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arXiv:0903.2865 · Physical Review B

Nonequilibrium Kondo model with voltage bias in a magnetic field

Peter Fritsch, Stefan K. Kehrein

Published Jan 21, 2010• 35 citations
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Abstract

We derive a consistent two-loop scaling picture for a Kondo dot in both equilibrium and nonequilibrium situations using the flow equation method. Our analysis incorporates the important decoherence effects from both thermal and nonequilibrium noise in a common setting. In the weak-coupling regime we calculate the spin-spin correlation function, the T-matrix, and the magnetization as functions of applied magnetic field, dc-voltage bias and temperature. In all these quantities we observe characteristic nonequilibrium features for a nonvanishing external voltage bias such as Kondo splitting and strongly enhanced logarithmic corrections.

DOI
10.1103/physrevb.81.035113

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We derive a consistent two-loop scaling picture for a Kondo dot in both equilibrium and nonequilibrium situations using the flow equation method. Our analysis incorporates the important decoherence effects from both thermal and nonequilibrium noise in a common setting. In the weak-coupling regime we calculate the spin-spin correlation function, the T-matrix, and the magnetization as functions of applied magnetic field, dc-voltage bias and temperature. In all these quantities we observe characteristic nonequilibrium features for a nonvanishing external voltage bias such as Kondo splitting and strongly enhanced logarithmic corrections.
We study the Kondo effect in quantum dots in an out-of-equilibrium state due to an applied dc-voltage bias. Using the method of infinitesimal unitary transformations ("flow equations"), we develop a perturbative scaling picture that naturally contains both equilibrium coherence and nonequilibrium decoherence effects. This framework allows one to study the competition between Kondo effect and current-induced decoherence, and it establishes a large regime dominated by single-channel Kondo physics for asymmetrically coupled quantum dots.
Using a nonequilibrium renormalization group method we study the real-time evolution of spin and current in the anisotropic Kondo model (both antiferromagnetic and ferromagnetic) at a finite magnetic field h(0) and bias voltage V. We derive analytic expressions for all times in the weak-coupling regime max{V, h(0),1/t} >> T(c) (T(c) is the strong-coupling scale). We find that all observables decay both with the spin relaxation and decoherence rates Gamma(1/2). Various V-dependent logarithmic, oscillatory, and power-law contributions are predicted. The low-energy cutoff of logarithmic terms is generically identified by the difference of transport decay rates. For small times t << max{V, h(0)}(-1), we obtain universal dynamics for spin and current.
The properties of a local spin S=1/2 coupled to K independent wires is studied in the presence of bias voltages which drive the system out of thermal equilibrium. For K≫1, a perturbative renormalization group approach is employed to construct the voltage-dependent scaling function for the conductance and the T matrix. In contrast to the single-channel case, the Kondo resonance is split even by bias voltages small compared to the Kondo temperature T(K), V≪T(K). Besides the applied voltage V, the current-induced decoherence rate Γ≪V controls the physical properties of the system. While the presence of V changes the structure of the renormalization group considerably, decoherence turns out to be very effective in prohibiting the flow towards new nonequilibrium fixed points even in variants of the Kondo model where currents are partially suppressed.
Correlations are a central ingredient of modern condensed matter physics. With the advent of mesoscopic physics, it became possible to study correlated systems in non-equilibrium situations, which can be reached by e.g. driving a constant current through a quantum dot. A basic question is how correlations on the one hand and a non-equilibrium current on the other hand influence each other. Here this interplay is studied for the model system of a quantum dot in the Kondo regime using the real time renormalization group (RTRG) method. This method is a functional renormalization group (RG) method allowing a non-perturbative description of correlated systems in non-equilibrium. For the Kondo quantum dot the method is generalized to the case of double vertices. The RTRG method not only provides equations for the calculation of the renormalized coupling between the quantum dot spin and the reservoir electrons, but also equations for the calculation of the non-equilibrium decoherence rate and the non-equilibrium current.The RG equations are derived to second order. This approximation is expected to be valid as long as the quantum dot does not enter the strong coupling regime, i.e. the effective coupling strength between the quantum dot spin and the electrons does not become too large. It is found that this is the case if the bias voltage applied to the dot or the temperature is much larger than the Kondo temperature.It is shown that the RG equations lead to coupling functions depending on the energy of the incident and the scattered electron. This dependence is simplified to the dependence on the average energy of the incident and the scattered electron in the weak coupling regime.A closer inspection of the RG equations for the couplings shows that they not only contain terms leading to a logarithmic increase of the coupling strength but also subleading terms which are unimportant in the weak coupling regime. The RG equations are solved analytically for the case where the subleading terms are neglected. It is found that the coupling functions develop resonances at energies associated with the chemical potentials of the reservoirs. The height of these resonances is determined by the decoherence rate. However, for bias voltages much larger than the Kondo temperature these resonances do not affect the conductance for a two terminal setup, so the conductance does not provide a measure for the decoherence rate in this case. It is found that the conductance for the two terminal setup is a universal function of voltage/Kondo temperature and temperature/Kondo temperature in the weak coupling regime. This function is calculated analytically in the limits of vanishing voltage or temperature. Furthermore, it is found that for zero temperature the decoherence rate is proportional to the non-equilibrium current flowing through the system. This supports the physical picture that a non-equilibrium current is associated with a decoherence rate, which cuts the logarithmic scaling of the coupling strength.Finally, the RG equations are solved numerically for zero temperature including the subleading corrections found before. As expected, the subleading corrections have no impact on the decoherence rate and the conductance in the weak coupling regime. On the other hand, it is found that they lead to strong deviations from the results of the analytic leading logarithmic approximation if the bias voltage approaches the Kondo temperature. Generally, it is found that the subleading corrections lead to a saturation of the decoherence rate and the conductance to some finite value. However, for the numerical solution a numerical regularization parameter has to be introduced. It turns out that the saturation value found depends on its value, so no statements on the precise value of the decoherence rate and the current can be made in this regime.
We study nonequilibrium transport through a spin-1 Kondo dot in a local magnetic field. To this end we perform a two-loop renormalization group analysis in the weak-coupling regime yielding analytic results for (i) the renormalized magnetic field and the ggg factor, (ii) the time evolution of observables and the relevant decay rates, (iii) the magnetization and anisotropy, and (iv) the current and differential conductance in the stationary state. In particular, we find that compared to a spin-1/2 Kondo dot there exist three additional decay rates resulting in an enhanced broadening of the logarithmic features observed in stationary quantities. Additionally, we study the effect of anisotropic couplings between reservoir and impurity spin.
We study the single-impurity Anderson model out of equilibrium under the influence of a bias voltage ϕ and a magnetic field B . We investigate the interplay between the shift ( ) of the Kondo peak in the spin-resolved density of states (DOS) and the one ( ) of the conductance anomaly. In agreement with experiments and previous theoretical calculations we find that, while the latter displays a rather linear behavior with an almost constant slope as a function of B down to the Kondo scale, the DOS shift first features a slower increase reaching the same behavior as only for . Our auxiliary master equation approach yields highly accurate nonequilibrium results for the DOS and for the conductance all the way from within the Kondo up to the charge fluctuation regime, showing excellent agreement with a recently introduced scheme based on a combination of numerical renormalization group with time-dependent density matrix renormalization group.