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Physical Review B

Transport properties for a Luttinger liquid wire in the presence of a time-dependent impurity

Fang Cheng, Guanghui Zhou

Published Mar 24, 2006• 17 citations
Publisher

Abstract

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 $V$. 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 $V$. However, the final current-noise relation is similar to that in the case of a static point impurity.

DOI
10.1103/physrevb.73.125335

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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 study the transport properties of a Luttinger liquid in the presence of several time-dependent weak pointlike impurities. Our starting point is the bosonized form of the Luttinger liquid Hamiltonian with a potential introduced by the impurities. We find the correction to the total current due to the backscattering of the electrons by the impurities in first order of perturbation theory for a general time-dependent impurity potential. We then apply this result to a single time-dependent impurity, and show that it reproduces the result obtained by Feldman and Gefen [Phys. Rev. B 67, 115337 (2003)], in particular that the current enhancement occurs only for strong repulsive electron-electron interactions. For the case of two impurities oscillating with the same frequency, we find that the dc component of the backscattering current is positive even for weak repulsive interactions, due to the presence of the interference term induced by spatial correlations in the Luttinger liquid. We show that the impurity frequencies at which the enhancement takes place form a bandlike structure. In the case of several impurities, the bands become broader, because of interference effects.
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.
The inhomogeneous Tomonaga-Luttinger liquid model describing an interacting quantum wire adiabatically coupled to noninteracting leads is analyzed in the presence of a weak impurity within the wire. Due to strong electronic correlations in the wire, the effects of impurity backscattering, finite bias, finite temperature, and finite length lead to characteristic nonmonotonic parameter dependencies of the average current. We discuss oscillations of the nonlinear current voltage characteristics that arise due to reflections of plasmon modes at the impurity and quasi-Andreev reflections at the contacts, and show how these oscillations are washed out by decoherence at finite temperature. Furthermore, the finite-frequency current noise is investigated in detail. We find that the effective charge extracted in the shot noise regime in the weak backscattering limit decisively depends on the noise frequency $\ensuremath{\omega}$ relative to vF∕gL{v}_{F}∕gLvF​∕gL, where vF{v}_{F}vF​ is the Fermi velocity, ggg the Tomonaga-Luttinger interaction parameter, and LLL the length of the wire. The interplay of finite bias, finite temperature, and finite length yields rich structure in the noise spectrum which crucially depends on the electron-electron interaction. In particular, the excess noise, defined as the change of the noise due to the applied voltage, can become negative and is nonvanishing even for noise frequencies larger than the applied voltage, which are signatures of correlation effects.
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.
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.
The nonlinear transport in a Luttinger liquid with finite-range interaction is considered for a general time-dependent driving electric field, in the presence of an impurity. In the static limit, the nonlinear current-voltage characteristic depends only on the integral of the driving field. For a time-dependent field, a relation between the currents at different sites is derived which depends only on the nonlocal ac conductivity of the Luttinger system without impurity, but is independent of the strength and the shape of the electric field. Experimental consequences are discussed.