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arXiv:cond-mat/9805268 · Physical review. B, Condensed matter

Applying voltage sources to a Luttinger liquid with arbitrary transmission

Reinhold Egger, Hermann Grabert

Published Oct 15, 1998• 62 citations
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Abstract

The Landauer approach to transport in mesoscopic conductors has been generalized to allow for strong electronic correlations in a single-channel quantum wire. We describe in detail how to account for external voltage sources in adiabatic contact with a quantum wire containing a backscatterer of arbitrary strength. Assuming that the quantum wire is in the Luttinger liquid state, voltage sources lead to radiative boundary conditions applied to the displacement field employed in the bosonization scheme. We present the exact solution of the transport problem for arbitrary backscattering strength at the special Coulomb interaction parameter $g=1/2.$

DOI
10.1103/physrevb.58.10761

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The Landauer approach to transport in mesoscopic conductors has been generalized to allow for strong electronic correlations in a single-channel quantum wire. We describe in detail how to account for external voltage sources in adiabatic contact with a quantum wire containing a backscatterer of arbitrary strength. Assuming that the quantum wire is in the Luttinger liquid state, voltage sources lead to radiative boundary conditions applied to the displacement field employed in the bosonization scheme. We present the exact solution of the transport problem for arbitrary backscattering strength at the special Coulomb interaction parameter g=1/2.g=1/2.g=1/2.
We analyze the transport properties of a Luttinger liquid with an embedded impurity of explicitly time-dependent strength. We employ a radiative boundary condition formalism to describe the coupling to the voltage sources. Assuming the impurity time dependence to be oscillatory, we present a full analytic perturbative result in impurity strength for arbitrary interaction parameter calculated with the help of Coulomb gas expansion (CGE). Moreover, a full analytic solution beyond the above restriction is possible for a special nontrivial interaction strength which has been achieved independently by full resummation of CGE series as well as via refermionization technique. The resulting nonlinear current-voltage characteristic turns out to be very rich due to the presence of the additional energy scale associated with the impurity oscillation frequency. In accordance with the previous studies, we also find an enhancement of the linear conductance of the wire to values above the unitary limit G0=2e2∕h{G}_{0}=2{e}^{2}∕hG0​=2e2∕h.
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.
We evaluate tunneling rates into/from a voltage (U)(U)(U) biased quantum wire containing a weak backscattering defect. Interacting electrons in such a wire form a true nonequilibrium state of the Luttinger liquid (LL). This state is created due to inelastic electron backscattering leading to the emission of nonequilibrium plasmons with typical frequency $\ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\le}U$. The tunneling rates are split in two edges. The tunneling exponent at the Fermi edge is positive and equals that of the equilibrium LL, while the exponent at the side edge ${E}_{F}\ensuremath{-}U$ is negative if Coulomb interaction is not too strong.
We study theoretically the transport properties of an interacting quantum wire in the presence of a time-dependent point impurity. Using the Keldysh formalism within the framework of Luttinger liquid model, we calculate and analyze the dc contribution of the backscattered current and the shot noise of the system with arbitrary Luttinger interaction strength and impurity position. From some numerical examples, we find that the time-dependent point impurity in the wire can enhance the total current in the strongly interaction regime, and the total current enhancement is dependent on both the impurity potential frequency and the equivalent frequency in relation to the drain-source voltage VVV. The dc contribution of the backscattered current oscillates as a function of impurity frequency with the period depending on the Luttinger interaction strength. Furthermore, the oscillation of the dc contribution of the backscattered current is not a finite-length effect, which is different from that in the case of system with a static point impurity, and the oscillation characteristic is also connected with VVV. However, the final current-noise relation is similar to that in the case of a static point impurity.
We (re)consider in this paper the problem of tunnelling through an impurity in a quantum wire with arbitrary Luttinger interaction parameter. By combining the integrable approach developed in the case of quantum Hall edge states with the introduction of radiative boundary conditions to describe the adiabatic coupling to the reservoirs, we are able to obtain the exact equilibrium and non-equilibrium current. One of the most striking features observed is the appearance of negative differential conductances out of equilibrium in the strongly interacting regime g ⩽0.2. In spite of the various charging effects, a remarkable form of duality is still observed. New results on the computation of transport properties in integrable impurity problems are gathered in the appendices. In particular, we prove that the TBA results satisfy a remarkable relation, originally derived using the Keldysh formalism, between the order T 2 correction to the current out of equilibrium and the second derivative of this current at T = 0 with respect to the voltage: this is the first direct link between the TBA approach out of equilibrium and the traditional Keldysh one