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arXiv:cond-mat/0605544

On universality and non-universality for a quantum dot in the Kondo regime

Artur F. Izmaylov, Ali Goker, Peter J. Nordlander, Barry Friedman

Published May 22, 2006• 0 citations
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Abstract

The time-dependent non-crossing approximation is employed for the single-electron transistor to calculate the transient response of the conductance for a variety of temperatures and biases. We consider the case when the dot-lead tunneling constant is suddenly changed such that the Kondo effect is present in the final state. In the fast non-universal timescale, which was previously identified, we see rapid oscillations. The frequency of these oscillations is equal to the dot level and their amplitude is modulated by the initial and final tunneling constants. To study the slow universal timescale, we develop a new numerical scheme. We compute the conductance for two systems which have different Kondo temperatures down to a fraction of $T_{K}$ in infinitesimal bias with this scheme. We conclude that universality is preserved as a function of $T/ T_{K}$. We also investigate the decay rate of recently identified SKP oscillations down to zero temperature and compare it with the previous analytical results obtained with the perturbative renormalization group.

DOI
10.48550/arxiv.cond-mat/0605544

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The time-dependent non-crossing approximation is employed for the single-electron transistor to calculate the transient response of the conductance for a variety of temperatures and biases. We consider the case when the dot-lead tunneling constant is suddenly changed such that the Kondo effect is present in the final state. In the fast non-universal timescale, which was previously identified, we see rapid oscillations. The frequency of these oscillations is equal to the dot level and their amplitude is modulated by the initial and final tunneling constants. To study the slow universal timescale, we develop a new numerical scheme. We compute the conductance for two systems which have different Kondo temperatures down to a fraction of TKT_{K}TK​ in infinitesimal bias with this scheme. We conclude that universality is preserved as a function of T/TKT/ T_{K}T/TK​. We also investigate the decay rate of recently identified SKP oscillations down to zero temperature and compare it with the previous analytical results obtained with the perturbative renormalization group.
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 present a theory for the Kondo spin-12\frac{1}{2}21​ effect in strongly correlated quantum dots. The theory is applicable at any temperature and voltage. It is based on a quadratic Keldysh effective action parametrized by a universal function. We provide a general analytical form for the tunneling density of states through this universal function for which we propose a simple microscopic model. We apply our theory to the highly asymmetric Anderson model with $U=\ensuremath{\infty}$ and describe its strong-coupling limit, weak-coupling limit, and crossover region within a single analytical expression. We compare our results with a numerical renormalization group in equilibrium and with a real-time renormalization group out of equilibrium and show that the universal shapes of the linear and differential conductance obtained in our theory and in these theories are very close to each other in a wide range of temperatures and voltages. In particular, as in the real-time renormalization group, we predict that at the Kondo voltage the differential conductance is equal to 2/32/32/3 of its maximum.
We study finite-frequency transport properties of the double dot system recently constructed to observe the two-channel Kondo effect [R. M. Potok et al., Nature 446, 167 (2007)]. We derive an analytical expression for the frequency-dependent linear conductance of this device in the Kondo regime. We show how the features characteristic of the two-channel Kondo quantum critical point emerge in this quantity, which we compute using the results of conformal field theory as well as numerical renormalization group methods. We determine the universal crossover functions describing non-Fermi-liquid vs Fermi-liquid crossovers and also investigate the effects of a finite magnetic field.
We examine the properties of a dc-biased quantum dot in the Coulomb blockade regime. For voltages V that are large compared to the Kondo temperature T(K), the physics is governed by the scales V and gamma, where gamma approximately V/ln(2)(V/T(K)) is the nonequilibrium decoherence rate induced by the voltage-driven current. Based on scaling arguments, self-consistent perturbation theory, and perturbative renormalization group, we argue that due to the large gamma the system can be described by renormalized perturbation theory in 1/ln(V/T(K))<<1. However, in certain variants of the Kondo problem, two-channel Kondo physics is induced by a large voltage V.
In the present work, we calculate the conductance through a single quantum dot weakly coupled to metallic contacts. We use the spin-1/2 Anderson model to describe the quantum dot, while considering a finite Coulomb repulsion. We solve the interacting system using the noncrossing approximation (NCA) and the one-crossing approximation (OCA). We obtain the linear response conductance as a function of temperature and energy position of the localized level. From the comparison of both approximations we extract the role of the vertex corrections, which are introduced in the OCA calculations and neglected in the NCA scheme. As a function of the energy position, we observe that the diagrams omitted within the NCA are really important for appropriately describing transport phenomena in Kondo systems as well as in the mixed valence regime. On the other hand, as a function of temperature, the corrections introduced by the OCA partly recover the universal scaling properties known from numerical approaches such as the numerical renormalization group.
We present a study of spin-1/2 Kondo singlets in single electron transistors under a microwave frequency bias excitation. We compare time-averaged conductance GGG to predicted universal response with respect to microwave frequency, oscillation amplitude, and the Kondo temperature and find a nonadiabatic response when the microwave photon energy hfhfhf is comparable to the Kondo temperature kBTK{k}_{B}{T}_{K}kB​TK​. We show that our measurements are qualitatively consistent with the predictions for the radiation-induced decoherence rate of the Kondo spin.