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arXiv:1009.5052 · Physical Review B

Rectification in Y-junctions of Luttinger liquid wires

Chenjie Wang, Dima E. Feldman

Published Jan 20, 2011• 10 citations
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Abstract

We investigate rectification of a low-frequency ac bias in Y-junctions of one-channel Luttinger liquid wires with repulsive electron interaction. Rectification emerges due to three scatterers in the wires. We find that it is possible to achieve a higher rectification current in a Y-junction than in a single wire with an asymmetric scatterer at the same interaction strength and voltage bias. The rectification effect is the strongest in the absence of the time-reversal symmetry. In that case, the maximal rectification current can be comparable to the total current $~$${e}^{2}V/h$ even for low voltages, weak scatterers, and modest interaction strength. In a certain range of low voltages, the rectification current can grow as the voltage decreases. This leads to a bump in the $I$-$V$ curve.

DOI
10.1103/physrevb.83.045302

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We investigate rectification of a low-frequency ac bias in Y-junctions of one-channel Luttinger liquid wires with repulsive electron interaction. Rectification emerges due to three scatterers in the wires. We find that it is possible to achieve a higher rectification current in a Y-junction than in a single wire with an asymmetric scatterer at the same interaction strength and voltage bias. The rectification effect is the strongest in the absence of the time-reversal symmetry. In that case, the maximal rectification current can be comparable to the total current ~$${e}^{2}V/h even for low voltages, weak scatterers, and modest interaction strength. In a certain range of low voltages, the rectification current can grow as the voltage decreases. This leads to a bump in the III-VVV curve.
Asymmetric current-voltage [I(V)][I(V)][I(V)] curves, known as the diode or rectification effect, in one-dimensional electronic conductors can have their origin from scattering off a single asymmetric impurity in the system. We investigate this effect in the framework of the Tomonaga-Luttinger model for electrons with spin. We show that electron interactions strongly enhance the diode effect and lead to a pronounced current rectification even if the impurity potential is weak. For strongly interacting electrons and not too small voltages, the rectification current ${I}_{r}=[I(V)+I(\ensuremath{-}V)]∕2$, measuring the asymmetry in the current-voltage curve, has a power-law dependence on the voltage with a negative exponent, ${I}_{r}\ensuremath{\sim}{V}^{\ensuremath{-}\ensuremath{\mid}z\ensuremath{\mid}}$, leading to a bump in the current-voltage curve.
We calculate the conductances of a three-way junction of spinless Luttinger-liquid wires as functions of bias voltages applied to three independent Fermi-liquid reservoirs. In particular, we consider the setup that is characteristic of a tunneling experiment, in which the strength of electron-electron interactions in one of the arms of the junction (``tunneling tip'') is different from that in the other two arms (which together form a ``main wire''). The scaling dependence of the two independent conductances on bias voltages is determined within a fermionic renormalization-group approach in the limit of weak interactions. The solution shows that, in general, the conductances scale with the bias voltages in an essentially different way compared to their scaling with the temperature TTT. Specifically, unlike in the two-terminal setup, the nonlinear conductances cannot be generically obtained from the linear ones by simply replacing TTT with the ``corresponding'' bias voltage or the largest one. Remarkably, a finite tunneling bias voltage prevents the interaction-induced breakup of the main wire into two disconnected pieces in the limit of zero TTT and a zero source-drain voltage.
We study charge transport through NNN-lead junctions ($N\ensuremath{\ge}3$) of spinless Luttinger-liquid wires with bias voltages applied to Fermi-liquid reservoirs. In particular, we consider a Y junction, which is a setup characteristic of the tunneling experiment. In this setup, the strength of electron-electron interactions in one of the arms (``tunneling tip'') is different from that in the other two arms (which form together the ``main wire''). For a generic single-particle SSS matrix of the junction, we find that the bias voltage VVV applied, even symmetrically, to the main wire generates a current proportional to ∣V∣|V|∣V∣ in the tip wire. We identify two mechanisms of this nonequilibrium-induced ``emergent chirality'' in a setup characterized by the time-reversal- and parity-symmetric Hamiltonian of the junction. These are (i) the emergence of an effective magnetic flux, which breaks time-reversal symmetry, and (ii) the emergence of parity-breaking asymmetry of the setup, both proportional to the interaction strength and the sign of the voltage. The current in the tip wire generated by mechanism (i) is reminiscent of the Hall current in the linear response of a system the Hamiltonian of which breaks time-reversal symmetry; however, in the absence of any magnetic field or a local magnetic moment. Similarly, mechanism (ii) can be thought of as an emergent ``photogalvanic effect,'' however, in the presence of inversion symmetry within the main wire. The nonequilibrium chirality implies a rectification of the current in the tip when the main wire is biased by ac voltage.
We calculate both the dc and the ac Josephson current through a one-dimensional system of interacting electrons, connected to two superconductors by tunnel junctions. We treat the (repulsive) Coulomb interaction in the framework of the one-channel, spin-1/2 Luttinger model. The Josephson current is obtained for two geometries of experimental relevance: a quantum wire and a ring. At zero temperature, the critical current is found to decay algebraically with increasing distance d between the junctions. The decay is characterized by an exponent which depends on the strength of the interaction. At finite temperatures T, lower than the superconducting transition temperature Tc{\mathit{T}}_{\mathit{c}}Tc​, there is a crossover from algebraic to exponential decay of the critical current as a function of d, at a distance of the order of \ensuremath{\Elzxh}vF{\mathit{v}}_{\mathit{F}}vF​/kBT{\mathit{k}}_{\mathit{BT}}kBT​. Moreover, the dependence of critical current on temperature shows nonmonotonic behavior. If the Luttinger liquid is confined to a ring of circumference L, coupled capacitively to a gate voltage and threaded by a magnetic flux, the Josephson current shows remarkable parity effects under the variation of these parameters. For some values of the gate voltage and applied flux, the ring acts as a \ensuremath{\pi} junction. These features are robust against thermal fluctuations up to temperatures on the order of \ensuremath{\Elzxh}vF{\mathit{v}}_{\mathit{F}}vF​/kBL{\mathit{k}}_{\mathit{BL}}kBL​. For the wire geometry, we have also studied the ac-Josephson effect. The amplitude and the phase of the time-dependent Josephson current are affected by electron-electron interactions. Specifically, the amplitude shows pronounced oscillations as a function of the bias voltage due to the difference between the velocities of spin and charge excitations in the Luttinger liquid. Therefore, the ac-Josephson effect can be used as a tool for the observation of spin-charge separation. \textcopyright{} 1996 The American Physical Society.
We study transport through two Luttinger liquids (one-dimensional electrons interacting through a Coulomb repulsion in a metal) coupled together at two points. External voltage biases are incorporated through boundary conditions. We include density-density couplings as well as single-particle hops at the contacts. For weak repulsive interactions, transport through the wires remains undisturbed by the interwire couplings, which renormalize to zero. For strong repulsive interactions, the interwire couplings become strong. For symmetric barriers and no external voltage bias, a single gate voltage is sufficient to tune for resonance transmission in both wires. However, for asymmetric couplings or for finite external biases, the system is insulating.
The electronic transport through a single impurity in a repulsive Luttinger liquid (LL) has been theoretically studied. It has been found that the direct current Ī above the threshold voltage related to the strength of the impurity potential is accompanied by coherent oscillations with frequency f = Ī / e . There is an analogy with the Josephson junctions: the well-known regime of the power-law I − V curves in the LL corresponds to the damping of the Josephson current below the critical one, while the oscillatory regime in the LL can be compared with the Josephson oscillations above the critical current.