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arXiv:cond-mat/0007037 · Journal of Physics A Mathematical and General

External voltage sources and tunnelling in quantum wires

A. Koutouza, F. Siano, Hubert Saleur

Published Jun 27, 2001• 6 citations
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Abstract

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

DOI
10.1088/0305-4470/34/27/303

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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
We have recently calculated exact nonequilibrium quantum transport properties through a point contact in a Luttinger liquid. Using a particular quasiparticle basis of the Hilbert space dictated by integrability, we here compute explicitly the exact I(V) characteristic and conductance out of equilibrium as a function of driving voltage V and temperature T. These are described by universal scaling functions of two variables, the scaled point-contact interaction strength, and V/T. The differential-conductance curve as a function of the interaction strength broadens significantly as V/T is increased, and develops a pronounced maximum at a (universal) critical value (eV/kB{\mathit{k}}_{\mathit{B}}kB​T)=7.188 68. . .. In addition, we derive an exact duality between strong and weak backscattering. The theory presented here has recently been realized experimentally in resonant tunneling-transport experiments between edge states in fractional quantum Hall effect devices. In this context the exact duality is between electron tunneling and Laughlin-quasiparticle tunneling.
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 study the transport properties of a quantum wire, described by the Tomonaga-Luttinger model, in the presence of a backscattering potential provided by several extended time-dependent impurities (barriers). Employing the B\"uttiker-Landauer approach, we first consider the scattering of noninteracting electrons (g=1)(g=1)(g=1) by a rectangularlike barrier and find an exact solution for the backscattering current, as well as a perturbative solution for a weak static potential with an arbitrary shape. We then include electron-electron interactions and use the Keldysh formalism combined with the bosonization technique to study oscillating extended barriers. We show that the backscattering current off time-dependent impurities can be expressed in terms of the current for the corresponding static barrier. Then we determine the backscattering current for a static extended potential, which, in the limit of noninteracting electrons (g=1)(g=1)(g=1), coincides with the result obtained using the B\"uttiker-Landauer formalism. In particular, we find that the conductance can be increased beyond its quantized value in the whole range of repulsive interactions 0<g<10<g<10<g<1 already in the case of a single oscillating extended impurity, in contrast to the case of a pointlike impurity, where this phenomenon occurs only for 0<g<1∕20<g<1∕20<g<1∕2.
We describe microscopic theory for the quantum transport through finite interacting systems connected to noninteracting leads. It can be applied to small systems such as quantum dots, quantum wires, atomic chain, molecule, and so forth. The Keldysh formalism is introduced to study the nonlinear current-voltage characteristics, and general properties of the nonequilibrium Green's functions are provided. We apply the formulated to an out-of-equilibrium Anderson model that has been used widely for the Kondo effect in quantum dots. In the linear-response regime the Kubo formalism still has an advantage, because it relates the transport coefficients directly to the correlation functions defined with respect to the thermal equilibrium, and it has no ambiguities about a profile of the inner electric field. We introduce a many-body transmission coefficient, by which the dc conductance can be expressed in a Landauer-type form, quite generally. We also discuss transport properties of the Tomonaga-Luttinger liquid, which has also been an active field of research in this decade.
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 derive the transport properties of a quantum impurity system, often realized as a quantum dot, subject to a source-drain bias voltage at zero temperature and magnetic field. Using the Scattering Bethe Anstaz(SBA), a generalization of the traditional Thermodynamic Bethe Ansatz(TBA) to open systems out of equilibrium, we derive some transport results for the small quantum dot, described by Anderson impurity Hamiltonian. Exact dot occupation out of equilibrium and the nonlinear conductance are obtained by introducing phenomenological spin- and charge-fluctuation distribution functions in the computation of the current. The current and dot occupation as a function of voltage are evaluated numerically. We also vary the gate voltage and study the transition from the mixed valence to the Kondo regime in the presence of a non-equilibrium current. For the larger quantum dot the Interacting Resonance Level Model is used to describe the strog correlation problem in this system. We show some current vs voltage results and compare with other theoretical results. We conclude with the difficulties we encountered by using this method and suggest some future extensions related to this SBA approach.