well known that while forward scattering has no effect on the conductance of one-dimensional systems, backscattering off a static impurity suppresses the current. We study the effect of a time-dependent point impurity on the conductance of a one-channel quantum wire. At strong repulsive interaction (Luttinger liquid parameter g<1/2), backscattering renders the conductance greater than its value e^2/h in the absence of the impurity. A possible experimental realization of our model is a constricted Hall bar at fractional filling factors ν=1/(2n+1) with a time-dependent voltage at the constriction.
We investigate how the conductance GGG through a clean interacting quantum wire is affected by the presence of contacts and noninteracting leads. The contacts are defined by a vanishing two-particle interaction to the left and a finite repulsive interaction to the right or vice versa. No additional single-particle scattering terms (impurities) are added. We first use Bosonization and the local Luttinger liquid picture and show that within this approach GGG is determined by the properties of the leads regardless of the details of the spatial variation of the Luttinger liquid parameters. This generalizes earlier results obtained for steplike variations. In particular, no single-particle backscattering is generated at the contacts. We then study a microscopic model applying the functional renormalization group and show that the spatial variation of the interaction produces single-particle backscattering, which in turn leads to a reduced conductance. We investigate how the smoothness of the contacts affects GGG and show that for decreasing energy scale its deviation from the unitary limit follows a power law with the same exponent as obtained for a system with a weak single-particle impurity placed in the contact region of the interacting wire and the leads.
We study the current produced in a Tomonaga-Luttinger liquid by an applied bias and by weak, pointlike impurity potentials which are oscillating in time. We use bosonization to perturbatively calculate the current up to second order in the impurity potentials. In the regime of small bias and low pumping frequency, both the dc and ac components of the current have power-law dependences on the bias and pumping frequencies with an exponent $2K\ensuremath{-}1$ for spinless electrons, where KKK is the interaction parameter. For K<1∕2K<1∕2K<1∕2, the current grows large for special values of the bias. For noninteracting electrons with K=1K=1K=1, our results agree with those obtained using Floquet scattering theory for Dirac fermions. We also discuss the cases of extended impurities and of spin-1∕21∕21∕2 electrons.
We study a homogeneous Tomonaga-Luttinger liquid with backscattering potential. A perturbative computation of the conductance at and near resonance is given. We find that the backscattering of one electron dominates that of two electrons for an interaction parameter KKK\ensuremath{\geqslant}13\frac{1}{3}31​ and that the resonance point depends on temperature. Our results may be relevant for recent experiments on shot noise in the fractional quantum Hall effect, where the charge 13\frac{1}{3}31​ and not 23\frac{2}{3}32​ is measured on resonance.
In one-dimensional conductors, Coulomb interactions result in correlated electronic systems called Tomonaga-Luttinger liquids (TLL). The TLL physics also applies to other many-body phenomena, providing complementary viewpoints while benefiting from the powerful TLL framework. One such phenomenon is the low-energy conductance suppression of a quantum coherent conductor embedded in a dissipative circuit, an effect called dynamical Coulomb blockade. Here we investigate the basic class of mesoscopic circuits constituted by a short single-channel quantum conductor in series with a resistance R. Remarkably, such circuits can be mapped on a TLL of interaction parameter 1/(1+Re^2/h), with an impurity. From this mapping, generalized to realistic dissipative circuits, a scaling law for the suppressed conductance is derived at R=h/e^2, and small deviations computed for R \neq h/e^2. We find that the scaling law is obeyed by our data for arbitrary quantum channels, emulated by a Ga(Al)As quantum point contact, and by the recent data of Mebrahtu et al. [Nature 488, 61 (2012)] obtained using a carbon nanotube. This demonstrates a highly tunable test-bed for TLL physics, and consolidates a recently proposed phenomenological expression for the conductance of a quantum channel in a linear circuit.