Kondo effect and the fate of bistability in molecular quantum dots with strong electron-phonon coupling

Abstract

We investigate the properties of the molecular quantum dot (Holstein-Anderson) model using numerical and analytical techniques. Path integral Monte Carlo simulations for the cumulants of the distribution function of the phonon coordinate reveal that at intermediate temperatures the effective potential for the oscillator exhibits two minima rather than a single one, which can be understood as a signature of a bistability effect. A straightforward adiabatic approximation turns out to adequately describe the properties of the system in this regime. Upon lowering the temperature the two potential energy minima of the oscillator merge to a single one at the equilibrium position of the uncoupled system. Using the parallels to the X-ray edge problem in metals we derive the oscillator partition function. It turns out to be identical to that of the Kondo model, which is known to possess a universal low energy fixed point characterized by a single parameter -the Kondo temperature TK . We derive an analogon of TK for the molecular quantum dot model, present numerical evidence pointing towards the appearance of the Kondo physics and discuss experimental implications of the discovered phenomena. In view of the recent progress in the field of microelectronic fabrication, which produces ever smaller electronic circuitry elements, it is reasonable to assume that the basic building blocks of the future nanoelectronics would be individual molecules.1 Contrary to the solid-state based systems their internal degrees of freedom play a principal role. The most important ones are the vibrational degrees of freedom.2,3 Although it is possible to model their effects with the help of a rather simple model -the molecular quantum dot (sometimes also referred to as HolsteinAnderson model ), its properties are still not understood in full detail [4]. One of the reasons is that the problem in general is not exactly solvable and many of the interesting regimes are not accessible analytically. In particular, some time ago it was predicted that when the electronphonon coupling is sufficiently strong, such a molecular dot might possess a bistability regime.5-7 A subsequent numerical analysis of systems under nonequilibirum conditions (with a finite voltage bias applied across the dot) has revealed some signatures of this phenomenon. However, so far the numerics were not able to supply conclusive evidence about the lifetime of the system in different conformational states of the molecule.8,9 On the other hand, there are several arguments against a bistable behaviour of such systems at low energies.10-12 The purpose of this paper is to reconsider the problem, trying to settle the open issues outlined above for systems in equilibrium. By doing this we have made a twofold progress. Firstly, we report path integral Monte Carlo (PIMC) simulations for the coordinate distribution functions of the localized vibrational degree of freedom, which are especially convenient for measurements in future experimental realizations of the model with the help of ultr...

Bi-stability in a two-level quantum dot with attracting e–e interaction

“…In recent years there have been a long debate about the existence of bi-stability in QDs, mostly in the presence of electron-phonon interaction. Several numerical approaches [3,4,5,6,7,8,9] and experimental results [10,11,12] confirm the appearance of this phenomenon in different QD systems, while there exist some arguments against bi-stability in some special cases [13,14,7].…”

Transient dynamics in interacting nanojunctions within self-consistent perturbation theory

“…For instance, it has been shown that the displacement fluctuation spectrum of a nanomechanical oscillator strongly coupled to electronic transport, either in the regime of semiclassical phonons [101,102], or for a quantum nanomechanical oscillator entering the Franck-Condon regime [103] bears clear signatures of a transition to a bistable regime. Moreover, by making a mapping to the Kondo problem, the bistability was shown to be destroyed in equilibrium conditions by quantum fluctuations if the temperature is lower than a phonon mediated Kondo temperature [53]. Notice, however, that this phonon displacement bistability does not correspond necessarily to a bistable behavior for the charge or the current, as predicted by the mean field approximation.…”

“…The most interesting and general coherentinteracting regime constitutes a great theoretical challenge. This regime has been addressed using several complementary approaches: diagrammatic techniques [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47], quantum Monte-Carlo (MC) [48][49][50][51][52][53][54][55], timedependent NRG [56][57][58][59][60][61][62][63], time-dependent DFT [64][65][66][67][68][69][70] among others [71][72][73][74][75].However, all of these techniques as they are actually implemented have some limitations. For instance, numerically exact methods like quantum MC are strongly time-consuming, require finite temperature and typically do not allow to reach long time scales.…”

Hierarchical quantum master equation approach to electronic-vibrational coupling in nonequilibrium transport through nanosystems: Reservoir formulation and application to vibrational instabilities

“…Recently, an imaginary-time formulation [93][94][95] as well as a Wigner-space representation 95,96 has been proposed. Other numerically exact methods to simulate vibrationally coupled charge transport in nanosystems include iterative path integral approaches, [97][98][99] diagrammatic quantum Monte Carlo simulations, [100][101][102][103][104][105] the numerical renormalization group technique, 60,61,106,107 the multilayer multiconfiguration timedependent Hartree method. 10,[108][109][110][111][112] In this paper, a hierarchical quantum master equation (HQME) approach is formulated to study vibrationally coupled charge transport.…”