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arXiv:1205.0876 · Physical Review B

Finite-frequency noise properties of the nonequilibrium Anderson impurity model

Christoph P. Orth, Daniel F. Urban, Andreas Komnik

Published Sep 28, 2012• 15 citations
arXivHTMLPDF

Abstract

We analyze the spectrum of the electric-current autocorrelation function (noise power) in the Anderson impurity model biased by a finite transport voltage. Special emphasis is placed on the interplay of nonequilibrium effects and electron-electron interactions. Analytic results are presented for a perturbation expansion in the interaction strength $U$. Compared to the noninteracting setup we find a suppression of noise for finite frequencies in equilibrium and an amplification in nonequilibrium. Furthermore, we use a diagrammatic resummation scheme to obtain nonperturbative results in the regime of intermediate $U$. At finite voltage, the noise spectrum shows sharp peaks at positions related to the Kondo temperature instead of the voltage.

DOI
10.1103/physrevb.86.125324

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We analyze the spectrum of the electric-current autocorrelation function (noise power) in the Anderson impurity model biased by a finite transport voltage. Special emphasis is placed on the interplay of nonequilibrium effects and electron-electron interactions. Analytic results are presented for a perturbation expansion in the interaction strength UUU. Compared to the noninteracting setup we find a suppression of noise for finite frequencies in equilibrium and an amplification in nonequilibrium. Furthermore, we use a diagrammatic resummation scheme to obtain nonperturbative results in the regime of intermediate UUU. At finite voltage, the noise spectrum shows sharp peaks at positions related to the Kondo temperature instead of the voltage.
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The theoretical description of strongly correlated quantum systems out of equilibrium presents several challenges and a number of open questions persist. In this paper we focus on nonlinear electronic transport through an interacting quantum dot maintained at finite bias using a concept introduced by Hershfield [Phys. Rev. Lett. 70, 2134 (1993)] whereby one can express such nonequilibrium quantum impurity models in terms of the system's Lippmann-Schwinger operators. These scattering operators allow one to reformulate the nonequilibrium problem as an effective equilibrium problem associated with a modified Hamiltonian. In this paper we provide a pedagogical analysis of the core concepts of the effective equilibrium theory. First, we demonstrate the equivalence between observables computed using the Schwinger-Keldysh framework and the effective equilibrium approach, and relate the Green's functions in the two theoretical frameworks. Second, we expound some applications of this method in the context of interacting quantum impurity models. We introduce a novel framework to treat effects of interactions perturbatively while capturing the entire dependence on the bias voltage. For the sake of concreteness, we employ the Anderson model as a prototype for this scheme. Working at the particle-hole symmetric point, we investigate the fate of the Abrikosov-Suhl resonance as a function of bias voltage and magnetic field.
We investigate nonequilibrium properties of the single-impurity Anderson model by means of the functional renormalization group (fRG) within Keldysh formalism. We present how the level broadening $\ensuremath{\Gamma}/2$ can be used as flow parameter for the fRG. This choice preserves important aspects of the Fermi-liquid behavior that the model exhibits in case of particle-hole symmetry. An approximation scheme for the Keldysh fRG is developed which accounts for the frequency dependence of the two-particle vertex in a way similar but not equivalent to a recently published approximation to the equilibrium Matsubara fRG. Our method turns out to be a flexible tool for the study of weak to intermediate on-site interactions $U\ensuremath{\lesssim}3\ensuremath{\Gamma}$. In equilibrium we find excellent agreement with numerical RG results for the linear conductance at finite gate voltage, magnetic field, and temperature. In nonequilibrium, our results for the current agree well with time-dependent density-matrix RG. For the nonlinear conductance as function of the bias voltage, we propose reliable results at finite magnetic field and finite temperature. Furthermore, we demonstrate the exponentially small scale of the Kondo temperature to appear in the second-order derivative of the self-energy. We show that the approximation is, however, not able to reproduce the scaling of the effective mass at large interactions.
In this report, we describe a recent development in a Fermi liquid theory for the Kondo effect in quantum dots under a finite bias voltage VVV. Applying the microscopic theory of Yamada and Yosida to a nonequilibrium steady state, we derive the Ward identities for the Keldysh Green's function, and determine the low-energy behavior of the differential conductance dI/dVdI/dVdI/dV exactly up to terms of order (eV)2(eV)^2(eV)2 for the symmetric Anderson model. These results are deduced from the fact that the Green's function at the impurity site is a functional of a nonequilibrium distribution feff(ω)f_{\text{eff}}(ω)feff​(ω), which at eV=0eV=0eV=0 coincides with the Fermi function. Furthermore, we provide an alternative description of the low-energy properties using a renormalized perturbation theory (RPT). In the nonequilibrium state the unperturbed part of the RPT is determined by the renormalized free quasiparticles, the distribution function of which is given by feff(ω)f_{\text{eff}}(ω)feff​(ω). The residual interaction between the quasiparticles U~\widetilde{U}U, which is defined by the full vertex part at zero frequencies, is taken into account by an expansion in the power series of U~\widetilde{U}U. We also discuss the application of the RPT to a high-bias region beyond the Fermi-liquid regime.
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 discuss the transient effects in the Anderson impurity model that occur when two fermionic continua with finite bandwidths are instantaneously coupled to a central level. We present results for the analytically solvable noninteracting resonant-level system first and then consistently extend them to the interacting case using the conventional perturbation theory and recently developed nonequilibrium Monte Carlo simulation schemes. The main goal is to gain an understanding of the full time-dependent nonlinear current-voltage characteristics and the population probability of the central level. We find that, contrary to the steady state, the transient dynamics of the system depends sensitively on the bandwidth of the electrode material.