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arXiv:1207.6222 · Physical Review B

Transient noise spectra in resonant tunneling setups: Exactly solvable models

Karsten Joho, Stefan Alexander Maier, A. Komnik

Published Oct 5, 2012• 14 citations
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Abstract

We investigate the transient evolution of finite-frequency current noise after the abrupt switching on of the tunneling coupling in two paradigmatic, exactly solvable models of mesoscopic physics: the resonant level model and the Majorana resonant level model, which emerges as an effective model for a Kondo quantum dot at the Toulouse point. We find a parameter window in which the transient noise can become negative, a property it shares with the transient current. However, in contrast to the transient current, which approaches the steady state exponentially fast, we observe an algebraic decay in time of the transient noise for a system at zero temperature. This behavior is dominant for characteristic parameter regimes in both models. At finite temperature the decay is altered from an algebraic to an exponential one with a damping constant proportional to temperature.

DOI
10.1103/physrevb.86.155304

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We investigate the transient evolution of finite-frequency current noise after the abrupt switching on of the tunneling coupling in two paradigmatic, exactly solvable models of mesoscopic physics: the resonant level model and the Majorana resonant level model, which emerges as an effective model for a Kondo quantum dot at the Toulouse point. We find a parameter window in which the transient noise can become negative, a property it shares with the transient current. However, in contrast to the transient current, which approaches the steady state exponentially fast, we observe an algebraic decay in time of the transient noise for a system at zero temperature. This behavior is dominant for characteristic parameter regimes in both models. At finite temperature the decay is altered from an algebraic to an exponential one with a damping constant proportional to temperature.
The Majorana resonant level model describes the universality class of the two-channel/terminal Kondo model at the Toulouse point as well as of a resonant level between two half-infinite Tomonaga-Luttinger liquids. We analyze the time evolution of the electric current and of the population function after an instantaneous switching on of the tunneling coupling. We find that the only time scale, which governs the relaxation of the initial dot preparation is the inverse contact transparency $\ensuremath{\Gamma}$, whatever the dot offset energy $\ensuremath{\Delta}$, applied bias voltage, or temperature. The voltage alone determines the superimposed oscillatory behavior of the observables for weak detuning $|\ensuremath{\Delta}|<\ensuremath{\Gamma}/2$. In the opposite case of strong detuning $|\ensuremath{\Delta}|>\ensuremath{\Gamma}/2$ a beating pattern emerges. For the current the finite temperature plays the similar role as the hybridization. The dot population function dynamics approaches that of a resonant $(\ensuremath{\Delta}=0)$ setup upon increasing the voltage or/and temperature.
Using a real-time renormalization group method we study the minimal model of a quantum dot dominated by charge fluctuations, the two-lead interacting resonant level model, at finite bias voltage. We develop a set of RG equations to treat the case of weak and strong charge fluctuations, together with the determination of power-law exponents up to second order in the Coulomb interaction. We derive analytic expressions for the charge susceptibility, the steady-state current, and the conductance in the situation of arbitrary system parameters, in particular away from the particle-hole symmetric point and for asymmetric Coulomb interactions. In the asymmetric situation we find that power laws can be observed for the current only as a function of the level position (gate voltage) but in general not as a function of the voltage except for extremely large voltages. Furthermore, we study the quench dynamics after sudden switch-on of the level-lead couplings. The time evolution of the dot occupation and current is governed by exponential relaxation accompanied by voltage-dependent oscillations and characteristic algebraic decay.
We calculate the finite-frequency current noise of a nonequilibrium resonance-level quantum dot close to a dissipative quantum phase transition of the Kosterlitz-Thouless (KT) type between a de-localized phase for weak dissipation and a localized phase for strong dissipation. The resonance-level is coupled to two spinless fermionic baths with a finite bias voltage and an Ohmic boson bath representing the dissipative environment. The system is equivalent to an effective anisotropic Kondo model out of equilibrium. To compute the finite-frequency noise, we combine two recently developed Functional Renormalization Group (FRG) approaches in Refs.[17,22] and in Ref.[23]. The nonequilibrium current noise at zero-temperature and finite frequencies shows a singular dip in the de-localized phase for the magnitude of frequencies equal to the bias voltage; while the dip is smeared out as the system moves to the localized phase. The corresponding peak-to-dip crossover is found in the AC conductance for the magnitude of frequencies equal to the bias voltage. The relevance and applications of our results for the experiments and for tunnelings between Fractional Quantum Hall Edge (FQHE) states and chiral Luttinger liquids are discussed.
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.
We discuss the conductance of a Luttinger liquid interrupted by a quantum dot containing a single resonant level. Using bosonization and refermionization methods, we find a mapping to a Kondo-type problem which possesses a nontrivial Toulouse-type solvable point. At this point, we obtain an analytic expression for the nonlinear current-voltage characteristics and analyze the differential conductance and the width of the resonance peak as functions of bias and gate voltages, temperature, and barrier asymmetry. We also determine the exact scaling function for the linear conductance.
Quantum dot devices may be tuned to Kondo models which have non-interacting limits such as the Toulouse limit. At sufficiently low temperatures the Toulouse limit can be used to make predictions for charge transport in the devices driven away from equilibrium, e.g. with an ac bias. We investigate non-equilibrium transport in a resonant level model with a single lead which may be mapped onto the Toulouse limit of the one-channel Kondo model. The main results are expressions for the charge current that are exact in the tunnelling couplings, while remaining in the linear response regime of the time-dependent voltage bias. We also show how the line of enquiry directly extends to the tunnelling junction studied by A.~Schiller and S.~Hershfield in which a Kondo impurity interacts separately with two leads.